An Introduction to Formal Logic with Philosophical Applications 9780199386499, 0199386498

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Introduction to Formal Logic with Philosophical Applications

Introduction to Formal Logic with Philosophical Applications RUSSELL MARCUS Hamilton College

The question of logic is: Does the conclusion certainly follow if the premises be true? AUGUSTUS DE MORGAN Formal Logic: Or, The Calculus of Inference, Necessary and Probable (1847)

New York  Oxford OXFORD UNIVERSITY PRESS

Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and certain other countries. Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America. © 2018 by Oxford University Press For titles covered by Section 112 of the US Higher Education Opportunity Act, please visit www.oup.com/us/he for the latest information about pricing and alternate formats. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by license, or under terms agreed with the appropriate reproduction rights organization. Inquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not circulate this work in any other form and you must impose this same condition on any acquirer. Library of Congress Cataloging-in-Publication Data Names: Marcus, Russell, 1966– author. Title: An introduction to formal logic with philosophical applications/  ​Russell Marcus. Description: New York: Oxford University Press, 2017. | Includes  ​bibliographical references and index. | Identifiers: LCCN 2017038173 (print) | LCCN 2017040131 (ebook) | ISBN  ​9780199386499 (Ebook) | ISBN 9780199386482 (pbk.) Subjects: LCSH: Logic—Textbooks. Classification: LCC BC71 (ebook) | LCC BC71 .M33 2017 (print) | DDC   160—dc23 LC record available at https://lccn.loc.gov/2017038173 9 8 7 6 5 4 3 2 1 Printed by LSC Communications, United States of America

Contents Preface ix

Chapter 1: Introducing Logic  1 1.1: Defining ‘Logic’  1 1.2: Logic and Languages  3 1.3: A Short History of Logic  5 1.4: Separating Premises from Conclusions  9 1.5: Validity and Soundness  17 Key Terms  21

Chapter 2: Propositional Logic:

Syntax and Semantics  22

2.1: Logical Operators and Translation  22 2.2: Syntax of PL: Wffs and Main Operators  43 2.3: Semantics of PL: Truth Functions  46 2.4: Truth Tables  60 2.5: Classifying Propositions  68 2.6: Valid and Invalid Arguments  78 2.7: Indirect Truth Tables  84 Key Terms  106

Chapter 3: Inference in Propositional Logic  107 3.1: Rules of Inference 1  107 3.2: Rules of Inference 2  118 3.3: Rules of Equivalence 1  129 3.4: Rules of Equivalence 2  140 v

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3.5: Practice with Derivations  150 3.6: The Biconditional  158 3.7: Conditional Proof  168 3.8: Logical Truths  178 3.9: Indirect Proof  185 3.10: Chapter Review  197 Key Terms  205

Chapter 4: Monadic Predicate Logic  206 4.1: Introducing Predicate Logic  206 4.2: Translation Using M  213 4.3: Syntax for M  227 4.4: Derivations in M  232 4.5: Quantifier Exchange  248 4.6: Conditional and Indirect Proof in M  257 4.7: Semantics for M  267 4.8: Invalidity in M  274 Key Terms  293

Chapter 5: Full First-Order Logic  294 5.1: Translation Using Relational Predicates  294 5.2: Syntax, Semantics, and Invalidity in F  312 5.3: Derivations in F  321 5.4: The Identity Predicate: Translation  335 5.5: The Identity Predicate: Derivations  354 5.6: Translation with Functions  364 5.7: Derivations with Functions  374 Key Terms  384

Chapter 6: Beyond Basic Logic  385 6.1: Notes on Translation with PL  385 6.2: Conditionals  392 6.3: Three-Valued Logics  406 6.4: Metalogic  425

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6.5: Modal Logics  431 6.6: Notes on Translation with M  445

Chapter 7: Logic and Philosophy  455 7.1: Deduction and Induction  455 7.2: Fallacies and Argumentation  466 7.3: Logic and Philosophy of Mind: Syntax, Semantics, and the Chinese Room  478 7.4: Logic and the Philosophy of Religion  488 7.5: Truth and Liars  503 7.6: Names, Definite Descriptions, and Logical Form  515 7.7: Logicism  523 Appendix on the Logical Equivalence of the Rules of Equivalence  533 Terms 539 Solutions to Selected Exercises  541 Glossary/Index 595

Preface “Who needs another logic textbook?” I got asked a lot as I was writing this one. The answer is: students and teachers who want to explore both the techniques of formal logic and their connections to philosophy and other disciplines. The best approach to thinking about those connections is both to learn the tools of logic, in careful detail, and to write about them. So this is a formal logic textbook for people who want to learn or teach the core concepts of classical formal logic, and to think and write about them. Introduction to Formal Logic with Philosophical Applications (IFLPA) and Introduction to Formal Logic (IFL) are a pair of new logic textbooks, designed for students of formal logic and their instructors to be rigorous, yet friendly and accessible. Unlike many other logic books, IFLPA and IFL both focus on deductive logic. They cover syntax, semantics, and natural deduction for propositional and predicate logics. They emphasize translation and derivations, with an eye to semantics throughout. Both books contain over two thousand exercises, enough for in-class work and homework, with plenty left over for extra practice, and more available on the Oxford website (see page xv). Since logic is most often taught in philosophy departments, special attention is given to how logic is useful for philosophers, and many examples use philosophical concepts. But the examples in the text are accessible to all students, requiring no special interest in or knowledge of philosophy, and there are plenty of exercises with no philosophical content, too. IFLPA also contains two chapters of stand-alone essays on logic and its application in philosophy and beyond, with writing prompts and suggestions for further readings. These essays help instructors and students to reflect on their formal work, to understand why logic is important to philosophers, and how it can be applied to other disciplines, and to students’ own lives and studies.

WHY THIS LOGIC BOOK? Introduction to Formal Logic with Philosophical Applications is the product of both my growing unease, over years of teaching logic in philosophy departments, and a sudden imprudent decision. My unease derived from the ways in which even excellent ix

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students finished my logic course without a good understanding of why philosophers studied the topic. I wanted to show my students connections between formal deductive logic and broader philosophical topics. I began teaching Philosophy Fridays in my logic classes at Hamilton College, putting aside the technical material on truth tables and derivations, and talking about nonstandard topics, ones that appeared only in cursory fashion, if at all, in most textbooks. Every other Friday, I would assign a philosophy reading relating to the course material—for example, Goodman’s The Problem of Counterfactual Conditionals, a selection from Aristotle’s De Interpretatione, Quine’s “On What There Is,” Searle’s “Minds, Brains, and Programs”—and spend a class talking with my students about how logic and philosophy connect. Each student would write a short paper on some topic raised in a Philosophy Friday. Students responded well to Philosophy Fridays, but the readings I assigned were often too difficult. As at many schools, Symbolic Logic at Hamilton attracts students from departments across the college. Most of these students were too unfamiliar with philosophy to work comfortably with the material I found most interesting. I received many fine papers and was convinced that my students were leaving the course with a greater awareness of why we philosophers study logic. But unfortunate numbers of students let me know that they were not enjoying the difficult, obscure (to them) readings. My sudden imprudent decision happened during a logic class, fall 2010, when I mentioned that if anyone wanted to spend time writing logic problems for a summer, while I wrote my own essays for Philosophy Fridays, I would try to find some funding for the project. A student volunteered, and we spent the summer working. The result, IFLPA, revised and expanded over the years, has two parts. The first part, chapters 1–5, is a nuts-and-bolts introductory formal deductive logic textbook. Chapter 1 is a brief introduction. Chapter 2 covers propositional semantics, leading to the standard truth-table definition of validity. Chapter 3 covers natural deductions in propositional logic. Chapter 4 covers monadic predicate logic. Chapter 5 covers full first-order logic, including identity and functions. This material is straight logic, and it does not discuss distracting questions in the philosophy of logic. The central innovation of IFLPA is in its second part. Chapters 6 and 7 contain stand-alone introductory essays on enrichment topics, along with discussion questions which can serve as essay prompts and suggestions for further reading. The first five chapters may come off as dogmatic, in places. For example, in chapter 2, I introduce the truth table for the material conditional without discussing the deep questions about that interpretation of ‘if . . . then . . .’ statements. I introduce bivalence without discussing other options, like three-valued logics. But nearly each section of the first five chapters contains a set of questions (“Tell Me More”), pointing to places in which such questions are explored in chapters 6 and 7. Chapter 6, “Beyond Basic Logic,” contains discussions of subtleties arising from thinking about logic, and it includes some auxiliary formal material. There are essays

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on the nature of conditionals; some metalogical concepts (completeness, soundness, decidability); three-valued logics; modal logic; and more intricate questions arising in translation. Chapter 7, “Logic and Philosophy,” mainly contains more-philosophical essays aimed at showing both the connections between formal logic and other areas of philosophy, and some of the ways in which logic can actually help treat philosophical problems. I discuss philosophy of mind, philosophy of science, philosophy of language, metaphysics, truth, and mathematics. I have found these essays to be especially engaging for students eager to move beyond more formal work.

SPECIAL FEATURES Each section of IFLPA contains a Summary. Sections in chapters 1 through 5 contain a list of important points to Keep in Mind. Key terms are boldfaced in the text and defined in the margins, and are listed at the end of each chapter. In addition, all terms are defined in the glossary/ index at the end of the book. There are over two thousand exercises in the book. Exercises are presented progressively, from easier to more challenging. Translate-and-derive exercises are available in every section on derivations, helping to maintain students’ translation skills. Translation exercises are supplemented with examples for translation from formal languages into English. Regimentations and translations contain both ordinary and philosophical themes. Solutions to exercises, about 20 percent of total, are included at the back of the book. Solutions to translate-and-derive exercises appear in two parts: first, just the translation, and then the derivation. Solutions to all exercises are available for instructors. IFLPA contains several formal topics and exercise types not appearing in standard logic texts: Seven rules for biconditionals, parallel to the standard rules for conditionals. Exercises asking students to interpret and model short theories. Two sections on functions at the end of chapter 5. Exercises asking students to determine whether an argument is valid or invalid, or whether a proposition is a logical truth or not, and then to construct either a derivation or a counterexample. These sections are perfect for stronger students, while easily skipped by others.

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Sections of enrichment material on philosophical applications in chapters 6 and 7 all contain essay prompts (“For Further Research and Writing”) and suggested readings. Tell Me More entries at the ends of sections, pointing to essays in chapters 6 and 7, encourage students to explore the importance of their formal work.

INTRODUCTION TO FORMAL LOGIC OR INTRODUCTION TO FORMAL LOGIC WITH PHILOSOPHICAL APPLICATIONS? T WO BOOKS—YOUR CHOICE As IFLPA went through the review process at Oxford, it became clear that some instructors were interested mainly in the first five chapters, which focused on formal logic, and did not see a use for the enrichment material in chapters 6 and 7. While I have found that the enrichment material has increased my students’ enjoyment and understanding of the formal work, different instructors have different goals. So we have also produced an abbreviated version of this book. Introduction to Formal Logic (IFL) is a nuts-and-bolts introductory formal deductive logic textbook, mainly just the first five chapters of IFLPA, though it also contains the two sections on subtleties of translation from IFLPA, as well as the section on fallacies and argumentation. The formal material is the same in IFL and IFLPA: the same examples, the same exercises, and the same numberings. Instructors and students may work together with either version and move freely between the two books.

USING INTRODUCTION TO FORMAL LOGIC WITH PHILOSOPHICAL APPLICATIONS The first five chapters of IFLPA proceed sequentially through standard formal logic. Mostly, instructors will move through as much material as they desire, from the start. There are three possible exceptions. First, the first three sections of chapter 1 are mainly background and not really necessary to cover; there are no exercises in those sections. Second, one of my goals for IFLPA was a better discussion of semantics, especially for predicate logic, a topic many logic books elide or exclude. Instructors who wish to skip this material (especially sections 4.7 and 5.2) will need to support students for the further work on distinguishing valid from invalid arguments in sections 4.8 and 5.3, or just skip the relevant exercises in those sections. Third, section 3.6 contains seven natural deduction rules governing inferences with biconditionals that do not appear in standard logic texts. This section can be skipped, though instructors might want to be careful in assigning subsequent exercises that use biconditionals. All later inferences will be derivable, but some exercises will be more difficult than they would be with the extra rules from section 3.6.

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Philosophical Applications: The Enrichment Material in Chapters 6 and 7 The sections of enrichment material on philosophical applications in chapters 6 and 7 are truly independent, from chapters 1–5, and from each other. This allows instructors to choose the sections that interest them or that they believe will most interest their students. Sections in both chapters include writing prompts (“For Further Research and Writing”) and additional reading suggestions. Essays on more technical topics include exercises. Both chapters are written to help students start to see the role, implicit and explicit, of logic in contemporary philosophy. While I have tried to be even-handed, my choices of topics do not reflect the discipline as a whole, and my choices of how to cover those topics may seem idiosyncratic, since I have omitted or elided some prominent and important views. These results are inevitable for short introductory essays on topics on which there are no settled views. My goal is to find ways to stimulate my students’ interests and to help them start to think and to write about why we study logic, not to cover everything. Due to space constraints, we are not able to include all of the sections I have written for chapters 6 and 7. Omitted sections, also standing alone, are available as supplements to the text on Oxford’s website. For chapter 6, the supplementary material includes a treatment of adequate sets of propositional operators, alternative notations, alternative methods of proof, rules of passage and prenex normal form, and secondorder logic. For chapter 7, the online supplementary material includes more work on logic and the philosophy of science, infinity, quantification and ontological commitment, and the color incompatibility problem. Instructors may print and distribute those sections to their students. There are different ways in which one can use chapters 6 and 7. I cover some sections, such as sections 6.1 on subtleties of translation or 6.2 on conditionals, at the same time that I cover the formal work and assign students the readings as preparatory for classes. Alternatively, I use some of the sections as biweekly pauses, my Philosophy Fridays, during which I try to seed student interests in various topics. I have students write a short essay, due toward the end of the course, in which they are asked to make connections between formal logic and the outside world. Philosophy Fridays motivate my students to think about their essay topics. Since there is more enrichment material in the last two chapters than can be covered comfortably in a semester, I teach some different topics each year. The enrichment sections may also be assigned to strong students as independent work. For example, some mathematics or computer science students who come to logic, perhaps looking for a humanities course within their comfort zones, find working independently on the sections on three-valued logics or Hilbert-style systems edifying and satisfying. To help students and teachers figure out when best to read and discuss the enrichment material, I have provided two tools. First, I have included suggestions (“Tell Me

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More”) at the end of most sections. These suggestions point to sections in chapter 6 or 7, or to the supplementary material on the Oxford website, that can help illuminate the concepts of the relevant technical material in chapters 1–5. Second, I have prepared the following chart of guidelines. These guidelines indicate which formal topics are presupposed in each enrichment section, so instructors can know at a glance whether a section is appropriate for students at any point in the course. I have included in the chart suggestions for integrating both the sections in the book and the supplementary sections available only on Oxford’s website. When you are considering assigning a section of enrichment material to your class, a quick glance at the relevant row of this chart may be useful. Suggestions for Integrating Enrichment Material on Philosophical Applications Section

Suggested Placement

Notable Presuppositions

Other Comments

6.1: Notes on Translation with PL

After 2.5

Logical equivalence

The section uses the truth-table definition of equivalence to justify some translations in PL.

6.2: Conditionals

After 2.5

Logical equivalence and tautology

You can do this section a bit earlier, but you do need the basic truth tables.

6.3: Three-Valued Logics

After 2.6

Basic truth tables and the truth-table definition of validity

The section mentions logical truths, and the discussion of validity might make more sense after 3.5.

6.4: Metalogic

Anywhere

6.5: Modal Logics

After 3.3

Truth tables, truth functions, and some natural deduction (modus ponens)

Focuses exclusively on propositional modal logics

6.6: Notes on Translation with M

After 4.7

Translation and semantics for M

This section is about translation in M but uses the derivation rules through 4.6 and the semantics in 4.7.

6S.7: The Propositions of Propositional Logic

After 2.5

Tautology

There are mentions of disjunctive syllogism, quantification, and languages of predicate logic.

6S.8: Adequacy

After 2.6

Basic truth tables

It’s useful, but not necessary, to have discussed validity.

6S.9: Logical Truth, Analyticity, and Modality

After 2.5

Tautology and logical truth

The section mentions logical truths of F, but not in detail. Still, it might be better after 3.8 and 3.9 when students prove logical truths using natural deductions.

6S.10: Alternative Notations

After 4.1

All logical operators, including quantifiers, and the truth tables

You could do it earlier if you gloss over the quantifiers; there’s not much about them.

6S.11: Axiomatic Systems

After 3.8

Logical truths and proofs of them

The section mentions predicate logic but doesn’t use it.

Mentions proof and truth, but doesn’t presuppose the technical work on it

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6S.12: Rules of Passage

After 5.3

Rules for F

6S.13: Second-Order Logic and Set Theory

After 5.5

Identity theory

7.1: Deduction and Induction

Anywhere

7.2: Fallacies and Argumentation

Anywhere

Arguments and validity

Deductions in both PL and M are mentioned in passing, early.

7.3: Logic and Philosophy of Mind

After 4.7

Syntax and semantics

I usually do this section earlier, emphasizing the syntax and semantics of PL, but it makes more sense after the semantics of M or F.

7.4: Logic and the Philosophy of Religion

After 1.5

Regimentation into premise/ conclusion form; validity and soundness

7.5: Truth and Liars

After 4.6

Proof in M, including indirect proof

7.6: Names, Definite Descriptions, and Logical Form

After 5.4

Identity theory, specifically definite descriptions

7.7: Logicism

After 5.4

Identity theory

7S.8: Logic and Science

After 3.5

Natural deductions, especially MP and MT

7S.9: Infinity

Anywhere

7S.10: Quantification and Ontological Commitment

After 4.7

Quantification, especially quantifier exchange; semantics for M and F

7S.11: Atomism and Color Incompatibility

After 4.7

M

I do this section after functions and connect it to 7.7: Logicism. The discussion of inference makes more sense after work in chapter 3.

Discussions of functions (5.6) and second-order logic (6S.13) can be helpful in setting up the project.

The section introduces and uses some very basic set theory, and invokes the general concepts of formalization.

There is discussion of the limits of firstorder logic, but really only M is needed.

STUDENT AND INSTRUCTOR RESOURCES A rich set of supplemental resources is available to support teaching and learning in this course. These supplements include Instructor Resources on the Oxford University Press Ancillary Resource Center (ARC); intuitive, auto-graded assessments and other student resources on Dashboard by Oxford University Press; a free Companion Website for students; and downloadable Learning Management System Cartridges. For access to these resources, please visit www.oup.com/us/ marcus.

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The ARC houses a wealth of Instructor Resources:

• A customizable, auto-graded Computerized Test Bank • An Instructor’s Manual, which includes the following: ■■ A Microsoft Word document containing the questions from the Computerized Test Bank

■■ A traditional “Pencil-and-Paper” version of the Test Bank, containing the

same questions as the Computerized Test Bank, but converted for use in hard-copy exams and homework assignments, including open-ended questions that allow students to develop extended analysis, such as completing truth tables and doing proofs ■■ Complete answers to every set of exercises in the book—over 2,000 exercises ■■ Bulleted Chapter Summaries, which allow the instructor to scan the important aspects of each chapter quickly and to anticipate section discussions ■■ PowerPoint Lecture Outlines to assist the instructor in leading classroom discussion ■■ Sample syllabi • Downloadable Course Cartridges which allow instructors to import the computerized test bank and student resources from the Companion Website into their school’s Learning Management System Dashboard at www.oup.com/us/dashboard delivers a wealth of Student Resources and auto-graded activities in a simple, intuitive, and mobile device–friendly format. Dashboard contains:

• A fully-integrated eBook version of the text • Level-One and Level-Two Quizzes, autograded and linked to the Learning Ob-

jectives for easy instructor analysis of each student’s topic-specific strengths and weaknesses. • A Proof-Checking Module for solving symbolic proofs that allows students to enter proof solutions, check the their validity, and receive feedback, both by line and as a whole, as well as Truth Table Creation Modules, all feeding automatically into a Gradebook that offers instructors the chance to view students’ individual attempts • Quiz Creation Capability for instructors who wish to create original quizzes in multiple-choice, true/false, multiple-select, long-answer, short- answer, ordering, or matching question formats, including customizable answer feedback and hints • A built-in, color-coded Gradebook that allows instructors to monitor student progress from virtually any device • Chapter Learning Objectives adapted from the book’s chapter headings

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• Interactive Flashcards of Key Terms and their definitions from the book • Tools for student communication, reference, and planning, such as messaging and spaces for course outlines and syllabi

Access to Dashboard can be packaged with Introduction to Formal Logic with Philosophical Applications at a discount, stocked separately by your college bookstore, or purchased directly at www.oup.com/us/dashboard. The free Companion Website found at www.oup.com/us/marcus contains supplemental Student Resources:

• Student Self-Quizzes • Interactive Flashcards of Key Terms and their definitions from the book • Bulleted Chapter Summaries • Additional content to supplement Chapters 6 and 7, including: ■■ 6.7 Laws of Logic ■■ 6.8 Adequacy ■■ 6.9 Logical Truth Analyticity and Modality ■■ 6.10 Alternate Notations ■■ 6.11 Axiomatics ■■ 6.12 Rules of Passage ■■ 6.13 Second Order Logic ■■ 7.8 Scientific Explanation and Confirmation ■■ 7.9 Infinity ■■ 7.10 Quantification and Commitment ■■ 7.11 Color Incompatibility To find out more information or to order Dashboard access or a Course Cartridge for your Learning Management System, please contact your Oxford University Press representative at 1 800-280-0280.

ACKNOWLEDGMENTS The first draft of this book was written in the summer of 2011. I worked that summer alongside my student Jess Gutfleish, with support of a Class of 1966 Faculty Development Award from the Dean of Faculty’s Office at Hamilton College, in the archaeology teaching lab at Hamilton. I wrote the text, and she worked assiduously and indefatigably writing exercises; I had difficulty keeping up with her. I am ineffably grateful to Jess for all of her hard work and the mountain of insidiously difficult (as well as more ordinary) logic problems she devised. Jess worked on more problems in spring 2014, and Spencer Livingstone worked with her. Deanna Cho helped enormously with the section summaries and glossary, supported by the philosophy department at Hamilton College. Spencer Livingstone and Phil Parkes worked during summer 2015, helping me with some research and writing still further exercises.

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Sophie Gaulkin made many editing suggestions and Reinaldo Camacho assisted me with new exercises. Jess, Spencer, and Rey were all indescribably supportive and useful as teaching assistants and error-seeking weapons. Students in my logic classes at Hamilton, too numerous to mention, found many typos. Andrew Winters, using a draft of the text at Slippery Rock University in 2016, sent the errors he and his students discovered, and made many helpful suggestions. At the behest of Oxford, the following people made helpful comments on drafts of the book, and I am grateful for their work: Joshua Alexander, Siena College Brian Barnett, St. John Fisher College Larry Behrendt, Mercer County Community College Thomas A. Blackson, Arizona State University Dan Boisvert, University of North Carolina, Charlotte Jeff Buechner, Rutgers University, Newark Eric Chelstrom, Minnesota State University Chris Dodsworth, Spring Hill College Michael Futch, University of Tulsa Nathaniel Goldberg, Washington and Lee University Nancy Slonneger Hancock, Northern Kentucky University Brian Harding, Texas Woman’s University Reina Hayaki, University of Nebraska, Lincoln Marc A. Hight, Hampden Sydney College Jeremy Hovda, KU Leuven Gyula Klima, Fordham University Karen Lewis, Barnard College Leemon McHenry, California State University, Northridge John Piers Rawling, Florida State University Reginald Raymer, University of North Carolina, Charlotte Ian Schnee, Western Kentucky University Aeon Skoble, Bridgewater State University Michael Stoeltzner, University of South Carolina Harold Thorsrud, Agnes Scott College Mark Tschaepe, Prairie View A&M University Andrew Winters, Slippery Rock University of Pennsylvania I am grateful also to Robert Miller, Executive Editor at Oxford, and Alyssa Palazzo, Associate Editor, for supporting both IFL and IFLPA. Thank you to Margaret Gentry and the Dean of Faculty’s office at Hamilton. I am grateful to Nathan Goodale and Tom Jones for letting us have their lab in which to work, summer 2011. I also owe thanks to the many students who have helped me construct an innovative Logic course, and for the constant, unwavering support of me and my course by the Hamilton College philosophy department. Thanks to Marianne Janack for example 4.2.27 and to Alan Ponikvar for example 7.2.10.

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More remotely, I am deeply grateful to authors of the logic books I’ve studied and with which I’ve taught, especially Irving Copi’s Symbolic Logic, Geoffrey Hunter’s Metalogic, Elliott Mendelson’s Introduction to Mathematical Logic, Patrick Hurley’s A Concise Introduction to Logic, John Nolt’s Logics, and Graham Priest’s An Introduction to Non-Classical Logic. Elliott Mendelson and Melvin Fitting were especially influential logic teachers of mine; they made logic elegant and beautiful. I studied Copi’s logic with Richard Schuldenfrei, whose encouragement I appreciate. And I am grateful to Dorothea Frede, into whose Ancient Philosophy class I brought my excitement about logic, for her patience as I discovered (by regimenting his arguments through the term) that, no, Plato wasn’t making simple logical errors. Most importantly, I am grateful to my wife, Emily, and my children, Marina and Izzy, who suffered through many summers that could have been more fun for them so that I could have the logic book I wanted.

Chapter 1 Introducing Logic

1.1: DEFINING ‘LOGIC’ An Introduction to Formal Logic and Its Application to Philosophy is a textbook in formal deductive logic and its relation to philosophy. If you work through the material in the first five chapters, you can gain a good sense of what philosophers and logicians call deductive arguments. In the sixth and seventh chapters, you can read about some connections between logic and philosophy. Let’s start by trying to characterize to what the terms ‘logic’ and ‘argument’ refer. Consider the following claims that someone might use to define those terms. 1.1.1

Logic is the study of argument. Arguments are what people who study logic study.

Two aspects of the pair of sentences in 1.1.1 are worth noticing. First, they provide a circular definition that makes the characterizations nearly useless. If you do not understand the terms ‘logic’ and ‘argument’, then the sentences in 1.1.1 are not going to help you, except for showing that the two terms are related. Second, the circularity of this pair of definitions is a formal result that can be seen in other pairs of purported definitions, like the pairs of sentences in 1.1.2 and 1.1.3. 1.1.2

Sheep are the things that shepherds tend. Shepherds are things that tend sheep.

1.1.3

Glubs are extreme cases of wizzles. Wizzles are ordinary forms of glubs.

In 1.1.2, you might not notice the problem of the formal property of circularity because you already know the meanings of the terms involved. In 1.1.3, the problem should be obvious. Without knowing what glubs and wizzles are, 1.1.3 is useless, and its uselessness is a product of its poor form. This textbook is about such formal results.

1

2    C h apter 1  Intro d u c i ng L og i c

Returning to the definitions of ‘logic’ and ‘argument’, notice that in contrast to 1.1.1, 1.1.4 is not formally circular. 1.1.4

Logic is the study of argument. An argument is a set of statements, called premises, together with a claim, called the conclusion, which the premises are intended to support or establish.

1.1.4 explains the meaning of one term, ‘ logic’, by using other ones, like ‘statement’ and ‘establish’. If such a definition is to be informative, these other terms should be more familiar. If not, we can continue the process, as at 1.1.5. 1.1.5

To establish a claim is to justify or provide evidence for it. A statement is a declarative sentence that has a truth value. Truth values include truth and falsity. Some interesting logics have other truth values: three (e.g., truth, falsity, and indeterminacy) or infinitely many. In this book we will focus on just truth and falsity.

Pairing 1.1.4 and 1.1.5, we see a characterization of logic as the rules of what follows from what, of which consequences derive from which assumptions. We make inferences all the time: if I buy this book, I won’t have enough money for the cup of coffee I wanted; if I make a turn here, I’ll end up in Waterville; she must be angry with me because she hasn’t returned my email. When we think about the consequences of our actions or the reasons some event has occurred, we are using logic. Good logic is thus a precondition for all good reasoning. Some inferences are better than others. I am well justified in inferring that it is after dawn from the light peeking through my window shades. I am not well justified in believing that it is nine in the morning from the fact that it was six in the morning an hour ago; that’s an error. This book is devoted to some general principles of evaluating certain kinds of arguments, called deductive arguments. Deductive arguments are contrasted with inductive arguments, though the difference between them is difficult to specify both precisely and briefly. Roughly, in a deductive argument, the conclusion follows without fail, necessarily, from the premises. The conclusions of inductive arguments are supported by their premises, more or less depending on the argument, but not guaranteed. Inductive arguments are often (though not always) generalizations from particular experiences and can be undermined by further evidence. Logic and mathematics are largely characterized by their uses of deduction, though statistical inferences are not purely deductive. Sciences involve both deduction and induction, broadly speaking, though there are other methods of inference, like inference to the best explanation. The best way to understand the difference between deduction and induction is to work through the material in chapters 1–5 and contrast that kind of reasoning with others. When evaluating an argument, we can perform two distinct steps. First, we can see whether the conclusion follows from the assumptions. An argument whose conclusion follows from its premises is called valid. Chapter 2 is dedicated to constructing

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a precise notion of deductive validity, of what follows, for propositional logic. Indeed, the notion of validity is the central topic of the book. A second step in evaluating an argument is to see whether the premises are true. In a valid deductive argument, if the premises are true, then the conclusion must be true. This result is what makes deductive logic interesting and is, in a sense, the most important sentence of this entire book: in a valid deductive argument, if the premises are true, then the conclusion must be. An Introduction to Formal Logic and Its Application to Philosophy is dedicated to the first step in the process of evaluating arguments. The second step is not purely logical, and it is largely scientific. Roughly speaking, we examine our logic to see if our reasoning is acceptable and we examine the world to see if our premises are true. Although we prefer our arguments both to be valid and to have true premises, this book is dedicated mainly to the form of the argument, not to its content. You might wonder whether the logic in this book, formal deductive logic, represents how we actually reason or whether it sets out rules for proper reasoning. Is logic descriptive or prescriptive? Before we can start to answer this question, we have to see what our logic looks like. The nature of some elementary systems of formal logic is the focus of the first five chapters of this book. In the sixth and seventh chapters, I discuss a variety of philosophical questions arising from or informed by the study of formal logic. The sections of these chapters may be read along with the formal material in the first five chapters. TELL ME MORE  • How does deductive logic differ from inductive logic? See 7.1: Deduction and Induction.

1.2: LOGIC AND LANGUAGES There are (at least) three kinds of languages in this book. First, most of the book is written in a natural language, English. Other natural languages include Spanish and Swahili. Second, there are the formal languages that we will discuss in careful detail. As these formal languages are our main objects of study, we can call them the object languages. Between formal and natural languages is a third kind of language made of elements of the other two and used to study a formal language. This metalanguage is mostly English. You might not even think of it as a language separate from English, and for the most part you need not think about the metalanguage too carefully. But it includes some technical terms that do not occur in ordinary English. For example, the rules of inference we will examine in chapter 3 are written using Greek letters. They are parts of the metalanguage we use to tell us how to work in the object language. We can add these same meta-linguistic rules to any natural language to form a metalanguage made mostly out of Spanish or Swahili. Our metalanguage thus differs from

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any particular natural language. I will not specify the metalanguage as precisely as the object languages. It is customary to give names to object languages. Chapters 2 and 3 focus on one object language that I will call PL, for propositional logic. Chapters 4 and 5 discuss three further formal languages: M F FF

Monadic (first-order) predicate logic Full (first-order) predicate logic Full (first-order) predicate logic with functors

For each formal language we study, we will specify a syntax and a semantics. The syntax gives the vocabulary of the language, series of symbols like letters and terms like ∨, ⊃, and ∃, along with rules for forming formulas. The semantics allows us to interpret the language, to understand it as meaningful, rather than just an empty set of squiggles. There are different possible interpretations of the symbols just as there are different meanings to most words or different languages using the same letters. We specify an interpretation of an object language by thinking of ourselves as stepping outside of those languages into metalanguages. We might say, for example, that we will use the letter ‘P’ in the object language to stand for the statement expressed in English by ‘Prunes are dried plums’. We will also study derivations (or proofs) in each language. There are advantages to both natural languages and formal languages. Natural languages are excellent for ordinary communication. Formal languages are excellent for precision, especially for clarifying ambiguities. Much of the formal material in this book is based on Frege’s Begriffsschrift (1879); Begriffsschrift means ‘concept writing’. In his preface, Frege compared natural languages and formal languages to an eye and a microscope, respectively: I believe I can make the relationship of my Begriffsschrift to ordinary language clearest if I compare it to that of the microscope to the eye. The latter, due to the range of its applicability, due to the flexibility with which it is able to adapt to the most diverse circumstances, has a great superiority over the microscope. Considered as an optical instrument, it admittedly reveals many imperfections, which usually remain unnoticed only because of its intimate connection with mental life. But as soon as scientific purposes place great demands on sharpness of resolution, the eye turns out to be inadequate. The microscope, on the other hand, is perfectly suited for such purposes.

Many students, when they begin to study logic, find it to be an amusing toy. There are careful rules for working in the object language. Once you learn those rules, it can be fun to play with them. When I started studying logic, in college, I couldn’t believe that one could earn credit for filling out truth tables, translating English into formal languages, and constructing derivations. It was like getting credit for eating candy. I love puzzles and games; logic seemed to be too much fun to be serious or important.

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But to many students, especially many philosophy students, logic seems too abstract and mathematical to be fun. We study philosophy because we want to think about metaphysics or morality or truth or beauty. Logic prides itself on its lack of content. Moreover, there are rules in logic that can be violated. You can get problems wrong. The solutions to problems are not always obvious. My advice to students who have difficulty with the computational or mathematical portions of the text is to practice, frequently. Do a lot of exercises, and do not let the work pile up. It is far better to work just a little each day than to try to pick up logic skills in long cramming sessions. The key ideas of the formal logic in this book were developed in the late nineteenth and early twentieth centuries. But logic is a much older discipline. Before starting our formal work, let’s look briefly at the history of logic and how the contemporary notion of logical consequence was developed.

1.3: A SHORT HISTORY OF LOGIC Aristotle, who lived in the fourth century b.c.e., famously described some fundamental logical rules, called categorical syllogisms. The categorical syllogisms described relations among four kinds of statements, known since the early Middle Ages as A, E, I, and O, and which we see in 1.3.1. 1.3.1

A E I O

All Fs are Gs. No Fs are Gs. Some Fs are Gs. Some Fs are not Gs.

In categorical logic, the fundamental elements are portions of assertions. The Fs and Gs of Aristotelian syllogisms stand for general terms, like ‘people’ or ‘Martians’ or ‘red’ or ‘mortal’. We will look at the modern version of term ‘logic’, called predicate or quantificational logic, in chapters 4 and 5. In the third century b.c.e., the stoic philosopher Chrysippus developed a propositional logic, in which the fundamental elements are complete assertions rather than terms. Some complete assertions are simple; others are complex. Complex assertions are composed of simple assertions combined according to logical rules. In chapters 2 and 3, we will look at the rules of propositional logic. Through the Middle Ages, although there were some major advances in logic, the structure of the discipline was generally stable. After the scientific revolution, philosophers started paying more attention to human psychological capacities. This focus, which we can see in Descartes, Locke, and Hume, culminated in the late eighteenthcentury work of Kant, and the early nineteenth-century work of Hegel. Kant’s logic was essentially a description of how human beings create their experiences by imposing, a priori, conceptual categories on an unstructured manifold given in sensation. The term ‘a priori’ indicates that Kant believed that some of our intellectual activity

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occurs prior to, or independent of, experience. Although he distinguished these a priori capacities from purely subjective psychological processes, logic, for Kant, still concerned human reasoning rather than objective rules of consequence. Moreover, according to Kant, logic, as a discipline, was complete: We shall be rendering a service to reason should we succeed in discovering the path upon which it can securely travel, even if, as a result of so doing, much that is comprised in our original aims, adopted without reflection, may have to be abandoned as fruitless. That logic has already, from the earliest times, proceeded upon this sure path is evidenced by the fact that since Aristotle it has not required to retrace a single step, unless, indeed, we care to count as improvements the removal of certain needless subtleties or the clearer exposition of its recognised teaching, features which concern the elegance rather than the certainty of the science. It is remarkable also that to the present day this logic has not been able to advance a single step, and is thus to all appearance a closed and completed body of doctrine. (Critique of Pure Reason B17)

In the nineteenth century, several developments led mathematicians to worry about logical entailments and to call Kant’s claims about logic, its completeness and its psychological status, into question. Because these mathematical worries led directly to the logic in this book, I will take a short detour to discuss two of them: the problem of infinity and non-Euclidean geometries. For nearly two hundred years, mathematicians had worked with the calculus of Newton and Leibniz. The calculus allowed mathematicians to find the area under a curve by dividing the area into infinitely many infinitely small areas. Working with infinity, both small and large, seemed problematic, even if the resulting calculations were successful. An infinitely small region seemed indistinguishable from an empty region, one with zero size. The sum of the sizes of any number of empty regions should be zero. To make matters worse, Cantor, in the mid-nineteenth century, discovered a proof that there are different sizes of infinity—indeed, there are infinitely many different sizes of infinity. Infinite size had long been identified with God, one of the divine properties in contrast to our human finitude. Cantor’s proof struck many mathematicians as absurd, even heretical, but they could not find a flaw in his logic. Developments in geometry raised similar worries about mathematical inferences. Consider the first four axioms, or postulates, of Euclidean geometry, at 1.3.2. 1.3.2

The First Four Axioms of Euclidean Geometry 1. Between any two points, one can draw a straight line. 2. Any straight line segment can be extended indefinitely, to form a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent.

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Euclid relied on a commonsense interpretation of the terms in these axioms, especially terms for concepts like ‘straight’ and ‘right angle’. Given those ordinary concepts, it seemed obvious that the parallel postulate, Euclid’s fifth postulate, would also hold. 1.3.3 Euclid’s Fifth Axiom, the Parallel Postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

The parallel postulate is equivalent to Playfair’s postulate (after John Playfair, the Scottish mathematician who proposed his version in 1795), 1.3.4, which may be easier to visualize. 1.3.4 Given a line, and a point not on that line, there exists a single line that passes through the point and is parallel to the given line.

In the two millennia between Euclid and the early nineteenth century, geometers tried in vain to prove 1.3.3 or 1.3.4. They did so mainly by trying to find that some contradiction would arise from the denials of one or the other. They supposed that there was more than one parallel line through the given point. They supposed that there were no parallel lines through the given point. Both suppositions led to odd kinds of spaces. But neither supposition led to an outright contradiction. By the early nineteenth century, some mathematicians realized that instead of leading to contradiction, the denials of 1.3.3 and 1.3.4 lead to more abstract conceptions of geometry, and exciting new fields of study. Riemann and others explored the properties of elliptical geometries, those that arise when adding the claim that there are no parallel lines through the given point mentioned in Playfair’s postulate to the first four axioms. Lobachevsky, Gauss, and others explored the properties of hyperbolic geometries, which arise when adding the claim that there are infinitely many parallel lines through the given point in 1.3.4 to the first four axioms. In both elliptical and hyperbolic geometries, the notions of straightness and right-angularity, among others, have to be adjusted. Our original Euclidean conceptions had been smuggled in to the study of geometry for millennia, preventing mathematicians from discovering important geometric theories. These geometric theories eventually found important applications in physical science. The parallel postulate is also equivalent to the claim that the sum of the angles of a triangle is 180°. Consider an interstellar triangle, formed by the light rays of three stars, whose vertices are the centers of those stars. The sum of the angles of our interstellar triangle will be less than 180° due to the curvatures of space-time corresponding to the gravitational pull of the stars and other large objects. Space-time is not Euclidean, but hyperbolic. As in the case of Cantor’s work with infinity, mathematicians considering the counterintuitive results of non-Euclidean geometries worried that the laws of logical

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consequence were being flouted. Mathematicians and philosophers began to think more carefully about the notion of logical consequence. In the late nineteenth century, Gottlob Frege argued that hidden premises, like the assumption that there is only one size of infinity, or that all space must conform to the parallel postulate, had undermined mathematical progress. Frege wanted to ensure that all branches of mathematics, indeed all of human reasoning, were not liable to similar problems. He thus formalized the study of logical consequence, turning logic into a mathematical subject. In 1879, Frege published Begriffsschrift, or Concept-­Writing, a logical calculus that subsumed both Aristotle’s term logic and the stoics’ propositional logic. Frege’s logic extended and refined the rules of logic, generalizing results. The preface to Frege’s Begriffsschrift makes his motivation clear: So that nothing intuitive could intrude [into our concept of logical consequence] unnoticed, everything had to depend on the chain of inference being free of gaps. In striving to fulfill this requirement in the strictest way, I found an obstacle in the inadequacy of language: however cumbersome the expressions that arose, the more complicated the relations became, the less the precision was attained that my purpose demanded. . . . The present Begriffsschrift . . . is intended to serve primarily to test in the most reliable way the validity of a chain of inference and to reveal every presupposition that tends to slip in unnoticed, so that its origin can be investigated.

In this book, by separating the syntax of logic, its formation and derivation rules, from its semantics, its interpretations and our ascriptions of truth and falsity, we are attempting to fulfill Frege’s dream of a secure theory of logical consequence. Frege’s work, while not immediately recognized as revolutionary, spurred fifty years of intense research in the logical foundations of mathematics and reasoning generally. Perhaps the culmination of this flurry of research came in the early 1930s, with Alfred Tarski’s work on truth and Kurt Gödel’s incompleteness theorems. Frege’s logic, in a neater and more perspicuous form, is mainly the focus of this textbook. Frege, like Whitehead and Russell in their Principia Mathematica, used an axiomatic, or what is now known as a Hilbert-style, inferential system, after the eminent mathematician and logician David Hilbert. Their work was in the service of a view, called logicism, that arithmetic is really just logic in complex disguise. This book uses a now more common style called natural deduction, developed independently in the 1930s by the Polish logician Stanislaw Jaśkowski and the German logician Gerhard Gentzen. The brief history I just sketched is of course, in its brevity, highly misleading. Many others contributed to the history of logic, especially in the late Middle Ages. Frege was not the only logician to develop modern logic. Charles Sanders Peirce, for example, independently developed much of what made Frege’s logic innovative, his work extending and generalizing Aristotle’s categorical logic to include relations. Augustus De Morgan, even earlier than Peirce and Frege, had worked on relational logic. But Frege’s larger logicist project, coming mainly as a response to Kant’s philosophy

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and that of the early nineteenth-century idealists, is especially interesting to contemporary philosophers. Indeed, Frege produced seminal work not only in logic and philosophy of mathematics, but in philosophy of language, epistemology, and metaphysics. But enough about this engaging history. Let’s get started with the formal work. TELL ME MORE  • How does the natural deduction of this book differ from Frege’s original system of logic? See 6S.11: Axiomatic Systems.

1.4: SEPARATING PREMISES FROM CONCLUSIONS Given that the central goal of this book is a better understanding of logical consequence, of what follows from what, our first formal task is to look at the ways in which deductive inferences are structured. Compare a disorganized heap of stones with the same pile of stones arranged into the form of a house. The stones are the same. The difference between the two collections is the organizational structure of the latter collection. We want to examine the organizational structure of our inferences. The basic medium for inference is called an argument. An argument is a set of statements, or propositions, one of which is called the conclusion, and the others of which are called premises. The premises are used to support or establish the conclusion. Our first task, then, is to analyze arguments, separating premises from conclusions. When we analyze an argument, we represent it in a way that reveals its structure. I will call this process and its results regimentation. The term regimentation can indicate either of two different processes. First, we can regiment by putting an argument into numbered premise-conclusion form, a process we will explore in this section. Second, we can regiment by translating an argument into one of the formal languages in this book, a process we will explore in chapters 2–5. Let’s consider the argument 1.4.1 in order to regiment it into numbered premiseconclusion form. 1.4.1 We may conclude that texting while driving is wrong. This may be inferred from the fact that texting is distracting. And driving while distracted is wrong.

The conclusion of this argument is that texting while driving is wrong. The premises are that texting is distracting and that driving while distracted is wrong. Notice that the premises, together, are reasons that entail or support the conclusion. In addition to the words used to make the assertions in 1.4.1, there are premise and conclusion indicators. ‘We may conclude that’ is used to indicate a conclusion. ‘This may be inferred from the fact that’ is used to indicate a premise. ‘And’ is also used to indicate a premise. When we regiment an argument, we eliminate those indicators.

Argumentsare collections of propositions, called premises, together with a claim, called the conclusion, that the premises are intended to support.

A propositionis a statement, often expressed by a sentence. A regimentationof an argument helps reveal its logical structure, either by putting the argument into numbered premiseconclusion form, or by translating the argument into a formal language.

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Here are some premise and conclusion indicators: Premise Indicators

Conclusion Indicators

since because for in that may be inferred from given that seeing that for the reason that inasmuch as owing to

therefore we may conclude that we may infer that entails that hence thus consequently it follows that implies that as a result

Although these lists are handy, they should not be taken as exhaustive or categorical. Natural languages like English are inexact and non-formulaic. Not all sentences in an argument will contain indicators. ‘And’ often indicates the presence of an additional premise, but it can also be used to indicate the extension of a conclusion. Often you will have to judge from the content of an argument which propositions are premises and which are conclusions. The best way to identify premises and conclusions is to determine what the main point of an argument is, and then to see what supports that point. Once we have determined what the conclusion of an argument is, and which propositions are the premises, we can regiment the argument into numbered premiseconclusion form, identifying each of the premises (P1, P2, etc.) and indicating the conclusion with a ‘C’. Thus we can regiment 1.4.1 as the perspicuous 1.4.2, eliminating premise and conclusion indicators, and placing the conclusion at the end. 1.4.2

P1. Texting is distracting. P2. Driving while distracted is wrong. C. Texting while driving is wrong.

When regimenting an argument, the order of premises is unimportant. 1.4.3 would be just as good a regimentation as 1.4.2. 1.4.3

P1. Driving while distracted is wrong. P2. Texting is distracting. C. Texting while driving is wrong.

Similarly, the number of premises is not very important. You can combine or separate premises, though it is often useful to keep the premises as simple as possible. 1.4.4 is logically acceptable but not as perspicuous as 1.4.2 or 1.4.3. 1.4.4

P1. Driving while distracted is wrong, and texting is distracting. C. Texting while driving is wrong.

The most important task when first analyzing an argument is to determine its conclusion. The most serious mistake you can make in this exercise is to confuse premises and conclusions. Argument 1.4.5 is derived from Leibniz’s work.

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1.4.5 God is the creator of the world. If this world is not the best of all possible worlds, then either God is not powerful enough to bring about a better world or God did not wish this world to be the best. So, this world is the best of all possible worlds, because God is both omnipotent and all-good.

1.4.6 is a poor and misleading regimentation of 1.4.5, merely listing the assertions in the order in which they appear in 1.4.5. 1.4.6

P1. God is the creator of the world. P2. If this world is not the best of all possible worlds, then either God is not powerful enough to bring about a better world or God did not wish this world to be the best. P3. This world is the best of all possible worlds. C. God is both omnipotent and all-good.

The main problem with 1.4.6 is that it switches a premise and the conclusion. The central claim of 1.4.5 is that this is the best of all possible worlds. The “so” at the beginning of the last sentence is a hint to the conclusion. Thinking about the content of the argument should produce the same analysis. A proper regimentation would switch P3 and C, as in 1.4.7. 1.4.7

P1. God is the creator of the world. P2. If this world is not the best of all possible worlds, then either God is not powerful enough to bring about a better world or God did not wish this world to be the best. P3. God is both omnipotent and all-good. C. This world is the best of all possible worlds.

Sometimes it is not easy to determine how to separate premises from conclusions. Often, such discrimination requires broad context. For example, some single sentences contain both a premise and a conclusion. Such compound sentences must be divided. 1.4.8 is derived from Locke’s work. 1.4.8

Words must refer either to my ideas or to something outside my mind. Since my ideas precede my communication, words must refer to my ideas before they could refer to anything else.

A good regimentation of 1.4.8 divides the last sentence, as in 1.4.9. 1.4.9

P1. Words must refer either to my ideas or to something outside my mind. P2. My ideas precede my communication. C. Words must refer to my ideas before they could refer to anything else.

Some arguments contain irrelevant, extraneous information. When constructing an argument, it is better to avoid extraneous claims, lest you distract or mislead a reader. But when regimenting someone else’s argument, it is usually good practice to

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include all claims, even extraneous ones. Then, when you are evaluating an argument, you can distinguish the important premises from the extraneous ones. Lastly, some arguments contain implicit claims not stated in the premises. These arguments are called enthymemes. 1.4.10 is enthymemic. 1.4.10

P1. Capital punishment is killing a human being. C. Capital punishment is wrong.

Again, when regimenting an argument, we ordinarily show just what is explicitly present in the original. When evaluating an argument, we can mention suppressed premises. For instance, we can convert 1.4.10 into a more complete argument by inserting a second premise. 1.4.11

P1. Capital punishment is killing a human being. P2. Killing a human being is wrong. C. Capital punishment is wrong.

Notice that P2 here is contentious. Is it always wrong to kill a human being? What if you are defending yourself from a raging murderer? Or what if you are fighting a just war? Some people believe that euthanasia is acceptable for people suffering from terminal illnesses and in great pain. The contentiousness of P2 might explain why someone defending 1.4.10 might suppress it. Still, filling out an enthymeme is a job for later, once you have become confident regimenting arguments as they appear. Nothing in our logic will determine an answer to the interesting questions around claims like P2, but logic will help us understand the structures of arguments that contain or suppress such premises. KEEP IN MIND

The first step in analyzing arguments is to identify a conclusion and separate it from the premises. There are often indicators for premises and conclusions.

TELL ME MORE  • How can putting an argument into premise-conclusion form help me better understand a philosophy text? See 7.4: Logic and the Philosophy of Religion.

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EXERCISES 1.4 Regiment each of the following arguments into premiseconclusion form. The inspiration for each argument is noted; not all arguments are direct quotations. 1. Statements are meaningful if they are verifiable. There are mountains on the other side of the moon. No rocket has confirmed this, but we could verify it to be true. Therefore, the original statement is significant. (A. J. Ayer, Language, Truth, and Logic) 2. The workingman does not have time for true integrity on a daily basis. He cannot afford to sustain the manliest relations to men, for his work would be minimized in the market. (Henry David Thoreau, Walden) 3. The passage from one stage to another may lead to long-continued different physical conditions in different regions. These changes can be attributed to natural selection. Hence, the dominant species are the most diffused in their own country and make up the majority of the individuals, and often the most well marked varieties. (Charles Darwin, On the Origin of Species) 4. We must be realists about mathematics. Mathematics succeeds as the language of science. And there must be a reason for the success of mathematics as the language of science. But no positions other than realism in mathematics provide a reason. (Hilary Putnam) 5. Local timelines are temporally ordered. The faster you go, the quicker you get to your destination. As you go faster, time itself becomes compressed. But it is not possible to go so fast that you get there before you started. (Albert Einstein, Relativity) 6. The sphere is the most perfect shape, needing no joint and being a complete whole. A sphere is best suited to enclose and contain things. The sun, moon, planets, and stars are seen to be of this shape. Thus, the universe is spherical. (Nicolaus Copernicus, The Revolution of the Celestial Orbs) 7. The happiest men are those whom the world calls fools. Fools are entirely devoid of the fear of death. They have no accusing consciences to make them fear it. Moreover, they feel no shame, no solicitude, no envy, and no love. And they are free from any imputation of the guilt of sin. (Desiderius Erasmus, In Praise of Folly) 8. It is impossible for someone to scatter his fears about the most important matters if he knows nothing about the universe, but gives credit to myths. Without the study of nature, there is no enjoyment of pure pleasure. (Epicurus of Samos, Sovran Maxims)

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9. If understanding is common to all mankind, then reason must also be common. Additionally, the reason which governs conduct by commands and prohibitions is common to us. Therefore, mankind is under one common law and so are fellow citizens. (Marcus Aurelius, Meditations) 10. Rulers define ‘justice’ as simply making a profit from the people. Unjust men come off best in business. But just men refuse to bend the rules. So, just men get less and are despised by their own friends. (Plato, Republic) 11. We must take non-vacuous mathematical sentences to be false. This is because we ought to take mathematical sentences at face value. If we take some sentences to be non-vacuously true, then we have to explain our access to mathematical objects. The only good account of access is the indispensability argument. But the indispensability argument fails. (Hartry Field) 12. Labor was the first price, in that it yielded money that was paid for all things. But it is difficult to ascertain the proportion between two quantities of labor. Every commodity is compared with other exchanged commodities rather than labor. Therefore, most people better understand the quantity of a particular commodity than the quantity of labor. (Adam Smith, The Wealth of Nations) 13. Authority comes from only agreed conventions between men. Strength alone is not enough to make a man into a master. Moreover, no man has natural authority over his fellows and force creates no right. (Jean Jacques Rousseau, The Social Contract) 14. Just as many plants only bear fruit when they do not grow too tall, so in the practical arts, the theoretical leaves and flowers must not be constructed to sprout too high, but kept near to experience, which is their proper soil. (Carl von Clausewitz, On War) 15. The greatest danger to liberty is the omnipotence of the majority. A democratic power is never likely to perish for lack of strength or resources, but it may fall because of the misdirection of this strength and the abuse of resources. Therefore, if liberty is lost, it will be due to an oppression of minorities, which may drive them to an appeal to arms. (Alexis de Tocqueville, Democracy in America) 16. There is no distinction between analytic and synthetic claims. If there is an analytic/synthetic distinction, there must be a good explanation of synonymy. The only ways to explain synonymy are by interchangeability salva veritate or definition. However, interchangeability cannot explain synonymy. And definition presupposes synonymy. (W. V. Quine) 17. The object of religion is the same as that of philosophy; it is the internal verity itself in its objective existence. Philosophy is not the wisdom of the world, but the knowledge of things that are not of this world. It is not the knowledge of external mass, empirical life and existence, but of the eternal, of the nature of God, and all which flows from his nature. This nature ought to manifest and develop

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itself. Consequently, philosophy in unfolding religion merely unfolds itself and in unfolding itself it unfolds religion. (Georg Wilhelm Friedrich Hegel, The Philosophy of Religion) 18. Every art and every inquiry, and similarly every action and pursuit, is thought to aim at some good; and for this reason the good has rightly been declared to be that at which all things aim. (Aristotle, Nicomachean Ethics) 19. By ‘matter’ we are to understand an inert, senseless substance, in which extension, figure, and motion do actually subsist. But it is evident from what we have already shown that extension, figure, and motion are only ideas existing in the mind, and that an idea can be like nothing but another idea, and that consequently neither they nor their archetypes can exist in an unperceiving substance. Hence it is plain that the very notion of what is called matter, or corporeal substance, involves a contradiction in it. (George Berkeley, A Treatise Concerning the Principles of Human Knowledge) 20. Reading challenges a person more than any other task of the day. It requires the type of training that athletes undergo, and with the same life-long dedication. Books must be read as deliberately and reservedly as they were written. Thus, to read well, as in, to read books in a true spirit, is a noble exercise. (Henry David Thoreau, Walden) 21. The only course open to one who wished to deduce all our knowledge from first principles would be to begin with a priori truths. An a priori truth is a tautology. From a set of tautologies alone, only further tautologies can be further deduced. However, it would be absurd to put forward a system of tautologies as constituting the whole truth about the universe. Therefore, we cannot deduce all our knowledge from first principles. (A. J. Ayer, Language, Truth, and Logic) 22. Men, in the state of nature, must have reached some point when the obstacles maintaining their state exceed the ability of the individual. Then the human race must either perish or change. Men cannot create new forces, only unite and direct existing ones. Therefore, they can preserve themselves only by combining forces great enough to overcome resistance. (Jean Jacques Rousseau, On the Social Contract) 23. Physics can be defined as the study of the laws that regulate the general properties of bodies regarded en masse. In observing physics, all senses are used. Mathematical analysis and experiments help with observation. Thus in the phenomena of physics man begins to modify natural phenomena. (Auguste Comte, The Course in Positive Philosophy) 24. There are not two indiscernible individuals in our world. If there were two indiscernible individuals in our world then there must be another possible world in which those individuals are switched. God could have had no reason for choosing one of these worlds over the other. But God must have a reason for acting as she does. (Leibniz)

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25. In aristocratic countries, great families have enormous privileges, which their pride rests on. They consider these privileges as a natural right ingrained in their being, and thus their feeling of superiority is a peaceful one. They have no reason to boast of the prerogatives that everyone grants to them without question. So, when public affairs are directed by an aristocracy, the national pride takes a reserved, haughty, and independent form. (Alexis de Tocqueville, Democracy in America) 26. It must be some one impression that gives rise to every real idea. But self or person is not any one impression, but that to which our several impressions and ideas are supposed to have a reference. If any impression gives rise to the idea of self, that impression must continue invariably the same through the whole course of our lives, since self is supposed to exist after that manner. But there is no impression constant and invariable. Pain and pleasure, grief and joy, passions and sensations succeed each other and never all exist at the same time. It cannot, therefore, be from any of these impressions or from any other that the idea of self is derived, and, consequently, there is no idea of the self. (David Hume, A Treatise of Human Nature) 27. Every violent movement of the will, every emotion, directly agitates the body. This agitation interferes with the body’s vital functions. So, we can legitimately say that the body is the objectivity of the will. (Arthur Schopenhauer, The World as Will and Idea) 28. The work of the defensive forces of the ego prevents repressed desires from entering the conscious during waking life, and even during sleep. The dreamer knows just as little about the meaning of his dreams as the hysteric knows about the significance of his symptoms. The technique of psychoanalysis is the act of discovering through analysis the relation between manifest and latent dream content. Therefore, the only way to treat these patients is through the technique of psychoanalysis. (Sigmund Freud, The Origin and Development of Psychoanalysis) 29. Either mathematical theorems refer to ideal objects or they refer to objects that we sense. If they refer to ideal objects, the radical empiricist cannot defend our knowledge of them, since we never sense such objects. If they refer to objects that we sense, they are false. So, for the radical empiricist, mathematical theorems are either unknowable or false. In either case, the radical empiricist cannot justify any proof of a mathematical theorem. (John Stuart Mill) 30. My mind is distinct from my body. I have a clear and distinct understanding of my mind, independent of my body. I have a clear and distinct understanding of my body, independent of my mind. Whatever I can clearly and distinctly conceive of as separate can be separated by God and so are really distinct. (René Descartes, Meditations on First Philosophy)

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1.5: VALIDITY AND SOUNDNESS Consider the following three arguments. 1.5.1

P1. All philosophers are thinkers. P2. Socrates is a philosopher. C. Socrates is a thinker.

1.5.2

P1. All persons are fish. P2. Alexander Hamilton is a person. C. Alexander Hamilton is a fish.

1.5.3

P1. All mathematicians make calculations. P2. Padmé Amidala makes calculations. C. Padmé Amidala is a mathematician.

1.5.1 is a good argument for two reasons. First, the conclusion follows from the premises. Second, the premises are true. 1.5.2 and 1.5.3 are both bad arguments, but for different reasons. In 1.5.2, the conclusion follows from the premises, but the first premise is false. In 1.5.3, the premises are true, we can suppose, but the conclusion does not follow from the premises. We call arguments like 1.5.3 invalid. 1.5.2 is valid, but unsound. The validity of an argument depends on its form. The conclusion of a valid argument follows logically from the premises. In this book, we will develop precise definitions of validity, and thus logical consequence, for formal languages. We will see that certain forms of argument are valid and certain forms are invalid. We will explore rigorous methods for distinguishing between valid and invalid arguments. In contrast, the soundness of an argument, as I will use the term, depends both on its formal structure and on the truth of its premises. A sound argument is a valid argument with true premises. Only valid arguments can be sound. A sound argument has both a valid structure and true premises. Valid arguments are important because in deductive logic, if the form of an argument is valid and the premises are all true, then the conclusion must be true. The previous sentence is the most important sentence of this book. The power of deductive logic is simply that if the premises of an argument in a valid form are true, then, on pain of contradiction, the conclusion of the argument must be true. In invalid arguments, the premises can be true at the same time that the conclusion is false. The central theme of this book, then, is to identify the valid forms of argument. The validity of an argument is independent of the truth of the premises of an argument. As we saw, 1.5.1 is both valid and sound, while 1.5.2 is valid but unsound. An argument, for example 1.5.3 or 1.5.4, can also have all true premises while being invalid. 1.5.4

P1. 2 + 2 = 4. P2. The sky is blue. C. Kant wrote Critique of Pure Reason.

An argument is valid when the conclusion is a logical consequence of the premises.

A valid argument is sound if, and only if, all of its premises are true. A valid argument is unsound when at least one of its premises is false.

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Although the soundness of an argument depends on the truth of the premises, validity is more closely related to possibility. An argument is sound only if the premises are actually true. An argument is valid if it is impossible to make the conclusion false while the premises are true by substituting alternative sentences of the same logical form. This last claim will become a little clearer once we have looked more carefully at the nature of logical form. The arguments 1.5.5–1.5.7 share a logical form. 1.5.5

Either the stock market will rise or unemployment will go up. The market will not rise. So, unemployment will increase.

1.5.6

You will get either rice or beans. You do not get the rice. So, you will have the beans.

1.5.7

The square root of two is either rational or irrational. It is not rational. So, it’s irrational.

We can represent this common logical form by replacing the specific propositions in the argument with variables, using the same variable each time a particular proposition is repeated. 1.5.8

Either P or Q. Not P. So, Q.

Just as an architect, when building a building, focuses on the essential structures, so a logician looks mainly at the form of an argument, ignoring the content of the sentences. ‘P’ and ‘Q ’, above, are variables, standing for statements and allowing us to see the logical form of the argument more clearly. We call the form displayed at 1.5.8 disjunctive syllogism. In chapter 3 we will identify some basic valid forms, and use them to determine whether any argument is valid. To start the process of identifying valid forms, in this section we rely on our intuitive judgments about whether or not some sample inferences are valid. The main purpose of chapter 2 is to develop a rigorous method to determine whether any form is valid. In our study of propositional logic, we will use capital English letters to stand for simple, positive propositions. Simple propositions are often of subject-predicate form, but not necessarily. They are the shortest examples of statements; they cannot be decomposed further in propositional logic. In predicate logic, chapters 4 and 5, we work beneath the surface of propositions.

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KEEP IN MIND

In deductive logic, if the form of an argument is valid, and the premises are all true, then the conclusion must be true. An argument is valid if the conclusion follows logically from the premises. The validity of an argument depends on its form and is independent of the truth of its premises. A valid argument is sound if all of its premises are true. Only valid arguments can be sound.

TELL ME MORE  • W hat is the difference between deduction and induction? See 7.1: Deduction and Induction.

EXERCISES 1.5 Determine whether each of the following arguments is intuitively valid or invalid. For valid arguments, determine whether they are sound (if you can). 1. Archaeologists are anthropologists. Anthropologists are social scientists. It follows that archaeologists are social scientists. 2. All trees are tall. All tall things are hard to climb. So, all trees are hard to climb. 3. Frankfort is the capital of Kentucky. Trenton is the capital of New Jersey. Phoenix is the capital of Arizona. It follows that Raleigh is the capital of North Carolina. 4. All princesses are women. Kate Middleton is a princess. Therefore, Kate Middleton is a woman. 5. All horses are mammals. All horses have four legs. So, all mammals have four legs. 6. All unicorns are pink. All unicorns have horns. So, if something is pink and has a horn, it is a unicorn. 7. Either some cats are black or all cats are fluffy. All cats are black. So, some cats are fluffy. 8. Some cats are black. Some cats are fluffy. So, some cats are black and fluffy. 9. Some cats are fluffy. All cats have whiskers. So, all fluffy cats have whiskers. 10. All doctors went to medical school. All medical students took chemistry. So, all doctors have taken chemistry.

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11. All circles are shapes. All shapes have distinct sides. Therefore, a circle has distinct sides. 12. All musicians play piano. Some musicians sing opera. So, some musicians play piano and sing opera. 13. Some percussionists play piano. All pianists are musicians. Therefore, all percussionists are musicians. 14. Plants need sunlight to grow. Plants need water to grow. So, plants need two things to grow. 15. Thunder occurs only when it is raining. It never rains in February. Hence, there is never any thunder in February. 16. All windows are made of glass. Glass is transparent. So, all windows must be transparent. 17. Either it is raining or it is sunny, but not both. It is not raining. So, it is sunny. 18. Either I stop smoking or I risk getting ill. If I stop smoking, then I will have withdrawal symptoms. If I get ill, then I risk death. So, either I have withdrawal symptoms or I risk death. 19. Some fish live in the Atlantic Ocean. The Atlantic Ocean is a body of water. So, some fish live in water. 20. All rats have tails. Some rats are white. So, all rats are white and have tails. 21. All rats have tails. Some rats are white. Therefore, some white things have tails. 22. All squares are rectangles. All rectangles are parallelograms. All parallelograms are quadrilaterals. Therefore, all squares are quadrilaterals. 23. All professional singers are classically trained. Some classically trained singers are Italian. So, some professional singers are Italian. 24. Kangaroos live in Australia. Sydney is in Australia. Hence, kangaroos live in Sydney. 25. All logicians are philosophers. All philosophers study Kant. It follows that all logicians study Kant. 26. If mathematical objects exist, then either we have mathematical intuition or we can’t know about them. We don’t have mathematical intuition. So, mathematical objects don’t exist. 27. Either only the present is real or time is four-dimensional. Time is four-dimensional. So, only the present is real. 28. Logic is a priori if, and only if, mathematics is. Mathematics is a priori if, and only if, metaphysics is. So, logic is a priori if, and only if, metaphysics is. 29. Nietzsche believes in eternal recurrence, but Spinoza does not. If Heidegger believes in the reality of time, then Spinoza believes in eternal recurrence. So, Heidegger does not believe in the reality of time.

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30. Objective morality is either consequentialist or deontological. If objective morality is deontological then Aristotle is a relativist. So, Aristotle is not a relativist. 31. All logical empiricists are verificationists. Some verificationists are holists. So, some holists are logical empiricists. 32. Either Plato taught Aristotle or Aristotle taught Plato. But Aristotle taught Alexander, and Alexander was not taught by Plato. So, Plato taught Aristotle. 33. Descartes corresponded with Elisabeth of Bohemia and Queen Christina of Sweden. So, Queen Christina and Elisabeth corresponded with each other. 34. If Hegel was influenced by Kant, then Marx was influenced by Hegel. Marx was influenced by Hegel if, and only if, Nietzsche was influenced by Marx. So, if Hegel was influenced by Kant, then Nietzsche was influenced by Marx. 35. There is a difference between correlation and causation only if we have knowledge of the laws of nature. But the laws of nature are obscured to us. So, correlation is causation. 36. All ravens are black. But black is a color. And nothing has color. So, there are no ravens. 37. All humans have some virtues. Not all humans have all virtues. So, some humans lack some virtues, but no humans lack all virtues. 38. If infinity is actual, then Achilles cannot catch the tortoise. If infinity is potential, then Achilles can catch the tortoise. Infinity is either actual or potential. So, Achilles can catch the tortoise. 39. If I am my body, then the self is constantly changing and does not persist through time. If the self does not persist through time, then the person who borrows money is not the one who returns it. So, if the person who borrows money is the person who returns it, then I am not my body. 40. If knowledge is justified true belief, then Gettier cases are not counterexamples. But Gettier cases are counterexamples, and there are others, too. So, knowledge is justified true belief with a causal connection between the knower and the object of knowledge.

KEY TERMS argument, 1.4 conclusion, 1.4 premise, 1.4

proposition, 1.4 regimentation, 1.4 sound argument, 1.5

unsound argument, 1.5 valid argument, 1.5

Chapter 2 Propositional Logic Syntax and Semantics

2.1: LOGICAL OPERATORS AND TRANSLATION

Logical operatorsare tools for combining propositions or terms.

The subjects of chapters 2 and 3 are the syntax and semantics of a formal language of propositional logic, which I will call PL. Propositional logic is the logic of propositions and their inferential relations. It is not easy to define ‘proposition’, but propositions are often taken to be whatever it is that we call true or false. We might say that a proposition is a statement, often expressed by a sentence. Some people take propositions to be just sentences. Others take them to be sets or types of sentences. Still others take them to be the meanings of sentences, though the nature of the meaning of a sentence is another controversial question. These controversies need not get in the way of our work with PL. In the first two sections of this chapter, to construct the language of PL, we will specify its syntax. In section 2.3, to interpret the language, we will specify its semantics. By the end of this chapter, in sections 2.6 and 2.7, we will have worked through a formal definition of validity, a way to determine, for any inference in PL, whether or not it is valid. In chapter 3, we will look more deeply at inferences using PL. Natural languages like English, as well as many formal languages, have a finite, if very large, stock of simple sentences. From these we can construct indefinitely many, perhaps infinitely many, grammatically correct complex sentences. To produce complex sentences from simple ones, we use what the logician calls operators or connectives. In natural language, we usually find it convenient to write or speak in short sentences. In logic, we assume an unrestricted ability to construct sentences of any length. Even in natural languages, though, we assume something like an ability to compose longer sentences indefinitely. This passage is from a much longer story composed of a single sentence: Now they’re going to see who I am, he said to himself in his strong new man’s voice, many years after he had seen the huge ocean liner without lights and without any sound which passed by the village one night like a great

22

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uninhabited palace, longer than the whole village and much taller than the steeple of the church, and it sailed by in the darkness toward the colonial city on the other side of the bay that had been fortified against buccaneers, with its old slave port and the rotating light, whose gloomy beams transfigured the village into a lunar encampment of glowing houses and streets of volcanic deserts every fifteen seconds . . . (Gabriel García Márquez, “The Last Voyage of the Ghost Ship,” emphases added)

Grammarians often bristle at long, run-on sentences like this one. But from a logical point of view, we can build sentences of indefinite length by repeated applications of operators like the ‘and’ in Márquez’s story. Such operators, including ‘or’ and ‘not’, are often all called conjunctions in grammar, though in logic we reserve the term ‘conjunction’ for just the operator for which we use ‘and’. The system of propositional logic that we will study uses five operators, which we identify by their syntactic properties, or shapes: Tilde Dot Vel Horseshoe Triple bar

∼ ∙ ∨ ⊃ ≡

These operators are used to represent logical operations on sentences. We will consider five basic logical operations, though systems of logic can be built from merely one or two operations. We could also introduce other, less intuitive logical operations. These five operators are standard for propositional logic: Negation Conjunction Disjunction Material implication The biconditional

∼ ∙ ∨ ⊃ ≡

We read or write sentences of English from left to right, and we might think of them as being composed in that way. But the logical structure of a complex sentence is grounded in its simple parts and the operators used, like bricks and mortar. We think of complex sentences, as we will see in the next section, as being composed or built up from smaller parts using the operators. Along with the assumption of our ability to construct sentences of indefinite length, we presume a principle, called compositionality, that the meaning of the longer sentences is determined by the meanings of the shorter sentences, along with the meanings of the conjunctions or other logical operators. The compositionality of our logic allows us to understand the properties of even very long sentences as long as we understand the nature of the logical operators. This section is a detailed explication of each of our five operators.

Compositionality: the meaning of a complex sentence is determined by the meanings of its component parts.

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Negation Negation, ∼, is the logical operator used for ‘it is not the case that’ and related terms.

A unary operatorapplies to a single proposition. Binary operators

relate or connect two propositions.

Negation is a unary operator, applying to one propositional variable. The other four operators are all binary. Some English indicators of negation include the following: not it is not the case that it is not true that it is false that

2.1.2–2.1.4 each express a negation of 2.1.1. 2.1.1 2.1.2 2.1.3 2.1.4

John will take the train. John won’t take the train. It’s not the case that John will take the train. John takes the train . . . not!

We can represent 2.1.1 as ‘P’ and each of 2.1.2–2.1.4 as the negation ‘∼P’. 2.1.5–2.1.7 are all negations, too. 2.1.5 2.1.6 2.1.7

∼R ∼(P ∙ Q) ∼{[(A ∨ B) ⊃ C] ∙ ∼D}

2.1.5 is built out of a simple sentence ‘R’ and a negation in front of it. 2.1.6 is built out of two simple sentences, conjoined and then negated. 2.1.7 is the negation of a conjunction of a conditional and another negation, though now we’re getting a little bit ahead of ourselves. Negation is a fairly simple logical operator to translate, though some subtleties are worth considering. Ordinarily, when we translate natural language into logical language, we want to reveal as much logical structure as we can so that we can see the logical relations among sentences. We use single capital letters to represent simple, positive sentences, so that we can show the logical operation of negation on those simple sentences. For example, we symbolize ‘Pedro has no beard’ as ‘∼P’, where ‘P’ stands for ‘Pedro has a beard’. For some sentences, it is not clear whether to use a negation when symbolizing. 2.1.9 has a negative feel to it. 2.1.8 2.1.9

Kant affirms that arithmetic is synthetic a priori. Kant denies that arithmetic is synthetic a priori.

It would be misleading to represent 2.1.9 as the negation of 2.1.8, though. Denying is not the negation of affirming. There are two ways to fail to affirm P. First, one can deny P. Second, one can remain silent. Denying is an activity that is related to affirming, but it is not, strictly, the negation of affirming. For similar reasons, rejecting, disputing, and dissenting are not negations of accepting or affirming. We want our simple sentences to be positive, if possible, but not at the expense of the meaning of the original. Sometimes a negative verb can represent a positive act, or anyway not the logical negation of any simple act.

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Conjunction These are some English indicators of a logical conjunction:

Conjunction, ∙, is the logical operator used for ‘and’ and related terms. The formulas joined by a conjunction are called conjuncts.

and still but moreover also although however nevertheless yet both

2.1.10–2.1.13 are English sentences that we can represent as conjunctions. 2.1.10 2.1.11 2.1.12 2.1.13

Angelina walks the dog, and Brad cleans the floors. Although Beyonce walks the dog, Jay cleans the floors. Key and Peele are comedians. Carolina is nice, but Emilia is really nice.

A∙B B∙J K∙P C∙E

Although the logical operator in each of 2.1.10–2.1.13 is a conjunction, the tone of the conjunction varies. Logicians often distinguish between the logical and pragmatic properties of language. ‘And’ and ‘but’ are both used to express conjunctions even though they have different practical uses. We use conjunctions to combine complete sentences. In English, 2.1.12 is short for a more complete sentence like 2.1.14. 2.1.14

Key is a comedian and Peele is a comedian.

Sometimes, sentences using ‘and’ are not naturally rendered as conjunctions. 2.1.15

Key and Peele are brothers.

2.1.15 is most naturally interpreted as expressing a relation between two people, and not a conjunction of two sentences. Of course, 2.1.15 could also be used to express the claim that both Key and Peele are monks, in which case it would best be represented in logic as a conjunction. In propositional logic, we regiment the most natural sense of 2.1.15 merely as a simple letter: ‘P’, say. We will see how to represent the sibling relation more finely in chapter 5. The difference between the two interpretations cannot be found in the sentence itself. It has to be seen from the use of the sentence in context. Many sentences are ambiguous when seen out of context. In symbols, 2.1.16–2.1.18 are all conjunctions. 2.1.16 P ∙ ∼Q 2.1.17 (A ⊃ B) ∙ (B ⊃ A) 2.1.18 (P ∨ ∼Q) ∙ ∼[P ≡ (Q ∙ R)]

Disjunction Disjunction is sometimes called alternation. Some English indicators of disjunction include the following: or either unless

Disjunction, ∨, is the logical operator used for ‘or’ and related terms. The formulas joined by a disjunction are called disjuncts.

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Unlessis represented as a disjunction in PL .

Most disjunctions use an ‘or’, though ‘unless’ is also frequently used for disjunction. 2.1.19–2.1.21 are English sentences that we can represent as disjunctions. 2.1.19 2.1.20 2.1.21

Either Paco makes the website or Matt does. Jared or Rene will go to the party. Tomas doesn’t feed the kids unless Aisha asks him to.

P∨M J∨R ∼T ∨ A

In symbols, 2.1.22–2.1.24 are all disjunctions. 2.1.22 ∼P ∨ Q 2.1.23 (A ⊃ B) ∨ (B ⊃ A) 2.1.24 (P ∨ ∼Q) ∨ ∼[P ≡ (Q ∙ R)] Not bothP and Q is represented in PL as

∼(P ∙ Q ).

Standard combinations of negations with disjunctions and conjunctions are useful to learn. The negation of a conjunction is simply ‘not both’, as in 2.1.25. 2.1.25

It is not the case that both Adam goes to the movies and Bianca works on her paper.

2.1.25 is the denial that both claims hold, leaving open the possibility that one (but not the other) holds. Such a claim is best translated as 2.1.26, which (as we will see in section 3.3) is logically equivalent to the form at 2.1.27. 2.1.26 2.1.27 NeitherP nor Q is represented in PL as

∼(P ∨ Q ).

∼(A ∙ B) ∼A ∨ ∼B

Not both A and B Not both A and B

In parallel, the negation of a disjunction is just the common structure of ‘neither’, short for ‘not either’. 2.1.28 is both a denial that Caleb takes ethics and a denial that Danica does. 2.1.28

Neither Caleb nor Danica takes ethics.

2.1.28 is most directly translated as 2.1.29, the negation of a disjunction. 2.1.30, the conjunction of two negations, is logically equivalent to both, and also acceptable. 2.1.29 2.1.30

∼(C ∨ D) ∼C ∙ ∼D

Neither C nor D Neither C nor D

‘Neither’ and ‘not-both’ sentences are not logically equivalent to each other, so it is important not to confuse the two.

Material Implication (the Conditional) Material implication, ⊃, is the logical operator used for conditionals, ‘if . . . then . . . statements’, and related terms.

The operator that we call material implication (or the material conditional) is most closely associated with ‘if . . . then . . .’ of natural language. We will use it to translate sentences with that structure, and related ones, though it is not a perfect representation of the natural language conditional. Briefly, some aspects of the natural-language conditional are simply not represented by the logic in this book. Here are some English indicators of material implication: if entails only if means only when provided that

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is a necessary condition for is a sufficient condition for implies

given that on the condition that in case

The order of the parts of a material implication is important in ways unlike the order of disjunctions and conjunctions. ‘A ∙ B’ and ‘B ∙ A’ are logically equivalent; so are ‘A ∨ B’ and ‘B ∨ A’. But ‘A ⊃ B’ and ‘B ⊃ A’ must be carefully distinguished. We even have different names for the different sides of the ⊃: the antecedent precedes the horseshoe and is the ‘if ’ clause; the consequent follows the horseshoe and is the ‘then’ clause. In ‘A ⊃ B’, ‘A’ is the antecedent and ‘B’ is the consequent. In ‘B ⊃ A’, ‘B’ is the antecedent and ‘A’ is the consequent. 2.1.31–2.1.39 are some examples of natural-language conditionals and their usual translations into PL, using ‘A’ to stand for ‘Marina dances’ and ‘B’ to stand for ‘Izzy plays tennis’. Make sure to recognize the placement of antecedents and conditionals. 2.1.31 If Marina dances, then Izzy plays tennis.

If A then B

A⊃B

2.1.32 Marina dances if Izzy plays tennis.

If B then A

B⊃A

2.1.33 Marina dancing entails (implies, A entails (implies, means) B A ⊃ B means) that Izzy plays tennis. 2.1.34 Marina dances given (provided, A given (provided, on the on the condition) that Izzy    condition) that B plays tennis.

B⊃A

2.1.35 Marina dances only if (only when) Izzy plays tennis.

A⊃B

A only if (only when) B

2.1.36 Marina dancing is a necessary A is necessary for B condition for Izzy playing tennis.

B⊃A

2.1.37 Marina dancing is a sufficient A is sufficient for B condition for Izzy playing tennis.

A⊃B

2.1.38 A necessary condition of Marina B is necessary for A dancing is Izzy playing tennis.

A⊃B

2.1.39 A sufficient condition for Marina B is sufficient for A dancing is Izzy playing tennis.

B⊃A

Note that in both 2.1.31 and 2.1.32, whatever follows the ‘if ’ is the antecedent. Conditions that entail, imply, or mean, as in 2.1.33, and conditions that are given or provided, as in 2.1.34, are also antecedents. In contrast, whatever follows an ‘only if ’ or an ‘only when’, as in 2.1.35, is a consequent. If I write an essay only when the deadline is looming, then if I’m writing an essay, the deadline is looming. But even if the deadline is looming, I might not be writing! Note in 2.1.36–2.1.39 that necessary conditions are consequents, whereas sufficient conditions are antecedents. The case of sufficient conditions is fairly easy to

In a conditional, the formula that precedes the ⊃ is called the antecedent; the formula that follows the ⊃ is called the consequent .

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understand. If some condition in the antecedent is met, then the consequent follows; the condition is sufficient to entail the consequent. Necessary conditions are trickier. If A is necessary for B, then if B is true, we can infer that A must also be true. For example, oxygen is necessary for burning. So, if something is burning, there must be oxygen present; the necessary condition is in the consequent. Given that the presence of oxygen is a necessary condition for something burning, we cannot infer from the presence of oxygen to something burning. Oxygen is not sufficient to cause a fire; it’s just one of various necessary conditions. To remember that sufficient conditions are antecedents and necessary conditions are consequents, we can use the mnemonic ‘SUN’. Rotating the ‘U’ to a ‘⊃’ we get ‘S ⊃ N’. In symbols, 2.1.40–2.1.42 are all conditionals. 2.1.40 ∼P ⊃ Q 2.1.41 (A ⊃ B) ⊃ (B ⊃ A) 2.1.42 (P ∨ ∼Q) ⊃ ∼[P ≡ (Q ∙ R)]

While we’re defining terms, we can define three conditionals using traditional names that you might run into. The names of the conditionals 2.1.44–2.1.46 are all relative to the original conditional at 2.1.43. 2.1.43 2.1.44 2.1.45 2.1.46

The conditional Its converse Its inverse Its contrapositive

A⊃B B⊃A ∼A ⊃ ∼B ∼B ⊃ ∼A

A statement and its contrapositive, 2.1.43 and 2.1.46, are logically equivalent. The inverse and the converse of a conditional, 2.1.44 and 2.1.45, are logically equivalent to each other. But a conditional is not equivalent to either its inverse or its converse. These names are holdovers from the traditional, Aristotelian logic, and ‘inverse’ especially is not much used in modern logic. I will explain what ‘logical equivalence’ means in more detail in section 2.5.

The Biconditional The biconditional, ≡, is the logical operator used for ‘if and only if ’ and related terms.

Our final propositional operator, the biconditional, is really the conjunction of a conditional with its converse. We see biconditionals in definitions, which give both necessary and sufficient conditions. Some English indicators of a biconditional include the following: if and only if when and only when just in case is a necessary and sufficient condition for

The biconditional ‘A ≡ B’ is short for ‘(A ⊃ B) ∙ (B ⊃ A)’, to which we will return, once we are familiar with truth conditions. ‘If and only if ’ statements often indicate definitions. For example, something is water if, and only if, it is H 2O. Thus, if

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something is water, then it is H2O. And, if something is H 2O, then it is water. ‘If and only if ’ is sometimes abbreviated ‘iff ’. 2.1.47 and 2.1.48 are English examples of biconditionals. 2.1.47 2.1.48

You’ll be successful just in case you work hard and are lucky. S ≡ (W ∙ L) Something is a bachelor if, and only if, it is unwed and a man. B ≡ (∼W ∙ M)

In symbols, 2.1.49–2.1.51 are all biconditionals. 2.1.49 ∼P ≡ Q 2.1.50 (A ⊃ B) ≡ (B ⊃ A) 2.1.51 (P ∨ ∼Q) ≡ ∼[P ≡ (Q ∙ R)]

Translation and Ambiguity When translating between English and propositional logic, make sure to resolve or avoid ambiguities. 2.1.52 can be translated as either 2.1.53 or 2.1.54, as it stands. 2.1.52 You may have salad or potatoes and carrots. 2.1.53 (S ∨ P) ∙ C 2.1.54 S ∨ (P ∙ C)

There is an important difference between the two translations. In the first case, you are having carrots and either salad or potatoes. In the second case, you are either having one thing (salad) or two things (potatoes and carrots). To avoid ambiguities, look for commas and semicolons. 2.1.55 2.1.56

You may have salad or potatoes, and carrots. You may have salad, or potatoes and carrots.

With commas, 2.1.55 is clearly best translated as 2.1.53, while 2.1.56 is clearly best translated as 2.1.54. Still, not all sentences of English, or any natural language, are unambiguous. In the real world, when we want to disambiguate, we might ask the speaker what she or he means, or try to determine the meaning of an ambiguous sentence from context. In this book, I follow a convention of using commas or semicolons to assist in clarity.

Arguments and Numbered Premise-Conclusion Form The theme of this chapter is validity for arguments of PL. To that end, we will consider not just individual propositions, but complete arguments, premises, and conclusions. Thus, in the exercises, you will be asked to regiment arguments like 2.1.57 into numbered premise-conclusion form in PL. 2.1.57

Morality is backward-looking. For, if morality is possible, then it is either forward-looking or backward-looking. But we can be moral. And morality is not forward-looking.

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Remember that we can regiment in two ways, by putting an argument into numbered premise-conclusion form, or by translating the argument into a formal language. Here we will do both. For the first step, remember that in chapter 1, we indicated premises with ‘P’ and a number, and conclusions with a ‘C’. Here, we will adjust that form slightly, omitting the ‘P’s and ‘C’s, and using a ‘/’ to indicate the separation between the premises and the conclusion. Thus, we can write the argument 2.1.57 as 2.1.58. 2.1.58

1. I f morality is possible, then it is either forward-looking or backward-looking. 2. We can be moral. 3. Morality is not forward-looking. / Morality is backward-looking.

There are alternatives to the ‘/’ to indicate conclusions. We could just use ‘so’, or some other simple English conclusion indicator. Some people use ‘∴’ to indicate a conclusion. Sometimes logicians draw a horizontal line between the premises and conclusions, as at 2.1.59. 2.1.59

1. I f morality is possible, then it is either forward-looking or backward-looking. 2. We can be moral. 3. Morality is not forward-looking. Morality is backward-looking.

Neither 2.1.58 nor 2.1.59 is regimented into PL, which is our goal here. To regiment it, we need to choose propositional letters for the simple English sentences. I’ll use ‘P’ for ‘morality is possible’ and ‘we can be moral’, since I take those to be the same proposition expressed slightly differently. ‘F’ can stand for ‘morality is forwardlooking’ and ‘B’ for ‘morality is backward-looking.’ The result is 2.1.60. 2.1.60 1. P ⊃ (F ∨ B) 2. P 3. ∼F

/B

Notice that I put the conclusion on the same line as the last premise, rather than on a different line. This form will be useful later on, and it makes the argument just a bit more compact.

Summary Now that you have seen each of the five operators and their English-language approximations, you can start to translate both simple and complex English sentences into our artificial language, PL. Given a translation key, you can also interpret sentences of PL into English sentences. Translation is an art. In this section, I presented some guidelines for translating English terms, like ‘and’ and ‘if . . . then . . .’, into our precise formal language. These guidelines are not hard and fast rules. Natural language is flexible and inexact, which

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is part of why formal languages are useful. The indicators of conditionals are particularly liable to misconstrual. You must be careful to distinguish antecedents and consequents. Be especially wary of confusing ‘only if ’ with ‘if ’, and with ‘if and only if.’ Certain uses of the indicators are not even properly translated as logical operators. ‘Means’ is a conditional in “this means war” and “Beth’s deciding to join us means that Kevin will be uncomfortable.” But ‘means’ is not a conditional in “Josie means to go to the party tonight” and “ ‘querer’ means ‘to love’ in Spanish.” Sometimes the indicators can be quite misleading; we will even see, in section 4.2, instances of  ‘and’ that are best translated using disjunction! But the indicators provided are generally good hints about where to start with a translation, and the guidelines in this section should be violated only for good reasons. As you develop greater facility with the logical languages in the book, you will come to a better feel of how best to translate. And there are many acceptable alternatives to any translation, as we will see better after more discussion of logical equivalence. KEEP IN MIND

Our language PL uses five operators, which we identify by their syntactic properties. The five propositional logical operations are negation, conjunction, disjunction, material implication, and the biconditional. The operators always apply to complete propositions, whether simple or complex. The rules for translating conditionals are particularly tricky and require carefully distinguishing between antecedents and consequents.

TELL ME MORE  • Why is ‘unless’ translated as a disjunction? See 6.1: Notes on Translation with PL. • Why is material implication controversial? See 6.2: Conditionals. • W hat are the propositions of propositional logic? See 6S.7: The Propositions of Propositional Logic. • A re there other ways of expressing the logical operators? See 6S.10: Alternative Notations.

EXERCISES 2.1a Identify the antecedents and consequents of each of the following sentences. 1. If Abhishek studies religion, then Bima majors in sociology. 2. Cora working for her state senator entails that Danilo takes a job at the law office.

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3. Edwidge’s thinking about graduate school implies that she takes the GREs. 4. Fernanda will apply to medical school provided that organic chemistry goes well. 5. Gita plays lacrosse if her financial aid comes through. 6. Henry’s joining the history club is a necessary condition for Isabel to join. 7. Janelle becoming president of the robotics club is sufficient for Kyrone becoming treasurer. 8. Lisette joins the activities board on the condition that the board revises its funding rules. 9. Mercedes manages mock trial only if Nana is too busy to do it. 10. Orlando organizes peer tutoring when Percy rounds up volunteers. 11. Aristotle distinguishes actual from potential infinity if Parmenides argues for the One. 12. If Bergson denies time, then so does McTaggart. 13. Camus encouraging authenticity means that Sartre does too. 14. Davidson defending anomalous monism is sufficient for Spinoza’s being correct about parallelism. 15. Emerson bails out Thoreau on the condition that Thoreau pays his taxes. 16. Fanon writing Black Skin, White Masks is a necessary condition for Freire writing Pedagogy of the Oppressed. 17. Grice analyzes pragmatics on the condition that Austin follows Wittgenstein. 18. Foot discussing trolley cases and philosophers reflecting on her work entail that there will be more thought experiments. 19. The Churchlands denying mental states is sufficient for Dennett denying qualia and Chalmers emphasizing the hard problem of consciousness. 20. When Singer is a utilitarian, no one else is.

EXERCISES 2.1b Translate each sentence into propositional logic using the propositional variables given after the sentence. 1. Andre likes basketball. (A) 2. Andre doesn’t like soccer. (A)

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3. Pilar and Zach are logicians. (P, Z) 4. Sabrina wants either a puppy or a kitten. (P, K) 5. Kangaroos are marsupials, and they live in Australia. (M, A) 6. José cooks only when his mother comes over for dinner. (C, M) 7. Martina doesn’t like shopping unless Jenna comes with her. (S, J) 8. The world will end just in case alien kittens invade. (E, A) 9. It is safe to swim when, and only when, the water is calm or a lifeguard is on duty. (S, C, L) 10. Logic is challenging and fun given that you pay attention in class. (C, F, P) 11. Cars are eco-friendly if they are hybrids or run on low-emission fuel. (E, H, L) 12. Cara will go horseback riding only if it doesn’t rain, and she has a helmet. (C, R, H) 13. The restaurant served chicken, and either peas or carrots. (C, P, T) 14. Making butter is a necessary condition for the farmer to go to the market and make a profit. (B, M, P) 15. Patrons may have corn and potatoes if, and only if, they do not order carrots. (C, P, T) 16. If the restaurant runs out of cheesecake, then you can have a meal of chicken and pie and ice cream. (C, K, P, I) 17. A farmer keeps goats in a pen and sheep in a pen only if the dogs and cat are kept inside. (G, S, D, C) 18. Either the farmer shears the sheep and milks the cows, or he slops the pigs and walks the dogs. (S, C, P, D) 19. If the farmer shears the sheep, then he makes wool, and if he milks the cows, then he makes butter. (S, W, C, B) 20. If the farmer goes to the market, then she makes a profit, and her wife is happy. (M, P, W) 21. Plato believed in the theory of forms, and Aristotle held that there are four kinds of causes, but Parmenides thought that only the one exists. (P, A, R) 22. If Thales reduced everything to water, then Democritus was an atomist if and only if Heraclitus claimed that the world is constantly in flux. (T, D, H) 23. If Plato believed in the theory of forms or Democritus was an atomist, then Aristotle held that there are four kinds of causes or Parmenides thought that only the one exists. (P, D, A, R)

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24. Democritus was not an atomist if and only if Plato didn’t believe in the theory of forms, and Thales didn’t reduce everything to water. (D, P, T) 25. Either Heraclitus claimed that the world is constantly in flux or Thales reduced everything to water, and either Aristotle held that there are four kinds of causes or Parmenides thought that only the one exists. (H, T, A, R) 26. Smart believes that minds are brains, and Skinner thinks that inner states are otiose, unless Descartes argues that the mind and body are distinct. (M, K, D) 27. Either Putnam claims that minds are probabilistic automata, or the Churchlands deny that there are any minds and Turing believes that machines can think. (P, C, T) 28. Searle rejects the possibility of artificial intelligence if, and only if, Smart believes that minds are brains and Turing believes that machines can think. (E, M, T) 29. Either Putnam doesn’t claim that minds are probabilistic automata and the Churchlands don’t deny that there are any minds, if Skinner thinks that inner states are otiose, or Searle rejects the possibility of artificial intelligence and Descartes doesn’t argue that the mind and body are distinct. (S, P, C, R, D) 30. Either Turing believes that machines can think or Smart doesn’t believe that minds are brains, and the Churchlands deny that there are any minds. (T, S, C)

EXERCISES 2.1c Translate each argument into propositional logic using the letters provided. D: Descartes defended libertarian free will. E: Elisabeth complained that free will makes virtue independent of luck. S: Spinoza defended determinism. H: Hume developed compatibilism. 1. Descartes defended libertarian free will and Elisabeth complained that free will makes virtue independent of luck. If Elisabeth complained that free will makes virtue independent of luck, then Spinoza defended determinism. Hume developed compatibilism. So, Spinoza defended determinism and Hume developed compatibilism. 2. Descartes defended libertarian free will if, and only if, Elisabeth complained that free will makes virtue independent of luck. Descartes does not defend libertarian free will. If Spinoza defended determinism, then Elisabeth complained

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that free will makes virtue independent of luck. Therefore, Spinoza does not defend determinism or Hume developed compatibilism. 3. If Descartes defended libertarian free will, then Elisabeth complained that free will makes virtue independent of luck. If Elisabeth complained that free will makes virtue independent of luck, then Hume developed compatibilism. Descartes defending libertarian free will is sufficient for either Hume not developing compatibilism or Spinoza defending determinism. So, Descartes does not defend libertarian free will or Spinoza defended determinism. 4. Descartes defended libertarian free will just in case Spinoza defended determinism. If Spinoza defended determinism, then either Hume developed compatibilism or Elisabeth complained that free will makes virtue independent of luck. Descartes defended libertarian free will. Hume does not develop compatibilism. Therefore, Descartes defended libertarian free will and Elisabeth complained that free will makes virtue independent of luck; also Spinoza defended determinism. 5. Descartes defended libertarian free will. Hume developed compatibilism if either Descartes defended libertarian free will or Spinoza defended determinism. Elisabeth complained that free will makes virtue independent of luck if Hume developed compatibilism. Elisabeth complaining that free will makes virtue independent of luck and Descartes defending libertarian free will are sufficient conditions for Spinoza not defending determinism. So, Spinoza does not defend determinism. A: Annas wrote on Aristotle and friendship. J: Sorabji works on Aristotle’s commentators. B: Broadie looks at Aristotle’s ethics in contemporary work. M: Sim compares Aristotle and Confucius. 6. It is not the case that if Broadie looks at Aristotle’s ethics in contemporary work, then Sorabji works on Aristotle’s commentators. Sorabji works on Aristotle’s commentators if Sim compares Aristotle and Confucius. Sim compares Aristotle and Confucius unless Annas wrote on Aristotle and friendship. Hence, Annas wrote on Aristotle and friendship. 7. It is not the case that both Broadie looks at Aristotle’s ethics in contemporary work and Sim compares Aristotle and Confucius. Broadie does not look at Aristotle’s ethics in contemporary work only if Annas did not write on Aristotle and friendship. If Sorabji works on Aristotle’s commentators, then both Sim compares Aristotle and Confucius and Annas wrote on Aristotle and friendship. So, Sorabji does not work on Aristotle’s commentators. 8. Sorabji working on Aristotle’s commentators is a sufficient condition for Sim comparing Aristotle and Confucius. Sorabji working on Aristotle’s

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commentators is a necessary condition for Sim comparing Aristotle and Confucius. Annas wrote on Aristotle and friendship. If Broadie looks at Aristotle’s ethics in contemporary work and Annas wrote on Aristotle and friendship, then Sorabji does not work on Aristotle’s commentators. Therefore, Sim comparing Aristotle and Confucius entails that Broadie does not look at Aristotle’s ethics in contemporary work. 9. If Sorabji does not work on Aristotle’s commentators, then Broadie does not look at Aristotle’s ethics in contemporary work. Annas wrote on Aristotle and friendship and Sim compares Aristotle and Confucius, if Sorabji works on Aristotle’s commentators. Broadie looks at Aristotle’s ethics in contemporary work. Annas writing on Aristotle and friendship and Sim comparing Aris­ totle and Confucius are necessary and sufficient for both Sorabji working on Aristotle’s commentators and Annas writing on Aristotle and friendship. So, Sorabji works on Aristotle’s commentators and Annas writes on Aristotle and friendship. 10. If Annas does not write on Aristotle and friendship, then Sorabji works on Aristotle’s commentators, and Sim compares Aristotle and Confucius if Broadie does not look at Aristotle’s ethics in contemporary work. It is not the case that Annas wrote on Aristotle and friendship and Broadie looks at Aristotle’s ethics in contemporary work. Sorabji does not work on Aristotle’s commentators. Sim compares Aristotle and Confucius just in case either Annas wrote on Aristotle and friendship or Sorabji works on Aristotle’s commentators. If Sorabji works on Aristotle’s commentators, then Broadie looks at Aristotle’s ethics in contemporary work, if, and only if, Sim compares Aristotle and Confucius. Hence, Broadie not looking at Aristotle’s ethics in contemporary work is necessary and sufficient for Sim comparing Aristotle and Confucius. F: Foot developed the trolley problem. T: Thomson introduced the fat man scenario. K: Kamm presents the looping trolley case. 11. Foot developed the trolley problem. Thomson introduced the fat man scenario. Kamm presents the looping trolley case, if Foot developed the trolley problem and Thomson introduced the fat man scenario. Therefore, Kamm presents the looping trolley case; however, Thomson introduced the fat man scenario. 12. Foot developing the trolley problem is sufficient for Thomson introducing the fat man scenario. Kamm presents the looping trolley case if Thomson introduced the fat man scenario. Foot developed the trolley problem. So, Kamm presents the looping trolley case. 13. Either Foot developed the trolley problem or Thomson introduced the fat man scenario. Thomson introduces the fat man scenario unless Kamm does not present the looping trolley case. Foot developing the trolley problem is

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necessary and sufficient for Kamm presenting the looping trolley case. Either Kamm presents the looping trolley case or Foot developed the trolley problem, given that Thomson introduced the fat man scenario. So, Thomson introduced the fat man scenario. 14. Foot developed the trolley problem unless Thomson does not introduce the fat man scenario. Kamm presents the looping trolley case, if, and only if, Foot developing the trolley problem is a necessary condition for Thomson introducing the fat man scenario. If either Foot developed the trolley problem or Thomson introduced the fat man scenario, then Kamm does not present the looping trolley case. Hence, Foot did not develop the trolley problem. 15. Either Foot developed the trolley problem or Kamm presents the looping trolley case. Foot developed the trolley problem unless Thomson introduced the fat man scenario. It is not the case that Foot developed the trolley problem, and Thomson introduced the fat man scenario. If Thomson introduced the fat man scenario, then Kamm presents the looping trolley case. Both Foot did not develop the trolley problem and Thomson introduced the fat man scenario if Kamm presents the looping trolley case. So, Foot did not develop the trolley problem and Thomson introduced the fat man scenario. F: Field is a fictionalist. B: Bueno is a nominalist. W: Wright is a neo-logicist. L: Leng is an instrumentalist. M: Maddy is a naturalist. 16. If Field is a fictionalist, then Leng is an instrumentalist and Wright is a neo-­ log­icist. Maddy is a naturalist and Field is a fictionalist. If   Wright is a neo-­­­logicist, then Bueno is a nominalist. So, Bueno is a nominalist. 17. Maddy is a naturalist only if Wright is a neo-logicist. Wright is a neo-logicist only if Field is a fictionalist and Bueno is a nominalist. Leng is an instrumentalist, but Maddy is a naturalist. Hence, Field is a fictionalist and Bueno is a nominalist. 18. Maddy is a naturalist, if, and only if, Bueno is not a nominalist. Maddy is a naturalist unless Leng is an instrumentalist. Leng being an instrumentalist is a sufficient condition for Field being a fictionalist. Wright is a neo-logicist, yet Bueno is a nominalist. Therefore, Field is a fictionalist. 19. Bueno is a nominalist unless both Wright is a neo-logicist and Maddy is a naturalist. Leng being an instrumentalist is a necessary condition for Bueno not being a nominalist entailing that Wright is a neo-logicist. Leng being an instrumentalist entails that Field is a fictionalist. Bueno is not a nominalist; however, Maddy is a naturalist. Thus, Field is a fictionalist.

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20. Leng is an instrumentalist given that Field is not a fictionalist. If Bueno is a nominalist, then Leng is not an instrumentalist. Either Field is not a fictionalist and Bueno is a nominalist, or Maddy is a naturalist. Maddy is a naturalist just in case Wright is a neo-logicist. Wright is a neo-logicist only if Bueno is not a nominalist. So, Bueno is not a nominalist. R: Rawls is a deontologist. V: Hursthouse is a virtue ethicist. A: Anscombe defends the doctrine of double effect. U: Hardin is a utilitarian. 21. If Hardin is a utilitarian, then Rawls is a deontologist. Rawls is a deontologist only if Hursthouse is not a virtue ethicist. Hardin is a utilitarian. Consequently, Anscombe defends the doctrine of double effect if Hursthouse is a virtue ethicist. 22. Hardin is a utilitarian and Rawls is not a deontologist only if  both Hursthouse is a virtue ethicist and Anscombe defends the doctrine of double effect. Hursthouse is not a virtue ethicist unless Anscombe does not defend the doctrine of double effect. Hardin is a utilitarian. Hence, Rawls is a deontologist. 23. Hursthouse being a virtue ethicist is a necessary condition for Hardin not being a utilitarian. Hardin being a utilitarian is a sufficient condition for Rawls not being a deontologist. Rawls is a deontologist. Anscombe defends the doctrine of double effect if Hardin is not a utilitarian. So, Anscombe defends the doctrine of double effect and Hursthouse is a virtue ethicist. 24. If Anscombe defends the doctrine of double effect, then Hardin is a utilitarian. Either Hursthouse is a virtue ethicist or Rawls is a deontologist. Hursthouse is not a virtue ethicist. Rawls is not a deontologist if  Hardin is a utilitarian. Consequently, Anscombe does not defend the doctrine of double effect. 25. Hardin is not a utilitarian if, and only if, Rawls is not a deontologist. Rawls is a deontologist. Anscombe defends the doctrine of double effect if Hardin is a utilitarian. Hursthouse is a virtue ethicist. If Hursthouse is a virtue ethicist and Anscombe defends the doctrine of double effect, then it is not the case that either Hardin is not a utilitarian or Anscombe defends the doctrine of double effect. So, Hardin is a utilitarian and Anscombe does not defend the doctrine of double effect. F: Freire is a liberation theologist. G: Gutiérrez is influenced by Lascasianism. V: Vaz integrates logic and pragmatism. 26. Gutiérrez is influenced by Lascasianism if, and only if, Vaz integrates logic and pragmatism. Vaz does not integrate logic and pragmatism. Freire is a liberation

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theologist only if Gutiérrez is influenced by Lascasianism. Thus, Freire is not a liberation theologist. 27. Freire is a liberation theologist only if Vaz integrates logic and pragmatism. Gutiérrez is influenced by Lascasianism if Vaz integrates logic and pragmatism. Gutiérrez is not influenced by Lascasianism. So, Freire is not a liberation theologist. 28. Freire being a liberation theologist is a sufficient condition for Gutiérrez not being influenced by Lascasianism, and Vaz integrating logic and pragmatism entails that Freire is a liberation theologist. Either Freire is a liberation theologist or Vaz integrates logic and pragmatism. If Gutiérrez is not influenced by Lascasianism unless Freire is a liberation theologist, then Freire is a liberation theologist. Vaz integrates logic and pragmatism. Therefore, Freire is a liberation theologist and Vaz integrates logic and pragmatism. 29. Freire being a liberation theologist is a necessary and sufficient condition for both it not being the case that Gutiérrez is influenced by Lascasianism and Vaz integrating logic and pragmatism. Gutiérrez is not influenced by Lascasianism. Vaz integrates logic and pragmatism. If Freire is a liberation theologist, then Vaz does not integrate logic and pragmatism just in case Gutiérrez is influenced by Lascasianism. So, it is not the case that Vaz integrates logic and pragmatism if, and only if, Gutiérrez is influenced by Lascasianism. 30. Freire is a liberation theologist or Gutiérrez is influenced by Lascasianism. Gutiérrez is influenced by Lascasianism unless Vaz integrates logic and pragmatism. Gutiérrez is influenced by Lascasianism given that Freire is a liberation theologist. Vaz integrates logic and pragmatism if Freire is a liberation theo­logist. Vaz does not integrate logic and pragmatism. So, Gutiérrez is influenced by Lascasianism. C: Chisholm is a foundationalist. L: Lehrer is a coherentist. G: Goldman is a reliabilist. U: Unger is a skeptic. Z: Zagzebski is a virtue epistemologist. 31. Zagzebski is a virtue epistemologist if, and only if, either Goldman is a reliabilist or Chisholm is a foundationalist. Zagzebski is a virtue epistemologist, but Unger is a skeptic. Lehrer is a coherentist and Chisholm is not a foundationalist. Thus, Goldman is a reliabilist. 32. If Unger is a skeptic, then Lehrer is a coherentist and Zagzebski is a virtue epistemologist. Chisholm being a foundationalist is sufficient for both Goldman being a reliabilist and Lehrer not being a coherentist. Chisholm is a foundationalist; still, Goldman is a reliabilist. So, Unger is not a skeptic.

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33. Chisholm is a foundationalist, or Unger is a skeptic only if Lehrer is a coherentist. Zagzebski is a virtue epistemologist given that Goldman is a reliabilist. If Goldman is a reliabilist, then Chisholm is not a foundationalist. Goldman is a reliabilist, yet Unger is a skeptic. So, Lehrer is a coherentist. 34. Chisholm is a foundationalist just in case both Goldman is a reliabilist and Unger is a skeptic. Either it is not the case that Goldman is a reliabilist or Unger is not a skeptic. If Chisholm is not a foundationalist, then Lehrer is a coherentist. Lehrer is a coherentist only if Zagzebski is a virtue epistemologist. Therefore, Zagzebski is a virtue epistemologist. 35. It is not the case that if Zagzebski is a virtue epistemologist, then Chisholm is a foundationalist. Chisholm being a foundationalist is a necessary condition for Unger being a skeptic. Lehrer is a coherentist or Unger is a skeptic. Lehrer is a coherentist if, and only if, Goldman is a reliabilist. If Goldman is a reliabilist, then Unger is not a skeptic. Hence, Goldman is a reliabilist. S: Searle is a descriptivist. K: Kripke is a direct reference theorist. N: Neale is a metalinguistic descriptivist. 36. Searle is a descriptivist or Neale is a metalinguistic descriptivist. If Neale is a metalinguistic descriptivist, then Kripke is a direct reference theorist. Searle is a descriptivist if Kripke is a direct reference theorist. So, Searle is a descriptivist. 37. Either Searle is a descriptivist or Kripke is a direct reference theorist. Kripke is not a direct reference theorist. Searle is a descriptivist only if Neale is a metalinguistic descriptivist. Therefore, Neale is a metalinguistic descriptivist and Searle is a descriptivist. 38. Searle is a descriptivist given that Neale is a metalinguistic descriptivist. It is not the case that Neale is a metalinguistic descriptivist unless Kripke is a direct reference theorist. Either Searle is a descriptivist or Kripke is not a direct reference theorist. If it is not the case that both Searle is a descriptivist and Kripke is a direct reference theorist, then Neale is a metalinguistic descriptivist. Thus, it is not the case that Searle is a descriptivist and Neale is not a metalinguistic descriptivist. 39. Searle being a descriptivist is sufficient for Kripke being a direct reference theorist. Kripke being a direct reference theorist is necessary and sufficient for Neale not being a metalinguistic descriptivist. If it is not the case that Kripke is a direct reference theorist, then Searle is a descriptivist and Neale is a metalinguistic descriptivist. Searle is a descriptivist. So, it is not the case that Neale is a metalinguistic descriptivist. 40. Either Kripke is a direct reference theorist or Neale is a metalinguistic descriptivist, just in case Searle is not a descriptivist. Neale is not a metalinguistic

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descriptivist unless Kripke is a direct reference theorist. Kripke is not a direct reference theorist; still, Searle is a descriptivist. Searle being a descriptivist is a sufficient condition for it not being the case that either Kripke is a direct reference theorist or Neale is a metalinguistic descriptivist. Hence, both it is not the case that Kripke is a direct reference theorist and Neale is not a metalinguistic descriptivist. K: Kant defended cosmopolitan right. H: Hegel criticized Kant’s views. D: Du Bois integrated pan-Africanism. A: Appiah emphasizes universality plus difference. 41. Kant defended cosmopolitan right and Appiah emphasizes universality plus difference, only when Du Bois integrated pan-Africanism. Either Kant defended cosmopolitan right or Hegel criticized Kant’s views. Hegel did not criticize Kant’s views. So, Appiah emphasizes universality plus difference only when Du Bois integrated pan-Africanism. 42. Hegel criticized Kant’s views if, and only if, both Appiah emphasizes universality plus difference and Du Bois integrated pan-Africanism. Hegel criticized Kant’s views; however, Kant defended cosmopolitan right. If either Du Bois integrated pan-Africanism or Appiah emphasizes universality plus difference, then Kant defended cosmopolitan right and Appiah emphasizes universality plus difference. Therefore, Kant defended cosmopolitan right and Appiah emphasizes universality plus difference. 43. If both Kant defended cosmopolitan right and Du Bois integrated pan-­ Africanism, then Hegel criticized Kant’s views. Either Appiah emphasizing universality plus difference or Du Bois not integrating pan-Africanism are necessary conditions for Hegel criticizing Kant’s views. Kant defended cosmopolitan right, but Hegel criticized Kant’s views. Appiah does not emphasize universality plus difference. Consequently, it is not the case that both Kant defended cosmopolitan right and Du Bois integrated pan-Africanism. 44. Kant defended cosmopolitan right. Hegel criticized Kant’s views. Kant defending cosmopolitan right and Hegel criticizing Kant’s views are necessary and sufficient for either Du Bois integrating pan-Africanism or Appiah emphasizing universality plus difference. Du Bois did not integrate pan-Africanism. Thus, Appiah emphasizes universality plus difference. 45. Hegel did not criticize Kant’s views. If Kant defended cosmopolitan right, then Hegel criticized Kant’s views. Either Kant defended cosmopolitan right or Du Bois integrated pan-Africanism. Du Bois integrated pan-Africanism just in case Appiah emphasizes universality plus difference. So, Appiah emphasizes universality plus difference and Hegel didn’t criticize Kant’s views.

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EXERCISES 2.1d Interpret the following sentences of propositional logic using the given translation key. Strive for elegance in your English sentences. A: Willa teaches English. B: Willa teaches history. C: Willa teaches in a middle school. D: Willa has a master’s degree. E: Javier teaches English. F: Ahmed teaches English. 1. C ⊃ (B ∨ A) 2. A ∙ ∼B 3. A ⊃ (E ∙ F) 4. ∼D ⊃ ∼(A ∨ B) 5. ∼(E ∨ F) ⊃ B G: Suneel majors in philosophy. H: Suneel majors in physics. I: Suneel majors in psychology. J: Suneel is a college student. K: Marjorie is a philosophy professor. L: Marjorie teaches logic. 6. (K ∙ L) ⊃ G 7. J ⊃ (G ∙ I) 8. ∼(G ∙ I) ∨ ∼H 9. ∼(K ∙ L) ⊃ (I ∨ H) 10. G ≡ ( J ∙ K) M: Carolina plants vegetables. N: Carolina plants flowers. O: Carolina has a garden. P: Carolina’s plants grow. Q: Carolina sprays her plants with pesticides. R: Deer eat the plants. 11. O ⊃ (M ∙ N) 12. (O ∙ P) ⊃ R

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13. [(N ∙ P) ∙ Q ] ⊃ ∼R 14. [(M ∨ N) ∙ P] ⊃ (Q ∨ R) 15. ∼P ≡ ∼Q

2.2: SYNTA X OF PL: WFFS AND MAIN OPERATORS To this point, we have been rather casual about the language of propositional logic. I will now be more rigorous in setting up the syntax of our first formal language, PL, one of many different possible languages for propositional logic. To specify a formal language, we start with a list of the vocabulary of the language, its symbols. For our purposes in this chapter and the next, the following thirty-seven different symbols will suffice. Capital English letters, used as propositional variables Five operators Punctuation

The syntaxof a logical language is the definition of its vocabulary and rules for making formulas.

A . . . Z ∼, ∙, ∨, ⊃, ≡ (, ), [, ], {, }

Notice that PL contains only twenty-six propositional variables. More flexible systems of propositional logic can accommodate infinitely many propositional variables. One way to include more propositional variables is by including the prime symbol among our vocabulary and allowing iterated repetitions of it. P, P′, P″, P‴, P⁗, P⁗′ . . .

Since we won’t need so many variables, we will just use English letters with no primes. We saw the five operators in the previous section. The punctuation comes in pairs and helps us to avoid ambiguity. Once we have specified the vocabulary of a formal language, we can combine our symbols into formulas. Some ways of combining the symbols are acceptable, while others are not. Consider the two combinations of English words 2.2.1 and 2.2.2. 2.2.1 2.2.2

Sky blue is the. The sky is blue.

2.2.2 is a well-formed English expression, a grammatical sentence, and 2.2.1 is not, even for Yoda. Analogously, in our language of propositional logic, only some strings of symbols are well formed. We call strings of logical symbols that are constructed properly well-formed formulas, or wffs, or just formulas (since any string must be well formed to be a formula). ‘Wff’ is pronounced like ‘woof ’, as if you are barking. 2.2.3 and 2.2.4 are wffs, while 2.2.5 and 2.2.6 are not wffs. 2.2.3 P ∙ Q 2.2.4 (∼P ∨ Q) ⊃ ∼R 2.2.5 ∙P Q 2.2.6 Pq ∨ R∼

In English, there are indefinitely many grammatical sentences, composed of a large, though finite, stock of words and grammatical conjunctions. In propositional logic,

A wff is a well-formed formula of a formal language.

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Formation rulessay

how to combine the vocabulary of a language into well-formed formulas.

An atomic formulais formed by a single use of PL1. All other wffs are complex formulas.

there are infinitely many wffs, constructed by applying a simple set of rules, called formation rules. Formation Rules for Wffs of PL PL1. A single capital English letter is a wff. PL2. If α is a wff, so is ∼α. PL3. If α and β are wffs, then so are: (α ∙ β) (α ∨ β) (α ⊃ β) (α ≡ β) PL4. These are the only ways to make wffs. The simplest wffs, which we call atomic, are formed by a single use of PL1. Complex wffs are composed of atomic wffs, using any of the other rules. The Greek letters α and β in the formation rules are metalinguistic variables; they can be replaced by any wffs of the object language to form more complex wffs. We add the punctuation in PL3 to group any pair of wffs combined using a binary operator. By convention, we drop the outermost punctuation of a wff. That punctuation must be replaced when a shorter formula is included in a more complex formula. As wffs get longer, it can become difficult to distinguish nested punctuation. For readability, I use square brackets, [ and ], when I need a second set of parentheses, and braces, { and }, when I need a third. The three kinds of punctuation are interchangeable. 2.2.7 provides an example of how one might construct a complex wff using the formation rules, starting with simple letters. 2.2.7 W X ∼W ∼W ∙ X (∼W ∙ X) ≡ X ∼[(∼W ∙ X) ≡ X]

The last operator added to a wff according to the formation rules is called the main operator.

By PL 1 By PL 1 By PL2 By PL3, and the convention for dropping brackets By PL3, putting the brackets back By PL2

The order of construction of a wff is especially important because it helps us determine the main operator. The main operator of a wff is important because we characterize wffs by their main operators: negations, conjunctions, disjunctions, conditionals, or biconditionals. In the next few sections, we will learn how to characterize wffs further. We can determine the main operator of any wff of PL by analyzing the formation of that wff, as I do at 2.2.8. 2.2.8 (∼M ⊃ P) ∙ (∼N ⊃ Q) ‘M’, ‘P’, ‘N’, and ‘Q’ are all wffs, by PL1. ‘∼M’ and ‘∼N’ are wffs by PL2. ‘(∼M ⊃ P)’ and ‘(∼N ⊃ Q)’ are then wffs by PL3. Finally, the whole formula is a wff by PL3 and the convention for dropping brackets.

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As I mentioned in the previous section, I use commas and semicolons in English to disambiguate ambiguous sentences; these will often appear at the main operators. Also, given our convention for dropping brackets, main operators are usually not enclosed in brackets.

Summary In this section, we examined the syntax of the language PL, its vocabulary and rules for constructing well-formed formulas. We saw that the main operator of a complex formula is the final operator added when the formula is built according to the formation rules. We identify formulas with their main operators, and, as we leave syntax to study the semantics of PL in the next three sections, we will classify formulas according to the truth conditions at their main operators.

KEEP IN MIND

We can use the formation rules to distinguish wffs from non-well-formed strings. We can also use the formation rules to identify the main operator of any wff.

TELL ME MORE  • Are there other logical operators? See 6.5: Modal Logics.

EXERCISES 2.2 Are the following formulas wffs? If so, which operator is the main operator? (For the purposes of this exercise, consider formulas without their outermost punctuation as well formed, according to the convention mentioned in this section.) 1. C ⊃ D ∙ E

7. ∼[A ∙ B ⊃ D ∨ E]

2. (T ∙ V)∼W

8. ∼Z ⊃ X ∙ Y

3. ( J ∨ ∼J) ⊃ K

9. (A ∨ B) ∙ C

4. ∼[(A ∨ B) ⊃ C]

10. M ≡ [(L ∙ N) ⊃ ∼O ∨ P]

5. ∼(A ∙ B) ⊃ C ∨ D

11. [(Q ⊃ R) ∨(S ∨ ∼T)] ≡ (T ∙ Q ∨ R)

6. ∼D ⊃ E ≡ C

12. (W ∨ X ∙ ∼Y) ⊃ [Z ≡ (Y ∨ W)]

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13. (D ∨ E) ⊃ ∼(E ∙ F)

20. (S ∨ L) ⊃ C ⊃ (Q ∙ R)

14. [G ⊃ (H ∙ I)] ∨ ∼[I ≡ (H ∨ G)]

21. (X ∙ Y∼Z) ⊃ [(F ∨ ∼G) ≡ ∼H]

15. (P ∙ Q ∨ R) ⊃ ∼S

22. ∼[ J ⊃ (K ∨ ∼L)] ≡ [(L ∙ M) ≡ ∼K]

16. ∼(K ⊃ N) ⊃ (L ∙ M)

23. ∼{[N ≡ (∼O ⊃ P)] ∨ [∼P ∙ (Q ∨ ∼O)]}

17. ∼K [M ≡ (N ∙ O)]

24. ∼{(P ∙ Q ) ⊃ [(P ∙ R) ∨(R ⊃ Q )]}

18. (D ∨ E) ≡ ∼[(F ⊃ G) ∙ H)]

25. [(T ∨ U) ∙ (U ∨ V)] ⊃ [(V ∙ W) ∨ (T ∙ W)]

19. [D ⊃ (E ∙ F)] ∨ (F ≡ D)

2.3: SEMANTICS OF PL: TRUTH FUNCTIONS

The semanticsof a formal language are the rules for interpreting the symbols and formulas of the language.

In our bivalent logic, every statement is either true or false, and not both. True and false are called truth values.

When constructing a formal system of logic, we start with a language like PL. In section 2.2, I provided formation rules, or a syntax, for that language. Once we have specified the language, there are two ways that we can use it. First, we can interpret the language, providing a semantics for it, which tells us how to understand the symbols and formulas. Second, we can use the language in a deductive system by introducing inference rules. Both the semantics and the inference rules will help us characterize logical consequence: what follows from what. We will study inference rules in chapter 3. In this section, we will look at the interpretations, or semantics, of our language. In the remainder of chapter 2, we will use our semantics to characterize different kinds of propositions and provide a formal test for the validity of an argument. Informally, we can interpret our propositional variables as particular English propositions. For example, we might take ‘P’ to stand for ‘It is raining in Clinton, NY’ and ‘Q’ to stand for ‘It is snowing in Clinton, NY’. Then ‘P ∙ Q’ would stand for ‘It is both raining and snowing in Clinton, NY’. More formally, and more generally, in PL and all standard propositional logics, we interpret propositional variables by assigning truth values to them. The truth value of a wff is a characteristic of the proposition, whether true, false, or something else. In nearly all of this book, we use a bivalent logic, on which every statement is either true or false, but not both. Other systems of logic use three or more truth values, with a third truth value of unknown, or undetermined, or indeterminate. We have carefully defined our language PL, and it does not contain tools for doing the interpretation. To interpret our formal language, we use a metalanguage. Our metalanguage will be English, supplemented with some specific symbols used with specific intents. For example, we will use “1” to represent truth and “0” to represent

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falsity. We specify rules of our metalanguage less formally than we do the rules of our object language. We will start our study of the semantics of propositional logic by looking at how we calculate the truth value of a complex proposition on the basis of the truth values of its component sentences. We can calculate the truth value of any complex proposition using the truth values of its component propositions and the basic truth tables for each propositional operator, which we will see shortly. The fact that the truth values of complex propositions are completely determined by the truth values of the component propositions is called truth-functional compositionality, a basic presupposition of our logic. Consider a complex proposition like 2.3.1 and its translation into PL 2.3.2. 2.3.1

If either the Beatles made The White Album or Jay-Z didn’t make The Black Album, then Danger Mouse did not make The Grey Album or Jay-Z did make The Black Album. 2.3.2 (W ∨ ∼J) ⊃ (∼G ∙ J)

We can easily determine the truth values of the component, atomic propositions, W, J, and G. In this case, all of the atomic propositions are true: the Beatles made The White Album, Jay-Z made The Black Album, and Danger Mouse made The Grey Album. But what is the truth value of the whole complex proposition 2.3.1? To determine the truth value of a complex proposition, we combine the truth values of the component propositions using rules for each operator. These rules are summarized in basic truth tables, one for each propositional operator. The basic truth table for each logical operator defines the operator by showing the truth value of the operation, given any possible distribution of truth values of the component propositions. Once we combine these truth tables, our semantics, with our translations of natural languages into PL, can cause certain problems to arise. Not all of our natural-language sentences conform precisely to the semantics given by the truth tables. Difficulties arise for the conditional, in particular. In this section, we’ll look at the details of the truth tables for each operator before returning to 2.3.1 to see how to use the basic truth tables.

Negation Negation is the simplest truth function. When a statement is true, its negation is false; when a statement is false, its negation is true. 2.3.3 2.3.4 2.3.5 2.3.6

Two plus two is four. Two plus two is not four. Two plus two is five. Two plus two is not five.

2.3.3 is true, and its negation, 2.3.4, is false. 2.3.5 is false, and its negation, 2.3.6, is true.

The truth value of a complex proposition is

the truth value of its main operator.

The basic truth table for each logical operator shows the truth value of a complex proposition, given the truth values of its component propositions.

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We generalize these results using the basic truth table for negation. In the first row of the truth table, we have an operator, the tilde, and a Greek metalinguistic letter, α. The column under the ‘α’ represents all possible assignments of truth values to a single proposition. We could use ‘T’ for ‘true’ and ‘F’ for ‘false’ in the truth table. I use ‘1’ for true and ‘0’ for false in this book, largely because ‘1’s and ‘0’s are very easy to tell apart. The column under the ‘∼’ represents the values of the negation of the proposition in each row. ∼

α

0

1

1

0

Basic Truth Table for Negation

The truth table for a complex proposition containing one variable has two lines, since there are only two possible assignments of truth values. This truth table says that if the value of a propositional variable is true, the value of its negation is false, and if the value of a propositional variable is false, the value of its negation is true.

Conjunction Conjunctions are true only when both conjuncts are true; otherwise they are false. 2.3.7

Esmeralda likes logic and metaphysics.

2.3.7 is true if ‘Esmeralda likes logic’ is true and ‘Esmeralda likes metaphysics’ is true. It is false otherwise. Note that we need four lines to explore all the possibilities of combinations of truth values of two propositions: when both are true, when one is true and the other is false (and vice versa), and when both are false. α

∙

β

1

1

1

1

0

0

0

0

1

0

0

0

Basic Truth Table for Conjunction

Our basic truth tables all have either two lines or four lines, since all of our operators use either one or two variables. Truth tables for more-complex sentences can be indefinitely long.

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Disjunction Disjunctions are false only when both disjuncts are false. 2.3.8

Kareem will get an A in either history or physics.

We’ll take 2.3.8 as expressing our optimism that Kareem will do very well in at least one of the named courses. If he gets an A in neither course, then our optimism will have proven to be unfounded; our statement will have been false. But as long as he gets an A in either history or physics, the statement will have been shown to be true. And if he gets an A in both of those classes, our optimism will have been shown to be more than called for. This interpretation of the ‘∨’ is slightly contentious, and is called inclusive disjunction. On inclusive disjunction, 2.3.8 is false only when both component statements are false.

α

∨

β

1

1

1

1

1

0

0

1

1

0

0

0

Basic Truth Table for Disjunction

There is an alternative use of ‘or’ on which a disjunction is also false when both component propositions are true, which we can call exclusive disjunction. 2.3.9 is most naturally interpreted as using an exclusive disjunction. 2.3.9

You may have either soup or salad.

Uses of 2.3.9 are usually made to express that one may have either soup or salad, but not both. Thus it seems that some uses of ‘or’ are inclusive and some uses of ‘or’ are exclusive. One way to manage the problem of the different senses of ‘or’ would be to have two different logical operators, one for inclusive ‘or’ and one for exclusive ‘or.’ This would give us more operators than we need, since we can define either one in terms of the other, along with other logical operators. So, it is mainly arbitrary whether we take inclusive or exclusive disjunction as the semantics of ‘∨’. We just need to be clear about what we mean when we are regimenting sentences into our formal logic. We will thus (traditionally, but also sort of arbitrarily) use inclusive disjunction, the ∨, to translate ‘or’.

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Material Implication To interpret English-language conditionals, we use what is called the material interpretation on which a conditional is false only when the antecedent is true and the consequent is false. α

⊃

β

1

1

1

1

0

0

0

1

1

0

1

0

Basic Truth Table for Material Implication

To understand the material interpretation, consider when 2.3.10 will be falsified. 2.3.10

If you paint my house, then I will give you five thousand dollars.

It is true in the first row, when both the antecedent and consequent are true: you paint my house and I give you five thousand dollars. It is false in the second row, when the antecedent is true and the consequent is false. You’ve painted my house, but I don’t pay you. The third and fourth rows of the truth table for the conditional, when the antecedent is false, are controversial. Notice that in those two cases, when you don’t paint my house, 2.3.10 is unfalsified whether or not I give you five thousand dollars. The only case in which 2.3.10 is clearly false is when you paint my house and I fail to give you the money; that’s the second row of the truth table for ‘⊃’. Since we have only two truth values, and we don’t want to call the statement false if you haven’t painted my house, we seem forced to call the sentence true in the last two rows. The conditional is the trickiest operator, in large part because many of our uses of ‘if . . . then . . .’ are not truth-functional. In other words, the truth value of many complex sentences that use conditionals are not exclusively dependent on the truth values of their components. Imagine that I have a sugar cube, a hunk of steel, and a pot of boiling water. 2.3.11 2.3.12

If this sugar cube is dropped into a pot of boiling water, then it will dissolve. If this hunk of steel is dropped into a pot of boiling water, then it will dissolve.

We naturally believe that 2.3.11 is true and 2.3.12 is false. But there is no logical difference between the sentences. They are both conditionals. The difference in our estimation of the two sentences depends on the laws of physics; sugar dissolves in

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boiling water while steel does not. That is not a logical difference, though, and the two sentences have the same truth conditions as far as ⊃ is concerned. Some uses of conditionals in English are truth-functional, and we are going to use ‘⊃’ to regiment conditionals into PL despite worries about sentences like 2.3.11 and 2.3.12.

The Biconditional A biconditional is true if the component statements share the same truth value. It is false if the components have different values.

2.3.13

α

≡

β

1

1

1

1

0

0

0

0

1

0

1

0

Basic Truth Table for the Biconditional

Supplies rise if, and only if, demand falls.

If supplies rise and demand falls, 2.3.13 is true. If supplies don’t rise and demand doesn’t fall, then 2.3.13 is true as well. But if one happens without the other, then 2.3.13 is false. The biconditional is thus really a shorthand for two material conditionals: if α then β and if β then α. The result is that ≡ works like an equals sign for propositions: it will be true if, and only if, the truth values of the components are the same.

Truth Values of Complex Propositions The basic truth tables can be used to evaluate the truth value of any proposition of any complexity, given truth values for all the simple component propositions. Remember, the truth value of a complex proposition is the truth value of its main operator. Method for Determining the Truth Value of a Proposition 1. Assign truth values to each atomic formula. 2. Evaluate any negations of those formulas. 3. Evaluate any binary operators for which both values are known. 4. Repeat steps 2 and 3, working inside out, until you reach the main operator. Let’s see how to use this method with the example 2.3.14.

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2.3.14 (A ∨ X) ∙ ∼B

Let’s arbitrarily assume that A and B are true and X is false. If we were starting with an English sentence, we might be able to determine appropriate truth values of the component sentences. First, assign the assumed values to the atomic formulas A, B, and X. ∨

(A

∙

X)

1

∼

B

0

1

Next, evaluate the negation of B. ∨

(A

∙

X)

1

0

∼

B

0

1

Since we know the values of the disjuncts, we can next evaluate the disjunction. (A

∨

X)

1

1

0

∙

∼

B

0

1

Finally, we can evaluate the main operator, the conjunction. (A

∨

X)

∙

∼

B

1

1

0

0

0

1

2.3.14 is thus false for the values we arbitrarily assumed. Let’s return to 2.3.1. We already regimented it as 2.3.2. Now we can assign the values we know to W, J, and G. (W 1

∨

∼

J) 1

⊃

(∼

G 1

∙

J) 1

Then we can use our method for determining the truth value of a complex proposition, first evaluating the negations.

2 . 3 : S e m ant i cs o f P L : T r u t h F u nct i ons   5 3

(W

∨

1

∼

J)

0

1

⊃

(∼

G

0

1

∙

J) 1

Now we can evaluate the disjunction on the left and the conjunction on the right. (W

∨

∼

J)

1

1

0

1

⊃

(∼

G

∙

J)

0

1

0

1

Finally, we can find the truth value of the main operator, the horseshoe. (W

∨

∼

J)

⊃

(∼

G

∙

J)

1

1

0

1

0

0

1

0

1

2.3.1 and 2.3.2 are thus false. 2.3.15 and 2.3.16 are a bit more complex. 2.3.15 A ⊃ (∼X ∙ ∼Y)

where A is true and X and Y are false.

Start by assigning truth values to the atomic propositions. A

⊃

(∼

1

X

∙

∼

0

Y) 0

Next, evaluate the negations of X and Y. A

⊃

1

(∼

X

1

0

(∼

X

1

0

∙

∼

Y)

1

0

∙

∼

Y)

1

1

0

Then the conjunction. A 1

⊃

5 4    C h apter 2   P ropos i t i onal L og i c

Finally, the conditional, the main operator. A

⊃

(∼

X

∙

∼

Y)

1

1

1

0

1

1

0

2.3.15 is thus true for our assumed values. 2.3.16 [(A ∙ B) ⊃ Y] ⊃ [A ⊃ (C ⊃ Z)] where A, B, and C are true; Y and Z are false.

First the atomic propositions. [(A

∙

1

B)

⊃

1

Y]

⊃

0

[A

⊃

1

(C

⊃

1

Z)] 0

Next, the formulas in parentheses. [(A

∙

B)

1

1

1

⊃

Y]

⊃

0

[A

⊃

1

(C

⊃

Z)]

1

0

0

Now we can evaluate both the antecedent and the consequent of the main operator. [(A

∙

B)

⊃

Y]

1

1

1

0

0

⊃

[A

⊃

(C

⊃

Z)]

1

0

1

0

0

Finally, the main operator. [(A

∙

B)

⊃

Y]

⊃

[A

⊃

(C

⊃

Z)]

1

1

1

0

0

1

1

0

1

0

0

2.3.16 is true for the given assignments of truth values.

Complex Propositions with Unknown Truth Values We have seen how to calculate the truth value of a complex proposition when the truth values of the components are known. But sometimes you don’t know truth values of one or more component variable. It may still be possible to determine the truth

2 . 3 : S e m ant i cs o f P L : T r u t h F u nct i ons   5 5

value of the complex proposition. If the truth values of the whole proposition are the same whatever values we assign to the unknown propositions, then the statement has that truth value. If the values come out different in different cases, then the truth value of the complex statement is really unknown. Let’s look at a few of these cases, and suppose that A, B, C are true; X, Y, Z are false; and P and Q are unknown for the remainder of the section. We’ll start with 2.3.17. 2.3.17 P ∙ A

If P were true, then the truth value of 2.3.17 would be true. P

∙

A

1

1

1

If P were false, then 2.3.17 would be false. P

∙

A

0

0

1

Since the truth value of 2.3.17 depends on the truth value of P, it too is unknown. In contrast, 2.3.18 has a determinable truth value even though one of the atomic propositions in it is unknown. 2.3.18 P ∨ A

If P is true, then 2.3.18 is true. P

∨

A

1

1

1

P

∨

A

0

1

1

If P is false, then 2.3.18 is true too!

The truth value of 2.3.18 is true in both cases. In our bivalent logic, these are the only cases we have to consider. Thus, the value of that statement is true, even though we didn’t know the truth value of one of its component propositions.

5 6    C h apter 2   P ropos i t i onal L og i c

We have seen that the truth value of a complex proposition containing a component proposition with an unknown truth value may be unknown and it may be true. Sometimes the truth value of such a complex proposition will come out false, as in 2.3.19. 2.3.19 Q ∙ Y

If Q is true, then 2.3.19 is false. Q

∙

Y

1

0

0

Q

∙

Y

0

0

0

If Q is false, then 2.3.19 is also false.

Since the truth value of the complex proposition is false in both cases, the value of 2.3.19 is false. Lastly, we can have more than one unknown in a statement. If there are two unknowns, we must consider four cases: when both propositions are true; when one is true and the other is false; the reverse case, when the first is false and the second is true; and when both are false, as in 2.3.20. 2.3.20 (A ⊃ P) ∨ (Q ⊃ A)

where A is true.

(A

⊃

P)

∨

(Q

⊃

A)

1

1

1

1

1

1

1

P and Q are true

1

1

1

1

0

1

1

P is true and Q is false

1

0

0

1

1

1

1

P is false and Q is true

1

0

0

1

0

1

1

P and Q are false

Since all possible substitutions of truth values for ‘P’ and ‘Q’ in 2.3.20 yield a true statement, the statement itself is true.

2 . 3 : S e m ant i cs o f P L : T r u t h F u nct i ons   5 7

Summary In this section I introduced all of the basic truth tables, one for each of the five propositional operators. The basic truth tables are mostly intuitive, and so not very difficult to reconstruct if you forget one or other of the lines. Remember, especially, that our disjunction is inclusive, and the material conditional is false only in the second row, when the antecedent is true and the consequent is false. The basic truth tables are useful in evaluating the truth value of a complex proposition on the basis of the truth values of the component, atomic propositions. We can even sometimes evaluate propositions for which we do not know all of the truth values of the atomic propositions.

KEEP IN MIND

The five basic truth tables give the semantics of PL. We interpret propositional variables by assigning truth values to them. The truth value of a complex proposition is the truth value of its main operator. The negation of a statement is the opposite truth value of the statement. The conjunction of two statements is true only when both conjuncts are true. The disjunction of two statements is false only when both disjuncts are false. The material conditional is false only when the antecedent is true and the consequent is false. The biconditional is true if the component statements share the same truth value. The basic truth tables can be used to evaluate the truth value of any proposition built using the same formation rules.

TELL ME MORE  • W hat is the relation between inclusive and exclusive disjunction, and how do we define one in terms of the other? See 6.1: Notes on Translation with PL. • A re there other interpretations of the natural-language conditional? See 6.2: Conditionals. • Are there alternatives to bivalence? See 6.3: Three-Valued Logics. • How do philosophers use the distinction between syntax and semantics? See 7.3: Logic and Philosophy of Mind. • A re there other operators and other truth tables in bivalent logics? Why do we use these? See 6S.8: Adequacy.

5 8    C h apter 2   P ropos i t i onal L og i c

EXERCISES 2.3a

EXERCISES 2.3b

Assume A, B, C are true and X, Y, Z are false. Evaluate the truth values of each complex expression.

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.

1. X ∨ Z

1. Q ∙ ∼Q

2. A ∙ ∼C

2. Q ⊃ B

3. ∼C ⊃ Z

3. P ∙ ∼C

4. (A ∙ Y) ∨ B

4. P ≡ ∼P

5. (Z ≡ ∼B) ⊃ X

5. P ∨ (X ∙ Y)

6. (A ⊃ B) ∨ ∼X

6.  ∼(Z ∙ A) ⊃ P

7. (Z ∙ ∼X) ⊃ (B ∨ Y)

7. Q ∨ ∼(Z ∙ A)

8. (B ≡ C) ⊃ ( A ⊃ X)

8. (P ⊃ A) ∙ (Z ∨ B)

9. (A ∙ Z) ∨ ∼(X ∙ C)

9. (P ≡ B) ∨ (Y ⊃ C)

10. (Z ∙ A) ∨ (∼C ∙ Y)

10. [(Z ⊃ C) ∙P] ≡ (A ∨ X)

11. X ∙ [A ⊃ (Y ∨ Z)]

11.  ∼[(P ∙ Z) ⊃ Y] ≡ (Z ∨ X)

12. (B ∨ X) ⊃ ∼(Y ≡ C)

12. [Q ∙ (B ≡ C)] ∙ ∼Y

13. (∼B ⊃ Z) ∙(A ≡ X)

13. [(A ∨ X) ⊃ (Y ∙ B)] ≡ ∼Q

14. ∼(A ≡ C) ⊃ (X ∙ Y)

14.  ∼(A ∨ P) ≡ [(B ∙ X) ⊃ Y]

15. ∼(A ∨ Z) ≡ (X ∙ Y)

15.  ∼P ⊃ [∼(A ∙ B) ∨ (Z ∙ Y)]

16. (C ⊃ Y) ∨ [(A ∙ B) ⊃ ∼X]

16. [∼Z ∙ (P ⊃ A)] ∨ [X ≡ ∼(B ⊃ Y)]

17. [(C ∙ Y) ∨ Z] ≡[∼B ∨ (X ⊃ Y)]

17.  ∼(X ∨ C) ∙[(P ⊃ B) ⊃ (Y ∙ Z)]

18. [(X ∙ A) ⊃ B] ≡[C ∨ ∼(Z ⊃ Y)]

18. [∼P ⊃ (A ∨ X)] ⊃ [(B ∨ P) ≡ (Y ⊃ Z)]

19. [(A ∙ B) ≡ X] ⊃ [(∼Z ∙ C) ∨ Y]

19. [(P ∙ A) ∨ ∼B] ≡{∼A ⊃ [(C ∨ X) ∙Z]}

20. [X ⊃ (A ∨ B)] ≡ [(X ∙ Y) ∨ (Z ∙ C)]

20. [(Q ∨ ∼C) ⊃ Q ] ≡ ∼[Q ≡ (A ∙ ∼Q )]

2 . 3 : S e m ant i cs o f P L : T r u t h F u nct i ons   5 9

EXERCISES 2.3c As in Exercises 2.3b, assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. 1. Q ⊃ (A ∨ P) 2. (P ⊃ C) ≡ [(B ∨ Q ) ⊃ X] 3. (A ∙ Q ) ∙ (X ∙ P) 4. (P ∙ Q ) ⊃ (X ∨ A) 5. (Q ⊃ P) ∙(Z ∨ ∼Y) 6. (P ∙ Z) ⊃ (Q ∨ A) 7. (P ∨ Q ) ∨ (∼A ≡ Y) 8. (P ∙ Z) ⊃ ∼(Q ≡ C) 9. ∼(Y ∨ Q ) ∨ [(P ⊃ B) ≡ A] 10. (X ∙ P) ≡ [(Q ∨ B) ⊃ (Z ≡ A)] 11. ∼{[P ⊃ (Q ⊃ C)] ∙ Z} 12. (Q ∙ P) ∨ (∼Q ∨ ∼P) 13. [(Q ⊃ (P ⊃ Z)] ∨ ∼(∼X ∨ C) 14. {Z ⊃ [P ⊃ (Q ⊃ A)]} ⊃ (X ∙ Q ) 15. [(Q ⊃ B) ∙ (X ∨ ∼Z)] ≡ [P ⊃ (Q ⊃ ∼Y)] 16. ∼{[(P ⊃ A) ∙ X] ≡ [(Q ∨ ∼Q ) ⊃ ∼B]} 17. ∼{[Q ⊃ (B ≡ (A ∨ P)]} ∙ {[∼C ⊃ (Z ≡ Y) ∨ (X ∙ P)]} 18. [Q ∨ (X ⊃ P)] ≡ [A ∙ ∼(Z ⊃ Q )] 19. [Z ∨ (X ∙ Q )] ≡ ∼[(Y ⊃ P) ⊃ Z] 20. [A ⊃ (Z ∙ ∼P)] ∨ [(Q ≡ X) ⊃ C]

6 0    C h apter 2   P ropos i t i onal L og i c

2.4: TRUTH TABLES

A truth tableshows the truth value for a complex proposition given any truth values of its component propositions.

As we saw in the previous section, when we are given a complex proposition and we know the truth values of the component propositions, we can calculate the truth value of the longer statement. When we are given a complex proposition and some of the truth values of the component propositions are unknown, sometimes we can still determine the truth value of the complex proposition. But sometimes the best we can do is to describe how the truth value of the whole complex proposition varies with the truth value of its parts. To do so, we construct truth tables for complex propositions using the basic truth tables as guides. We can construct truth tables for any proposition, with any number of component propositions of unknown truth values. Such truth tables summarize the distributions of all possible truth values of the complex propositions. I start this section by describing the method of constructing truth tables for complex propositions. To use this method comfortably, you should have memorized the basic truth tables and mastered their applications. Then, in the next section, we’ll use the method to identify some interesting properties of propositions and to describe some relations among propositions. In the following section, we will use truth tables to distinguish valid from invalid arguments, the central task of this book. We construct truth tables for wffs of PL in three steps. Method for Constructing Truth Tables Step 1. Determine how many rows are needed. Step 2. Assign truth values to the component variables. Step 3. Working from the inside out according to the order of construction, evaluate each operator, placing each column directly beneath that operator until you reach the main operator and complete the table. For step 1, the number of rows of a truth table is a function of the number of variables in the wff. With one propositional variable, we need only two rows, as in the basic truth table for negation: one for when the variable is true and one for when it is false. With two propositional variables, we need four rows, as in the basic truth tables for all the binary operators. Each additional variable doubles the number of rows needed: the number of rows needed for the simpler table when the new variable is true and the same number again when the new variable is false. Determining the Size of a Truth Table 1 variable: 2 rows 2 variables: 4 rows 3 variables: 8 rows 4 variables: 16 rows n variables: 2 n rows

2 . 4 : T r u t h T a b les   6 1

For step 2, it is conventional and useful to start truth tables in a systematic way, assigning a set of truth values that depends on the size of the truth table to the first variable in the proposition, a different set to the next variable, and so on. These conventions are constrained by two requirements:

• The truth table must contain every different combination of truth values of the component propositions.

• The assignments of truth values to any particular propositional variable must

be consistent within the truth table: if the third row under the variable ‘P’ has a 1 in one column, the third row under the variable ‘P’ must have a 1 in every column.

Our conventional method for constructing truth tables, which I’ll describe in the remainder of this section, can be adapted to construct a truth table for any wff of  PL. First, I’ll introduce columns on the left of the table, one for each variable in the wff. Then, I’ll use a conventional method for assigning truth values to each variable. The method is the same for each wff with the same number of variables and expands in a natural way for longer formulas. There are other ways of presenting the same information, the truth conditions for any proposition, but I’ll use this one method consistently throughout the book. For wffs with only one variable, we only need to consider what happens when that variable is true and when it is false: two rows. We’ll consider what happens when the variable is true in the first row and what happens when it is false in the second row. Here is a two-row truth table, for ‘P ⊃ P’:

P

P

⊃

P

1

1

1

1

0

0

1

0

Notice that the left side of the truth table contains a column for the only variable, ‘P’. The values in that column are exactly the same under every instance of ‘P’ in the table. The column under the ⊃, the main operator, contains the values of the whole wff, which we calculate using the values of ‘P’ and the basic truth table for the material conditional. Also notice that, to make things a little easier to read, I highlight the values of the main operator. Some students like to use highlighters for the values of the main operator, or even different colored highlighters for different columns as one constructs the table.

6 2    C h apter 2   P ropos i t i onal L og i c

Below example 2.4.1 is the beginning of a four-row truth table. 2.4.1 (P ∨ ∼Q) ∙ (Q ⊃ P)

P

Q

1

1

1

0

0

1

0

0

(P

∨

∼

Q)

∙

(Q

⊃

P)

Since the wff at 2.4.1 has two variables, the left side of the truth table has two columns. The assignments of truth values to the variables ‘P’ and ‘Q’ use the conventional method I mentioned; it would be good to memorize this pair of columns. All four-row truth tables ordinarily begin with this set of assignments, though it does not matter which variable gets which column at first. To continue to complete the truth table for 2.4.1, we copy the values from the left side of the truth table to columns under each propositional variable on the right side, making sure to assign the same values to any particular variable each time it occurs.

∨

∼

Q)

∙

(Q

⊃

P

Q

(P

P)

1

1

1

1

1

1

1

0

1

0

0

1

0

1

0

1

1

0

0

0

0

0

0

0

To complete the truth table, we have to fill in the column under the main operator, the conjunction. We work toward it in the order described in the formation rules of section 2.2, first evaluating the negations of any formulas whose columns are already complete, then evaluating binary operators whose two sides are complete. Let’s continue our example 2.4.1. First complete the column under the tilde.

2 . 4 : T r u t h T a b les   6 3

Q)

1

0

1

1

1

0

1

1

0

0

1

0

1

0

0

1

1

0

0

0

0

1

0

0

0

Q

(P

1

1

1

∨

∙

∼

P

(Q

⊃

P)

Then we can complete the columns under the disjunction and the conditional. P

Q

(P

∨

∼

Q)

1

1

1

1

0

1

0

1

1

0

1

0

0

0

0

∙

(Q

⊃

P)

1

1

1

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

0

Finally, we can complete the truth table by completing the column under the main operator, the conjunction, using the columns for the disjunction and the conditional. P

Q

(P

∨

∼

Q)

∙

(Q

⊃

P)

1

1

1

1

0

1

1

1

1

1

1

0

1

1

1

0

1

0

1

1

0

1

0

0

0

1

0

1

0

0

0

0

0

1

1

0

1

0

1

0

Thus, 2.4.1 is false when P is false and Q is true, and true otherwise. Ordinarily, we write out the truth table only once, as in the last table in this demonstration. Some people choose not to use the left side of the truth table, just assigning values to variables directly. This has the short-term advantage of making your truth tables shorter, but the long-term disadvantage of making them more difficult to read.

6 4    C h apter 2   P ropos i t i onal L og i c

Here is the start to an eight-line truth table, for 2.4.2. 2.4.2 [(P ⊃ Q) ∙ (Q ⊃ R)] ⊃ (P ⊃ R) P

Q

R

1

1

1

1

1

0

1

0

1

1

0

0

0

1

1

0

1

0

0

0

1

0

0

0

[(P

⊃

Q)

∙

(Q

⊃

R)]

⊃

(P

⊃

R)

To proceed, first copy the values of the component propositions, P, Q , and R into the right side of the table. ⊃

Q)

∙

(Q

⊃

R)]

⊃

(P

⊃

P

Q

R

[(P

R)

1

1

1

1

1

1

1

1

1

1

1

0

1

1

1

0

1

0

1

0

1

1

0

0

1

1

1

1

0

0

1

0

0

0

1

0

0

1

1

0

1

1

1

0

1

0

1

0

0

1

1

0

0

0

0

0

1

0

0

0

1

0

1

0

0

0

0

0

0

0

0

0

2 . 4 : T r u t h T a b les   6 5

Now work inside out, determining the truth values of the operators inside parentheses. P

Q

R

[(P

⊃

Q)

1

1

1

1

1

1

1

0

1

1

0

1

1

0

0

∙

(Q

⊃

R)]

1

1

1

1

1

1

1

0

0

0

1

0

1

1

0

0

1

0

0

0

0

0

⊃

(P

⊃

R)

1

1

1

1

0

0

1

0

0

0

1

1

1

1

1

0

0

1

0

1

0

0

1

1

1

1

1

0

1

1

0

1

1

1

0

0

0

1

0

1

0

1

0

0

1

1

0

1

1

0

0

1

0

0

1

0

0

1

0

(P

⊃

R)

Next, we evaluate the conjunction, inside the square brackets. P

Q

R

[(P

⊃

Q)

∙

(Q

⊃

R)]

⊃

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

1

1

0

1

0

0

1

0

0

1

0

1

1

0

0

0

0

1

1

1

1

1

1

0

0

1

0

0

0

0

1

0

1

0

0

0

1

1

0

1

1

1

1

1

1

0

1

1

0

1

0

0

1

1

0

1

0

0

0

1

0

0

0

1

0

1

0

1

0

1

1

0

1

1

0

0

0

0

1

0

1

0

1

0

0

1

0

6 6    C h apter 2   P ropos i t i onal L og i c

Finally, we reach the main operator. P

Q

R

[(P

⊃

Q)

∙

(Q

⊃

R)]

⊃

(P

⊃

R)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

1

1

0

1

0

0

1

1

0

0

1

0

1

1

0

0

0

0

1

1

1

1

1

1

1

0

0

1

0

0

0

0

1

0

1

1

0

0

0

1

1

0

1

1

1

1

1

1

1

0

1

1

0

1

0

0

1

1

0

1

0

0

1

0

1

0

0

0

1

0

1

0

1

0

1

1

1

0

1

1

0

0

0

0

1

0

1

0

1

0

1

0

1

0

You may notice that 2.4.2 has an interesting property: it is true in every row. Not every proposition is true in all cases! We will return to this property, and others, in the next section.

Summary The goal of this section is to show you how to construct truth tables for any proposition, of any length, at least in principle. It will be helpful to memorize the method for assigning truth values to variables for propositions with one, two, and three, and even four variables. But there is, of course, a general method that you could learn for propositions of any number of variables. Constructing Truth Tables for Propositions with Any Number of Variables Consider the atomic propositional variables, in any order. The first variable is assigned 1 in the top half of the table and 0 in the bottom half. The second variable is assigned 1 in the top quarter, 0 in the second quarter, 1 in the third quarter, and 0 in the bottom quarter. The third variable is assigned 1 in the top eighth, 0 in the second eighth, 1 in the third either, 0 in the fourth eighth . . . . . .  The final variable is assigned alternating instances of 1 and 0.

2 . 4 : T r u t h T a b les   6 7

Thus, in a 128-row truth table (7 variables), the first variable would get 64 1s and 64 0s, the second variable would get 32 1s, 32 0s, 32 1s, and 32 0s, the third variable would alternate 1s and 0s in groups of 16, the fourth variable would alternate 1s and 0s in groups of 8s . . . , and the seventh variable would alternate single instances of 1s and 0s. It does not matter which variables we take as first, second, third, and so on; I usually choose the order from the appearances of the variables in the formula, from left to right. Remember that every instance of the same variable letter gets the same assignment of truth values. While we can, in theory, construct truth tables for propositions of any number of variables or any complexity (measured by the number of operators), we ordinarily restrict our studies to propositions with fewer than five variables. The exercises at the end of this section contain propositions with at most four variables, requiring truth tables of no more than sixteen rows. We will want to work with propositions with more variables and greater complexity later, but by then we will have other methods to do so.

KEEP IN MIND

Truth tables summarize the distributions of truth values of simple and complex propositions. We construct truth tables for wffs of PL in three steps: 1. Determine how many rows we need. 2. Assign truth values to the component variables. 3. Work inside out until we reach the main operator. The number of rows of a truth table is a function of the number of variables in the wff. Assignments of truth values to a propositional variable must be consistent within the truth table. If completed correctly, the truth table will contain every different combination of truth values of the component propositions.

TELL ME MORE  • How do truth tables differ when we have more truth values? See 6.3: Three-Valued Logics. • How do we construct truth tables for other operators? See 6S.8: Adequacy.

6 8    C h apter 2   P ropos i t i onal L og i c

EXERCISES 2.4 Construct truth tables for each of the following propositions. 1. A ⊃ ∼A

21. A ∙ (B ∨ ∼C)

2. B ⊃ (∼B ⊃ B)

22. (D ∙ E) ⊃ ∼F

3. (C ∙ ∼C) ⊃ C

23. ∼G ≡ (H ∨ I)

4. (D ∨ ∼D) ≡ D

24. (M ⊃ N) ⊃ ∼(N ∨ O)

5. E ≡ ∼E

25. (P ⊃ Q ) ∨ [R ≡ (∼Q ∙ P)]

6. ∼[(I ∙ ∼I) ⊃ ∼I]

26. (S ∨ ∼T) ⊃ [(T ∙ ∼U) ≡ S]

7. ∼E ⊃ F 8. G ≡ ∼H 9. (K ≡ L) ⊃ L 10. ∼(M ∨ N) ≡ N 11. (M ∙ N) ∨ ∼M

27. [L ⊃ (M ∨ N)] ≡ L 28. [∼O ∙ (P ⊃ O)] ∨ Q 29. (∼R ∨ S) ∙(∼T ⊃ R) 30. [U ⊃ (V ⊃ W)] ∙ (V ∨ W) 31. [∼X ≡ (Y ∙ Z)] ⊃ (X ∨ Z)

12. (K ∙ L) ⊃ ∼K

32. (A ⊃ B) ∨ (C ≡ D)

13. (O ∨ ∼P) ⊃ (P ∙ ∼O)

33. [I ⊃ (J ∙ K)] ∨ (L ≡ I)

14. ∼[(Q ∨ R) ≡ ∼R]

34. (∼W ≡ X) ∨ (Z ⊃ ∼Y)

15. (S ∙ ∼T) ∨ (T ⊃ S)

35. ∼(A ⊃ B) ∙ (C ∨ D)

16. ∼(W ⊃ X) ∙(X ≡ W)

36. (A ∙ C) ⊃ [(B ∨ C) ≡ D]

17. (U ∙ ∼V) ⊃ (V ∨ U)

37. ∼(G ∙ F) ≡ [E ⊃ (H ∨ F)]

18. ∼[(W ∨ X) ∙ ∼X] ⊃ W

38. [(I ∨ J) ∙ (K ⊃ L)] ∨ ( J ∙ K)

19. [(∼Y ∙ Z) ⊃ Y] ∨ (Y ≡ Z)

39. (∼M ⊃ N) ∨ [(N ≡ O) ∙ P]

20. (A ≡ ∼B) ⊃ [(B ∨ ∼B) ∙ A]

40. [(∼M ∙ N) ∨ (O ⊃ P)] ≡ M

2.5: CLASSIFYING PROPOSITIONS The technical work of constructing truth tables for propositions of any length allows us to classify individual propositions and their relations in a variety of interesting ways. As in section 2.3, where the truth value of a complex proposition is its truth value at its main operator, the truth conditions for a proposition are the truth conditions at its main operator.

2 . 5 : C lass i f y i ng P ropos i t i ons   6 9

2.5.1, which you should recognize from the last section, is what we call tautologous; it is true in every row of its truth table. 2.5.1 [(P ⊃ Q) ∙ (Q ⊃ R)] ⊃ (P ⊃ R) P

Q

R

[(P

⊃

Q)

∙

(Q

⊃

R)]

⊃

(P

⊃

R)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

1

1

0

1

0

0

1

1

0

0

1

0

1

1

0

0

0

0

1

1

1

1

1

1

1

0

0

1

0

0

0

0

1

0

1

1

0

0

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1

1

0

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1

1

1

1

1

1

0

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0

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1

1

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1

0

0

0

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0

Tautologies are important because they are the logical truths of PL, propositions that are true on any interpretation, for any values of its component premises. While there are infinitely many tautologies of PL, most wffs are not tautologies. 2.5.2 is true in some cases, false in others; its truth value is contingent on the truth values of its component propositions.

A tautologyis a proposition that is true in every row of its truth table.

Logical truthsare propositions which are true on any interpretation.

2.5.2 P ∨ ∼Q P

Q

P

∨

∼

Q

1

1

1

1

0

1

1

0

1

1

1

0

0

1

0

0

0

1

0

0

0

1

1

0

Contingencies are true in at least one row of their truth table and false in at least one row. In ordinary language, we say that an event is contingent if it is possible that

A contingency is a proposition that is true in some rows of its truth table and false in others.

7 0    C h apter 2   P ropos i t i onal L og i c

A proposition which is false in every row of its truth table is a contradiction .

it happens and possible that it doesn’t happen; logical contingency is similarly neither certainly true nor certainly false. Some propositions are false in every row. We call such statements contradictions. 2.5.3 and 2.5.4 are contradictions. 2.5.3 P ∙ ∼P P

P

∙

∼

P

1

1

0

0

1

0

0

0

1

0

2.5.4 (∼P ⊃ Q) ≡ ∼(Q ∨ P) P

Q

(∼

P

⊃

Q)

≡

∼

(Q

∨

P)

1

1

0

1

1

1

0

0

1

1

1

1

0

0

1

1

0

0

0

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1

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0

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0

In addition to helping us characterize individual propositions, truth tables give us tools to characterize relations among two or more propositions. Propositions can have the same values or opposite values. Consider the tautology 2.5.5. 2.5.5 (A ∨ B) ≡ (∼B ⊃ A) A

B

(A

∨

B)

≡

(∼

B

⊃

A)

1

1

1

1

1

1

0

1

1

1

1

0

1

1

0

1

1

0

1

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1

0

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0

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0

2 . 5 : C lass i f y i ng P ropos i t i ons   7 1

Now consider the two sides of the biconditional in 2.5.5, as separate propositions, 2.5.6 and 2.5.7, and let’s look at the truth table for just the main operators of each. A ∨ B

2.5.6 ∨

A

∼B ⊃ A

2.5.7 ∼

B

⊃

B

1

1

1

1

1

1

0

0

A

Notice that 2.5.6 and 2.5.7 have the same truth values in each row; that’s what makes the biconditional between them a tautology. This property of propositions, having identical truth conditions, is called logical equivalence. The concept of logical equivalence has many uses. It is important in part because it shows a limit to the expressibility of truth-functional languages like PL: there are many equivalent ways of saying the same thing, of expressing the same truth conditions. For example, notice that the truth conditions of any statement made using the biconditional are identical to those made with a conjunction of two conditionals. That is, a statement of the form ‘α ≡ β’ is logically equivalent to a statement that uses only other operators, a statement of the form ‘(α ⊃ β) ∙ (β ⊃ α)’. α

β

α

≡

β

(α

⊃

β)

∙

(β

⊃

α)

1

1

1

1

1

1

1

1

1

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0

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1

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1

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1

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1

0

We can thus see the biconditional as a superfluous element of our logical language. Other operators can be shown to be superfluous in similar ways. When constructing languages for propositional logic, we have choices of which operators to use and how many operators to use. The study of the relations among the different operators, and

Two or more propositions are logically equivalent when they have the same truth values in every row of their truth tables.

7 2    C h apter 2   P ropos i t i onal L og i c

which operators are adequate for propositional logic, is a topic in metalogic. Meta­ logic is the study of logical systems. When evaluating the relations among two or more propositions, make sure to assign the same truth conditions to the same variables throughout the exercise. To compare the two propositions, the column under the A in 2.5.8 should be the same as the column under the A in 2.5.9, and similarly for the B, even though the B comes first, reading left to right, in the latter proposition. 2.5.8 A ∨ ∼B A

B

A

∨

∼

B

1

1

1

1

0

1

1

0

1

1

1

0

0

1

0

0

0

1

0

0

0

1

1

0

A

B

B

∙

∼

A

1

1

1

0

0

1

1

0

0

0

0

1

0

1

1

1

1

0

0

0

0

0

1

0

2.5.9 B ∙ ∼A

Two propositions with opposite truth values in all rows of the truth table are contradictory.

2.5.8 and 2.5.9 have opposite truth values in each row; we call this pair of propositions a contradiction. Notice that just as a biconditional connecting logically equivalent statements is a tautology, a biconditional connecting two contradictory statements will be a contradiction. Also notice that contradiction is a relation between exactly two propositions, where logical equivalence can hold for indefinitely many propositions. Most pairs of statements, like 2.5.10 and 2.5.11, are neither logically equivalent nor contradictory.

2 . 5 : C lass i f y i ng P ropos i t i ons   7 3

2.5.10 E ⊃ D

∼E ∙ D

2.5.11

E

D

E

⊃

D

E

D

∼

E

∙

D

1

1

1

1

1

1

1

0

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We can see that 2.5.10 and 2.5.11 are not contradictory in rows 2 and 4; they have the same truth values in those two rows. We can see that they are not logically equivalent in rows 1 and 4, where they have opposite truth values. Still, there are ways to characterize their relation. 2.5.10 and 2.5.11 are called consistent propositions, since while they are not equivalent, they still may be true together. There is at least one row of the truth tables in which both propositions are true. In consistent propositions, there are values of the component variables that will make both propositions true in the same conditions. 2.5.10 and 2.5.11 are both true in row 3. Thus, someone who uttered both propositions would be speaking truthfully if E is false and D is true. This assignment of truth values to component propositions is called a valuation. When you determine that two or more propositions are consistent, you can thus describe a consistent valuation by stating the values of the component variables in the row in which both full propositions are true. If two statements are neither logically equivalent nor contradictory, they may thus be consistent or inconsistent. Inconsistency is just the negation of consistency; like contradictoriness, inconsistency holds only among pairs of propositions. 2.5.12 and 2.5.13 are an inconsistent pair. 2.5.12 E ∙ F

∼(E ⊃ F)

2.5.13

E

F

E

∙

F

E

F

∼

(E

⊃

F)

1

1

1

1

1

1

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0

1

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Two or more propositions that are true in at least one common row of their truth tables are consistent.

A valuation i s an assignment of truth values to simple component propositions.

74    C h apter 2   P ropos i t i onal L og i c

In an inconsistent pairof propositions, there is no row of the truth table in which both statements are true; there is no consistent valuation.

Notice that the conjunction of two inconsistent statements is a self-contradiction. The difference between two sentences that are inconsistent and two sentences that are contradictory is subtle. In both cases, the pair of sentences cannot be true together. The difference is whether the pair can be false in the same conditions. Contradictory pairs always have opposite truth values. Inconsistent pairs may have truth conditions in which they are both false. When we are making assertions, and aiming at the truth, it is generally just as bad to make inconsistent assertions as it is to make contradictory assertions.

Summary In this section, we used the technique of completing truth tables for complex propositions that I described in section 2.4 to characterize both individual propositions and complex propositions. I identified three characteristics of individual propositions; they may be tautologies, contingencies, or contradictions. I identified four characteristics of comparisons between propositions: they may be logically equivalent or contradictory, or they may be consistent or inconsistent. In this section, we compared only pairs of propositions. Only pairs of propositions may be contradictory. But consistency, inconsistency, and logical equivalence are properties that can hold among sets of propositions of any size: two, three, or more propositions may be consistent or not, or logically equivalent. In section 2.7, we will explore a shortcut method for testing consistency and work with larger sets of propositions.

KEEP IN MIND

Remember to assign truth values consistently through the truth tables. In a single truth table, give each propositional variable the same distribution of truth values each time that letter appears in the formula. When comparing two propositions, assign the same truth values to each instance of each propositional variable throughout the two truth tables. Most pairs of statements are neither logically equivalent nor contradictory. Contradictory statements always have opposite truth values, whereas inconsistent pairs may have truth conditions in which they are both false. When comparing two propositions, first look for the stronger conditions: logical equivalence and contradiction. Then, if these fail, look for the weaker conditions: consistency and inconsistency.

2 . 5 : C lass i f y i ng P ropos i t i ons   7 5

TELL ME MORE  • How does the concept of logical equivalence help with translation? See 6.1: Notes on Translation with PL. • How are bivalence and logical truth related? See 6.3: Three-Valued Logics. • How does the need for consistency affect our ability to express truth in logic? See 7.5: Truth and Liars. • How can we use the concept of logical equivalence to explore other options for operators? See 6S.8: Adequacy. • Why are tautologies important? See 6S.9: Logical Truth, Analyticity, and Modality. • What is the role of consistency in theories, especially scientific theories? See 7S.8: Logic and Science.

EXERCISES 2.5a Construct truth tables for each of the following propositions and then classify each proposition as tautologous, contingent, or contradictory. 1. A ∨ ∼A

13. (∼M ∙ N) ∙ (N ⊃ M)

2. B ≡ ∼B

14. (A ⊃ B) ≡ (∼A ∨ B)

3. ∼C ⊃ ∼C

15. (∼E ⊃ F) ∨ (∼E ∙ ∼F)

4. ∼(A ∨ ∼A)

16. (M ⊃ ∼N) ∙ (M ∙ N)

5. ∼(B ∙ ∼B)

17. (Q ⊃ R) ≡ (∼R ∙ Q )

6. ∼C ≡ (C ∨ ∼C)

18. (S ⊃ ∼T) ∨ (T ≡ S) 19. (U ∙ ∼V) ⊃ (V ≡ ∼U)

7. B ≡ (A ∙ ∼B)

20. (∼P ≡ Q ) ∙ ∼[Q ⊃ (P ∨ Q )]

8. (C ∨ D) ∙ ∼(D ⊃ C)

21. (T ⊃ U) ∨ (U ⊃ T)

9. (E ⊃ F) ≡ ∼(F ∨ ∼E)

22. (D ⊃ F) ∨ (E ⊃ D)

10. (G ∙ ∼H) ∨ (H ⊃ ∼G)

23. (O ≡ P) ≡ [(∼O ∨ P) ⊃ (P ∙ ∼O)]

11. ∼(I ∙ J) ≡ (∼I ∨ ∼J)

24. ∼[W ⊃ (X ∨ ∼W)]

12. (K ⊃ L) ≡ (K ∙ ∼L)

25. (∼Y ⊃ ∼Z) ∙ (Z ∨ Y)

7 6    C h apter 2   P ropos i t i onal L og i c

26. ∼C ≡ (A ∨ ∼B) 27. (G ∙ H) ⊃ (G ∨ I) 28. (J ∙ ∼K) ∙ ∼(L ∨ J) 29. (N ∨ O) ⊃ (M ∙ O) 30. ∼(P ∙ Q ) ∨ (Q ⊃ R) 31. ∼{A ⊃ [(B ∙ C) ≡ ∼A]} 32. [(G ∙ H) ⊃ (I ∨ ∼H)] ≡ ∼(G ∙ H) 33. [(J ∙ K) ⊃ L] ≡ [L ∨ (∼J ∨ ∼K)] 34. [M ⊃ (∼N ∙ ∼O)] ∙ [(M ∙ N) ∨ (M ∙ O)] 35. [∼A ∨ (∼B ∙ ∼C)] ≡ [(A ∙ B) ∨ (A ∙ C)] 36. [D ∨ (E ∙ F)] ≡ [(D ∨ E) ∙ (D ∨ F)] 37. (G ∨ H) ∨ (I ∨ J) 38. (T ∙ U) ⊃ ∼(V ⊃ W) 39. [K ∙ (L ⊃ M)] ∨ (N ≡ K) 40. [P ⊃ (Q ∙ R)] ⊃ [∼S ≡ (P ∨ R)] 41. [(W ∙ X) ⊃ (Y ∨ Z)] ∨ [(∼Z ∙ X) ∙ (W ∙ ∼Y)] 42. [(A ∨ B) ⊃ (∼D ∙ C)] ≡ {∼[(A ∨ B) ∙ D] ∙ [(A ∨ B) ⊃ C]} 43. [(E ∙ F) ∨ (∼E ∨ ∼F)] ⊃ [(∼G ∙ H) ∙ (∼G ⊃ ∼H)] 44. [(J ⊃ ∼I) ∙ (∼K ⊃ ∼L)] ∙[(L ∙ ∼K) ∨ (I ∙ J)] 45. [M ≡ (∼N ∙ O)] ⊃ [(P ∙ N) ⊃ M]

EXERCISES 2.5b Construct truth tables for each of the following pairs of propositions. Then, for each pair of propositions, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. 1. ∼E ⊃ ∼F

and E ∨ F

2. G ⊃ H

and ∼H ∙ G

2 . 5 : C lass i f y i ng P ropos i t i ons   7 7

3. K ≡ L

and ∼(L ⊃ K)

4. ∼(M ∨ N)

and ∼M ∙ ∼N

5. ∼O ⊃ P

and O ∨ P

6. ∼Q ≡ R

and Q ∙ R

7. (S ∨ T) ∙ ∼S

and T ⊃ S

8. ∼Y ⊃ Z

and ∼Z ⊃ Y

9. ∼(A ∙ B)

and ∼A ⊃ B

10. C ⊃ (D ∙ C)

and ∼D ∙ C

11. (E ∨ F) ∙ E and ∼(E ∨ F) 12. (G ∙ H) ∨ ∼G and ∼H ⊃ (G ≡ H) 13. I ∨ (J ∙ ∼J)

and ( J ≡ ∼I) ∙ J

14. (∼M ∙ ∼N) ≡ N

and (N ∨ M) ∙ ∼M

15. (O ∨ P) ⊃ O and ∼O ≡ (P ∙ O) 16. (Q ∨ R) ∙ S

and (Q ⊃ S) ∙ R

17. T ∨ (U ∙ W)

and (T ∨ U) ∙ (T ∨ W)

18. (X ∙ Y) ∨ Z

and (∼X ∨ ∼Y) ∙ ∼Z

19. (A ∙ B) ⊃ C

and A ⊃ (B ⊃ C)

20. ∼(G ∨ H) ∙ I

and (I ⊃ G) ∙ H

21. ( J ≡ K) ∙ L

and [(∼L ∨ ∼K) ∙(L ∨ K)] ∨ ∼L

22. (M ⊃ N) ∨ (N ∙ ∼O) and (M ∙ ∼N) ∙ (∼N ∨ O) 23. (X ∙ Y) ⊃ Z

and (X ∙ Y) ∙ ∼Z

24. (A ⊃ B) ∙ C

and (∼B ⊃ ∼A) ∙ C

25. (∼D ⊃ ∼E) ∨ (F ≡ E) and (∼D ∙ E) ∙[(∼F ∨ ∼E) ∙ (F ∨ E)] 26. (∼K ⊃ L) ∙ ∼M

and M ≡ (L ∨ K)

27. (∼M ≡ N) ∙ (O ≡ P)

and ∼{(∼M ≡ N) ⊃ [(O ∙ P) ∨ (∼P ∙ O)]}

28. ∼[(W ∙ X) ⊃ (Y ⊃ Z)] and [(Y ⊃ W ) ⊃ (Y ∙ ∼X)] ∙ [(Z ∨ W) ⊃ (∼Z ∙ ∼X)] 29. (A ∨ B) ⊃ (C ∙ D)

and [∼C ⊃ (∼A ∙ ∼B)] ∙ [(A ∨ B) ⊃ D]

30. ( J ⊃ K) ⊃ (L ∨ I)

and (∼L ∙ ∼I) ∙ (K ∨ ∼J)

7 8    C h apter 2   P ropos i t i onal L og i c

2.6: VALID AND INVALID ARGUMENTS The central task of this book is to characterize logical consequence, to distinguish valid from invalid inferences. We have thus far used truth tables to characterize individual propositions and relations among propositions. We will now use truth tables to define validity in PL. 2.6.1 is a valid argument. 2.6.1

1. If God exists then every effect has a cause. 2. God exists. So, every effect has a cause.

2.6.1 is valid because there is no way for the premises to be true while the conclusion is false. Whether the premises are true, whether 2.6.1 is a sound argument, is a separate question, which we set aside here. 2.6.1 has the form 2.6.2. 2.6.2

α⊃β α / β

2.6.2 is a valid argument form known as modus ponens. We will see a lot of valid argument forms in this book; there are infinitely many. We will give names to eleven valid forms of argument for PL in chapter 3, as well as fourteen rules of equivalence and three different proof techniques. We write the valid forms which we name in the metalanguage, using Greek letters to indicate that any consistent substitution of wffs of PL for the metalinguistic variables (replacing each α with the same wff and each β with the same wff) yields a valid inference. We write the premises on sequential lines, and the conclusion on the same line as the final premise, following a single slash. In contrast to 2.6.1, 2.6.3 is not valid, since the premises could be true while the conclusion is false; the conclusion fails to follow from the premises. 2.6.3

A counterexample t o an argument is a valuation that makes the premises true and the conclusion false.

1. If God exists then every effect has a cause. 2. Every effect has a cause. So, God exists.

To show that 2.6.3 is invalid, we could assign truth values to the component propositions which yield true premises and a false conclusion. If ‘God exists’ were false and ‘every effect has a cause’ were true, then the conclusion would be false, but each of the premises would be true. (The first premise is vacuously true according to the basic truth table for the material conditional.) This assignment of truth values, or valuation, is called a counterexample to argument 2.6.3. The argument in example 2.6.3 has the form at 2.6.4. 2.6.4

α⊃β β / α

2 . 6 : Val i d an d In v al i d A rg u m ents   7 9

In deductive logic, an invalid argument is called a fallacy. In informal or inductive contexts, the term ‘fallacy’ has a broader meaning. Arguments of the form 2.6.4 are fallacies that are so well known that they have a name: affirming the consequent. It is logically possible for its premises to be true while its conclusion is false. A counterexample is generated when the wff that replaces α is false and the wff that replaces β is true. This fallacy is a formal result having nothing to do with the content of the propositions used in the argument. We need a rigorous method for distinguishing valid argument forms like 2.6.2 from invalid ones like 2.6.4. The truth table method for determining if an argument is valid is both rigorous and simple.

Method of Truth Tables to Test for Validity Step 1. Set up one set of truth tables for the whole argument by determining how many rows are needed: how many variables appear in the premises or conclusion? Assign truth values to the component variables on the left side of the truth table. Step 2. Line up premises and conclusion horizontally, separating premises with a single slash and separating the premises from the conclusion with a double slash. Step 3. Construct truth tables for each premise and the conclusion, using the assignments to component variables from the left side of the truth table consistently throughout the whole set of truth tables. Step 4. Look for a counterexample: a row in which the premises are all true and the conclusion is false. • If there are one or more counterexamples, the argument is invalid. Specify at least one counterexample. • If there is no counterexample, the argument is valid.

Recall that in a valid argument, if the premises are true then the conclusion must be true. An invalid argument is one in which it is possible for true premises to yield a false conclusion. In such cases, the conclusion contradicts the premises. By focusing on valid arguments, we can make sure that if all our premises are true, our conclusions will be true as well. False premises are irrelevant to counterexamples. Let’s examine the argument 2.6.5 to determine whether it is valid. 2.6.5 P ⊃ Q P

/Q

A valid argument h as no row of its truth table in which the premises are true and the conclusion is false. An invalid argument has at least one counterexample.

8 0    C h apter 2   P ropos i t i onal L og i c

First, we construct our truth table for the argument, assigning values to the two propositional variables on the left and then carrying them over to the appropriate columns on the right. Then, we evaluate all the main operators of the premises and the conclusions, which in this case involves only evaluating the conditional in the first premise; the values of the second premise and the conclusion are just the values of P and Q , respectively. P

Q

P

⊃

Q

1

1

1

1

1

1

1

1

0

1

0

0

1

0

0

1

0

1

1

0

1

0

0

0

1

0

0

0

/

P

//

Q

Now that our truth table is complete, we can search for a counterexample. Notice that in no row are the premises true and the conclusion false. There is thus no counterexample. 2.6.5 is a valid argument. In contrast, both 2.6.6 and 2.6.7 are invalid arguments. To show that they are invalid, we specify a counterexample. Some arguments will have more than one counterexample; demonstrating that one counterexample is sufficient to show that an argument is invalid. 2.6.6 P ⊃ Q Q

/P

P

Q

P

⊃

Q

1

1

1

1

1

1

1

1

0

1

0

0

0

1

0

1

0

1

1

1

0

0

0

0

1

0

0

0

/

Q

//

P

Argument 2.6.6 has a counterexample in the third row, when P is false and Q is true.

2 . 6 : Val i d an d In v al i d A rg u m ents   8 1

2.6.7 (P ∙ Q) ⊃ R ∼P ∨ R Q ∨ R / R ∙ Q P

Q

R

(P

∙

Q)

⊃

R

1

1

1

1

1

1

1

1

1

0

1

1

1

1

0

1

1

0

1

0

0

1

0

1

1

0

1

0 0

∼

P

∨

R

1

0

1

1

1

0

0

0

1

0

0

0

1

1

0

1

1

1

0

0

1

0

0

1

0

0

0

0

1

1

1

1

0

1

1

0

0

0

1

1

0

1

0

1

0

0

1

0

0

0

1

1

1

0

1

1

0

0

0

0

0

1

0

1

0

1

0

//

R

∙

Q

/

/

Q

∨

R

1

1

1

1

1

1

1

1

0

0

0

1

0

1

1

1

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

0

0

0

1

0

1

1

1

0

0

0

0

0

0

0

0

8 2    C h apter 2   P ropos i t i onal L og i c

Argument 2.6.7 has a counterexample in row 3, where P and R are true and Q is false. There is another counterexample in row 6, where Q is true but P and R are false. Again, one needs only a single counterexample to demonstrate that an argument is invalid.

Summary The method of determining the validity of an argument of PL in this section is the most important item in this chapter. It is a foundation of all of the work on derivations in chapter 3 and in principle could be used to test the validity of any argument of propositional logic. As we will see in the next section, and in the next chapter, this method can get unwieldy and there are alternative methods for determining the validity and the invalidity of longer, more complicated arguments. All of those methods, though, rely on this method for their justifications. KEEP IN MIND

To test an argument for validity, look for a counterexample, a valuation on which the premises are true and the conclusion is false. To look for a counterexample, construct one truth table for the entire argument, including all of the premises and the conclusion. Line up premises and conclusion horizontally, separating premises with a single slash and separating the premises from the conclusion with a double slash. Use consistent assignments to component variables throughout the whole truth table. If there is a counterexample, the argument is invalid. An invalid argument is one in which it is possible for true premises to yield a false conclusion.

TELL ME MORE  • How is validity related to bivalence? See 6.3: Three-Valued Logics. • What role does validity play in constructing logical systems? See 6.4: Metalogic. • What are fallacies? See 7.2: Fallacies and Argumentation.

EXERCISES 2.6 Construct truth tables to determine whether each argument is valid. If an argument is invalid, specify a counterexample. 1. A ⊃ ∼A ∼A

/A

2. ∼ ∼B ∨ (∼B ⊃ B)  / B

3. A ⊃ ∼A  ∼ ∼A / ∼A

2 . 6 : Val i d an d In v al i d A rg u m ents   8 3

4. B ∙ C C

/ ∼B

19. D ∨ E ∼D ∙ ∼F / ∼(E ∙ F)

5. C ∨ D ∼D

/ ∼C

20. G ≡ H H ∙ ∼I / ∼(I ∙ G)

6. E ∨ F ∼(E ∙ ∼F) / E ≡ F

21. J ⊃ ∼K K ⊃ L

7. G ≡ H ∼H

22. I ⊃ ( J ∙ K) I ∙ ∼K / J ∙ ∼K

/ ∼G

8. (K ∙ L) ∨ (K ∙ ∼L) ∼K /L 9. M ≡ ∼N M∨N M

/ ∼N ⊃ N

10. ∼P ⊃ Q Q ⊃ P

/ ∼P

11. A ⊃ B ∼B ∙ ∼A B

/ B ∨ ∼A

12. G ⊃ (H ∙ ∼G) H∨G ∼H / ∼G

23. O ⊃ P ∼P ∨ Q ∼Q

/ ∼(L ∙ J)

/ ∼O

24. (∼A ∨ B) ⊃ C A ∙ B /C 25. L ≡ (M ∨ N) L ∙ ∼N M ⊃ ∼L / ∼L 26. ∼R ∨ S ∼(∼T ∙ S) ∼T ∙ ∼R R ∨ S /T∙S 27. (U ∙ V) ∨ W (∼W ⊃ U) ⊃ V ∼V ∨ W ∼W ∨ U / U

13. J ⊃ K K ∼J ∨ K

/ ∼J

14. P ⊃ Q ∼Q ∨ P ∼Q

/P

28. (X ∙ Y) ≡ Z ∼Z ∙ X ∼X ⊃ Y /Y

15. R ≡ S ∼R ∨ S ∼S ⊃ ∼R

/R

/D

16. R ⊃ S S ∨ T

/R⊃T

17. X ∙ ∼Y Y ∨ Z

/ ∼Z

18. ∼(A ∙ B) B ⊃ C

/A

29. D ∨ ∼E ∼E ∙ F ∼D ⊃ F

30. (G ∙ H) ⊃ ∼I I∨G H ⊃ ∼G H ≡ I / ∼H ∨ G 31. T ⊃ (U ∙ V) T∙U ∼V U ⊃ ∼T / T

8 4    C h apter 2   P ropos i t i onal L og i c

32. M ∙ ∼N O⊃P P ∨ N

/ ∼M

33. Q ⊃ R S∨T T

/R

34. ∼W ⊃ (X ∨ Y) Y∙Z ∼(Z ⊃ X)

/W≡Y

35. ∼A ∙ (B ∨ C) C⊃A B ⊃ D

/ A ⊃ ∼D

36. E ∙ F G ⊃ (H ∨ ∼E) ∼F ∨ G

/H

38. ∼A ⊃ (B ∨ C) ∼C ∙ (∼B ∨ A) C ∨ ∼A A ≡ (B ⊃ ∼C) B / ∼A 39. (D ∙ G) ⊃ (E ∙ F) D∨E (G ∨ F) ≡ ∼E G ⊃ E / ∼G 40. ∼(H ⊃ K) K ⊃ (I ∙ J) I≡H H ⊃ ( J ∨ K) ∼J ∙ (H ∨ ∼K) / K

37. (W ⊃ X) ≡ Z ∼Z ∙ Y (Y ∙ W) ⊃ X X∙Z (W ∙ Y) ⊃ (∼Z ∙ X) / Z ∨ W

2.7: INDIRECT TRUTH TABLES We can use the truth table method of the previous section, a mechanical procedure for determining counterexamples, to determine the validity of any argument. But the method becomes unwieldy as the number of variables in an argument grows. With merely five variables, for example, a truth table is 32 lines. The truth table for an argument that contains ten propositional variables would require 1024 lines. Fortunately, there is a shortcut method for constructing counterexamples, the subject of this section. This method will also help us demonstrate whether a set of propositions is consistent or inconsistent. To show that an argument is valid, one must show that there is no row of the truth table with true premises and a false conclusion; we seem to have to examine every row. But we need only one row in order to demonstrate that an argument is invalid: a counterexample. Thus, to determine whether an argument is valid or invalid, we can try to construct a counterexample. If we find a counterexample, then we know the argument is invalid. If there are no counterexamples, then the argument is valid.

2 . 7 : In d i rect T r u t h T a b les   8 5

2.7.1 is an invalid argument, as we can show with the indirect, or shortcut, method. 2.7.1 G ≡ H G / ∼H

To show that 2.7.1 is invalid, first write it out, as you would a normal truth table for an argument. Just as I did for the truth tables, I’ll list all the component propositions on the left side of the table; that way, when we’re done, the valuation that generates a counterexample will be obvious. G

H

G

≡

H

/

G

//

∼

H

Next, we can assign the value true to H, in order to make the conclusion false. I’ll use the left side of the truth table to keep track of the valuation. G

H

G

≡

H

/

G

//

1

∼

H

0

1

∼

H

0

1

∼

H

0

1

Carry this value over to any other H in the argument. G

H

G

≡

1

H

/

G

//

1

Assign a value to G that makes the premises true. G

H

G

≡

H

1

1

1

1

1

/

G 1

//

2.7.1 is thus invalid since there is a counterexample when G is true and H is true. Note that an argument is either valid or invalid. If there is at least one counterexample, the argument is invalid. It is not merely invalid on that assignment of truth values; it is always invalid.

8 6    C h apter 2   P ropos i t i onal L og i c

Method of Indirect Truth Tables for Validity Line up your component variables, premises, and conclusions horizontally, as in the full truth table method, but do not create any further rows for the table. Try to assign values to component variables that make the conclusion false and all premises true. If such a valuation is possible, then the argument is invalid; specify the counter­ example. If no such valuation is possible, then the argument is valid. If there is a counterexample, this indirect method will be able to find it. But we have to make sure to try all possible valuations before we pronounce the argument valid. 2.7.2 is a valid argument. We will not be able to construct a counterexample. Let’s see how that goes. 2.7.2 C ⊃ (D ⊃ E) D ⊃ (E ⊃ F)

/ C ⊃ (D ⊃ F)

The only way to make the conclusion false is to assign true to C and to D, and false to F.

D

C

D

1

1

⊃

(E

E

F

C

⊃

(D

⊃

E)

/

//

C

⊃

(D

⊃

F)

1

0

1

0

0

⊃

(D

⊃

E)

/

0

⊃

F)

Carry these values over to the premises.

D 1

C

D

1

1

⊃

(E

E

⊃

F

C

0

1

F)

//

0

1

C

⊃

(D

⊃

F)

1

0

1

0

0

2 . 7 : In d i rect T r u t h T a b les   8 7

To make the first premise true, E must also be true.

D

C

D

E

F

C

⊃

(D

⊃

E)

1

1

1

0

1

1

1

1

1

⊃

(E

⊃

F)

//

C

⊃

(D

⊃

F)

1

0

1

0

0

1

1

0

/

But now the second premise is false. If we tried to make the second premise true by making E false, the first premise would come out false. There was no other way to make the conclusion false. So, there is no counterexample. 2.7.2 is thus valid. In some arguments, there is more than one way to make a conclusion false or to make premises true. You may have to try more than one. Once you arrive at a counterexample, you may stop. But if you fail to find a counterexample, you must keep going until you have tried all possible assignments. The argument at 2.7.3 has multiple counterexamples. 2.7.3 I ⊃ K K ⊃ J

/I∙J

There are three ways to make the conclusion of 2.7.3 false. We can try them in any order, but we have to remember that if our first attempts to construct true premises fail, we must try the others. I’ll write them all, which gives us (potentially) a threerow truth table to complete; it’s still fewer than the eight rows we would need in a full truth table. I

K

J

I

⊃

K

/

K

⊃

J

//

I

∙

J

1

0

1

0

0

0

1

0

0

1

0

0

0

0

0

In the first row, there is no way to assign a truth value to K that makes the premises true.

8 8    C h apter 2   P ropos i t i onal L og i c

I

K

I

∙

J

1

0

0

1

0

0

1

0

0

0

0

I

∙

J

0

1

0

0

1

0

0

1

0

0

0

J

I

1

0

1

0 0

⊃

K

/

?

K

⊃

?

J

//

0

We must move on to the second option. I

K

J

I

1

0

1

0

1

0

0

0

⊃

K

/

?

K

⊃

?

J

//

In the second row, we can assign either value to K and find a counterexample. So, 2.7.3 is shown invalid by the counterexample when I is false, J is true, and K is true; it is also shown invalid by the counterexample when I is false, J is true, and K is false. Since we found counterexamples in the second option, there is no need to continue with the third option. 2.7.4 requires more work. 2.7.4 T ⊃ (U ∨ X) U ⊃ (Y ∨ Z) Z ⊃ A ∼(A ∨ Y)

/ ∼T

Let’s start with the conclusion, making T true in order to make its negation false, carrying that assignment into the first premise. T

U

X

Y

Z

A

1

U

⊃

(Y

∨

Z)

/

T

⊃

(U

∨

X)

/

∨

Y)

//

∼

T

0

1

1

Z

⊃

A

/

∼

(A

2 . 7 : In d i rect T r u t h T a b les   8 9

From the first premise, ‘U ∨ X’ must be true, but there are three ways to assign values to make it so. Similarly, there are multiple ways to assign values for the second and third premises. But there is only one assignment that makes the fourth premise true, making A and Y false. T

U

X

1

U

Y

Z

0

⊃

(Y

∨

Z)

A

T

0

1

/

Z

⊃

(U

∨

X)

/

⊃

A

/

∼

(A

∨

Y)

1

0

0

0

//

∼

T

0

1

∼

T

0

1

∼

T

0

1

Let’s carry these assignments to the Y in the second premise and the A in the third. T

U

X

1

U

Y

Z

0

⊃

(Y

∨

Z)

A

T

0

1

/

Z

⊃

(U

∨

X)

/

⊃

A

/

∼

(A

∨

Y)

1

0

0

0

0

0

//

Inspecting the third premise, we can see that Z must also be false; we can carry this value to the second premise. T

U

X

1

U

⊃

(Y 0

⊃

(U

∨

X)

/

Z

⊃

A

/

∼

(A

∨

Y)

0

1

0

1

0

0

0

Y

Z

A

T

0

0

0

1

∨

Z)

/

0

//

9 0    C h apter 2   P ropos i t i onal L og i c

Since “Y ∨ Z” has now been made false, U must be made false in order to keep the second premise true.

U

⊃

(Y

∨

Z)

0

1

0

0

0

T

U

1

0

/

Z 0

X

⊃

(U

∨

X)

/

(A

∨

Y)

//

∼

T

0

0

0

0

1

Y

Z

A

T

0

0

0

1

⊃

A

/

∼

1

0

1

Carry the value of U to the first premise; we are now forced to make X true in order to make the first premise true.

U

⊃

(Y

∨

Z)

0

1

0

0

0

T

U

1

0

/

Z 0

X

⊃

Y

Z

A

T

(U

0

0

0

1

⊃

A

/

∼

(A

∨

Y)

1

0

1

0

0

0

∨

X)

/

//

∼

T

0

1

0

The counterexample is thus constructed. The argument is shown invalid when T and X are true and U, Y, Z, and A are all false.

U

⊃

(Y

∨

Z)

0

1

0

0

0

T

U

X

Y

Z

A

T

⊃

(U

∨

X)

1

0

1

0

0

0

1

1

0

1

1

/

Z

⊃

A

/

∼

(A

∨

Y)

//

∼

T

0

1

0

1

0

0

0

0

1

/

2 . 7 : In d i rect T r u t h T a b les   9 1

Consistency and the Indirect Method The most important use of the indirect truth table method is in determining whether an argument is valid. An argument is valid if there is no valuation, or assignment of truth values to the component propositional variables, such that the premises come out true and the conclusion comes out false. That condition for validity is the same as testing whether the negation of the conclusion is consistent with the premises: a set of propositions is consistent when there is a set of truth values that we can assign to the component variables such that all the propositions come out true. So, we can use the same method for determining whether a set of propositions is consistent as we used for determining whether an argument is valid. If we can find a valuation that makes all of the propositions in a set true, then we have shown them to be consistent; this assignment is called a consistent valuation. If no consistent valuation is possible, then the set is inconsistent. To determine if a set of propositions is consistent, line them up, just as we lined up the premises and conclusion in evaluating arguments. We use only single slashes between the propositions; since a set of sentences has no conclusion, there is no differentiation between premises and conclusion and we are just trying to make all propositions come out true. Let’s examine the set of propositions 2.7.5 to see if they are consistent.

A consistent valuation is an assignment of

truth values to atomic propositions that makes a set of propositions all true.

2.7.5 A ⊃ (B ∨ C) ∼B ∨ ∼C (A ∙ B) ⊃ C A ∙ D

Let’s start with the last proposition, since there is only one way to make it true. Remember, we are working now with bare sets of propositions, not premises and conclusions, trying to make all propositions true.

A

B

C

1

(A

D

A

⊃

(B

∨

C)

C

/

A

∙

D

1

1

1

1

∙

B)

⊃

/

∼

B

∨

∼

C

/

9 2    C h apter 2   P ropos i t i onal L og i c

I’ll carry the value of A through the rest of the set (there are no other Ds), but there are no other obvious, forced moves. A

B

C

1

D

A

1

1

⊃

(B

∨

C)

/

∼

B

∨

∼

C

/

(A

∙

B)

⊃

C

/

A

∙

D

1

1

1

1

The consequent in the first proposition must be true, but there are three ways to make it true (making B true, C true, or both). There are three ways to make any conditional, like that in the second proposition, true. And the antecedent of the third proposition may be either true or false, so we are not forced to assign a value to its consequent. We must arbitrarily choose a next place to work. I’ll choose to start with B, expanding the table to include a true value and a false value for B. If one does not work out, I will have to return to the other one. A

B

1 1

C

⊃

(B

∨

A

1

1

1

1

1

0

1

1

0

0

(A

C)

∙

/

∼

D

B)

⊃

B

C

∨

∼

C

/

/

A

∙

D

1

1

1

1

1

1

0

1

1

1

I’ll try the first line first. Assigning 1 to B makes the first proposition true, without constraining an assignment to C; so far so good. In the second proposition, if B is true,

2 . 7 : In d i rect T r u t h T a b les   9 3

then C must be false. But in the third proposition, if B is true, then the antecedent is true and so C must be true. D

A

⊃

(B

∨

1

1

1

1

1

1

1

0

1

1

(A

∙

B)

⊃

C

1

1

A

B

1

1

C

C)

/

∼

B

0

1

0

∼

C

/

C

/

0

A

∙

D

1

1

1

1

0

1

1

1

/

∨

There is thus no consistent valuation with B true. Let’s move to the second line, where B is false; I’ll cross off the values in the first row to remind us that we’re finished with it. With B false, in the first proposition, C must be true. D

A

⊃

(B

∨

1

1

1

1

1

1

1

1

1

0

1

1

∙

B)

⊃

C

/

A

∙

D

1

1

1

1

1

0

1

1

1

A

B

1

1

1

0

(A 1 1

C

C)

/

∼

B

0

1

∨

∼

0

But making B false makes the second proposition true without considering the value for C. And the third proposition is the same; once we make B false, the antecedent is false and so the proposition is true.

9 4    C h apter 2   P ropos i t i onal L og i c

A

B

1

1

1

0

C

1

D

A

⊃

(B

∨

1

1

1

1

1

1

1

1

0

1

1

(A

∙

B)

1

1

1

1

0

0

C)

/

∼

B

∨

0

1

1

0

1

⊃

C

/

1

∼

C

/

A

∙

D

1

1

1

1

1

1

We have thus found a consistent valuation. The set of propositions is shown consistent when A, C, and D are true and B is false. Method of Indirect Truth Tables for Consistency Line up your component variables on the left and all propositions on the right, separated by single slashes. Assign values to propositional variables to make each statement true. If you can make each statement true, then the set is consistent. Provide a consistent valuation. If it is not possible to make each statement true, then the set is inconsistent. Let’s look at another example. 2.7.6 A ≡ B (B ∨ ∼A) ⊃ C (A ∨ ∼B) ⊃ D D ⊃ E ∼F ∨ ∼D

First, we’ll line up the propositions, separating each by a single slash. Again, I’ll use two rows. A

B

C

D

E

F

A

≡

B

/

(B

∨

∼

A)

⊃

C

/

(A

∨

∼

B)

⊃

D

/

D

⊃

E

/

∼

F

∨

∼

D

2 . 7 : In d i rect T r u t h T a b les   9 5

There is no obvious place to start. There are three ways to make the conditionals in the second, third, and fourth propositions true and three ways to make the disjunction in the final proposition true. We might as well start with the first proposition, since there are only two ways to make it true: either both A and B are true or both A and B are false. Other options are available, and may even be better in the long run. In this example, I’ll work on both rows at the same time, carrying values for A and B throughout. A

≡

B

1

1

1

1

1

1

0

0

0

1

0

0

0

(A

∨

/

D

⊃

A

B

1

C

∼

D

B)

1

1

0

0

E

⊃

F

D

/

E

(B

/

∨

∼

∼

A)

⊃

C

F

∨

∼

D

⊃

C

∼

D

/

We can make some progress on the second and third propositions. A

≡

B

1

1

1

0

0

0

(A

∨

∼

B)

/

1

1

0

1

0

1

1

0

A

B

1

C

D

E

⊃

F

D

(B

∨

∼

A)

1

1

1

0

1

1

0

0

1

1

0

D

⊃

/

∼

F

∨

/

E

Looking at the second proposition, above, we see that C must be true in both rows, since the antecedent of the main operator is true in both rows. Similar reasoning holds for D. I’ll fill in the results for the second and third propositions and carry the values for D to the fourth and fifth.

/

9 6    C h apter 2   P ropos i t i onal L og i c

A

≡

B

1

1

1

1

1

0

(A

∨

∼

B)

⊃

D

1

1

0

1

1

1

1

1

0

1

1

0

1

1

1

1

A

B

C

D

1

1

1

0

0

E

F

(B

∨

∼

A)

⊃

C

1

1

1

0

1

1

1

1

0

0

1

1

0

1

1

/

D

E

/

∼

F

∨

∼

/

⊃

/

D

Now we can see from the fourth proposition that E must be true in both rows, too. We can also evaluate the negation in the fifth proposition.

A

≡

B

1

1

1

1

1

0

∨

∼

B)

⊃

D

1

1

0

1

1

0

1

1

0

1

A

B

C

D

E

1

1

1

1

0

0

1

(A

F

(B

∨

∼

A)

⊃

C

1

1

1

0

1

1

1

1

0

0

1

1

0

1

1

/

D

⊃

E

/

∼

F

∨

∼

D

1

1

1

1

0

1

1

1

1

1

0

1

/

/

Since we want the last proposition to be true, since we are working toward a consistent valuation, the negation of F must be true. But for the negation of F to be true, F must be false.

2 . 7 : In d i rect T r u t h T a b les   9 7

A

B

C

D

E

F

A

≡

B

(B

∨

∼

A)

⊃

C

1

1

1

1

1

0

1

1

1

1

1

0

1

1

1

0

0

1

1

1

0

0

1

0

0

1

1

0

1

1

(A

∨

∼

B)

⊃

D

/

D

⊃

E

/

∼

F

∨

∼

D

1

1

0

1

1

1

1

1

1

1

0

1

0

1

0

1

1

0

1

1

1

1

1

1

0

1

0

1

⊃

T)

/

/

We have thus found two consistent valuations for 2.7.6: when A, B, C, D, and E are all true and F is false; and when A, B, and F are false and C, D, and E are true. Remember, just as an argument is invalid if there is at least one counterexample, a set of propositions is consistent if there is at least one consistent valuation; we do not need the second one. If there is no consistent valuation, the set is inconsistent. Let’s look at one more example. 2.7.7 P ⊃ (Q ∙ R) Q ⊃ (S ⊃ T) R ⊃ (T ⊃ ∼S) P ∙ S

We have a clear place to begin: the fourth proposition. P and S must both be true. I’ll fill in those values through all four propositions. P

Q

R

1

R

S

T

1

⊃

(T

⊃

P

⊃

(Q

∙

R)

1

∼

S) 1

/

Q

⊃

(S 1

/

P

∙

S

1

1

1

/

9 8    C h apter 2   P ropos i t i onal L og i c

Looking at the first proposition next, we can see that the values of Q and R are also determined. I’ll fill those in throughout, finishing the first proposition, and evaluate the negation of S in the third proposition. P

Q

R

S

1

1

1

1

T

P

⊃

(Q

∙

R)

1

1

1

1

1

R

⊃

(T

1

/

Q

⊃

1

⊃

(S

⊃

T)

/

P

∙

S

1

1

1

1

∼

S)

0

1

/

Now we can turn our attention to the final component variable T. If we make T true, then the second proposition comes out true but the third proposition turns out false. If we make T false, then the third proposition comes out true but the second proposition comes out false. There are no other possibilities: our hand was forced at each prior step. There is no way to make all the propositions in the set true. 2.7.7 is thus an inconsistent set of propositions.

Summary The method of indirect truth tables is powerful when applied both to determining the validity of an argument and to determining the consistency of a set of propositions. (It can also be fun to use!) At root, it is the same method. But be careful to distinguish the two cases. When we want to know if a set of propositions is consistent, we try to make all the propositions true. When we want to know if an argument is valid, we look for a counterexample, a valuation on which the premises all come out true but the conclusion comes out false. And remember, some arguments are valid and some arguments are invalid, and some sets of propositions are consistent and some sets of propositions are inconsistent. So, even though you must try all possible valuations, you might not be able to find a counterexample or consistent valuation. We will use an extended version of this indirect truth table method for determining counterexamples to arguments again in chapters 4 and 5, in first-order logic. For now, there are two salient applications of the method. When determining if an argument is valid, the method, if used properly, will generate a counterexample if there is one. For sets of sentences, the method will yield a consistent valuation if there is one. Make sure to work until you have exhausted all possible assignments of truth values to the simple, component propositions.

2 . 7 : In d i rect T r u t h T a b les   9 9

KEEP IN MIND

To determine whether an argument is valid or invalid, try to construct a counterexample. Making the premises true and the conclusion false shows that an argument is invalid. To determine whether a set of propositions is consistent, try to construct a consistent valuation. Making all propositions true shows that the set is consistent.

TELL ME MORE  • How, other than being formally invalid, can an argument be bad? See 7.2: Fallacies and Argumentation.

EXERCISES 2.7a Determine whether each of the following arguments is valid. If invalid, specify a counterexample. 1. L ≡ (M ∙ N) L∙O (M ∙ O) ⊃ P

/P

2. A ⊃ (B ∨ C) C ∙ (∼D ⊃ A) E ∙ B

/E∙A

3. F ≡ (G ∨ H) I ⊃ (J ⊃ F) (I ∙ G) ∨ H

/J⊃G

4. (Z ∙ V) ⊃ (U ∨ W) X ∨ (∼Y ≡ W) Z ∙ Y / ∼U 5. A ∙ B B⊃C ∼B ∨ (D ⊃ ∼C) / ∼D

1 0 0    C h apter 2   P ropos i t i onal L og i c

6. ∼Y ≡ (∼X ∙ Z) Z ⊃ Y

/Z⊃X

7. J ∨ M L∙M K ⊃ L

/ ∼K ⊃ J

8. N ⊃ O O∙P P ≡ Q

/ ∼(Q ∨ N)

9. T ≡ S S∙U R ⊃ U

/R∨T

10. Z ∨ (X ∙ Y) W≡V Z ∙ V

/ W ⊃ (X ∨ Y)

11. S ⊃ (V ∙ T) U∨R ∼S ≡ (R ∨ T)

/T⊃U

12. E ⊃ (F ∨ H) (G ∙ H) ⊃ E ∼F ∙ ∼H E ⊃ ∼G

/ E ⊃ ∼H

13. W ⊃ (X ∙ Y) ∼(Z ⊃ X) X ∨ (W ∙ ∼Z)

/ Y ∙ ∼Z

14. A ∨ (D ∙ C) A ⊃ (B ∨ C) D ∙ (∼B ∙ ∼C) / D ∙ C 15. ∼N [(O ∨ P) ∨ Q ] ⊃ (N ∙ R) P ⊃ ∼Q (O ∙ R) ⊃ N /P 16. D ∨ ∼E (F ∙ G) ∙ ∼H D ⊃ (H ∨ I) ∼I

/ F ∙ ∼E

17. J ⊃ (K ∙ ∼L) ∼L ≡ (N ⊃ M) J ∨ ∼N K ∙ M

/ ∼N

2 . 7 : In d i rect T r u t h T a b les   1 0 1

18. ∼(P ⊃ Q ) R ≡ (S ∨ ∼T) P⊃R Q ∨ T

/S∨O

19. B ∙ (D ∨ C) D ⊃ (A ∨ E) ∼E ∨ (B ∙ C)

/ (A ⊃ E) ∨ C

20. (F ∨ G) ≡ (H ∙ J) (I ⊃ H) ∙ (J ∨ G) ∼G /I⊃F 21. K ⊃ (M ⊃ P) P ∙ ∼(N ∨ L) O ⊃ (K ≡ N)

/M

22. Q ⊃ (T ∙ S) R ≡ (U ∨ T) ∼[S ⊃ (T ⊃ Q )] / ∼U ∙ S 23. Y ⊃ (Z ≡ X) Y ∙ ∼W W ⊃ (Y ∨ Z)

/ ∼(X ⊃ ∼W)

24. L ⊃ (M ≡ ∼N) (M ∙ O) ∨ (∼P ∙ O) O∨L ∼M /N 25. S ∙ (T ∨ W) U ⊃ (W ∙ V) S ≡ ∼W T ⊃ V

/ ∼(S ∨ U)

26. ∼(X ∨ Y) ∼(Z ⊃ W) U∙Z X ⊃ V

/Y≡U

27. R ⊃ [U ∨ (S ∨ Q )] R ∙ ∼S ∼U ≡ T /T⊃Q 28. N ∙ (Q ⊃ P) M ∨ ∼L L⊃Q P ∨ M

/L≡M

1 0 2    C h apter 2   P ropos i t i onal L og i c

29. U ∙ ∼R (T ∨ ∼S) ≡ U R∨S T ⊃ (∼R ∨ V)

/ ∼V

30. E ∙ F E⊃G ∼G ∙ ∼H

/F≡H

31. N ∨ O N ⊃ (Q ⊃ O) (P ∨ Q ) ∨ R R ⊃ ∼R / ∼O ⊃ P 32. A ⊃ B B∨C D ≡ C

/ D ≡ ∼B

33. ∼(I ≡ J) K ⊃ ( J ∨ L) (I ∙ L) ⊃ K (L ∨ J) ∨ (K ⊃ I) / ∼( J ⊃ L) 34. Q ⊃ T (T ∙ S) ∨ R R≡Q ∼(S ∙ R)

/ ∼Q ≡ S

35. Q ⊃ (R ∙ ∼S) (T ∨ U) ∙ (V ⊃ W) (R ⊃ S) ∙ ∼U /Q∙T 36. F ⊃ (G ∙ H) ∼I ⊃ (G ⊃ ∼H) I ⊃ ( J ∙ K)

/ F ⊃ ( J ∙ K)

37. V ⊃ (Z ∙ W) X ∨ ∼Y Z⊃Y V ≡ Y

/ ∼W

38. B ∙ (C ⊃ E) B ⊃ (A ∙ F) D ⊃ (∼B ∨ C) E ⊃ D

/ A ≡ [F ∙ (E ⊃ C)]

39. E ≡ [(F ∙ G) ⊃ H] ∼H ∙ ∼F E∨G G ⊃ (F ∙ E) / H ∙ ∼E

2 . 7 : In d i rect T r u t h T a b les   1 0 3

40. ∼(E ∙ F) F ⊃ (G ∨ H) (H ∙ E) ≡ F G ∨ ∼F / ∼(E ∨ G) 41. I ∨ ( J ∙ K) (∼I ⊃ J) ⊃ L L ≡ (∼J ∨ ∼I)

/ (I ∨ K) ∙ (I ⊃ ∼J)

42. ∼(∼C ∙ B) A∨D D ≡ (∼B ∙ ∼A) C ⊃ ∼A /∼(B ∨ D) 43. ∼[I ⊃ ( J ∙ K)] J ∨ ∼L M ⊃ (K ∙ I) L ≡ M

/J≡K

44. (M ⊃ N) ∙ (O ⊃ P) N∨O (M ∨ P) ≡ (∼N ⊃ ∼O) (O ⊃ N) ⊃ (∼M ∙ ∼P) / ∼(M ∨ P) 45. (A ∙ ∼D) ∨ (∼B ∙ C) ∼C ⊃ ∼B (A ∨ E) ⊃ D A ≡ B

/ ∼(A ∙ ∼E) ∙ B

EXERCISES 2.7b Determine, for each given set of propositions, whether it is consistent. If it is, provide a consistent valuation. 1. A ∨ B B ∙ ∼C ∼C ⊃ D D≡A

3. D A⊃C (B ∙ ∼C) ∙ ∼A D ⊃ (A ∙ B)

2. D ⊃ F F ≡ (A ∙ E) D ∙ (B ∨ C) ∼A E∨C

4. B ∙ (C ⊃ A) D ∨ (E ∙ F) F ⊃ (C ∨ D) E ∙ ∼A

1 0 4    C h apter 2   P ropos i t i onal L og i c

5. ∼A ∙ ∼E (A ∨ B) ⊃ (D ∙ F) C ⊃ ( E ⊃ D) ∼A ∙ (C ∨ B)

14. D ⊃ (∼A ∙ ∼F) E ∨ (∼B ∨ C) E⊃C A ∙ (∼B ≡ D)

6. ∼[A ⊃ (F ∙ B)] B ∙ (E ∙ ∼D) E≡F D ⊃ (C ∙ A)

15. B ∨ (F ∙ D) E≡B ∼E ∙ ∼F D ⊃ (A ⊃ C) (C ∙ E) ∨ A

7. G ⊃ (H ∙ I) ∼J ⊃ (K ∨ L) L ∨ (G ⊃ J) (I ≡ K) ∨ H 8. (A ∙ C) ⊃ (D ∙ B) ∼(A ⊃ D) ∙ ∼(C ⊃ B) B ≡ ∼(D ∨ C) (A ∙ B) ⊃ ∼C 9. (W ∙ X) ⊃ Z (Y ∙ W) ≡ (X ∙ Z) W ∨ (Y ⊃ Z) (X ∙ Y) ⊃ (Z ∨ W) 10. (E ∙ F) ⊃ (G ∨ H) (E ∙ ∼H) ∙ (I ∨ J) (I ⊃ ∼H) ∙ (F ∙ ∼G) ( J ∙ I) ≡ ∼F

16. (O ∨ ∼P) ⊃ ∼Q R ∙ (∼S ∨ T) O ∙ ∼(R ⊃ Q ) P⊃S 17. O ≡ Q P ∙ (Q ∨ O) R ⊃ ∼(P ∙ S) (S ∨ O) ∙ ∼Q 18. T ∙ V U ⊃ (W ∙ X) Y ∙ (T ⊃ ∼V) (Z ∙ U) ≡ (W ∙ Y) X ⊃ (V ∨ W) 19. Q ⊃ (R ∨ S) T ≡ (U ∙ Q ) (∼S ∙ Q ) ∙ (R ∨ T) U ∨ (S ∙ T)

11. ∼F ∙ ∼G H ⊃ (I ∙ F) J ∙ (F ∨ G) ∼H ∨ (I ∙ J) H ≡ ∼F

20. ∼(J ⊃ I) I ∙ (K ∨ L) (L ∙ J) ≡ K (K ∙ I) ⊃ ∼( J ∨ L)

12. (F ∙ G) ≡ I (H ∨ J) ⊃ F K ≡ (G ∙ J) H ⊃ (K ≡ I)

21. ∼F (E ∙ G) ⊃ F (E ∙ H) ∙ G F≡H

13. C ≡ (D ∨ B) D ∙ (C ⊃ A) ∼A ∙ (E ∨ F) F ⊃ (B ∙ A)

22. ∼(M ⊃ K) ( J ∙ L) ⊃ K ( J ∨ M) ∙ (M ⊃ J) K∨L

2 . 7 : In d i rect T r u t h T a b les   1 0 5

23. ∼( J ⊃ N) N ⊃ (M ∙ ∼L) K ≡ ∼I J ∙ (K ∨ M) I∙L 24. K ⊃ (L ∙ M) N ∙ ∼M (K ∙ ∼L) ∨ (K ∙ ∼M) (N ⊃ K) ∨ (N ⊃ L) 25. (∼L ∙ M) ⊃ O ∼(N ∙ P) L ≡ (P ∨ O) (M ∙ P) ∨ (N ∙ O) 26. K L ⊃ (M ∙ N) (N ∙ K) ≡ ∼L (M ⊃ L) ∨ ∼K 27. S ⊃ [O ∙ (∼P ∙ R)] S ∨ (T ∙ ∼O) R ⊃ (P ≡ T) ∼S ∨ R

32. (∼J ∨ ∼K) ∙ L ∼I ∨ (M ∨ N) L ⊃ (I ∙ J) ( J ∙ M) ⊃ ∼N 33. ∼[A ⊃ (B ∙ C)] C ⊃ (D ≡ E) (B ∙ E) ∨ (A ∙ C) (D ∨ B) ⊃ ∼A 34. ∼[(T ∙ Z) ⊃ (W ≡ V)] U ⊃ (X ∙ Y) (X ∨ T) ⊃ [Y ⊃ (W ∙ U)] ∼(V ∙ Z) 35. ∼[(C ⊃ D) ∨ (A ∙ B)] B ⊃ (A ∙ C) ∼D ≡ (B ∙ C) (A ∙ D) ⊃ ∼B 36. M ≡ (N ⊃ P) (O ∨ Q ) ∙ (M ⊃ P) ∼[(N ∨ O) ∙ Q ] (P ⊃ Q ) ⊃ ∼M

28. P ⊃ [Q ⊃ (R ⊃ O)] ∼S ∙ T R ≡ (T ∙ P) ∼(O ∨ S)

37. R ⊃ ∼(S ∙ T) (U ∨ R) ⊃ (T ∙ V) V ≡ (S ⊃ R) (U ∨ T) ⊃ ∼(S ∙ R)

29. ∼(T ∙ S) ∙ (∼O ⊃ R) S ∙ (O ∨ ∼P) R ⊃ (Q ∙ P) T ∨ ∼O

38. (P ∨ Q ) ∙ (R ⊃ ∼S) (Q ⊃ R) ∙ (P ⊃ S) S ≡ (∼Q ∨ S) (R ∙ P) ⊃ Q

30. ∼[ J ⊃ (K ⊃ L)] (M ≡ N) ⊃ J (L ∨ N) ∙ K ∼(M ∙ K)

39. ∼[E ∨ (F ≡ H)] ∼[(G ∙ F) ∨ (H ∙ I)] I ⊃ (G ∙ E) ∼(I ∙ F) ⊃ H

31. ∼O (P ∙ Q ) ⊃ (R ∨ S) S≡O (P ⊃ R) ⊃ Q

40. L ∨ (K ∙ J) J ⊃ (M ∙ N) M ⊃ (I ∨ J) ∼[(N ⊃ K) ∙ L]

1 0 6    C h apter 2   P ropos i t i onal L og i c

KEY TERMS antecedent, 2.1 atomic formula, 2.2 basic truth table, 2.3 biconditional, 2.1 binary operator, 2.1 bivalent logic, 2.3 complex formula, 2.2 compositionality, 2.1 conditional, 2.1 conjunction, 2.1 consequent, 2.1 consistent, 2.5 consistent valuation, 2.7

contingency, 2.5 contradiction, 2.5 contradictory, 2.5 counterexample, 2.6 disjunction, 2.1 formation rules, 2.2 inconsistent pair, 2.5 logical truths, 2.5 logically equivalent, 2.5 main operator, 2.2 material implication, 2.1 negation, 2.1 neither, 2.1

not both, 2.1 operators, 2.1 semantics, 2.3 syntax, 2.2 tautology, 2.5 truth table, 2.4 truth value, 2.3 unary operator, 2.1 unless, 2.1 valid argument, 2.6 valuation, 2.5 wff, 2.2

Chapter 3 Inference in Propositional Logic

3.1: RULES OF INFERENCE 1 We have used truth tables, including the indirect method, to separate valid from invalid arguments. Our work was guided by a semantic definition of validity: an argument is valid if there are no assignments of truth values to the propositional variables on which the premises are true and the conclusion is false. The truth table method gets increasingly and prohibitively arduous as the complexity of an argument grows. The indirect method, while pleasant and effective, requires ingenuity and can be nearly as laborious as the complete truth table method. More importantly, while semantic tests of validity are effective in propositional logic, they become less useful than other methods in more sophisticated logical systems. This chapter explores one salient and enjoyable alternative method for determining valid inferences: the method of natural deduction. A natural deduction is sometimes called a derivation or a proof, though the use of the word ‘proof ’ for the derivations in this book is somewhat loose. Roughly, a derivation, or proof, is a sequence of formulas, every member of which is an assumed premise or follows from earlier wffs in the sequence according to specified rules; we will adjust this definition in section 3.9. The specified rules comprise a system of inference. Systems of inference are constructed first by specifying a language and then by adding rules governing derivations: how to get new wffs from the old ones. In addition, in some formal systems, some basic axioms are given. In formal systems of propositional logic, these axioms are ordinarily tautologies. Since tautologies are true in all cases, they can be added to any derivation. We will not use any axioms. Just as we named our languages, we can also name our systems of inferences. Any logical language may be used with various different systems of inference. But since we are mainly using only a single system of inference for each language, I won’t bother to confuse us with more names than we need. We’ll just continue to call our language PL and leave our one system of inference using PL unnamed. For natural deductions, our formal system will use the language of propositional logic, eleven rules of inference, fourteen rules of equivalence, and three proof

A derivation, or proof, is a sequence of formulas, every member of which is an assumed premise or follows from earlier formulas in the sequence. A system of inference is a set of rules for derivations.

107

1 0 8    C h apter 3  In f erence i n P ropos i t i onal L og i c

In a complete system of inference, every valid

argument and every logical truth is provable.

Rules of inferenceare

valid argument forms that are used to justify steps in an inference.

In a sound system of inference, every provable argument is semantically valid; every provable proposition is logically true.

methods. The rules are chosen so that our system is complete: every valid argument and logical truth will be provable using our rules. For PL, the logical truths are just the tautologies; we will expand our definition of logical truth for the logics in chapters 4 and 5. Our rules are chosen arbitrarily, in the sense that there are many different complete systems of rules—indeed, infinitely many. One can devise deductive systems with very few rules; the resulting proofs become very long. One can devise systems so that proofs become very short; in such systems the required number of rules can be unfeasibly large. I chose a moderate number of rules (twenty-five) so that there are not too many to memorize and the proofs are not too long. I also chose the rules and proof methods in our system of inference to mirror, at least loosely, natural patterns of inference. You are likely to find some of the rules to be easy and obvious, though the full collection of rules will include some inferences you may find awkward at first. The rules we choose are defined purely syntactically, in terms of their form, but they are justified semantically. A rule of inference must preserve truth: given true premises, the rules must never yield a false conclusion. A rule preserves truth if every argument of its form is valid. We can show that each of the rules of inference preserves truth using the indirect truth table method. We show that each rule of equivalence preserves truth using truth tables as well. This criterion for our rules, that they should preserve the truth of the premises, underlies our goal of soundness for a system of inference. I do not prove the metalogical results of soundness or completeness for the systems in this book; the proofs require more mathematics than we will use. Derivations begin with any number of premises and proceed by steps to a conclusion. A derivation is valid if every step is either a premise or derived from premises or previous steps using our rules. I introduce four rules of inference in this section and four more in the next section. I introduce ten rules of equivalence in the third and fourth sections of this chapter. In section 6, I introduce the seven remaining rules (three inference rules and four equivalence rules), all of which govern the biconditional.

Modus Ponens (MP) Let’s start to examine our first rules. Observe first that each of 3.1.1–3.1.3 are valid inferences; you can use truth tables to check them. 3.1.1 A ⊃ B A

/B

3.1.2 (E ∙ I) ⊃ D (E ∙ I)

/D

3.1.3

∼G ⊃ (F ∙ H) ∼G

/F∙H

3 . 1 : R u les o f In f erence 1    1 0 9

Notice that despite their differing complexity, 3.1.1–3.1.3 share a form. The first premise of each argument is a conditional. The second premise is the antecedent of that conditional. The conclusion is the consequent of the conditional. We can write this form at 3.1.4, using metalinguistic (Greek) variables. 3.1.4

α⊃β α /β

Modus Ponens

This form of argument is called modus ponens, abbreviated MP. We can apply 3.1.4 in our object language, PL, by constructing substitution instances of it, particular applications of the rule which match, syntactically, its form. In particular, the main operators of each formula in the substitution instance will be the same as the main operators in the rule. So, a substitution instance of MP will contain one wff whose main operator is a conditional and another that is precisely the antecedent of that conditional. The last wff of a substitution instance of MP will contain exactly the consequent of the conditional statement as a new wff in a derivation. Notice that any substitution instance of MP yields a valid argument. Logicians ordinarily prove the validity of rules by mathematical induction. Here, an informal argument should suffice: the only way to construct a counterexample would be on a line on which the main operator of the conclusion were false and the main operator of the second premise were true. Any such valuation would make the first premise false and so make the inference valid. (Remember, a counterexample requires true premises and a false conclusion.) Given that every substitution instance of MP will be valid, we can substitute simple or complex formulas for α and β in 3.1.4 and be sure that the resulting deduction is valid. 3.1.5 is another example of MP, with even greater complexity. 3.1.5 [(H ∨ G) ⊃ I] ⊃ (K ∙ ∼L) [(H ∨ G) ⊃ I]

Modus ponens (MP) is a

rule of inference of PL.

A substitution instance of a rule is a set of wffs of

PL that match the form of the rule.

/ (K ∙ ∼L)

Modus Tollens (MT) We can justify adopting other rules of inference just as we justified modus ponens. Consider modus tollens, at 3.1.6. 3.1.6

α⊃β ∼β / ∼α

Modus Tollens

Like modus ponens, our first premise is a conditional. But in modus tollens, we infer the denial of the antecedent of the conditional from the denial of its consequent. Let’s assume that if you receive your paycheck by Friday, then you will go out to dinner on Saturday. Then, if you don’t go out to dinner on Saturday, if the conditional holds, we can infer that you didn’t get your paycheck by Friday. For obvious reasons, we are mainly interested in valid rules of inference, like modus ponens and modus tollens. But it is sometimes useful to contrast the valid forms with invalid ones, like 3.1.7 and 3.1.8. Again, we can check them, using truth tables, or indirect truth tables.

Modus tollens (MT) is a rule of inference of PL.

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3.1.7

α ⊃ β β / α

Fallacy of Affirming the Consequent

3.1.8

α ⊃ β ∼α / ∼β

Fallacy of Denying the Antecedent

To show that 3.1.7 is invalid, we can assign false to α and true to β. The premises turn out true and the conclusion turns out false. The same set of assignments provides a metalinguistic counterexample for 3.1.8. Any substitution instance of these forms will thus be invalid and we can construct an object-language counterexample in the same way. Let’s look at a couple of concrete instances to get an intuitive sense of the difference between valid and invalid arguments. Let ‘P’ stand for ‘I study philosophy’ and ‘Q’ stand for ‘I write essays’. We can write the conditional ‘P ⊃ Q’ as 3.1.9. 3.1.9

If I study philosophy, then I write essays.

From 3.1.9 and the claim that I study philosophy, modus ponens licenses the inference that I write essays. From 3.1.9 and the claim that I do not write essays, modus tollens licenses the inference that I do not study philosophy. But I would commit the fallacy of affirming the consequent if I concluded, from 3.1.9 and the claim that I write essays, that I study philosophy. People write papers without studying philosophy. Similarly, from 3.1.9 and the claim that I do not study philosophy, it does not follow that I do not write papers; such an inference would be an instance of the fallacy of denying the antecedent.

Disjunctive Syllogism (DS) Disjunctive syllogism (DS) is a rule of inference

of PL.

Similar arguments thus show that the forms 3.1.10, disjunctive syllogism, and 3.1.11, hypothetical syllogism, are also all valid. 3.1.10

α∨β ∼α / β

Disjunctive Syllogism

We can check that each form is valid by using truth tables on the metalinguistic forms. For 3.1.10, if we make the conclusion false and the second premise true, then α and β are false and the first premise is false. If we try to make the first premise true, then either the second premise comes out false or the conclusion comes out true. Disjunctive syllogism captures the form of inference we make when, for example, we are faced with the choice between soup or salad: if we don’t have the soup, then we’re having the salad. Or, if you get to a fork in the road and you have only two options, if you don’t take the left fork, you’re going to take the right. We might be considering two hypotheses. When we discover evidence against one of them, we conclude the other. Often, when we are faced with options, we have more than two from which to choose. But if you have only two options and you don’t take one, you can infer the other by what is sometimes called a process of elimination.

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Hypothetical Syllogism (HS) 3.1.11

α⊃β β ⊃ γ / α ⊃ γ

Hypothetical Syllogism

In 3.1.11, hypothetical syllogism, if we try to make the conclusion false, we have to make α true and γ false. Then, to make the first premise true, we have to make β true; that makes the second premise false. If we try to make the second premise true by making β false, then we make the first premise false. In either case, we cannot construct a counterexample. Thus, any substitution instance of HS will be valid. Hypothetical syllogism captures the reasoning we perform when we are faced with a chain of inferences. If going to college entails taking loans, and taking loans entails having to work to pay them off, then going to college entails having to work to pay off one’s loans.

Using the Rules in Derivations We now have four rules of inference: MP, MT, DS, and HS. Let’s see how to combine uses of them to derive the conclusion of the argument 3.1.12, showing that it is a valid argument. 3.1.12

1. (X ⊃ Y) ⊃ T 2. S ∨ ∼T 3. U ⊃ ∼S 4. U

/ ∼(X ⊃ Y)

We could show that the argument is valid using truth tables, including the indirect truth table method. Instead, in this chapter, we will show that arguments are valid by constructing derivations of their conclusions, starting with assumed premises. We will string together inferences, using our rules one at a time, until we reach our desired conclusion. The rules are purely syntactic; we won’t use truth values in our derivations. But the rules are chosen to preserve validity, so any conclusion we reach will never be false if the premises are true. In other words, every inference will be valid. 3.1.13 is an example of the natural deductions we will use throughout the rest of the book. 3.1.13 1. (X ⊃ Y) ⊃ T 2. S ∨ ∼T 3. U ⊃ ∼S 4. U 5. ∼S 6. ∼T 7. ∼(X ⊃ Y)

/ ∼(X ⊃ Y) 3, 4, MP 2, 5, DS 1, 6, MT

(taking ‘U’ for α and ‘∼S’ for β) (taking ‘S’ for α and ‘∼T’ for β) (taking ‘X ⊃ Y’ for α and ‘T’ for β)

QED

Let’s notice some properties of the derivation 3.1.13. First, we number all of the premises as well as every wff that follows. While a derivation is really just the sequence

Hypothetical syllogism (HS) is a rule of inference

of PL.

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A justification i n a derivation includes the rule used and the earlier line numbers to which the rule is applied.

QEDis placed at the end of a derivation, to show that it is finished.

of wffs, we will write our deductions in the metalanguage, including line numbers and justifications in a second column. The line numbers allow us to keep track of our justifications. All steps except the premises require justification. The justification of any step includes the line numbers and rule of inference used to generate the new wff. For example, “3, 4, MP” on line 5 indicates that ‘∼S’ is derived directly from the wffs at lines 3 and 4 by a use of the rule of modus ponens. The explanations such as “taking ‘U’ for α and ‘∼S’ for β” are not required elements of the derivation, but they can be useful, especially when you are first learning to use natural deductions. The conclusion of the argument is initially written after a single slash following the last premise. The conclusion, like the justifications of every following step, is not technically part of the deduction. Importantly, you may not use it as part of your proof. It merely indicates what the last numbered line of your derivation should be. Lastly, QED at the end of the derivation stands for ‘Quod erat demonstrandum’, which is Latin for ‘that which was required to be shown’. ‘QED’ is a logician’s punctuation mark: “I’m done!” It is not essential to a proof, but looks neat and signals your intention to end the derivation. Rules of inference are to be used only on whole lines, not on portions of lines. In other words, the main operators of the propositions to which you are applying the rule must match the operators given in the rule. The inference at 3.1.14 violates this condition and so is illegitimate, even though valid. 3.1.14

1. P ⊃ (Q ⊃ R) 2. Q 3. P ⊃ R

1, 2, MP

Not Acceptable!

We’ll have other ways to make such valid inferences once our proof system is complete. 3.1.15 is an example of a longer derivation using our first four rules of inference. 3.1.15 1. ∼A ⊃ [A ∨ (B ⊃ C)] 2. (B ∨ D) ⊃ ∼A 3. B ∨ D 4. C ⊃ A 5. ∼A 6. A ∨ (B ⊃ C) 7. B ⊃ C 8. B ⊃ A 9. ∼B 10. D QED

/D 2, 3, MP 1, 5, MP 6, 5, DS 7, 4, HS 8, 5, MT 3, 9, DS

Summary In this section, we saw the first four of our rules of inference and how they can combine to form derivations of the conclusions of arguments. Constructing derivations can be intimidating at first. If you can, start with simple sentences or negations of

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simple negations. Plan ahead. Working backward from the conclusion on the side can be helpful. For example, in 3.1.15, we could start the derivation by observing that we could get the conclusion, ‘D’, by DS from line 3 if we had ‘∼B’. Then, both ‘∼B’ and ‘D’ are goals as we work forward through the proof. Don’t worry about introducing extraneous lines into your proof as long as they are the results of valid inferences. Especially as we introduce further rules, we are going to be able to infer statements that are not needed for the most concise derivation. But as long as every step is valid, the entire inference will be valid. It is not the case that every wff must be used after it is introduced into the deduction. Lastly, notice that some wffs may be used more than once in a derivation. In 3.1.15, the ‘∼A’ at line 5 was used first with premise 1 in a MP to yield the wff at line 6. Then, it is used immediately a second time, with the wff at line 6, to yield ‘B ⊃ C’ on line 7. Some students will have encountered proofs like these, perhaps in slightly less rigorous form, in a geometry class, or in other mathematics courses. For other students, natural deductions of this sort are new. Be patient, and practice. And practice some more. KEEP IN MIND

Our formal system for propositional logic will use eleven rules of inference, fourteen rules of equivalence, and three proof methods. We have seen four rules of inference: modus ponens (MP); modus tollens (MT); disjunctive syllogism (DS); hypothetical syllogism (HS). Every valid argument will be provable using our rules once our rule set is complete. Rules of inference preserve truth; given true premises, the rules never yield a false conclusion. Derivations begin with any number of premises and proceed by steps to a conclusion. A derivation is valid if every step is either a premise or derived from premises or previous steps using our rules. In derivations: Number all premises and every wff that follows. The conclusion of the argument is written after a single slash following the last premise. Justify all steps except the premises. A justification includes line numbers and the rule of inference used to generate the new wff. Use rules of inference only on whole lines, not on portions of lines. QED may be added to the end of a derivation to mark its conclusion.

Rules Introduced Modus Ponens (MP) α⊃β α / β

Modus Tollens (MT) α⊃β ∼β / ∼α

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Disjunctive Syllogism (DS) α∨β ∼α / β

Hypothetical Syllogism (HS) α⊃β β ⊃ γ / α ⊃ γ

TELL ME MORE  • What are soundness and completeness for logical systems? See 6.4: Metalogic. • What other kinds of fallacies are there? See 7.2: Fallacies and Argumentation. • How do logical systems with axioms differ from natural deduction systems? See 6S.11: Axiomatic Systems.

EXERCISES 3.1a Derive the conclusions of each of the following arguments using natural deduction. 1. 1. V ⊃ (W ⊃ X) 2. V 3. ∼X / ∼W

7. 1. P ⊃ R 2. Q ⊃ P 3. (Q ⊃ R) ⊃ S

2. 1. X ⊃ Y 2. ∼Y 3. X ∨ Z

8. 1. (P ∙ Q ) ∨ R 2. ∼(P ∙ Q  ) 3. R ⊃ ∼S / ∼S

/Z

3. 1. E ⊃ F 2. ∼F 3. ∼E ⊃ (G ∙ H) / G ∙ H 4. 1. I ⊃ J 2. J ⊃ K 3. ∼K / ∼I 5. 1. (I ∙ L) ⊃ (K ∨ J) 2. I ∙ L 3. ∼K /J 6. 1. (P ∨ Q  ) ⊃ R 2. R ⊃ S 3. P ∨ Q

/S

/S

9. 1. P ⊃ (Q ⊃ R) 2. (Q ⊃ R) ⊃ S 3. (P ⊃ S) ⊃ (T ⊃ P) / T ⊃ P 10. 1. P ⊃ (Q ∙ R) 2. ∼(Q ∙ R) 3. P ∨ (S ≡ T)

/S≡T

11. 1. (P ⊃ Q  ) ⊃ (P ⊃ R) 2. P ⊃ S 3. S ⊃ Q /P⊃R 12. 1. G ⊃ E 2. F ⊃ ∼E 3. H ∨ F 4. ∼H / ∼G

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13. 1. A ⊃ D 2. D ⊃ (B ⊃ C) 3. B 4. A

/C

23. 1. E ⊃ H 2. G ∨ ∼F 3. ∼G 4. H ⊃ F

/ ∼E

14. 1. L ∨ N 2. ∼L 3. N ⊃ (M ∨ O) 4. (M ∨ O) ⊃ (P ≡ Q ) / P ≡ Q

24. 1. J ⊃ L 2. L ⊃ (I ∙ M) 3. (I ∙ M) ⊃ K 4. ∼K / ∼J

15. 1. R ⊃ S 2. S ⊃ (T ∨ U) 3. R 4. ∼T

/U

25. 1. Q ⊃ (∼R ⊃ S) 2. T ∨ Q 3. ∼T 4. R ⊃ T /S

/X

26. 1. ∼Q ⊃ (N ∙ O) 2. (N ∙ O) ⊃ (P ⊃ Q ) 3. M ∨ ∼Q 4. ∼M / ∼P

/A

27. 1. (P ∨ Q ) ∨ (S ∨ ∼T) 2. R ⊃ ∼(P ∨ Q ) 3. (S ∨ ∼T) ⊃ ∼S 4. R / ∼T

16. 1. U ⊃ V 2. ∼V 3. U ∨ W 4. W ⊃ X 17. 1. X ⊃ Z 2. Z ⊃ Y 3. ∼Y 4. ∼X ⊃ A

18. 1. P ⊃ (Q ∙ ∼R) 2. S ⊃ ∼(Q ∙ ∼R) 3. T ∨ S 4. ∼T / ∼P 19. 1. ∼ ∼R ⊃ (∼P ⊃ Q ) 2. ∼R ⊃ P 3. ∼P

28. 1. (P ∙ ∼R) ⊃ (Q ∨ S) 2. Q ⊃ (S ≡ T) 3. ∼(S ≡ T) ⊃ (P ∙ ∼R) 4. ∼(S ≡ T) / S

20. 1. (P ≡ R) ∨ (Q ⊃ ∼R) 2. (P ≡ R) ⊃ S 3. Q 4. ∼S / ∼R

29. 1. A ⊃ (B ∙ C) 2. G ∨ ∼H 3. E ⊃ F 4. H ∨ E 5. (B ∙ C) ⊃ ∼G 6. D ∨ A 7. ∼D /F

21. 1. P ∨ (Q ⊃ R) 2. ∼Q ⊃ (S ∙ ∼T) 3. ∼P 4. ∼R

/ S ∙ ∼T

30. 1. C ⊃ (D ≡ ∼E) 2. (D ≡ ∼E) ⊃ (B ∨ A) 3. C ⊃ ∼B 4. C /A

/S

31. 1. V ⊃ (W ∨ U) 2. X ∨ V 3. X ⊃ Y 4. ∼Y 5. ∼Y ⊃ ∼W / U

22. 1. P ∨ [Q ∨ (∼R ∨ S)] 2. T ⊃ ∼P 3. T ⊃ ∼Q 4. T 5. ∼ ∼R

/Q

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32. 1. X ⊃ (Y ⊃ Z) 2. W ∨ X 3. W ⊃ Y 4. ∼Y 5. ∼W ⊃ Y / Z 33. 1. (H ∙ ∼G) ⊃ F 2. F ⊃ (G ∨ J) 3. I ∨ (H ∙ ∼G) 4. I ⊃ G 5. ∼G /J 34. 1. A ⊃ B 2. B ⊃ (C ⊃ D) 3. E ∨ C 4. E ⊃ F 5. ∼F 6. C ⊃ A /D 35. 1. (P ⊃ Q  ) ⊃ (R ∨ S) 2. ∼R ⊃ (∼R ⊃ Q  ) 3. P ⊃ ∼R 4. P /S

37. 1. (P ∙ ∼Q  ) ⊃ (R ⊃ S) 2. (P ∙ ∼Q  ) ∨ (R ≡ S) 3. (R ≡ S) ⊃ (P ∨ ∼Q  ) 4. ∼(R ⊃ S) / P ∨ ∼Q 38. 1. (A ∙ ∼B) ⊃ (C ⊃ D) 2. (A ∙ ∼B) ∨ (D ⊃ ∼C) 3. ∼(C ⊃ D) 4. ∼C ⊃ A / D ⊃ A 39. 1. (P ∨ R) ⊃ (∼P ⊃ R) 2. (Q ∨ S) ⊃ (P ∨ R) 3. T ∨ (Q ∨ S) 4. ∼R 5. ∼T / ∼ ∼P 40. 1. P ⊃ [∼(Q ⊃ R) ⊃ (Q ∨ ∼R)] 2. (Q ⊃ R) ∨ P 3. (Q ⊃ R) ⊃ T 4. ∼T 5. ∼Q / ∼R

36. 1. (P ⊃ Q ) ⊃ [R ⊃ (S ∨ T)] 2. P ⊃ (R ≡ ∼S) 3. (R ≡ ∼S) ⊃ Q 4. R 5. ∼S /T

EXERCISES 3.1b Translate each of the following paragraphs into arguments written in PL. Then, derive the conclusions of the arguments using the first four rules of our system of natural deduction. 1. If Allison doesn’t go grocery shopping, Billy will go. Allison goes grocery shopping only if Carla gets home from school early. Carla doesn’t get home early. Therefore, Billy goes grocery shopping. 2. Don Juan plays golf only if Edie makes a reservation. If Edie makes a reservation, then Frederique writes it on the calendar. Don Juan played golf. So, Frederique wrote it down on the calendar.

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3. If Gertrude mops the kitchen, then Hillary washes the dishes. Either Inez or Gertrude mops the kitchen. Inez doesn’t mop the kitchen. So, Hillary washes the dishes. 4. Katerina driving to practice is a necessary condition for Jelissa’s playing soccer. Katerina drives only if Liza puts gas in her car. Liza doesn’t put gas in the car. So, Jelissa doesn’t play soccer. 5. Nico skateboards if Mandy gives him lessons. If Nico skateboards, then either Olivia or Patricia will watch. Mandy gives skateboarding lessons. Olivia doesn’t watch. So, Patricia watches. 6. Jose will play either trombone or ukulele. If he plays trombone, then he’ll also play violin. If he plays ukulele, then he’ll also play a woodwind instrument. He doesn’t play violin. So, he plays a woodwind instrument. 7. If the corn doesn’t grow, dandelions will grow. If dandelions grow, then the apple tree will bloom. If the corn grows, then the badgers will eat the crops. The badgers don’t eat the crops. So, the apple tree blooms. 8. If the zoo has hippos, then it has yaks. If the zoo has yaks, then it has zebras. The zoo has either water buffalo or hippos. The zoo having water buffalo is a sufficient condition for their having turtles. But they don’t have turtles. So, the zoo has zebras. 9. If we are just, then we have settled the nature of justice. Either we are just or deliberation is useful. We haven’t settled the nature of justice. So, deliberation is useful. 10. If there are social points of view acceptable to all, then we can construct principles of justice. If we can construct principles of justice, then we are free equals. But we are not free equals. So, there are no social points of view acceptable to all. 11. If there are genocides, then we must develop schemes of humanitarian intervention. If we must develop schemes of humanitarian intervention, then being able to develop just war theory is a necessary condition for international cooperation being possible. There are genocides. International cooperation is possible. So, just war theory can be developed. 12. If all things are full of gods, then water is holy. If water is holy, then all things are made of water or all things are caused by water. If all things are made of or caused by water, then the world itself is divine. But the world is not divine. So, it is not the case that all things are full of gods. 13. Either our wills are free or responsibility is either meaningless or incomprehensible. If responsibility is either meaningless or incomprehensible, then I need not fret about my decisions. If responsibility is meaningless, then I do

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have to fret about my decisions. Our wills are not free. So, responsibility is incomprehensible. 14. If mathematics can be known a priori, then so can logic. If logic is knowable a priori, then human reason is not purely scientific. If the a priori knowability of mathematics entails that human reason is not purely scientific, then if logic can be known a priori, then there are eternal truths. So, if mathematics is knowable a priori, then there are eternal truths. 15. Either monadism is true just in case atomism is, or space is infinitely divisible if and only if the world is a plenum. If monadism entails atomism, then space is not infinitely divisible. Either space is infinitely divisible or it’s not the case that monadism is true just in case atomism is. If monadism is true, then there are elementary particles. But if there are elementary particles, then atomism is true. So, space is infinitely divisible if, and only if, the world is a plenum. 16. You’re befuddled. Either you are a necessitarian or you are not a proper apriorist. Either you are a contingentist or you are not a proper empiricist. If you’re a contingentist, then you don’t believe that logic is a priori. But it’s not the case that you do not believe that logic is a priori. You’re not a necessitarian. And if you aren’t a proper apriorist, then if you aren’t a proper empiricist, then you are befuddled.

3.2: RULES OF INFERENCE 2 In this section, I introduce and discuss four more valid rules of inference. While there are no new ideas about derivations in this section, each rule has its own characteristics that must be learned. As we add rules to our system, the proofs become more interesting and amusing, but also they can be more difficult.

Conjunction (Conj) and Addition (Add) Conjunction (Conj) is a rule of inference of PL.

Addition (Add) is a rule of

inference of PL.

The rule of inference at 3.2.1, conjunction, should be highly intuitive. 3.2.1

α β / α ∙ β Conjunction

Conjunction merely allows us to put two prior premises together on one line. It hardly seems like an inference worth making: if we have peas and we have carrots, then we have peas and carrots. But Conj will be useful in a variety of ways. In contrast, the rule of addition, at 3.2.2, uses a disjunction and requires only one premise. 3.2.2

α / α ∨ β Addition

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If some proposition is already established—say, that Nietzsche is a nihilist—then we can infer that either Nietzsche is a nihilist or Berkeley is an idealist. We can also infer that either Nietzsche is a nihilist or Berkeley is a materialist. Since the first disjunct in the conclusion of an inference licensed by addition is already taken as true, it doesn’t matter whether the second disjunct is true or false; a disjunction is true as long as one of the disjuncts is. Addition can thus license our adding a false proposition into a proof, as a subformula. This may seem odd. But as long as our argument is not inconsistent, the addition of such formulas has no ill effect on the system or its soundness. We’ll see what happens with inconsistent arguments in section 3.5. For now, make sure to distinguish conjunction from addition; these two rules are easy for beginners to confuse. We can add anything to an already established wff; conjunction works only with two formulas that have already appeared. The inference at 3.2.3 uses Add properly. 3.2.3 1. ∼M ∨ N 2. ∼∼M 3. N 4. N ∨ O QED

/N∨O 1, 2, DS 3, Add

Notice that ‘O’ never appears in the derivation until it is added in the last step. This oddity of Add is perfectly legitimate and useful. If a proposition is true, then its disjunction with any other proposition, no matter its truth value, will also be true. 3.2.4 is just a slightly longer derivation illustrating uses of addition and conjunction. 3.2.4 1. (∼A ∨ B) ⊃ (G ⊃ D) 2. (G ∨ E) ⊃ (∼A ⊃ F) 3. A ∨ G 4. ∼A 5. G 6. G ∨ E 7. ∼A ⊃ F 8. F 9. ∼A ∨ B 10. G ⊃ D 11. D 12. F ∙ D QED

/F∙D 3, 4, DS 5, Add 2, 6, MP 7, 4, MP 4, Add 1, 9, MP 10, 5, MP 8, 11, Conj

Simplification (Simp) Simplification, the rule of inference shown at 3.2.5, is like the reverse of conjunction, allowing you to infer the first conjunct of a conjunction. 3.2.5

α ∙ β / α Simplification

Simplification (Simp) i s a

rule of inference of PL.

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If you have peas and carrots, then you have peas. Notice that Simp does not license the derivation of ‘you have carrots’ from ‘you have peas and you have carrots’; a rule of equivalence in the next section will allow us to infer the second conjunct. For now, our list of rules is incomplete. We must leave the second conjunct alone. 3.2.6 is a sample derivation using conjunction and simplification. 3.2.6 1. A ⊃ B 2. F ⊃ D 3. A ∙ E 4. ∼D 5. A 6. B 7. ∼F 8. B ∙ ∼F QED

/ B ∙ ∼F 3, Simp 1, 5, MP 2, 4, MT 6, 7, Conj

Be careful to avoid the invalid inferences 3.2.7 and 3.2.8. 3.2.7 3.2.8

α / α ∙ β α ∨ β / α

Invalid! Invalid!

From a single proposition, 3.2.7, we cannot conclude the conjunction of two propositions unless the second appears earlier in our derivation. And from a disjunction, 3.2.8, we cannot conclude either disjunct unless the negation of the other appears earlier in our derivation.

Constructive Dilemma (CD) Constructive dilemma (CD) is a rule of inference

of PL.

The last of our four new rules of inference in this section, constructive dilemma, shown at 3.2.9, is different from our other rules in having three premises. (Add and Simp have one premise; the other rules all have two premises.) 3.2.9

α⊃β γ⊃δ α ∨ γ / β ∨ δ

Constructive Dilemma

Note the similarity of CD to modus ponens. In MP, we infer a consequent from a conditional and (separately) its antecedent. In CD, we conclude the disjunction of two consequents from two conditionals and (separately) the disjunction of their antecedents. 3.2.10 is a simple derivation using CD. Note that one of the disjuncts used in the inference, at line 3, is itself a conjunction; the antecedent of the wff at line 2 is the same conjunction. 3.2.10 QED

1. N ⊃ (O ∙ P) 2. (Q ∙ R) ⊃ O 3. N ∨ (Q ∙ R) 4. (O ∙ P) ∨ O

/ (O ∙ P) ∨ O 1, 2, 3, CD

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The derivation at 3.2.11 uses all of the rules of inference of this section. 3.2.11 1. P ∨ Q 2. Q ⊃ S 3. R ⊃ T 4. ∼P ∙ U 5. ∼P 6. Q 7. Q ∨ R 8. S ∨ T 9. Q ∙ (S ∨ T) QED

/ Q ∙ (S ∨ T) 4, Simp 1, 5, DS 6, Add 2, 3, 7, CD 6, 8, Conj

Summary The four new rules of inference in this section differ only in their details from the four rules of section 3.1. Any substitution instance of a rule yields a valid inference. You can check the validity of each form using the truth table method, or indirect truth table method, applied to the metalinguistic forms. Remember, we choose these rules on two bases: the completeness of the resulting logical system and the way in which they represent or reflect ordinary inferences. As our derivations become more complex, it will become increasingly important for you not only to be able to use the rules we have, but also to see substitution instances of the rules quickly and naturally. Constructing derivations requires not just understanding the rules, but knowing how to use them. It’s like riding a bicycle or cooking: you can’t just know how to do it in theory; you have actually to do it in order to get good at it. At the risk of redundancy: practice, practice, practice. KEEP IN MIND

We have seen four more rules of inference: conjunction (Conj); addition (Add); simplification (Simp); constructive dilemma (CD). We now have eight rules. Be especially careful not to confuse conjunction and addition.

Rules Introduced Conjunction (Conj) α β / α ∙ β

Addition (Add) α / α ∨ β

Simplification (Simp) α ∙ β / α

1 2 2    C h apter 3  In f erence i n P ropos i t i onal L og i c

Constructive Dilemma (CD) α⊃β γ⊃δ α ∨ γ / β ∨ δ

EXERCISES 3.2a For each of the following arguments, determine which, if any, of the eight rules of inference is being followed. Though there are many valid inferences other than our eight rules, in these exercises, if the inference is not in the form of one of the eight rules, it is invalid. The invalid inferences in these exercises are common errors that logic students make when learning the rules of inference, so it might be worth your time to study and understand the errors in order to avoid them yourself. 1. A ⊃ (B ∙ C) ∼(B ∙ C)

/ ∼A

2. (D ∨ E) ⊃ F F ⊃ (G ≡ H) (D ∨ E) ∨ F

/ F ∨ (G ≡ H)

3. I ⊃ ∼J K ⊃ I

/ K ⊃ ∼J

4. L ∼M ∙ N

/ ∼(M ∙ N) ∙ L

5. O

/ O ∙ ∼O

6. P

/ P ∨ [Q ≡ (R ∙ ∼P)]

7. S ∨ ∼T ∼ ∼T / ∼S 8. ∼U ≡ V (∼U ≡ V) ⊃ W

/W

9. X ⊃ ∼Y ∼Y ⊃ Z

/ (X ⊃ ∼Y) ∙ (∼Y ⊃ Z)

10. (A ∨ ∼B) ∨ ∼∼C / A ∨ ∼B

3 . 2 : R u les o f In f erence 2    1 2 3

11. ∼[D ⊃ (E ∨ F)] [D ⊃ (E ∨ F)] ∨ [G ⊃ (E ∙ ∼F)] / G ⊃ (E ∙ ∼F) 12. [(G ∨ H) ∙ I] ∙ (∼I ≡ K)

/ (G ∨ H) ∙ I

13. P

/P∨P

14. P ⊃ (Q ∙ R) Q ∙ R

/P

15. P ⊃ (Q ∨ R) ∼P

/ ∼(Q ∨ R)

16. ∼(P ∨ ∼Q ) ⊃ R ∼(P ∨ ∼Q )

/R

17. (P ∙ ∼Q ) ⊃ R P ∙ ∼Q

/R

18. P ⊃ Q R ⊃ ∼S P ∨ R

/ Q ∙ ∼S

19. P ⊃ ∼Q Q ⊃ ∼S

/ P ⊃ ∼S

20. P ∙ Q

/ (P ∙ Q ) ∨ ∼(P ∙ Q  )

EXERCISES 3.2b Derive the conclusions of each of the following arguments using the eight rules of inference. 1. 1. P ∙ Q 2. R

/P∙R

2. 1. P ⊃ ∼Q 2. ∼Q ⊃ R / (P ⊃ R) ∨ (S ⊃ T) 3. 1. (A ⊃ C) ⊃ D 2. ∼B ⊃ C 3. A ⊃ ∼B / D 4. 1. (E ∨ F) ⊃ ∼G 2. H ⊃ G 3. E / ∼H

1 2 4    C h apter 3  In f erence i n P ropos i t i onal L og i c

5. 1. I ∨ J 2. ∼I ∙ K

/J∨L

6. 1. W ⊃ X 2. ∼X ∙ Y

/ (∼W ∨ Z) ∙ ∼X

7. 1. T ∨ S 2. ∼T 3. U

/U∙S

8. 1. ∼P ⊃ ∼Q 2. ∼R ⊃ ∼S 3. T ∨ (∼P ∨ ∼R) 4. ∼T / ∼Q ∨ ∼S 9. 1. N ∨ ∼ ∼P 2. ∼N ∙ Q 3. ∼P ∨ Q

/ ∼∼P ∙ Q

10. 1. (P ≡ Q  ) ⊃ R 2. Q ∨ ∼R 3. ∼Q 4. ∼P ⊃ (P ≡ Q  )

/ ∼∼P

11. 1. P ⊃ Q 2. ∼R ⊃ S 3. P ∨ ∼R 4. ∼Q

/S

12. 1. P ⊃ Q 2. ∼Q ∙ R

/ ∼P ∨ R

13. 1. ∼P ∨ Q 2. ∼P ⊃ R 3. ∼R

/Q∨S

14. 1. P ∙ ∼Q 2. ∼Q ∙ R 3. (P ∙ ∼Q ) ⊃ S

/S∙P

15. 1. ∼P ⊃ Q 2. ∼Q ⊃ R 3. (∼P ∨ ∼Q ) ∙ S / Q ∨ R 16. 1. (P ∙ ∼Q  ) ∙ R 2. P ⊃ S 3. R ⊃ T

/S∨T

17. 1. (P ∨ Q ) ⊃ R 2. (P ∨ S) ⊃ T 3. P ∙ V

/R∙T

3 . 2 : R u les o f In f erence 2    1 2 5

18. 1. ∼P ⊃ Q 2. ∼R ⊃ S 3. Q ⊃ ∼R 4. ∼P / ∼R ∨ S 19. 1. (E ∨ I) ⊃ H 2. H ⊃ (F ∙ G) 3. E / (F ∙ G) ∙ E 20. 1. M ⊃ N 2. O ⊃ P 3. M ∙ Q

/N∨P

21. 1. ∼A ⊃ B 2. C ⊃ D 3. A ⊃ D 4. ∼D

/B∨D

22. 1. M ⊃ N 2. N ⊃ O 3. M ∙ P

/O∨P

23. 1. B ⊃ A 2. ∼A ∙ D 3. ∼B ⊃ C / C ∨ A 24. 1. D ∨ E 2. D ⊃ F 3. ∼F ∙ G

/ (E ∨ H) ∙ ∼F

25. 1. O ⊃ Q 2. Q ⊃ P 3. P ⊃ (R ∙ S) 4. O /R∙S 26. 1. (R ∨ T) ⊃ S 2. S ⊃ U 3. R /U∨T 27. 1. P ∙ Q 2. ∼P ∙ R

/S

28. 1. [(∼Q ∙ ∼P) ⊃ R] ∙ (S ∨ ∼T) 2. P ⊃ Q 3. ∼Q /R∨T 29. 1. P ⊃ (Q ⊃ R) 2. P ⊃ (R ⊃ ∼S) 3. Q ∨ R 4. P ∙ T / R ∨ ∼S

1 2 6    C h apter 3  In f erence i n P ropos i t i onal L og i c

30. 1. (∼P ∨ Q  ) ⊃ (S ⊃ T) 2. P ⊃ T 3. ∼T / ∼S 31. 1. (P ≡ Q  ) ⊃ (R ∨ T) 2. (P ≡ R) ∨ (P ≡ Q  ) 3. ∼(P ≡ R) 4. ∼R /T 32. 1. ∼P ∨ (R ∙ S) 2. ∼Q ∙ (R ∨ S) 3. ∼P ⊃ Q / R 33. 1. P ⊃ (Q ∨ ∼R) 2. Q ⊃ (S ∨ ∼T) 3. P ∨ Q 4. ∼(Q ∨ ∼R) 5. ∼S / ∼T 34. 1. W ⊃ Z 2. Z ⊃ (X ∨ Y) 3. W ∙ Y 4. X ⊃ U 5. Y ⊃ V /U∨V 35. 1. R ⊃ S 2. S ⊃ (T ⊃ U) 3. R 4. U ⊃ R /T⊃R 36. 1. P ⊃ (Q ⊃ R) 2. S ⊃ (P ⊃ T) 3. P ∨ S 4. ∼(Q ⊃ R) 5. ∼T /S 37. 1. ∼P ⊃ (Q ∨ S) 2. ∼R ⊃ ∼Q 3. P ⊃ R 4. ∼R /S 38. 1. (P ∨ Q  ) ∨ (R ∨ S) 2. (P ∨ Q  ) ⊃ T 3. R ⊃ T 4. ∼T /S

3 . 2 : R u les o f In f erence 2    1 2 7

39. 1. P ⊃ (Q ⊃ ∼U) 2. R ⊃ (Q ⊃ S) 3. (P ∨ R) ∙ T 4. ∼(Q ⊃ ∼U) 5. Q / S ∨ ∼U 40. 1. P ⊃ (Q ⊃ R) 2. S ⊃ (T ⊃ U) 3. W ⊃ X 4. ∼(Q ⊃ R) 5. P ∨ S 6. T ∨ W / U ∨ X

EXERCISES 3.2c Translate each of the following paragraphs into arguments written in PL. Then, derive the conclusions of the arguments using the eight rules of inference. 1. If Alessandro sings in the musical, then Beatriz will buy a ticket. Beatriz doesn’t buy a ticket and Carlo goes to watch the musical. So, Alessandro doesn’t sing in the musical and Beatriz doesn’t buy a ticket. 2. If  Don is an EMT, then everyone is saved. All girls are saved provided that Frank is an EMT. Helga’s being a doctor implies that Don is an EMT. Helga is a doctor; moreover, all girls are saved. So, either everyone is saved or all girls are saved. 3. If the classroom is quiet, then it is not rowdy. If the classroom isn’t rowdy, then it’s silent. The classroom is quiet and not tumultuous. So, the classroom is quiet and silent. 4. Having a thunderstorm is a sufficient condition for needing an umbrella. Either it is very cloudy or you don’t need an umbrella. It’s not very cloudy. So, either there aren’t thunderstorms or it’s windy. 5. Either elephants or flamingos eat nuts. If elephants eat nuts, then gorillas eat fruit. Gorillas don’t eat fruit, but hippos eat berries. So, either flamingos eat nuts or hippos eat berries. 6. Elia playing basketball is a necessary condition of her taking art. She’ll walk the dog on the condition that she takes ceramics. She doesn’t play basketball. She takes ceramics. So, she doesn’t take art, but she does walk the dog.

1 2 8    C h apter 3  In f erence i n P ropos i t i onal L og i c

7. Jaime either flies a kite or lies in the sun and listens to music. He doesn’t fly a kite, but he juggles. If he lies in the sun, then he juggles. So, he either juggles or listens to music. 8. If Xavier takes Spanish, then Yolanda tutors him. Zeke pays Yolanda if she tutors Xavier. Either Waldo or Xavier takes Spanish. Waldo doesn’t take Spanish; also Yolanda doesn’t tutor Xavier. So, Zeke pays Yolanda, but Waldo doesn’t take Spanish. 9. If God is either benevolent or omnipotent, then we have both freedom and knowledge. Either God is morally neutral or benevolent. But God is not morally neutral. So, we are free. 10. If I do not have sense experience of apples, then I do not know about apples. If I have an idea of an apple, then the apple is real. If you tell me about apples, then either I do not have sense experience of apples or I have an idea of an apple. You tell me about apples. It is not the case that I do not know about apples. So, an apple is real. 11. If we eat meat, then the environment is degraded. If we are vegetarians, then fewer livestock are raised. If humanity persists, then either we eat meat or are vegetarians. Humanity persists. So, either the environment is degraded or fewer livestock are raised. 12. Either art is dead or a new form will appear. If art is dead, then it is not the case that some sculpture by Botero is valuable. But the claim that it’s not the case that some sculpture by Botero is valuable is false. So, a new form will appear and art is not dead. 13. If Mill is right, then consequences have moral weight; also, I like Mill’s work. If Kant is right, then pleasure is not important; I’m not a fan of Kant’s work. Either Mill is right or Kant is. So, either consequences have moral weight or pleasure is not important. 14. If values are transcendent, then truth does not matter. Either values are transcendent or the world has no meaning. But it is not the case that truth does not matter. So, either the world has no meaning or truth is pleonastic. 15. If names are either purely referential or contain descriptive content, then both Mill and Frege are worth reading. Names are purely referential and do not contain descriptive content. So, Mill is worth reading and names are purely referential. 16. If there is a self, then I could be eternal. If I could be eternal, then I am not my body. If I could be eternal, then I am not my soul. Either there is a self or I could be eternal. So, either I am not my body or I am not a soul.

3 . 3 : R u les o f E q u i v alence 1    1 2 9

3.3: RULES OF EQUIVALENCE 1 Rules of inference allow you to derive new conclusions based on previously accepted premises or derivations. They are justified by appeal to the truth table definitions of validity: using a rule of inference can never lead you from true premises to a false conclusion. They must be used on whole lines only, when the main operators of the lines of  the derivation you wish to use match the operators that appear in the rules. Further, uses of the rules go only in one direction, from premises to conclusion. Rules of equivalence are pairs of logically equivalent forms of propositions; you may substitute a proposition of the form of one of the pair for a proposition of the other form of the pair. You may use rules of equivalence on parts of a proposition, too. The rules may be used in either direction; propositions of the form of either of any of the pairs may be substituted for propositions of the other. Since the rules of equivalence are based on truth table equivalences, we can check the legitimacy of the substitutions by looking at truth tables to see that the expressions are in fact logically equivalent. The appendix to this book has truth tables showing the non-obvious rules of equivalence. There are five rules of equivalence in this section, though some of the rules have multiple forms. Five more are discussed in the next section, and four more are covered in section 3.6. As with the rules of inference, I present the rules of equivalence in a metalanguage, using Greek letters to emphasize that any consistent substitution of wffs for the metalinguistic variables is acceptable. I introduce a new metalinguistic → ’, to mean ‘is logically equivalent to’. This symbol does not belong to PL symbol, ‘ ←  and is used only in formulating the rules, not in expressions of the object language.

A rule of equivalence is a pair of logically

equivalent proposition forms.

→ is a metalogical sym← bol used for ‘is logically equivalent to’.

De Morgan’s Laws (DM) De Morgan’s laws summarize the equivalences of propositions using ‘neither’ and ‘not both’. ∼(α ∙ β) ∼(α ∨ β)

→ ∼α ∨ ∼β ← → ∼α ∙ ∼β ←

De Morgan’s Laws

Notice that there are two versions of De Morgan’s law: one for the negation of a conjunction, and the other for the negation of a disjunction. We often think of the negation of the conjunction, the first form above, as a statement of the form ‘not both’. The negation of a disjunction is a ‘neither’ sentence. Since you can substitute any formula of the form of either side for a formula of the other, as with all rules of equivalence, you can go forward (left-to-right) or backward (right-to-left). A forward DM distributes the tilde to the components of the conjunction or disjunction, changing the operator inside the parentheses. A backward DM factors out the tilde. Both the forward and backward uses require the same justification. 3.3.1 contains a forward use of DM, while 3.3.2 contains a backward use.

De Morgan’s laws (DM) a re rules of equivalence

of PL.

1 3 0    C h apter 3  In f erence i n P ropos i t i onal L og i c

3.3.1 1. (A ∨ B) ⊃ E 2. ∼E 3. A ∨ D 4. ∼(A ∨ B) 5. ∼A ∙ ∼B 6. ∼A 7. D QED

/D 1, 2, MT 4, DM 5, Simp 3, 6, DS

3.3.2 1. G ⊃ ∼(H ∙ F) 2. ∼(∼H ∨ ∼F) / ∼G 3. ∼∼(H ∙ F) 2, DM 4. ∼G 1, 3, MT QED

Association (Assoc) Association (Assoc) a re

rules of equivalence of PL.

Association allows you to regroup series of conjunctions or disjunctions. → (α ∨ β) ∨ γ Association α ∨ (β ∨ γ) ← → (α ∙ β) ∙ γ α ∙ (β ∙ γ) ←

As with DM, Assoc has a version for conjunction and a version for disjunction. Unlike DM, Assoc requires no switching of operators. It merely allows you to regroup the component propositions; the two operators must be the same. Assoc is often used to organize a series of conjunctions before simplifying one of the conjuncts, or with DS, as in 3.3.3. 3.3.3 1. (L ∨ M) ∨ N 2. ∼L 3. (M ∨ N) ⊃ O 4. L ∨ (M ∨ N) 5. M ∨ N 6. O QED

/O 1, Assoc 4, 2, DS 3, 5, MP

Distribution (Dist) Distribution (Dist) a re rules of equivalence of PL.

The rules of distribution allow you to distribute a conjunction over a disjunction or to distribute a disjunction over a conjunction. → (α ∙ β) ∨ (α ∙ γ) Distribution α ∙ (β ∨ γ) ← → (α ∨ β) ∙ (α ∨ γ) α ∨ (β ∙ γ) ←

The main operator is always switched (between conjunction and disjunction) after a use of Dist. So, using Dist on a sentence whose main operator is a disjunction yields a conjunction from which you can simplify.

3 . 3 : R u les o f E q u i v alence 1    1 3 1

Notice that while the grouping of terms changes, the order of the first two operators remains after using Dist, with an extra operator of the first type added at the end (going left to right) or taken away (going right to left). So, ∙∨ becomes ∙∨∙ and ∨∙ becomes ∨∙∨ (and vice versa). Be careful to distinguish Dist from Assoc. Assoc is used when you have two of the same operators. Dist is used when you have a combination of conjunction and disjunction. 3.3.4 contains a forward use of Dist, while 3.3.5 contains a backward use. 3.3.4 1. H ∙ (I ∨ J) 2. ∼(H ∙ I) 3. (H ∙ I) ∨ (H ∙ J) 4. H ∙ J QED

/H∙J 1, Dist 3, 2, DS

3.3.5 1. (P ∨ Q) ∙ (P ∨ R) 2. ∼P 3. P ∨ (Q ∙ R) 4. Q ∙ R QED

/Q∙R 1, Dist 3, 2, DS

Commutativity (Com) Commutativity often combines with rules of inference to facilitate some obvious inferences that we could not yet make. α ∨ β α ∙ β

→ β ∨ α Commutativity ← → β∙α ←

In effect, Com doubles the rules DS, Simp, and Add. From a disjunction, we can now infer the first disjunct from the negation of the second, as at 3.3.6. From a conjunction, we can now infer the second conjunct using Simp, as at 3.3.7. And we can add a proposition in front of a given wff, as at 3.3.8. 3.3.6 1. P ∨ Q 2. ∼Q 3. Q ∨ P 4. P

1, Com 3, 4, DS

3.3.7

1. P ∙ Q 2. Q ∙ P 3. Q

1, Com 2, Simp

3.3.8

1. P 2. P ∨ Q 3. Q ∨ P

1, Add 2, Com

Commutativity (Com) a re rules of equivalence

of PL.

1 3 2    C h apter 3  In f erence i n P ropos i t i onal L og i c

Each of the three derivations 3.3.6–3.3.8 can be inserted into any derivation. 3.3.9 demonstrates the use of commutativity with simplification and disjunctive syllogism. 3.3.9 1. A ∙ B 2. B ⊃ (D ∨ E) 3. ∼E 4. B ∙ A 5. B 6. D ∨ E 7. E ∨ D 8. D QED

/D 1, Com 4, Simp 2, 5, MP 6, Com 7, 3, DS

Double Negation (DN) Double negation (DN) is a rule of equivalence of PL.

Double negation allows you to add two consecutive negations to formula or to remove two consecutive negations. → ∼∼α α ←

Double Negation

Be sure to use DN with two consecutive tildes. Do not remove two tildes separated by a parenthesis or other punctuation. And never insert one negation in front of punctuation and one after. You may add two consecutive tildes either inside or outside of a bracket. Just do not divide them around punctuation. Double negation is often used right-to-left as a way of clearing extraneous tildes. But be careful not to add or subtract single tildes. They must be added or removed in consecutive pairs. There are three ways to use DN to add two tildes to a statement with a binary operator. 3.3.10 can be transformed, in a single use of DN, into 3.3.11, 3.3.12, or 3.3.13. Two uses of DN yields 3.3.14. 3.3.10 P ∨ Q 3.3.11 ∼∼P ∨ Q 3.3.12 P ∨ ∼∼Q 3.3.13 ∼∼(P ∨ Q) 3.3.14 ∼∼(∼∼P ∨ Q)

by double-negating the ‘P’ by double-negating the ‘Q’ by double-negating the whole disjunction by double-negating both the ‘P’ and the disjunction

DN, like Com, allows us to expand our uses of other rules, as we can see in 3.3.15. 3.3.15 1. ∼F ⊃ ∼G 2. G 3. F ⊃ H 4. ∼∼G 5. ∼∼F 6. F 7. H QED

/H 2, DN 1, 4, MT 5, DN 3, 6, MP

3 . 3 : R u les o f E q u i v alence 1    1 3 3

Rules of Equivalence and Rules of Inference Be careful to distinguish the rules of equivalence, which we saw in this section, from the rules of inference, which we saw in the previous two sections. One difference is that each rule of equivalence can be used in two different directions. Another difference is that the rules of equivalence are justified by showing that expressions of each form are logically equivalent, which I have done for most of the rules of equivalence in the appendix. A third difference is that rules of equivalence apply to any part of a proof, not just to whole lines. Rules of inference must be used on whole lines, as we saw in example 3.1.15. In contrast, we can use any rule of equivalence on only a part of a line, as with DM in 3.3.16 and DN and DM in 3.3.17. 3.3.16 P ⊃ ∼(Q ∨ P) P ⊃ (∼Q ∙ ∼P) DM 3.3.17 S ⊃ (∼P ∙ Q) S ⊃ (∼P ∙ ∼∼Q) DN S ⊃ ∼(P ∨ ∼Q) DM

Summary Rules of equivalence are transformation rules that allow us to replace some formulas and subformulas with logical equivalents. These transformations help expand the applications of our rules of inference. They are also, in many cases, formal versions of natural-language equivalencies. We’ve seen five rules of equivalence in this section, though each rule has at least two different applications (in each direction), and some of the rules, like De Morgan’s laws and distribution, are actually two pairs of rules. We’ll see five more rules of equivalence in the next section, and four more in section 3.6. While the thirteen rules to this point are not very many to manage, they allow so many more derivations than just the first few rules that the proofs can be subtle and interesting. Even the strongest logic students should find some of the derivations in this section challenging. KEEP IN MIND

Rules of equivalence allow you to substitute one proposition or part of a proposition with a logically equivalent expression. We saw five rules of equivalence in this section: De Morgan’s laws (DM); association (Assoc); distribution (Dist); commutativity (Com); double negation (DN). Forward DM distributes a tilde to the components of a conjunction or disjunction. Backward DM factors out the tilde. All uses of DM switch a conjunction to a disjunction or a disjunction to a conjunction.

1 3 4    C h apter 3  In f erence i n P ropos i t i onal L og i c

Assoc is used when you have two conjunctions or two disjunctions. Dist is used when you have a combination of conjunction and disjunction. Differences between rules of equivalence and rules of inference include the following: Rules of inference are based on the truth table definition of validity; they are unidirectional. Rules of equivalence are based on truth table equivalences and may be used in either direction. Unlike rules of inference, rules of equivalence may be used on parts of lines or on whole lines.

Rules Introduced De Morgan’s Laws (DM) ∼(α ∙ β) ∼(α ∨ β)

→ ∼α ∨ ∼β ← → ∼α ∙ ∼β ←

Association (Assoc) α ∨ (β ∨ γ) α ∙ (β ∙ γ)

→ (α ∨ β) ∨ γ ← → (α ∙ β) ∙ γ ←

Distribution (Dist) α ∙ (β ∨ γ) α ∨ (β ∙ γ)

→ (α ∙ β) ∨ (α ∙ γ) ← → (α ∨ β) ∙ (α ∨ γ) ←

Commutativity (Com) α ∨ β α ∙ β

→ β∨α ← → β∙α ←

Double Negation (DN) → ∼∼α α ←

TELL ME MORE  • How are De Morgan’s laws useful in constructing empirical theories? See 7S.8: Logic and Science.

3 . 3 : R u les o f E q u i v alence 1    1 3 5

EXERCISES 3.3a Derive the conclusions of each of the following arguments using the rules of inference and the first five rules of equivalence. 1. 1. A ⊃ B 2. C ∙ A

/B

2. 1. ∼(P ∨ Q ) 2. R ⊃ P

/ ∼R

3. 1. H ∨ J 2. I ∙ ∼H

/J

4. 1. X ⊃ Y 2. Z ∙ ∼Y / ∼X ∙ Z 5. 1. R ∨ B 2. B ⊃ M 3. R ⊃ D 4. ∼M

/D

6. 1. Q ⊃ R 2. ∼(S ∨ T) 3. T ∨ Q

/R

7. 1. X ⊃ Y 2. (∼Y ∙ Z) ∙ T 3. X ∨ W

/W

8. 1. ∼A ∨ B 2. ∼[(∼A ∨ C) ∨ D] / B 9. 1. A ∨ (B ∙ C) 2. (C ∨ A) ⊃ ∼∼B

/B

10. 1. A ⊃ (C ∨ B) 2. ∼C ∙ A 3. B ⊃ D

/D

11. 1. (A ⊃ B) ∨ T 2. ∼T 3. B ⊃ C

/A⊃C

1 3 6    C h apter 3  In f erence i n P ropos i t i onal L og i c

12. 1. ∼A ⊃ C 2. B ∙ ∼C 3. A ⊃ D

/D∙B

13. 1. ∼D ∙ ∼E 2. (D ∨ F) ∨ E

/F

14. 1. E ∙ D 2. D ⊃ ∼A 3. (B ∨ A) ∨ C

/B∨C

15. 1. P ∨ (Q ∙ R) 2. P ⊃ S 3. R ⊃ T

/S∨T

16. 1. C ⊃ (∼A ∨ ∼C) 2. C ∙ D 3. D ⊃ B / ∼(A ∙ C) ∙ B 17. 1. (P ∨ Q  ) ⊃ R 2. R ⊃ S 3. ∼S / ∼(R ∨ P) 18. 1. R ∙ (S ∨ T) 2. ∼R ∨ ∼S

/T

19. 1. (A ∙ B) ∨ (A ∙ C) 2. D ⊃ ∼A / ∼D 20. 1. ∼(E ∨ F) ⊃ D 2. ∼∼G ∙ ∼F 3. E ⊃ ∼G

/D

21. 1. P ∙ (∼Q ∨ R) 2. ∼P ∨ Q

/P∙R

22. 1. I ∙ {∼[ J ∙ (K ∨ L)] ∙ M} 2. (∼J ∨ ∼L) ⊃ N / N 23. 1. ∼[(G ∙ H) ∙I] 2. G ∙ I

/ ∼H

24. 1. (K ∙ L) ∙M 2. K ⊃ N 3. N ⊃ ∼(O ∨ P) / ∼P 25. 1. [T ∙ (U ∨ V)] ⊃ W 2. W ⊃ ∼X 3. Y ∙ X / ∼(T ∙ U) ∙ ∼(T ∙ V)

3 . 3 : R u les o f E q u i v alence 1    1 3 7

26. 1. O ⊃ P 2. (O ∙ ∼Q  ) ∙ ∼R 3. P ⊃ [Q ∨ (R ∨ S)]

/S

27. 1. U ⊃ V 2. V ⊃ ∼(W ∙ X) 3. U ∙ (W ∙ Y)

/ ∼X ∙ Y

28. 1. A ⊃ D 2. D ⊃ ∼(A ∙ B) 3. A ∙ (B ∨ C)

/A∙C

29. 1. C ∨ (D ∙ B) 2. (C ∨ D) ⊃ ∼C

/D∙B

30. 1. E ∨ (F ∨ G) 2. ∼(∼∼G ∨ ∼H) 3. [(E ∨ F) ∙ ∼G] ⊃ A

/A

31. 1. ∼X ∙ (Y ∨ Z) 2. ∼Y ∨ ∼∼X 3. (∼X ∙ Z) ⊃ W

/T∨W

32. 1. (P ∨ Q  ) ∨ R 2. ∼P 3. Q ⊃ S 4. R ⊃ T 5. ∼S

/T

33. 1. J ⊃ K 2. K ⊃ [L ∨ (M ∙ N)] 3. ∼N ∙ J

/L

34. 1. [O ∨ (P ∙ Q )] ⊃ R 2. R ⊃ ∼S 3. P ∙ S

/ ∼Q

35. 1. A ⊃ B 2. ∼[(C ∙ D) ∨ (C ∙ B)] 3. C ∙ E / ∼A 36. 1. F ⊃ G 2. H ⊃ I 3. (J ∨ F) ∨ H 4. ∼J ∙ ∼G

/I

37. 1. ∼(A ∨ B) 2. D ⊃ B 3. A ∨ (∼E ∨ D) 4. [∼(∼C ∨ E) ⊃ F] ∙ C / F

1 3 8    C h apter 3  In f erence i n P ropos i t i onal L og i c

38. 1. A ∙ ∼C 2. ∼(C ∙ D) ⊃ E 3. ∼(F ∨ C) ⊃ ∼E / F ∙ E 39. 1. M ∨ (Q ⊃ ∼P) 2. (∼Q ∙ L) ⊃ (∼Q ⊃ ∼O) 3. (P ∨ M) ∙ (M ∨ L) 4. ∼M / ∼O 40. 1. (O ∙ P) ⊃ (Q ∙ R) 2. P ⊃ ∼Q 3. O ⊃ ∼R 4. P ∨ O / ∼P ∨ ∼O

EXERCISES 3.3b Translate each of the following paragraphs into arguments written in PL. Then, derive the conclusions of the arguments using the eight rules of inference and the first five rules of equivalence. 1. If Albert asks Bernice on a date, then she’ll say yes. Bernice doesn’t say yes to a date and her cat died, but her dog is still alive. So, Albert didn’t ask Bernice on a date. 2. Callie majors in English only if she reads Charles Dickens. Either Callie and Elisa major in English or Callie and Franz major in English. So, Callie reads Charles Dickens. 3. If there is a mouse in the house, then nuts were left out. The lights were turned off unless no nuts were left out. Neither the lights were turned off nor were the doors left open. So, there was no mouse in the house. 4. It is not the case that either there was a paper or both a quiz and recitation in French class. If there is no quiz, then the students are happy. If there is no recitation, the teacher is happy. So, either the students or the teacher is happy. 5. Roland will either go on the upside-down roller coaster, or the speedy vehicle or the water slide. He doesn’t go on the upside-down roller coaster and he doesn’t go on the speedy vehicle. If he goes on the tilt-a-whirl, then he won’t go on the water slide. So, he doesn’t go on the tilt-a-whirl. 6. If Luz doesn’t travel to Greece, then she’ll go to Haiti. She’ll go to Israel given that she travels to Haiti. She doesn’t go to either Greece or Jordan. So, she goes to Israel and not Jordan.

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7. It is not the case that either Ernesto and Francisco go to swim practice or Gillian or Hayden go to swim practice. Either Isaac or Joan goes to swim practice. If Isaac goes to swim practice, then Hayden will go to swim practice. So, Joan goes to swim practice. 8. If it’s not the case that both Katrina and Laetitia go to math class, then Ms. Macdonald will be angry. Ms. Macdonald is angry only when Nigel skips math class. It is not the case that either Olivia and Polly both skip math class, or Nigel does. Therefore, Laetitia goes to math class. 9. Time is not both dynamic and static. But time is both subjective and dynamic. So, time is not static. 10. Anaximander, Thales, or Pythagoras believes that everything is made of water. But neither Anaximander nor Pythagoras believes that everything is made of water. So, either Thales or Protagoras believes that everything is made of water. 11. If meaning is atomic and compositional, then there are no incompatible translation manuals. But there are incompatible translation manuals. And meaning is compositional. So, meaning is not atomic. 12. Either Sartre believes in freedom just in case Camus does, or existentialism is problematic. But existentialism is neither incoherent nor problematic. So, Sartre believes in freedom if, and only if, Camus does. 13. Descartes and either Spinoza or Leibniz defend the ontological argument. But if Descartes and Spinoza defend the ontological argument, then rationalism is not theistic. If Descartes and Leibniz defend the ontological argument, then rationalism is not libertarian. So, rationalism is not both theistic and libertarian. 14. If truth is not subjective, then there are universally valid principles of justice. If truth is not relative, then we can know the principles of justice. If truth is both subjective and relative, then there are no moral facts. But there are moral facts. So, either there are universally valid principles of justice or we can know the principles of justice. 15. Either morality is individualistic or Nietzsche is not right about morality. Either morality is individualistic or Thrasymachus is not right about morality. Nietzsche and Thrasymachus are not both wrong. So, morality is individualistic. 16. The self is either the soul or consciousness, or it’s irreducible or nonexistent. If the self is either the soul or consciousness, then empirical science is useless. If the self is irreducible, then it is really consciousness. Empirical science is not useless. So, neither empirical science is useless nor is the soul not nonexistent.

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3.4: RULES OF EQUIVALENCE 2 This section introduces the next five rules of equivalence. Once we have these five rules, we have nearly a complete set of rules for derivations in PL. The derivations in this section are among the most difficult in the book. In the next section, I will demonstrate a few short techniques that will be useful to learn. In the following section, we will discuss seven further rules (three rules of inference and four rules of equivalence) that govern the biconditional. Then, we will explore two additional proof methods, techniques that make derivations less challenging and complete our system.

Contraposition (Cont) Contraposition (Cont) is a

rule of equivalence of PL.

Contraposition is based on the equivalence of a conditional and its contrapositive. → ∼β ⊃ ∼α Contraposition α ⊃ β ←

In other words, the antecedent and consequent of a conditional statement may be exchanged if they are both negated (or, right-to-left, un-negated). Cont is often used with HS, as in 3.4.1. 3.4.1 1. A ⊃ B 2. D ⊃ ∼B 3. ∼∼B ⊃ ∼D 4. B ⊃ ∼D 5. A ⊃ ∼D QED

/ A ⊃ ∼D 2, Cont 3, DN 1, 4, HS

Cont can be tricky when only one formula is negated, as we can see in 3.4.2 and 3.4.3, which perform the same transformation in different orders. You can either add a negation to both the antecedent and consequent when you use Cont or you can take a tilde off of each of them. But you cannot mix-and-match. Thus, you often need to invoke DN together with Cont. 3.4.2 A ⊃ ∼B ∼∼B ⊃ ∼A B ⊃ ∼A

by Cont (left-to-right) by DN

3.4.3 A ⊃ ∼B ∼∼A ⊃ ∼B B ⊃ ∼A

by DN by Cont (right-to-left)

Material Implication (Impl) Material implication (Impl) is a rule of

equivalence of PL.

The rule of material implication allows you to change a disjunction to a conditional, or vice versa, showing the relation between implication and disjunction. → ∼α ∨ β α ⊃ β ←

Material Implication

It is often easier to work with disjunctions. From a disjunction, you may be able to use De Morgan’s laws to get a conjunction. You may be able to use distribution, which does not apply to conditionals. In contrast, sometimes you just want to work with

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conditionals, using hypothetical syllogism, modus ponens, or modus tollens. Proofs are overdetermined by our system: there are multiple ways to do them once we have all the rules. The rule of material implication gives us a lot of options. The rule of material implication also illustrates the underlying logic of the material conditional. It is just a way of saying that either the antecedent is false or the consequent is true. Unlike many natural-language conditionals, it says nothing about the connections between the antecedent and the consequent. The derivation 3.4.4 illustrates the use of Impl with HS. 3.4.4 1. G ⊃ ∼E 2. E ∨ F 3. ∼∼E ∨ F 4. ∼E ⊃ F 5. G ⊃ F QED

/G⊃F 2, DN 3, Impl 1, 4, HS

Material Equivalence (Equiv) The two versions of material equivalence are the first rules that govern inferences with the biconditional. We will look at seven more in section 3.6. α ≡ β α ≡ β

→ (α ⊃ β) ∙ (β ⊃ α) ← → (α ∙ β) ∨ (∼α ∙ ∼β) ←

Material Equivalence

The first option for unpacking a biconditional tends to be more useful since it yields a conjunction, both sides of which you can simplify, as in 3.4.5. 3.4.5 1. A ≡ B 2. ∼A 3. B ⊃ C 4. (A ⊃ B) ∙ (B ⊃ A) 5. (B ⊃ A) ∙ (A ⊃ B) 6. B ⊃ A 7. ∼B 8. A ⊃ B 9. A ⊃ C 10. ∼B ∙ (A ⊃ C)

/ ∼B ∙ (A ⊃ C) 1, Equiv 4, Com 5, Simp 6, 2, MT 4, Simp 8, 3, HS 7, 9, Conj

The second version of material equivalence reflects the truth table definition of the operator. Remember, a biconditional is true if either both components are true (first disjunct of Equiv) or both disjuncts are false (second disjunct of Equiv). 3.4.6 demonstrates an instance of the second use of the rule. 3.4.6 1. D ≡ E 2. ∼D / ∼D ∙ ∼E 3. (D ∙ E) ∨ (∼D ∙ ∼E) 1, Equiv 4. ∼D ∨ ∼E 2, Add 5. ∼(D ∙ E) 4, DM 6. ∼D ∙ ∼E 3, 5, DS

Material equivalence (Equiv) a re rules of

equivalence of PL.

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If you need to derive a biconditional, again the first version of the rule is often more useful. First, derive the two component conditionals. Then, conjoin them and use the rule. We will explore this method more carefully in sections 3.6, 3.7, and 3.9. For now, take a moment to see how the rule is used at 3.4.7. 3.4.7 1. ∼[(K ⊃ ∼H) ∙ (∼H ⊃ K)] 2. (I ∙ J) ⊃ (K ≡ ∼H) / ∼(I ∙ J) 3. ∼(K ≡ ∼H) 1, Equiv 4. ∼(I ∙ J) 2, 3, MT QED

Exportation (Exp) Exportation (Exp) i s a

rule of equivalence of PL.

Exportation allows you to group antecedents of nested conditionals either together as a conjunction (on the right) or separately (on the left). → (α ∙ β) ⊃ γ Exportation α ⊃ (β ⊃ γ) ←

According to Exp, a typical nested conditional like 3.4.8 can be translated as either 3.4.9 or 3.4.10. 3.4.8

If I get my paycheck today, then if you come with me, we can go to dinner. 3.4.9 P ⊃ (C ⊃ D) 3.4.10 (P ∙ C) ⊃ D

While 3.4.9 is the more natural reading of 3.4.8, the alternative 3.4.10 is also satisfying. A close English translation of 3.4.10, at 3.4.11, is intuitively equivalent to the original. 3.4.11

If I get my paycheck today and you come with me, then we can go to dinner.

Further, exportation, when combined with commutativity, allows us to switch antecedents. So, 3.4.9 is also equivalent to 3.4.12. A natural translation of that proposition into English is at 3.4.13. 3.4.12 C ⊃ (P ⊃ D) 3.4.13 If you come with me, then if I get my paycheck, we can go to dinner.

While 3.4.13 is not as intuitively satisfying as 3.4.11 as an equivalent of 3.4.8, they are all logically equivalent. The difference in tone or presupposition may arise from the awkwardness of representing natural-language conditionals, and their causal properties, with the material conditional. The rule of exportation sometimes allows you to get to MP or MT, as in 3.4.14.

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3.4.14 1. L ⊃ (M ⊃ N) 2. ∼N / ∼L ∨ ∼M 3. (L ∙ M) ⊃ N 1, Exp 4. ∼(L ∙ M) 3, 2, MT 5. ∼L ∨ ∼M 4, DM QED

When using exportation, be careful to distinguish propositions like 3.4.15 from propositions like 3.4.16. These are not equivalent. Remember that exportation allows us to group two antecedents, as in the former, not two consequents, as in the latter. Only 3.4.15 may be used with exportation. 3.4.15 A ⊃ (B ⊃ C) 3.4.16 (A ⊃ B) ⊃ C

Tautology (Taut) Tautology eliminates some redundancy. → α ∙ α Tautology α ← → α∨α α ←

The conjunction version of Taut is redundant on whole lines, right-to-left, since we can use Simp instead. The disjunction version is redundant on whole lines left-toright, since we can use Add instead. But Taut can be used on parts of lines, and the other directions can also be useful, especially for disjunction, as in 3.4.17. 3.4.17 1. O ⊃ ∼O / ∼O 2. ∼O ∨ ∼O 1, Impl 3. ∼O 2, Taut QED

Summary We have now seen eight rules of inference and ten rules of equivalence. It is a lot of rules to learn and master. The best way to learn how to use the rules is just to practice lots of derivations. With our eighteen rules, our proof system is almost complete. We’ll need at least one of the proof methods of sections 3.7 and 3.9 to finish. But we have plenty of interesting rules to learn and use, and the derivations of this section and the next are among the most difficult in the textbook. While it will take some work to learn the new proof techniques, they will, in the end, make derivations much simpler. Before we get to the new techniques, though, our next section, 3.5, features some hints and tricks that may be adapted for use in lots of longer derivations and should help make some of the more difficult derivations more manageable. In section 3.6, we will see a set of rules governing inferences using the biconditional.

Tautology (Taut) a re rules of equivalence of PL.

1 4 4    C h apter 3  In f erence i n P ropos i t i onal L og i c

KEEP IN MIND

We saw five further rules of equivalence in this section: contraposition (Cont), material implication (Impl), material equivalence (Equiv), exportation (Exp), and tautology (Taut). Cont displays the equivalence of a statement with its contrapositive. The rule of material implication is another way of saying that either the antecedent is false or the consequent is true. Equiv provides two ways to unpack or introduce a biconditional. Exp allows you to group the antecedents of some nested conditionals. Taut eliminates redundancy with conjunctions or disjunctions. We now have eighteen rules available for use in derivations.

Rules Introduced Contraposition (Cont)

→ ∼β ⊃ ∼α α ⊃ β ←

Material Implication (Impl) → ∼α ∨ β α ⊃ β ←

Material Equivalence (Equiv) α ≡ β α ≡ β

→ (α ⊃ β) ∙ (β ⊃ α) ← → (α ∙ β) ∨ (∼α ∙ ∼β) ←

Exportation (Exp)

→ (α ∙ β) ⊃ γ α ⊃ (β ⊃ γ) ←

Tautology (Taut) α α

→ α∙α ← → α∨α ←

EXERCISES 3.4a For each of the following inferences, determine which single rule of equivalence of sections 3.3 or 3.4 is used, if any. If the second formula does not result from a single application of a rule of equivalence to the first formula, write, ‘does not follow’. (Some of those inferences are valid, even if not immediately inferable in our system.) The inferences that do not immediately follow in these exercises are common errors that logic students make when

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learning the rules of equivalence. It might be worth your time to study and understand the errors, in order to avoid them yourself. 1. (P ⊃ Q  ) ⊃ ∼R / P ⊃ (Q ⊃ ∼R)

16. (∼P ∙ Q  ) ∨ (∼P ∙ R) / ∼P ∙ (Q ∨ R)

2. ∼P ∨ [Q ∙ (R ≡ S)] / (∼P ∙ Q  ) ∨ [∼P ∙ (R ≡ S)]

17. (P ∙ Q  ) ⊃ ∼R / ∼(P ∙ Q  ) ∨ R

3. ∼[P ∨ (∼Q ∙ R)] / ∼P ∙ ∼(∼Q ∙ R)

18. (P ⊃ ∼Q  ) ⊃ (∼Q ⊃ P) / P ≡ ∼Q

4. (P ∨ Q  ) ≡ ∼R / (Q ∨ P) ≡ ∼R

19. (P ∨ ∼Q  ) ⊃ ∼R / ∼ ∼R ⊃ ∼(P ∨ ∼Q  )

5. P ∙ ∼(Q ∨ R) / ∼ ∼P ∙ ∼(Q ∨ R)

20. P ⊃ (Q ⊃ ∼R) / (P ∙ Q  ) ⊃ ∼R

6. P ∨ [(S ∨ T) ∨ U] / [P ∨ (S ∨ T)] ∨ U

21. P ∙ (∼Q ∙ R) / (P ∙ ∼Q  ) ∨ (P ∙ ∼R)

7. R ≡ (S ∨ S) /R≡S

22. ∼ ∼(P ∨ Q  ) ∨ ∼R / ∼[∼(P ∨ Q  ) ∙ R]

8. ∼P ≡ Q / (∼P ∙ Q  ) ∨ (∼ ∼P ∙ ∼Q  )

23. P ∙ (Q ≡ ∼R) / ∼(Q ≡ ∼R) ∙ ∼P

9. P ⊃ ∼Q / ∼Q ⊃ P

24. P ∨ ∼Q / ∼(∼P ∨ ∼ ∼Q  )

10. ∼(∼P ∨ Q  ) /P∨Q

25. (P ∙ ∼Q  ) ∙ (R ∙ ∼S) / [(P ∙ ∼Q  ) ∙ R] ∙ ∼S

11. [(P ∙ Q  ) ∙ ∼R] ⊃ S / (P ∙ Q  ) ⊃ (∼R ⊃ S)

26. P ∨ ∼P /P

12. (P ∙ ∼Q  ) ∨ (R ∙ S) / [(P ∙ ∼Q  ) ∨ R] ∙ [(P ∙ ∼Q  ) ∨ S]

27. P ⊃ (Q ∨ ∼R) / ∼P ∨ (Q ∨ ∼R)

13. ∼P ∙ ∼(Q ∙ R) / P ∨ (Q ∙ R)

28. P ≡ [(Q ∨ S) ≡ R] / P ≡ {[(Q ∨ S) ⊃ R] ∙ [R ⊃ (Q ∨ S)]}

14. P ∙ ∼Q / ∼ ∼P ∙ ∼ ∼Q

29. P ∨ ∼Q / Q ∨ ∼P

15. ∼P ∙ (Q ∨ ∼R) / (∼P ∙ Q  ) ∨ ∼R

30. ∼(P ≡ ∼Q  ) ∨ ∼R / ∼ ∼[(P ≡ ∼Q  ) ∨ R]

1 4 6    C h apter 3  In f erence i n P ropos i t i onal L og i c

EXERCISES 3.4b Derive the conclusions of each of the following arguments using the rules of inference and equivalence. 1. 1. P ⊃ ∼Q 2. R ⊃ Q 3. (P ⊃ ∼R) ⊃ S

/S

2. 1. P ∨ Q 2. ∼Q ∨ R

/P∨R

3. 1. ∼(P ≡ ∼Q  ) 2. P

/Q

4. 1. ∼I ∨ J 2. J ≡ K 3. (I ∙ L) ∨ (I ∙ M) / K 5. 1. G ∨ H 2. ∼I ∙ ( J ∙ ∼G)

/ H ∨ ∼I

6. 1. P ∨ (Q ∙ R) 2. S ⊃ ∼R

/S⊃P

7. 1. ∼P ∨ (Q ∨ S) 2. ∼P ⊃ R

/ ∼R ⊃ (Q ∨ S)

8. 1. E ≡ F 2. ∼(G ∨ E)

/ ∼F

9. 1. A ∨ (B ∨ A) 2. ∼(B ∨ C) 3. A ⊃ D

/D

10. 1. (P ∙ Q  ) ⊃ R 2. (P ∙ S) ∨ (P ∙ T) / Q ⊃ R 11. 1. L ⊃ ∼(∼M ∨ K) 2. M ⊃ (∼K ⊃ N) 3. ∼N / ∼L 12. 1. D ≡ E 2. (E ∨ F) ⊃ G 3. ∼(G ∨ H)

/ ∼D

13. 1. (P ∙ Q  ) ∨ (R ∙ S) 2. ∼S /P 14. 1. (P ∙ Q  ) ⊃ R 2. ∼(R ∨ S)

/ P ⊃ ∼Q

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15. 1. P ⊃ (∼Q ⊃ R) 2. ∼(R ∨ S)

/P⊃Q

16. 1. ∼P ∨ Q 2. ∼R ⊃ ∼Q 3. S ∨ ∼R

/P⊃S

17. 1. ∼(P ∨ Q  ) ⊃ R 2. ∼P / ∼R ⊃ Q 18. 1. ∼(P ∙ Q  ) ⊃ R 2. ∼S ∨ ∼R

/S⊃P

19. 1. ∼Q ⊃ ∼P 2. ∼Q ∨ R 3. ∼(∼S ∙ R)

/ ∼S ⊃ ∼P

20. 1. P ≡ ∼Q 2. P ∨ R 3. Q

/R

21. 1. (P ∙ Q  ) ∨ ∼R 2. ∼R ⊃ S

/ ∼S ⊃ P

22. 1. ∼P ∨ Q 2. ∼Q ∨ (R ⊃ ∼S) / S ⊃ (∼P ∨ ∼R) 23. 1. (P ∙ Q  ) ⊃ R 2. ∼S ∨ P

/ (S ∙ Q  ) ⊃ R

24. 1. D ∨ (E ∨ F) 2. F ⊃ (G ∙ H) 3. ∼G

/D∨E

25. 1. Q ⊃ R 2. R ⊃ (S ⊃ T)

/ ∼T ⊃ (S ⊃ ∼Q  )

26. 1. (P ⊃ ∼Q  ) ∨ R / (∼R ∙ P) ⊃ ∼Q 27. 1. (P ≡ Q  ) ∨ P

/ P ∨ ∼Q

28. 1. ∼[(P ∨ Q  ) ∙ R] 2. R ∨ S /Q⊃S 29. 1. (P ≡ Q  ) ∨ ∼P

/P⊃Q

30. 1. ∼P ∨ Q 2. R ⊃ ∼Q 3. R ∨ ∼S 4. ∼T ⊃ S

/P⊃T

31. 1. (S ≡ T) ∙ ∼U 2. ∼S ∨ (∼T ∨ U)

/ ∼S

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32. 1. [V ∨ (W ∨ X)] ⊃ Y 2. Y ⊃ Z / Z ∨ ∼V 33. 1. F ⊃ (G ⊃ H) 2. G ∙ ∼H 3. J ⊃ F

/ ∼J

34. 1. N ⊃ O 2. P ⊃ Q 3. ∼(Q ∨ O)

/P≡N

35. 1. T ⊃ (U ⊃ V) 2. Q ⊃ (R ⊃ V) 3. (T ∙ U) ∨ (Q ∙ R) / V 36. 1. (P ∙ Q  ) ⊃ (R ∙ S) 2. Q / ∼S ⊃ ∼P 37. 1. (P ∙ ∼Q  ) ⊃ (R ∨ S) 2. P ∙ ∼S /Q∨R 38. 1. Q ⊃ ∼P 2. ∼Q ⊃ R 3. ∼R ∨ ∼S 4. S ∨ ∼P / ∼P 39. 1. P ≡ (Q ∙ R) 2. S ⊃ P 3. T ⊃ P 4. ∼S ⊃ T

/Q

40. 1. ∼(P ≡ ∼Q  ) 2. P ⊃ R 3. Q ∨ R

/ R

EXERCISES 3.4c Translate each of the following paragraphs into arguments written in PL. Then, derive the conclusions of the arguments using the rules of inference and equivalence. 1. There is a rainbow if, and only if, the sun is out. The sun is not out. So, there is no rainbow. 2. If there are alpacas on the farm, then there are beagles. If there are beagles, then there are cows. So, either there are cows or there are no alpacas.

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3. If there is a line, Marla must wait in it. If New England High School shows up, then there is a line if the organist attends. The organist attends and New England High School shows up. Therefore, Marla must wait in line. 4. Cecilia goes roller skating if, and only if, Denise comes with her. Denise and Elise go roller skating, and Felicia goes running. So, Cecilia goes roller skating. 5. Either Ana doesn’t like lemons or she likes mangoes. She likes lemons and nectarines, and oranges. She either doesn’t like mangoes or she likes plums. So, she likes plums. 6. Quincy takes the job just in case Miriam does not veto the move. Miriam vetoes the move. So, either Quincy does not take the job or she gets another offer. 7. I can be happy if, and only if, I have both friends and wealth. But I have no friends. So, I cannot be happy. 8. Either we act freely or we lack reasons to act. Either we conceive of ourselves as free or we do not act freely. So, either we conceive of ourselves as free or we lack reasons to act. 9. Either art does not presuppose a distinctive sort of experience or there is no unified essence for art. If art does not presuppose a distinctive sort of experience then there is a unified essence for art. So, art presupposes a distinctive sort of experience if, and only if, there is no unified essence for art. 10. Either there are moral facts or murder is not wrong. Either murder is wrong or we cannot know ethical principles. If there are moral facts then we can know ethical principles. So, there are moral facts if, and only if, we can know ethical principles. 11. If metaphysics is a priori, then if it is synthetic, then Hume is wrong about causation. If we cannot see gravity, then Hume is not wrong about causation. Therefore, if metaphysics is synthetic and a priori, then we can see gravity. 12. We are conscious if, and only if, not all facts are physical. If we are not conscious and we are zombies, then dualism is true. All facts are physical. So, if we are zombies, then dualism is true. 13. If there is a self, then the concept of the self is irreducible. If I am my conscious experience, then the concept of the self is not irreducible. If I do not have a soul, then I am my conscious experience. If I do have a soul, then I am not my body. So, if I am my body, then there is no self. 14. Consequences are morally important if, and only if, duties are not. Either consequences are morally important or duties are not. So, consequences are morally important and duties are not.

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15. If womanhood has an essence, then if there is a mystique of feminine fulfillment, then there is a monolithic patriarchy. But there is no monolithic patriarchy. So, if there is a mystique of femininity, then womanhood has no essence. 16. Either you are moderate and restrained, or you are not happy. Either you are not moderate but are restrained, or you are happy. Either it is not the case that you are happy if and only if you are moderate, or you are restrained. So, you are restrained.

3.5: PRACTICE WITH DERIVATIONS Our proof system is now fairly robust. With eighteen of our twenty-five rules available, some of the proofs you will be asked to derive now are long. Some are quite difficult. In the next section, I add seven more rules, all governing biconditionals. Toward the end of the chapter, I introduce two additional proof techniques which will make derivations easier. For now, the best way to improve your skill at constructing derivations is to practice. A lot. It’s a skill, like playing an instrument or riding a horse. You can’t learn it merely by reading about it, and you can’t get better without lots of practice. Practice constructing derivations improves your skill largely because you learn some simple tricks that recur in proofs. In this section, I show you some techniques, some mini-proofs, that can be applied in various different derivations.

Making Conditionals In 3.5.1, we infer from the negation of a wff that the wff (un-negated) entails anything. You just add the desired consequent and use the rule of material implication. 3.5.1 1. ∼A 2. ∼A ∨ B 3. A ⊃ B QED

/A⊃B 1, Add 2, Impl

In 3.5.2, we see that any wff entails a formula that is already assumed or proven. As in 3.5.1, you add a wff: this time, the negation of your desired antecedent. Again, a use of Impl ends the derivation. 3.5.2 1. E 2. E ∨ ∼F 3. ∼F ∨ E 4. F ⊃ E QED

/F⊃E 1, Add 2, Com 3, Impl

3 . 5 : P ract i ce w i t h Der i v at i ons   1 5 1

Switching Antecedents of a Nested Conditional 3.5.3 demonstrates how to switch the antecedents of a conditional whose consequent is another conditional, using exportation and commutativity. 3.5.3 QED

1. G ⊃ (H ⊃ I) 2. (G ∙ H) ⊃ I 3. (H ∙ G) ⊃ I 4. H ⊃ (G ⊃ I)

/ H ⊃ (G ⊃ I) 1, Exp 2, Com 3, Exp

Negated Conditionals Having the negation of a conditional in a proof can often be useful. Remember, the only way for a conditional to be false is for the antecedent to be true and the consequent to be false. So, if you have assumed or derived the negation of a conditional, you can also derive the antecedent conjoined with the negation of the consequent, as at 3.5.4. Then you can simplify either conjunct. 3.5.4 1. ∼(P ⊃ Q) 2. ∼(∼P ∨ Q) 3. ∼ ∼P ∙ ∼Q 4. P ∙ ∼Q QED

/ P ∙ ∼Q 1, Impl 2, DM 3, DN

Simplifying Antecedents and Consequents Examples 3.5.5 and 3.5.6 show how to simplify a conditional. In 3.5.5, you might be tempted to simplify either of the conjuncts in the conclusion of the premise. But Simp is a rule of inference and may not be used on a part of a line. Instead, we can use Impl to turn the main conditional into a disjunction, and distribute the first disjunct, creating a conjunction. Then, we can simplify either conjunct, using Com for the second one, and turn the resulting, simpler disjunction back into a conditional to finish. 3.5.5 1. O ⊃ (P ∙ Q) 2. ∼O ∨ (P ∙ Q) 3. (∼O ∨ P) ∙ (∼O ∨ Q) 4. ∼O ∨ P 5. O ⊃ P QED

/O⊃P 1, Impl 2, Dist 3, Simp 4, Impl

In 3.5.6, we use the same general technique, turning the conditional into a disjunction, distributing, and then simplifying either of the resulting conjuncts.

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3.5.6 1. (R ∨ S) ⊃ T 2. ∼(R ∨ S) ∨ T 3. (∼R ∙ ∼S) ∨ T 4. T ∨ (∼R ∙ ∼S) 5. (T ∨ ∼R) ∙ (T ∨ ∼S) 6. T ∨ ∼R 7. ∼R ∨ T 8. R ⊃ T QED

/R⊃T 1, Impl 2, DM 3, Com 4, Dist 5, Simp 6, Com 7, Impl

Be careful to note the contrast between 3.5.5 and 3.5.6. We can reduce a conditional with a conjunction in the consequent or a conditional with a disjunction in the antecedent. We cannot reduce a conditional with a conjunction in the antecedent, nor can we reduce a conditional with a disjunction in the consequent. If α entails β and γ, then α entails β and α entails γ. If either α or β entails γ, then α entails γ and β entails γ. But from α and β together entailing γ, one cannot conclude that either α or β on its own entails γ. And from α entailing either β or γ, one does not know whether β or γ is entailed.

Combining Conditionals 3.5.7 and 3.5.8 show techniques that are the reverse of those in 3.5.5 and 3.5.6, combining two conditionals that share a consequent (in the former) and combining two conditionals that share an antecedent (in the latter). 3.5.7 1. W ⊃ X 2. Y ⊃ X 3. (W ⊃ X) ∙ (Y ⊃ X) 4. (∼W ∨ X) ∙ (Y ⊃ X) 5. (∼W ∨ X) ∙ (∼Y ∨ X) 6. (X ∨ ∼W) ∙ (∼Y ∨ X) 7. (X ∨ ∼W) ∙ (X ∨ ∼Y) 8. X ∨ (∼W ∙ ∼Y) 9. (∼W ∙ ∼Y) ∨ X 10. ∼(W ∨ Y) ∨ X 11. (W ∨ Y) ⊃ X QED

/ (W ∨ Y) ⊃ X 1, 2, Conj 3, Impl 4, Impl 5, Com 6, Com 7, Dist 8, Com 9, DM 10, Impl

3.5.8 1. A ⊃ B 2. A ⊃ C 3. ∼A ∨ B 4. ∼A ∨ C 5. (∼A ∨ B) ∙ (∼A ∨ C) 6. ∼A ∨ (B ∙ C) 7. A ⊃ (B ∙ C) QED

/ A ⊃ (B ∙ C) 1, Impl 2, Impl 3, 4, Conj 5, Dist 6, Impl

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A Statement Entailing Its Own Negation If a statement entails its own negation, the statement is false, as 3.5.9 shows. 3.5.9 1. D ⊃ ∼D 2. ∼D ∨ ∼D 3. ∼D QED

/D 1, Impl 2, Taut

Explosion Lastly, let’s take a look at an important and curious inference that logicians call explosion. Explosion is a characteristic of inconsistent theories, given the rules of inference of classical logic. An inconsistent theory is one in which both a statement and its negation are derivable. In other words, inconsistent theories contain contradictions. In chapter 2, we saw that individual statements can be self-contradictory, if they are false in every row of the truth table. We also saw that pairs of statements can be contradictory, if they differ in truth value in each row of the truth table, and inconsistent, if they cannot be true together. For the purposes of our proof theory, we will henceforth take a more narrow view of the term ‘contradiction’, as the conjunction of any statement with its negation, any statement of the form α ∙ ∼α, for any wff α. Let’s look at explosion, starting with a contradiction, at 3.5.10. 3.5.10 1. P ∙ ∼P 2. P 3. P ∨ Q 4. ∼P ∙ P 5. ∼P 6. Q QED

1, Simp 2, Add 1, Com 4, Simp 3, 5, DS

Notice that the only premise for the explosive inference is the contradiction at line 1; Q never appears until it’s added at line 3. And then it is derived all by itself! From a contradiction, anything, and everything, follows. That’s why logicians call this property of logical systems explosion: every wff of the language can be derived from any contradiction. Classical systems explode. We will return to explosion and the importance of contradictions (and avoiding them in classical systems such as ours) in section 3.9. For now, just notice that if you ever find an argument in which a contradiction is provable, you can just insert a few lines, as in 3.5.10, to demonstrate any conclusion. (I ordinarily try to keep the premises of the arguments in the exercises consistent, but here I included a few contradictions—­see if you can find them!)

In derivations, a contradiction is any statement of the form: α ∙ ∼α.

Explosionis a property of classical systems of inference: from a contradiction, any statement can be derived.

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Summary The proofs of section 3.4 were often difficult due both to the number of rules we have to know and the complexities of the arguments whose conclusions we are now able to derive. One way to improve your abilities to construct complicated derivations is to know and recognize a variety of common techniques, such as the ones of this section. They are worth a little time studying, so that you can use them in future derivations. Moreover, some of the underlying concepts, such that if a statement is true then anything entails it or that a contradiction entails all other formulas, are central to PL, classical propositional logic. So, getting to know these techniques can help you better understand the logic you are using. KEEP IN MIND

If a statement is assumed or derived, its opposite entails any wff. If a statement is assumed or derived, any wff entails it. In a nested conditional with two antecedents, the order of the antecedents may be reversed. The negation of a conditional is your friend. Conditionals with conjunctions in their consequents can be simplified. Conditionals with disjunctions in their antecedents can be simplified. If a statement entails its own negation, the statement is false. A contradiction entails anything.

TELL ME MORE  • How does explosion relate to logical truths, especially the so-called paradoxes of material implication? See 6.2: Conditionals. • How does explosion relate to the logic of truth? See 7.5: Truth and Liars.

EXERCISES 3.5a Derive the conclusions of each of the following arguments using the rules of inference and equivalence. 1. 1. A ⊃ B 2. B ⊃ ∼B / ∼A 2. 1. ∼K ∨ L 2. L ⊃ ∼K / ∼K 3. 1. G ⊃ H 2. ∼(I ⊃ H) / ∼G

3 . 5 : P ract i ce w i t h Der i v at i ons   1 5 5

4. 1. (T ∙ U) ⊃ V 2. ∼(T ⊃ W)

/U⊃V

5. 1. ∼(P ⊃ Q  ) 2. ∼(R ⊃ S)

/ ∼(Q ∨ S)

6. 1. P ⊃ Q 2. P

/R⊃Q

7. 1. (P ∨ Q  ) ⊃ R 2. R ⊃ ∼S

/ P ⊃ ∼S

8. 1. (A ⊃ B) ⊃ C 2. ∼A ∨ (B ∙ D) / C 9. 1. W ⊃ (X ∙ Y) 2. (W ∙ ∼X) ∨ Z / Z 10. 1. N ⊃ (O ∙ P) 2. ∼N ⊃ Q

/ ∼O ⊃ Q

11. 1. ∼P ⊃ R 2. ∼Q ⊃ R 3. ∼R

/ S ⊃ (P ∙ Q  )

12. 1. P ≡ (Q ∙ R) 2. ∼Q / ∼P 13. 1. P ⊃ (∼Q ⊃ ∼R) 2. R /P⊃Q 14. 1. ∼[(P ∙ Q  ) ∙ R] 2. R / P ⊃ ∼Q 15. 1. (P ∙ Q  ) ⊃ (R ⊃ S) 2. Q ∙ R / ∼S ⊃ ∼P 16. 1. I ⊃ J 2. ∼J ∙ K 3. ∼J ⊃ L 4. ∼ ∼I

/K∙L

17. 1. ∼(P ≡ ∼Q  ) 2. P ⊃ ∼Q / ∼Q ∙ ∼P 18. 1. P ⊃ R 2. Q ⊃ R 3. S ⊃ (P ∨ Q  )

/S⊃R

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19. 1. R ∨ Q 2. ∼R ∨ ∼S 3. ∼(∼S ∙ ∼T) 4. ∼(P ⊃ U) 5. ∼(P ∙ Q  )

/ T ∙ ∼U

20. 1. (P ∙ Q  ) ⊃ (R ∨ S) / ∼R ⊃ [(Q ∙ P) ⊃ S] 21. 1. ∼(X ⊃ Y) 2. Y ∨ (Z ∙ A)

/Z≡A

22. 1. (H ∙ I) ⊃ J 2. H ∙ (I ∨ K)

/ ∼J ⊃ K

23. 1. (X ⊃ Y) ⊃ Z 2. W ⊃ ∼Z / ∼(W ∙ Y) 24. 1. ∼V ⊃ W 2. X ⊃ Y 3. V ⊃ Z 4. ∼W ∙ X 5. ∼Z ∙ Y

/ Y ∙ ∼V

25. 1. P ⊃ Q 2. P ⊃ R 3. (Q ∙ R) ⊃ ∼S / ∼P ∨ ∼S 26. 1. P ⊃ (Q ∨ R) 2. R ⊃ (S ∙ T) 3. ∼Q

/P⊃T

27. 1. ∼P ∨ Q 2. ∼R ∨ ∼Q 3. ∼R ⊃ (S ∙ T)

/P⊃S

28. 1. A ⊃ B 2. B ⊃ D 3. D ⊃ A 4. A ⊃ ∼D / ∼A ∙ ∼D 29. 1. (I ∙ E) ⊃ ∼F 2. F ∨ (G ∙ H) 3. I ≡ E

/I⊃G

30. 1. ( J ⊃ J) ⊃ (K ⊃ K) 2. (K ⊃ L) ⊃ ( J ⊃ J)

/K⊃K

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EXERCISES 3.5b Translate each of the following paragraphs into arguments written in PL. Then, derive the conclusions of the arguments using the rules of inference and equivalence. 1. If David quits the team, then Sandra watches the games provided that Ross joins the team. So, it is not the case that David quits the team, and Ross joins the team, and Sandra doesn’t watch the games. 2. If you are from the planet Orc, then you have pin-sized nostrils. But, things with pin-sized nostrils are not from Orc. Either you are from Orc or Quaznic, or you rode a long way in your spaceship. So, you are from Quaznic unless you rode a long way in your spaceship. 3. It is not the case that violets bloom only if they are watered. Either violets are watered or they undergo special treatment. So, they undergo special treatment. 4. If Francesca playing the xylophone entails that she yawns in class, then Zara gives a presentation in class. If Zara gives a presentation, then the woodwind players listen. So, either the woodwind players listen or Francesca plays xylophone. 5. Either experience eternally recurs unless there is no God, or suffering is the meaning of existence. If I can go under, then experience does not eternally recur. So, if I can go under and there is a God, then the suffering is the meaning of existence. 6. If life is suffering, then if you do not have compassion, then only the truth can save us. It is not the case that if life is suffering, then you have compassion. So, only the truth can save us. 7. If we explain events by reference to better-known phenomena, then explanations are not inferences. Explanations of events refer to better-known phenomena. So, we explain events by reference to better-known phenomena if, and only if, explanations are not inferences. 8. If God’s nonexistence entails her existence, then the existence of goodness entails that there is no goodness. God exists. So, there is no goodness. 9. If removing one’s glasses entails that the quality of experience changes, then the content of experience is subjective. But the content of experience is not subjective. So, if the quality of experience changes, then qualia fade. 10. If truth arises from societies with constraints and not from solitary freedom, then philosophy and politics are inextricably linked. Truth arising from socie­ ties with constraints does not entail that philosophy and politics are inextricably linked. So, truth arises from solitary freedom.

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11. If acting freely entails the existence of external causation, then we are aware of our freedom. If we are aware of our freedom, then we are unaware of our freedom. So, we act freely. 12. If slowness is not a property of a walker, then if it is a property of walking, then events exist. Either slowness is a property of walking and of running, or slowness is a property of walking and of thinking. So, if slowness is not a property of a walker, then events exist. 13. If moral theory is useful, then it should not serve only oneself. If either moral theory should not serve only oneself or self-interest is difficult to know, then we ought to consider the good of others. If moral theory is not useful, then we should not consider the good of others. So, a moral theory is useful if, and only if, we should consider the good of others. 14. If the general will is common interest, then if foreign powers see the state as an individual, then the general will involves total subjection and is sometimes misunderstood. If the general will is sometimes misunderstood, then to govern is to serve. Foreign powers see the state as an individual. So, if the general will is common interest, then to govern is to serve. 15. If sense experience is reliable, then mass is a real property and color is not. If Newtonian physics is true, then mass is a real property and teleology is not a physical concept. If Newtonian physics is not true, then sense experience is reliable. If mass is a real property and color is not, then teleology is a physical concept. So, color is a real property if, and only if, teleology is not a physical concept. 16. If there is a God, then there is goodness. But the existence of a God also entails that we are free. Either the nonexistence of God entails the nonexistence of goodness or we are not free. So, God exists if, and only if, there is goodness and we are free.

3.6: THE BICONDITIONAL Conditionals and biconditionals have different meanings and different truth conditions. These differences can be subtle and difficult to discern in natural language. For example, the inference 3.6.1 is not logically valid. 3.6.1

I’ll accompany you if you go to the movies. You don’t go to the movies. So, I don’t go with you.

Nevertheless, people sometimes make such fallacious inferences. One account of the fact that people make inferences like 3.6.1 is that they confuse the conditional and

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the biconditional. 3.6.2, which is identical to 3.6.1 except for the main operator of the first premise, is logically valid. 3.6.2

I’ll go with you if, and only if, you go to the movies. You don’t go to the movies. So, I don’t go with you.

In 3.6.1, I commit to joining you if you go to the movies, but I say nothing about what happens if you decide instead to go bowling. Perhaps I really like you and would join you no matter what you do. (And perhaps I utter the first premise of 3.6.1 in order not to appear overeager!) In contrast, in 3.6.2, I both commit to joining you if you go to the movies and not going with you if you do anything else. I join you only if you go to the movies, so if you go bowling, I’m out. Compounding the confusion, perhaps, is the fact that in many mathematical or logical contexts, people use conditionals where biconditionals are also (perhaps more) appropriate. For example, a mathematician might utter 3.6.3. 3.6.3

If a tangent to a circle intersects a chord at a right angle, the chord is a diameter.

While there’s nothing wrong with 3.6.3, the stronger 3.6.4 is also warranted. 3.6.4

A tangent to a circle intersects a chord at a right angle if, and only if, the chord is a diameter.

Since conditionals and biconditionals have different truth conditions, it is important to keep them distinct in your mind and regimentations. It will also be useful to have some more rules governing inferences using the biconditional. We have lots of rules governing the conditional. The only rule we have so far governing use of the biconditional is Equiv. So, the inference 3.6.5 is made in a single step. 3.6.5 QED

1. P ⊃ Q 2. P 3. Q

Premise Premise 1, 2, MP

In contrast, the parallel inference 3.6.6 has five lines. 3.6.6 QED

1. P ≡ Q 2. P 3. (P ⊃ Q) ∙ (Q ⊃ P) 4. P ⊃ Q 5. Q

Premise Premise 1, Equiv 3, Simp 4, 2, MP

Since our reasoning with biconditionals often parallels (with important differences) our reasoning with conditionals, it is useful to shorten some derivations by adopting some rules governing the biconditional that are parallel to those governing the conditional. Here are three rules of inference and four rules of equivalence. The validity of these rules of inference and the equivalence of the rules of equivalence

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are easily demonstrated using truth tables; tables for BDM and BInver appear in the appendix. Biconditional Rules of Inference Biconditional Modus Ponens (BMP) α ≡β α / β Biconditional Modus Tollens (BMT) α ≡β ∼α / ∼β Biconditional Hypothetical Syllogism (BHS) α ≡β β ≡ γ / α ≡ γ Biconditional Rules of Equivalence Biconditional De Morgan’s Law (BDM) → ∼α ≡ β ∼(α ≡ β) ← Biconditional Commutativity (BCom) → β ≡α α ≡ β ← Biconditional Inversion (BInver) → ∼α ≡ ∼β α ≡ β ← Biconditional modus ponens (BMP) is a rule of

inference of PL, parallel to modus ponens, but used with a biconditional.

Biconditional modus tollens (BMT)is a rule

of inference of PL. Unlike modus tollens, use BMT when you have the negation of the term which precedes the biconditional.

Biconditional commutativity (BCom)

is a rule of equivalence of PL which allows you to switch the order of formulas around a biconditional.

Biconditional Association (BAssoc) → (α ≡ β) ≡ γ α ≡ (β ≡ γ) ← Most of the biconditional rules are fairly intuitive if you have mastered the material in sections 3.1–3.5. Biconditional modus ponens shortens the inference at 3.6.6, as we see at 3.6.7. 3.6.7 QED

1. P ≡ Q 2. P 3. Q

Premise Premise 1, 2, BMP

Like modus tollens, the second, or minor, premise of biconditional modus tollens consists of the negation of one side of the first premise. But in BMT, we negate the left side of the biconditional. We can use biconditional commutativity in combination with BMT if we have the negation of the right side of the first premise, as in 3.6.8. 3.6.8 1. P ≡ ∼Q 2. ∼ ∼Q / ∼P 3. ∼Q ≡ P 1, BCom 4. ∼P 3, 2, BMT QED

3 . 6 : T h e B i con d i t i onal   1 6 1

Biconditional hypothetical syllogism facilitates natural chains of reasoning, ones that we often see in mathematics. It can be used effectively in combination with biconditional inversion, in which both sides of a biconditional are negated though they do not switch sides. 3.6.9 1. P ≡ ∼Q 2. Q ≡ ∼R 3. ∼Q ≡ ∼ ∼R 4. ∼Q ≡ R 5. P ≡ R QED

/P≡R 2, BInver 3, DN 1, 4, BHS

Biconditional De Morgan’s law allows you to take a negation inside brackets in which the main operator is a biconditional. But be careful with BDM. When you move the negation inside the parentheses, only one side of the resulting biconditional gets negated. 3.6.10 1. P ≡ (Q ≡ ∼R) 2. ∼P 3. ∼(Q ≡ ∼R) 4. ∼Q ≡ ∼R 5. Q ≡ R QED

/Q≡R 1, 2, BMT 3, BDM 4, BInver

Lastly, biconditional association helps with propositions containing multiple biconditionals, often in combination with other biconditional rules. 3.6.11 1. P ≡ (Q ≡ R) 2. ∼R ∙ ∼Q 3. (P ≡ Q) ≡ R 4. R ≡ (P ≡ Q) 5. ∼R 6. ∼(P ≡ Q) 7. ∼P ≡ Q 8. Q ≡ ∼P 9. ∼Q ∙ ∼R 10. ∼Q 11. ∼ ∼P 12. P QED

/P 1, BAssoc 3, BCom 2, Simp 4, 5, BMT 6, BDM 7, BCom 2, Com 9, Simp 8, 10, BMT 11, DN

Summary The biconditional rules of this section facilitate many inferences, though they do not allow you to convert biconditionals into other operators as Equiv does. They supplement, rather than supplant, the earlier rule.

Biconditional hypothetical syllogism (BHS) is a rule of

inference of PL, and works just like ordinary hypothetical syllogism.

Biconditional inversion (BInver)is a rule of

equivalence of PL. To use BInver, negate both sides of the biconditional.

Biconditional De Morgan’s lawis a rule of

equivalence of PL. When bringing a negation inside parentheses with BDM, make sure to negate only the formula on the left side of the biconditional.

Biconditional Association (BAssoc)is a rule of

equivalence of PL which allows you to regroup propositions with two biconditionals.

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KEEP IN MIND

We saw three new rules of inference in this section: biconditional modus ponens (BMP), biconditional modus tollens (BMT), and biconditional hypothetical syllogism (BHS). We saw four new rules of equivalence in this section: biconditional De Morgan’s law (BDM), biconditional commutativity (BCom), biconditional inversion (BInver), and biconditional association (BAssoc). BMT uses the negation of the left side of the biconditional. BHS often must be set up properly with BCom. BDM does not require changing an operator, only the punctuation. To use BInver, either add one negation to each side of a biconditional or remove one from each side. It is important, especially in future sections, not to confuse the biconditional rules with the parallel rules governing the conditional.

Rules Introduced Rules of Inference: Biconditional Modus Ponens (BMP) α≡β α / β

Biconditional Modus Tollens (BMT) α≡β ∼α / ∼β

Biconditional Hypothetical Syllogism (BHS) α≡β β ≡ γ / α ≡ γ

Rules of Equivalence: Biconditional De Morgan’s Law (BDM)

→ ∼α ≡ β ∼(α ≡ β) ←

Biconditional Commutativity (BCom)

→ β≡α α ≡ β ←

Biconditional Inversion (BInver)

→ ∼α ≡ ∼β α ≡ β ←

Biconditional Association (BAssoc) → (α ≡ β) ≡ γ α ≡ (β ≡ γ) ←

3 . 6 : T h e B i con d i t i onal   1 6 3

TELL ME MORE  Are the biconditional rules needed in our system of inference? See 6.4: Metalogic.

EXERCISES 3.6a Derive the conclusions of each of the following arguments using the eighteen standard rules and the new rules for the biconditional. Compare your derivations to those done in 3.4b without these new rules. 1. (3.4b.3) 1. ∼(P ≡ ∼Q  ) 2. P

/Q

2. (3.4b.4) 1. ∼I ∨ J 2. J ≡ K 3. (I ∙ L) ∨ (I ∙ M) / K 3. (3.4b.8)

1. E ≡ F 2. ∼(G ∨ E)

/ ∼F

4. (3.4b.12)

1. D ≡ E 2. (E ∨ F) ⊃ G 3. ∼(G ∨ H)

/ ∼D

5. (3.4b.20)

1. P ≡ ∼Q 2. P ∨ R 3. Q

/R

6. (3.4b.27)

1. (P ≡ Q  ) ∨ P

/ P ∨ ∼Q

7. (3.4b.29)

1. (P ≡ Q  ) ∨ ∼P

/P⊃Q

8. (3.4b.31)

1. (S ≡ T) ∙ ∼U 2. ∼S ∨ (∼T ∨ U)

/ ∼S

9. (3.4b.39)

1. P ≡ (Q ∙ R) 2. S ⊃ P 3. T ⊃ P 4. ∼S ⊃ T

/Q

10. (3.4b.40) 1. ∼(P ≡ ∼Q  ) 2. P ⊃ R 3. Q ∨ R

/R

1 6 4    C h apter 3  In f erence i n P ropos i t i onal L og i c

EXERCISES 3.6b Derive the conclusions of each of the following arguments using the rules of inference and equivalence, including the biconditional rules. 1. 1. A ≡ B 2. ∼B / ∼A 2. 1. ∼(E ≡ F) 2. F

/ ∼E

3. 1. G ≡ H 2. ∼H ≡ ∼I

/G≡I

4. 1. J ≡ K 2. K ≡ ∼L

/ L ≡ ∼J

5. 1. M ≡ (N ≡ O) 2. ∼O / ∼M ≡ N 6. 1. ∼(S ≡ T) 2. ∼(T ≡ U) / S ≡ U 7. 1. X ≡ (∼Y ∨ Z) 2. X ∙ ∼Z / ∼Y 8. 1. (A ≡ B) ≡ C 2. ∼B / ∼A ≡ C 9. 1. ∼[D ≡ (E ∙ F)] 2. ∼F /D 10. 1. (G ≡ H) ⊃ H 2. ∼H /G 11. 1. L ∙ M 2. M ≡ N

/L≡N

12. 1. (P ≡ Q  ) ∙ (P ∨ R) 2. ∼R /Q 13. 1. W ≡ (X ∨ Y) 2. Y ∨ Z 3. ∼W /Z 14. 1. (P ≡ Q  ) ⊃ ∼(R ≡ ∼S) 2. ∼(R ≡ S) / ∼P ≡ Q 15. 1. ∼P ≡ (Q ∙ R) 2. ∼Q /P

3 . 6 : T h e B i con d i t i onal   1 6 5

16. 1. ∼(P ≡ Q  ) 2. ∼(Q ≡ R) 3. ∼(R ≡ ∼S)

/S≡P

17. 1. P ⊃ (Q ≡ R) 2. ∼(P ⊃ ∼R) / Q 18. 1. P ≡ (Q ≡ R) 2. S ≡ (R ≡ T) / (P ≡ Q  ) ≡ (S ≡ T) 19. 1. ∼[A ≡ (B ≡ C)] 2. C / A ≡ ∼B 20. 1. D ≡ (E ∙ F) 2. ∼F / ∼D 21. 1. J ≡ K 2. ∼(L ≡ K) 3. M ≡ J

/ ∼L ≡ M

22. 1. ∼[P ≡ (Q ≡ R)] 2. P ∙ ∼R /Q 23. 1. S ≡ T 2. ∼T ≡ U 3. W ≡ ∼U 4. W ≡ ∼S / ∼S 24. 1. X ⊃ (Y ≡ Z) 2. X ≡ ∼Z 3. ∼Z / ∼Y 25. 1. (P ∙ Q  ) ≡ R 2. P ≡ S 3. R /S∙Q 26. 1. (P ∨ Q  ) ⊃ (R ≡ S) 2. ∼R ∙ S / ∼P ∙ ∼Q 27. 1. (P ≡ Q  ) ⊃ R 2. (Q ≡ S) ⊃ R 3. ∼R /P≡S 28. 1. (P ∙ Q  ) ≡ R 2. (Q ⊃ R) ⊃ S / P ⊃ S 29. 1. P ⊃ (Q ≡ R) 2. ∼Q ≡ S 3. R ≡ S / ∼P

1 6 6    C h apter 3  In f erence i n P ropos i t i onal L og i c

30. 1. P ≡ Q 2. ∼Q ≡ R 3. R ≡ P

/S

31. 1. (A ∙ B) ≡ C 2. (D ∙ ∼A) ∨ (D ∙ ∼B) 3. (C ≡ D) ≡ E / ∼E 32. 1. (J ∙ K) ≡ L 2. J ≡ M 3. K ≡ N 4. M ≡ N 5. M ∨ N

/L

33. 1. ∼X ∨ Y 2. X ∨ ∼Y 3. (Z ≡ A) ⊃ ∼(X ≡ Y) / ∼Z ≡ A 34. 1. ∼P ≡ Q 2. Q ≡ R 3. (R ∙ S) ≡ T 4. S ∙ ∼T

/P

35. 1. P ≡ (Q ∙ ∼R) 2. ∼S ≡ P 3. S ∙ ∼R

/Q≡R

36. 1. ∼P ≡ Q 2. ∼Q ≡ R 3. P ⊃ S 4. ∼R ⊃ S

/S

37. 1. B ≡ (C ∙ D) 2. E ≡ C 3. ∼D ≡ ∼E

/B≡E

38. 1. (F ∨ G) ⊃ H 2. (I ∨ J) ⊃ ∼H 3. ∼I ⊃ F

/ ∼(F ≡ I)

39. 1. P ∨ (Q ∙ R) 2. ∼(P ∙ Q  )

/ ∼(P ≡ Q  )

40. 1. P ≡ (Q ∨ R) 2. R ≡ S 3. Q ⊃ R

/P≡S

3 . 6 : T h e B i con d i t i onal   1 6 7

EXERCISES 3.6c Translate each of the following paragraphs into arguments written in PL. Then, derive the conclusions of the arguments using the rules of inference and equivalence. 1. Edye is patient when, and only when, she is not sleepy. She is sleepy when, and only when, her children are not happy. So, Edye’s children are happy when, and only when, Edye is patient. 2. Gustavo plays tennis if, and only if, he runs. But Gustavo doesn’t run. So, if Gustavo plays tennis, then Martians have landed on Earth. 3. Aardvarks eat ants just in case they don’t drink beer. Aardvarks drink beer just in case they do not chase chickens. But aardvarks do chase chickens. So, they eat ants. 4. Doug’s playing golf entails his eating a hearty dinner if, and only if, he either plays with Bob or he doesn’t eat at home. But it’s not the case that if Doug eats at home, then he plays with Bob. So, Doug plays golf but does not eat a hearty dinner. 5. Emily studies in Rome if, and only if, it is not the case that she prefers classes on campus and her funding does not fall through. Her preferring classes on campus does not entail that her funding falls through. So, she does not study in Rome. 6. I’ll work in the supermarket this summer just in case I need money for a new guitar. It’s not the case that I need money for a new guitar if, and only if, my band gets back together. My band gets back together if, and only if, the drummer drops out and the guitarist transfers back home. Neither the guitarist transfers back home nor the singer breaks up with her girlfriend. So, I will work in the supermarket this summer. 7. Isla stays in to study if, and only if, Christine goes to the party just in case Mercedes does not go to the movie. Either Mercedes goes to the movie or Kwadwo doesn’t hang around reading Wittgenstein. Kwadwo hangs around reading Wittgenstein if, and only if, Hunter is busy working on his paper. It’s not the case that Hunter’s being busy with his paper entails that Christine goes to the party. So, Isla stays in to study. 8. Genesis does research on Hobbes if, and only if, she gets a grant or finds other money for her work. She does not do research on Hobbes if, and only if, she takes a different job. So, if she gets a grant, then she will not take a different job.

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9. We are not free if, and only if, our wills are determined or our bodies are constrained. But we are free. So, if our wills are determined, then Leibniz is a libertarian. 10. We have moral responsibilities if, and only if, it is not the case that our wills are free just in case there are souls. But we do not have moral responsibilities and our wills are not free. So, there are no souls. 11. I am not mortal if, and only if, the self is a conceptual construct. I am mortal just in case either my body dies or the self is something physical. So, the self is a conceptual construction if, and only if, my body does not die and the self is not physical. 12. God is perfect if, and only if, there is no evil, just in case human intelligence is limited. If God is perfect, then there is no evil. If God is not perfect, then there is evil. So, human intelligence is limited. 13. Zombies are possible if, and only if, we are conscious just in case mental states are not physical. But mental states are physical and zombies are not possible. So, we are conscious. 14. I am altruistic, just in case I am not just if, and only if, I use the ring of Gyges. But I am just. So, I do not use the ring of Gyges if, and only if, I am altruistic. 15. Either arithmetic is synthetic or not a priori, if, and only if, it is not analytic. Arithmetic is synthetic just in case seven and five are not contained in twelve. Seven and five are contained in twelve if, and only if, arithmetic is not a priori. So, arithmetic is not analytic. 16. If color is real if, and only if, mass is, then philosophy is not independent of science. If philosophy is not independent of science, then there are synthetic a priori claims. If there are synthetic a priori claims, then arithmetic is synthetic. Either arithmetic is not synthetic or philosophy is independent of science. Color is not real. So, mass is.

3.7: CONDITIONAL PROOF There are three derivation methodsin this book. We have been working with direct proof. In this section we explore conditional proof. In section 3.9, we will look at indirect proof.

There are four more sections in this chapter on natural deductions in PL. This section introduces a derivation method, called conditional proof, that allows us to simplify many long, difficult proofs. It will also allow us to derive logical truths, or theorems of our system of logic, as we will see in the next section. In section 3.9, we will examine a third derivation method, indirect proof. At the end of the chapter, we will review our twenty-five rules and three derivation methods: conditional proof, indirect proof, and direct proof, the last of which is the method we have been using in this chapter so far.

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Conditional proof is useful when you want to derive a conditional conclusion. We assume the antecedent of the desired conditional, for the purposes of the derivation, taking care to indicate the presence of that assumption later. Consider the argument at 3.7.1, which has a conditional conclusion. 3.7.1

1. A ∨ B 2. B ⊃ (E ∙ D)

Conditional proof is a

derivation method useful for deriving conditional conclusions.

/ ∼A ⊃ D

Think about what would happen if we had the antecedent of the conditional conclusion, ‘∼A’, as another premise. First, we would be able to infer ‘B’ by DS with line 1. Then, since we would have ‘B’, we could use MP to infer ‘E ∙ D’ from line 2. Lastly, given ‘E ∙ D’ we could use Com and Simp to get ‘D’. So, ‘D’ would follow from ‘∼A’. The method of conditional proof formalizes this line of thought. Method of Conditional Proof 1. Indent, assuming the antecedent of your desired conditional. Justify the assumption by writing ‘ACP’, for ‘assumption for conditional proof ’. Use a vertical line to set off the assumption from the rest of your derivation. 2. Derive the consequent of desired conditional within an indented sequence. Continue the vertical line. Proceed as you would normally, using any propositions already established. 3. Discharge (un-indent). Write the first line of your assumption, a ‘⊃’, and the last line of the indented sequence. Justify the un-indented line with CP, and indicate the indented line numbers. The line of thought we took discussing 3.7.1 is thus formalized by using the indented sequence you see at 3.7.2. 3.7.2 1. A ∨ B 2. B ⊃ (E ∙ D) / ∼A ⊃ D 3. ∼A ACP Suppose ∼A. 4. B 1, 3, DS 5. E ∙ D 2, 4, MP 6. D ∙ E 5, Com 7. D 6, Simp Then D would follow. 8. ∼A ⊃ D 3–7, CP So, if ∼A were true, then D would be. QED

The purpose of indenting and using a vertical line is to create an indented sequence that marks the scope of your assumption. Any statements you derive within the scope of an assumption are not derived only from the premises, as in all the direct derivations we have done until now. They are derived from the premises with an additional

An indented sequence is a series of lines in a

derivation that do not follow from the premises directly, but only with a further assumption, indicated on the first line of the sequence.

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assumption like the one we made at line 3. Thus, after you discharge an assumption, you may not use statements derived within the scope of that assumption later in the proof. We could have discharged our assumption in 3.7.2 after any number of steps in the indented sequence: ‘∼A ⊃ (D ∙ E)’; ‘∼A ⊃ (E ∙ D)’; ‘∼A ⊃ B’; and even ‘∼A ⊃ ∼A’ are all valid inferences given the premises. But none of the consequents of those conditional statements are themselves validly inferred from the premises without assuming ‘∼A’. Conditional proof makes many of the derivations we have done earlier using the direct method significantly easier. To see a striking difference between the direct and conditional derivation methods, compare an argument proved directly, in 3.7.3, and conditionally, in 3.7.4. 3.7.3 Direct Method 1. (P ⊃ Q) ∙ (R ⊃ S) 2. P ⊃ Q 3. ∼P ∨ Q 4. (∼P ∨ Q) ∨ ∼R 5. ∼P ∨ (Q ∨ ∼R) 6. (R ⊃ S) ∙ (P ⊃ Q) 7. (R ⊃ S) 8. ∼R ∨ S 9. (∼R ∨ S) ∨ ∼P 10. ∼P ∨ (∼R ∨ S) 11. [∼P ∨ (Q ∨ ∼R)] ∙ [∼P ∨ (∼R ∨ S)] 12. ∼P ∨ [(Q ∨ ∼R) ∙ (∼R ∨ S)] 13. ∼P ∨ [(∼R ∨ Q) ∙ (∼R ∨ S)] 14. ∼P ∨ [∼R ∨ (Q ∙ S)] 15. P ⊃ [∼R ∨ (Q ∙ S)] 16. P ⊃ [R ⊃ (Q ∙ S)] 17. (P ∙ R) ⊃ (Q ∙ S) QED 3.7.4 Conditional Method 1. (P ⊃ Q) ∙ (R ⊃ S) 2. P ∙ R 3. P ⊃ Q 4. P 5. Q 6. (R ⊃ S) ∙ (P ⊃ Q) 7. R ⊃ S 8. R ∙ P 9. R 10. S 11. Q ∙ S 12. (P ∙ R) ⊃ (Q ∙ S) QED

/ (P ∙ R) ⊃ (Q ∙ S) 1, Simp 2, Impl 3, Add 4, Assoc 1, Com 6, Simp 7, Impl 8, Add 9, Com 5, 10, Conj 11, Dist 12, Com 13, Dist 14, Impl 15, Impl 16, Exp

/ (P ∙ R) ⊃ (Q ∙ S) ACP 1, Simp 2, Simp 3, 4, MP 1, Com 6, Simp 2, Com 8, Simp 7,9, MP 5, 10, Conj 2–11, CP

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Not only is the conditional method often much shorter, as in this case, it is also conceptually much easier. In this case, to see that one has to add what one needs at lines 4 or 9 in the direct version is not easy. The conditional proof proceeds in more obvious ways. You can use CP repeatedly within the same proof, whether nested or sequentially. 3.7.5 demonstrates a nested use of CP. 3.7.5 1. P ⊃ (Q ∨ R) 2. (S ∙ P) ⊃ ∼Q 3. S ⊃ P 4. S 5. P 6. Q ∨ R 7. S ∙ P 8. ∼Q 9. R 10. S ⊃ R 11. (S ⊃ P) ⊃ (S ⊃ R) QED

/ (S ⊃ P) ⊃ (S ⊃ R) ACP Now we want S ⊃ R. ACP Now we want R. 3, 4, MP 1, 5, MP 4, 5, Conj 2, 7, MP 6, 8, DS 4–9, CP 3–10, CP

Within an indented sequence, you can use any formula in which that sequence is embedded. So, in the sequence following line 4, you can use lines 1 and 2 as well as line 3. But once you discharge your assumption, as I do at line 10, any conclusions of that indented sequence are also put off limits. At line 10, the only lines I can use are lines 1–3. If you need any of the propositions derived within an indented sequence after you discharge the relevant assumption, you have to rederive them. Given this restriction, it is often useful to do as much work as you can before making an assumption. 3.7.6 shows how we can use CP sequentially to prove biconditionals. In such cases, you want ‘α ≡ β’ which is logically equivalent to ‘(α ⊃ β) ∙ (β ⊃ α)’. This method is not always the best one, but it is usually a good first thought. 3.7.6 1. (B ∨ A) ⊃ D 2. A ⊃ ∼D 3. ∼A ⊃ B 4. B 5. B ∨ A 6. D 7. B ⊃ D 8. D 9. ∼ ∼D 10. ∼A 11. B 12. D ⊃ B 13. (B ⊃ D) ∙ (D ⊃ B) 14. B ≡ D QED

/B≡D ACP 4, Add 1, 5, MP 4–6 CP ACP 8, DN 2, 9, MT 3, 10, MP 8–11 CP 7, 12, Conj 13, Equiv

A nested sequence is an assumption within another assumption.

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Notice that we start the second sequence at line 8 intending to derive ‘B’. We already have a ‘B’ in the proof at line 4. But that ‘B’ was a discharged assumption, and is off limits after line 6. Method for Proving a Biconditional Conclusion Assume α, derive β, discharge. Assume β, derive α, discharge. Conjoin the two conditionals. Use material equivalence to yield the biconditional. You may also use CP in the middle of a proof to derive statements that are not your main conclusion, as in 3.7.7. 3.7.7 1. P ⊃ (Q ∙ R) 2. (P ⊃ R) ⊃ (S ∙ T) 3. P 4. Q ∙ R 5. R ∙ Q 6. R 7. P ⊃ R 8. S ∙ T 9. T ∙ S 10. T QED

/T ACP 1, 3, MP 4, Com 5, Simp 3–6, CP 2, 7, MP 8, Com 9, Simp

Such uses are perhaps not common. But you can feel free to use a conditional proof at any point in a derivation if you need a conditional claim.

Summary We now have two derivation methods, a direct method and a conditional method. In direct proofs we ordinarily construct our derivations by looking at the premises and seeing what we can infer. Sometimes we work backward from our conclusions, figuring out what we need, but that kind of work is done on the side, not within a proof. When setting up conditional proofs, in contrast, we generally look toward our desired conditionals, assuming the antecedent of some conditional we want, rather than looking at what we have in the premises. We hope that our assumptions will work with our premises, of course, and we proceed, after our assumptions, to use the ordinary, direct methods. But in setting up our indented sequences, we focus on what we want, thinking about how our assumption will be discharged. As it was used in this section, the conditional derivation method is used within a direct proof, as a subsequence of formulas. In the next section, we’ll do some proofs completely by the conditional derivation method. In the following section, we’ll look at a third and final derivation method, indirect proof.

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KEEP IN MIND

When you want to derive a conditional conclusion, you can assume the antecedent of the conditional, taking care to indicate the presence of that assumption later. Conditional proofs are especially useful when the conclusion of the argument is a conditional or a biconditional.For biconditionals, assume one side to derive the other side and discharge; do a second CP for the reverse (if necessary); then conjoin the two conditionals. Indent and use a vertical line to mark the scope of an assumption. After you discharge an assumption, you may not use statements derived within the scope of that assumption later in the proof. It is often useful to do what you can with a proof before making an assumption so that the propositions you derive are available after you discharge your assumption. You can use conditional proof at any point during a proof and anytime you need a conditional statement, not just when the conclusion of the argument is a conditional.

EXERCISES 3.7a Derive the conclusions of each of the following arguments using the method of conditional proof where appropriate. 1. 1. (A ∨ C) ⊃ D 2. D ⊃ B

/A⊃B

2. 1. X ⊃ Y 2. Y ⊃ Z

/ X ⊃ (Y ∙ Z)

3. 1. R ⊃ ∼O 2. ∼R ⊃ [S ∙ (P ∨ Q  )] / O ⊃ (P ∨ Q  ) 4. 1. (E ∨ F) ∨ G 2. ∼F / ∼E ⊃ G 5. 1. L ⊃ M 2. L ⊃ N 3. (M ∙ N) ⊃ O

/L⊃O

6. 1. Q ⊃ (∼R ∙ S)

/ R ⊃ ∼Q

7. 1. ∼M ⊃ N 2. L ⊃ ∼N / ∼L ∨ M 8. 1. I ⊃ H 2. ∼I ⊃ J 3. J ⊃ ∼H

/ J ≡ ∼H

9. 1. ∼M ∨ N 2. P

/ (M ∨ ∼P) ⊃ (O ∨ N)

1 74    C h apter 3  In f erence i n P ropos i t i onal L og i c

10. 1. ∼(I ∨ ∼K) 2. L ⊃ J

/ (I ∨ L) ⊃ (K ∙ J)

11. 1. E ⊃ ∼(F ⊃ G) 2. F ⊃ (E ∙ H)

/E≡F

12. 1. H ∨ (I ∨ J) 2. H ⊃ K 3. J ⊃ K

/ ∼I ⊃ K

13. 1. (P ∨ Q  ) ⊃ R 2. S ⊃ ∼R 3. S ∨ P

/Q⊃P

14. 1. A ⊃ (B ≡ C) 2. ∼C

/ B ⊃ ∼A

15. 1. D ≡ E 2. F ∨ D

/ ∼E ⊃ F

16. 1. W ⊃ T 2. X ⊃ (T ∨ W) 3. X ∨ S

/T∨S

17. 1. P ∨ Q 2. ∼P ∨ ∼Q / ∼(P ≡ Q  ) 18. 1. R ⊃ (S ∨ W) 2. R ⊃ (T ∨ W) 3. ∼(W ∨ X)

/ R ⊃ (S ∙ T)

19. 1. P ⊃ (Q ∙ R) 2. (Q ∨ S) ⊃ ∼P / ∼P 20. 1. A ⊃ [(D ∨ B) ⊃ C] / A ⊃ (D ⊃ C) 21. 1. Z ⊃ ∼Y

/ (X ∙ Y) ⊃ (Z ⊃ W)

22. 1. ∼(U ∨ V) 2. W ⊃ X

/ (U ∨ W) ⊃ (V ⊃ X)

23. 1. E ⊃ (F ⊃ G) 2. ∼(I ∨ ∼E) 3. G ⊃ H

/F⊃H

24. 1. (T ⊃ ∼Q  ) ∙ ∼W 2. ∼Q ⊃ [(W ∨ S) ∙ (W ∨ T)] 3. ∼T ∨ (S ⊃ X) /T⊃X 25. 1. M ⊃ (∼K ∨ N) 2. N ⊃ L 3. M ∨ (K ∙ ∼L)

/ M ≡ (K ⊃ L)

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26. 1. A ≡ (B ∙ ∼C) 2. C ⊃ (D ∙ E) 3. (D ∨ F) ⊃ G

/ (∼A ∙ B) ⊃ G

27. 1. (H ∨ J) ⊃ K 2. (I ∨ L) ⊃ M

/ (H ∨ I) ⊃ (K ∨ M)

28. 1. J ⊃ K 2. L ⊃ ∼K 3. ∼J ⊃ M 4. N ⊃ ∼O 5. ∼N ⊃ I 6. ∼O ⊃ L

/M∨I

29. 1. D ⊃ (F ∨ G) 2. E ⊃ (F ∨ H) 3. I ⊃ ∼F 4. ∼H

/ (D ∨ E) ⊃ (I ⊃ G)

30. 1. (X ⊃ Y) ⊃ Z 2. (∼X ∨ Y) ≡ (A ∨ B) 3. ∼B ⊃ (D ⊃ A) / ∼Z ⊃ ∼D 31. 1. (K ∙ ∼L) ⊃ ∼M 2. M ∨ N 3. M ∨ O 4. ∼(N ∙ O) / ∼K ∨ L 32. 1. L ⊃ M 2. O ⊃ M 3. ∼N ⊃ (L ∨ O) 4. (M ∙ N) ⊃ K 5. ∼(J ⊃ K)

/ ∼M ≡ N

33. 1. I ⊃ (J ∨ K) 2. ∼J ∨ (∼I ∨ L) 3. L ⊃ ∼I

/I⊃K

34. 1. (A ⊃ B) ⊃ (C ⊃ B) 2. A ⊃ ∼(B ⊃ D) 3. (A ⊃ ∼D) ⊃ C / B 35. 1. A ⊃ (∼B ∨ C) 2. ∼A ⊃ (B ∨ C) 3. C ⊃ ∼C / ∼(A ≡ B) 36. 1. (A ∙ B) ⊃ (C ∙ D) 2. (A ∙ C) ⊃ (E ∨ ∼D) 3. F ⊃ (E ⊃ G) / A ⊃ [B ⊃ (F ⊃ G)]

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37. 1. (P ∙ Q  ) ∨ (R ∙ S) 2. ∼P ∨ T 3. ∼Q ∨ W 4. T ⊃ (W ⊃ S)

/ ∼R ⊃ S

38. 1. X ⊃ [(T ∨ W) ⊃ S] 2. (W ⊃ S) ⊃ (Y ⊃ R) 3. ∼Z ⊃ ∼R

/ X ⊃ (Y ⊃ Z)

39. 1. ∼R ⊃ S 2. S ⊃ (R ∨ ∼P) 3. ∼(R ∨ P) ⊃ (Q ⊃ ∼S) / (P ∨ Q  ) ⊃ R 40. 1. J ≡ (L ∨ M) 2. (M ∨ J) ≡ N 3. (L ⊃ N) ⊃ (K ≡ ∼K)

/ L ≡ (N ∨ K)

EXERCISES 3.7b Translate each of the following paragraphs into arguments written in PL. Then, derive the conclusions of the arguments. 1. If Raul doesn’t play lacrosse, then he plays tennis. So, if Raul doesn’t play lacrosse, then he plays either tennis or soccer. 2. It is not the case that either Polly or Ramon takes out the trash. So, if Owen cleans his room, then Polly takes out the trash only if Quinn clears the table. 3. If Adams and Barnes are translators, then Cooper is a reviewer. Evans is an editor if either Cooper or Durning are reviewers. Hence, Adams being a translator is a sufficient condition for Barnes being a translator only if Evans is an editor. 4. If it’s not the case that there are frogs in the pond, then George will go swimming. So, if Eloise goes swimming and George does not, then either there are frogs in the pond or hornets in the trees. 5. If Kip does well on his report card, then he will get ice cream. If Kip doesn’t do well on his report card, then he’ll be jealous of his brother. So, Kip will either get ice cream or be jealous. 6. If Lisa goes to Arizona, then she’ll go to Colorado. If she goes to Boulder, Colorado, then she’ll go to Dragoon, Arizona. So, if she goes to Arizona and Boulder, then she’ll go to Colorado and Dragoon.

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7. If the train doesn’t come, then it is not the case that Shanti and Ricardo go to New York. So, If Ricardo goes to New York, then Shanti goes to New York only if the train comes. 8. If Justin goes to Ikea, then Luke doesn’t go. Either Luke goes to Ikea or Kate sleeps on the floor. If either Kate or Madeline sleeps on the floor, then Justin goes to Ikea. So, Justin goes to Ikea if, and only if, Kate sleeps on the floor. 9. If Aristotle’s Physics is right, then motion is goal-directed and everything has a telos. But if everything is goal-directed, then other planets are unlike Earth. So, if Aristotle’s Physics is right, then other planets are unlike Earth. 10. If nothing is worse for society than anarchy, then if people are mutually hostile, then we need a central authority. But we do not need a central authority. So, if nothing is worse for society than anarchy, then people are not mutually hostile. 11. If meanings are abstract objects or mental states, then if I believe that cats are robots, then cats are robots. But cats are not robots. So, if meanings are mental states, then I don’t believe that cats are robots. 12. If being a platonist entails rejecting empiricism, then Quine is not a platonist. Being a platonist entails being an apriorist. Not rejecting empiricism entails not being an apriorist. So, Quine is not a platonist. 13. If the common interest is imposed on individuals, then they are alienated or not self-determining. But people are self-determining. So, if people are not alienated, then the common interest is not imposed. 14. Either it is not the case that nothing is certain or we have unmediated access to our mental states. If we have unmediated access to our mental states and our basic beliefs are not secure, then either our mental states are potentially misleading or we lack mental states. But if we lack mental states, then our basic beliefs are secure. So, if nothing is certain and our basic beliefs are not secure, then we have unmediated access to our mental states, but they are potentially misleading. 15. Either some objects are beautiful or we impose cultural standards on artifacts. It’s not the case that some particular proportions are best. So, if some objects being beautiful entails that some particular proportions are best, then if something is aesthetically moving, then we impose cultural standards on artifacts. 16. If suicide is not legal, then we lack autonomy and the least powerful people do not have self-determination. If education is universal and free, then the least powerful people have self-determination. If only the privileged are educated, then suicide is not legal. Either education is universal and free or only the privileged are educated. So, suicide is legal if, and only if, the least powerful people have self-determination.

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3.8: LOGICAL TRUTHS

A theory is a set of sentences, called theorems . A formal theory is a set of sentences of a formal language.

To this point, all of our derivations have started with some assumptions, or premises. Even in direct proofs, our derivations have been, in a sense, conditional: on the assumption of such and such premises, a conclusion follows. In this section, we will use the method of conditional proof to prove theorems of logic, constructing derivations with no premises at all. Such derivations may look strange at first, but they can really be quite simple once you learn how to construct them. A theory is a set of sentences, called theorems. A formal theory is a set of sentences of a formal language. The provable statements of a logical system of inference are theorems, just as provable geometric statements are theorems of, say, Euclidean geometry. As we saw in section 2.5, a logical theory is characterized by the set of its logical truths. In PL, the logical truths are the same as the derivable theorems. Thus, PL can be identified with either the logical truths, defined semantically, or its theorems, the derivable propositions. Those propositions are true no matter what assumptions we make about the world, or whatever we take to be the content of our propositional variables. They thus can be proved without assuming any premises. So, there are two ways to show that a wff is a logical truth of PL. Semantically, we can show, using truth tables, whether any wff is a tautology or not. We just see whether it is true in all rows of the truth table. In this section, we see how we can prove theorems using any premises, and even without any premises. Since our theory is sound, any statement that is provable without premises is a tautology. One way to derive a theorem with no premises, which we are not using, is to adopt a deductive system that takes certain wffs as axioms. Some theories, including most nonlogical theories, are axiomatic. Axiomatic logical theories normally take a few tautologies as axioms or axiom schemas. In such a system, any sentence of the form of an axiom or schema can be inserted into a derivation with no further justification. In our logical system so far, we have had no way to construct a derivation with no premises. Now we can use conditional proof to derive logical truths without any premises. We can just start our derivation with an assumption, as in 3.8.1, which shows that ‘[(P ⊃ Q  ) ∙ P] ⊃ Q’ is a logical truth. 3.8.1 1. (P ⊃ Q) ∙ P ACP 2. P ⊃ Q 1, Simp 3. P ∙ (P ⊃ Q) 1, Com 4. P 3, Simp 5. Q 2, 4, MP 6. [(P ⊃ Q) ∙ P] ⊃ Q 1–5, CP

In other words, from the assumption that P entails Q  , and P, Q follows. Our conclusion is conditional, but holds without any further assumptions than the antecedent of that conditional. Note that the last line of 3.8.1 is further un-indented than the first line, since the first line is indented. Lines 1–5 are all based on at least one assumption. But line 6 requires no assumption. It is a theorem of logic, a logical truth.

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Many proofs of logical truths involve nesting conditional proofs, as the derivation 3.8.2 does in showing that ‘(P ⊃ Q  ) ⊃ [(Q ⊃ R) ⊃ (P ⊃ R)]’ is a logical truth. 3.8.2 1. P ⊃ Q 2. Q ⊃ R 3. P ⊃ R 4. (Q ⊃ R) ⊃ (P ⊃ R) 5. (P ⊃ Q) ⊃ [(Q ⊃ R) ⊃ (P ⊃ R)] QED

ACP ACP 1, 2, HS 2–3, CP 1–4, CP

Again, the conclusion is a conditional statement, but one that requires no premises for its derivability. It is another logical truth. You can check that the theorems at 3.8.1 and 3.8.2 are logical truths by constructing truth tables for them, or for any of the logical truths of this section. They will all be tautologies. Derivations of logical truths can look awkward when you are first constructing and considering them. Remember, the logical truth we prove in 3.8.2 is conditional, and doubly so: if P entails Q  , then if Q entails R, then P entails R. So, while we have demonstrated a logical truth out of thin air, the nature of that logical truth should make the process seem less magical. When the logical truth has nested conditionals, as 3.8.2 does, setting up the assumptions can require care. But such logical truths are often simple to derive once they are set up properly. Be especially careful not to use the assigned proposition in the proof. The conclusion is not part of the derivation until the very end. 3.8.3 shows that ‘[P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q  ) ⊃ (P ⊃ R)]’ is a logical truth, using three nested conditional sequences. ACP (to prove (P ⊃ Q) ⊃   (P ⊃ R)) 2. P ⊃ Q ACP (to prove (P ⊃ R)) 3. P ACP (to prove R) 4. Q ⊃ R 1, 3, MP 5. Q 2, 3, MP 6. R 4, 5, MP 7. P ⊃ R 3–6 CP 8. (P ⊃ Q) ⊃ (P ⊃ R) 2–7, CP 9. [P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)] 1–8, CP QED 3.8.3

1. P ⊃ (Q ⊃ R)

A trivial, or degenerate, instance of CP can prove one of the simplest logical truths, at 3.8.4. 3.8.4 1. P 2. P ⊃ P QED

ACP CP, 1

Notice that the CP at 3.8.4 has only one line. The second line discharges the assumption; since the first and last line are the same, the antecedent and consequent of

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The law of the excluded middle is that any claim

of the form α ∨ ~α is a tautology, a logical truth of PL.

the discharging formula are the same. It should be no surprise that a statement entails itself. But a use of Impl and Com on that formula yields an instance of the standard form of the law of excluded middle, at 3.8.5, one of the characteristic logical truths. 3.8.5 P ∨ ∼P

The metalinguistic version of the law of the excluded middle is called bivalence, as we saw in section 2.3. Bivalence, that every proposition is either true or false, and not both, underlies the two-valued semantics of PL. (The middle that is excluded is any truth value other than truth or falsity.) Bivalence has long been a controversial claim. Consider the problem of future contingents; Aristotle’s example is, ‘there will be a sea battle tomorrow’. Since we do not know today whether there will be a sea battle tomorrow, we don’t know whether the statement is true or false and seem unable to assert either. We surely could look back on the day after tomorrow to assign a truth value to the claim, but as of today, it may not even have a truth value. Though even this simple logical truth is controversial, our uses of CP do not raise these problems. The problem comes from the semantics of PL, since every instance of the law of excluded middle is a tautology.

A Common Error to Avoid in Using CP to Derive Logical Truths A common error made by students just learning to use CP is to include the desired conclusion as a numbered line in the argument. This can be done in at least two ways, both wrong. In the first way, one puts the assumed formula as the first assumption for CP, as at 3.8.6 (reusing the example at 3.8.3 to show the error). 3.8.6

1. [P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)]

ACP (but not a good one!)

Although one can assume anything for CP, starting the CP in this way is unproductive for proving this logical truth. Remember, when you discharge an assumption for CP, the first line becomes the antecedent of the discharging formula; imagine what that formula would look like on the assumption at 3.8.6! Students sometimes follow errant assumptions like the one at 3.8.6 with assuming, further, the antecedent of that very formula, as at 3.8.7. 3.8.7 1. [P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)] 2. P ⊃ (Q ⊃ R)

ACP (again not useful!) ACP

Now, one could derive the consequent of the formula at line 1, using MP, as at 3.8.8, but then look at the resulting discharged formula. 3.8.8 1. [P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)] ACP (still not useful!) 2. P ⊃ (Q ⊃ R) ACP 3. (P ⊃ Q) ⊃ (P ⊃ R) 1, 2, MP 4. [P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)] 2–3, CP 5. {[P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)]} ⊃ {[P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)]} 1–4, CP

There is actually nothing wrong with the CP at 3.8.8, except that it doesn’t prove what one sets out to prove. Indeed, it merely long-windedly proves a complex instance of the law of the excluded middle, as at 3.8.4!

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The second version of the error, not properly setting up the CP, is to take the desired formula as a premise, as at 3.8.9. Then a CP can just prove the same formula you’ve already assumed. 3.8.9 1. [P ⊃ (Q ⊃ R)] ⊃[(P ⊃ Q) ⊃ (P ⊃ R)] 2. P ⊃ (Q ⊃ R) 3. (P ⊃ Q  ) ⊃ (P ⊃ R) 4. [P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)]

Premise ACP 1, 2, MP 2–3, MP

In 3.8.9, the conclusion certainly follows from the premise. Line 4 is just a restatement of line 1. Any statement entails itself! But what we want, as at 3.8.3, is a derivation of the logical truth with no premises at all. The assumptions of this subsection, from 3.8.6–3.8.9, are all errors to avoid in constructing conditional proofs to demonstrate logical truths. If you learn to set up your CPs correctly, indenting and assuming only the antecedent of your desired conditional, you can easily avoid these mistakes and the proofs tend to be quite simple.

Converting Ordinary Derivations into Logical Truths Until now, our derivations have required assumptions as premises. Such assumptions are often naturally interpreted as empirical claims, taken from observation, perhaps. Most of the premises of most of the arguments we have seen so far have been contingencies, though we can take any kind of premise, even a contradiction, as an assumption. Whatever their status, premises of arguments are generally not justified by the same methods that we use to justify our system of logic. Thus, our derivations before this section may be seen as not purely logical. They are not, as they stand, proofs of logical conclusions. They are merely derivations from assumed premises to conclusions. But for every valid argument requiring premises, we can create a proof of a purely logical truth. Neither of the premises of 3.8.10 are logical truths, for examples, being mere atomic wffs, both contingencies. 3.8.10 1. ∼A ⊃ B 2. ∼A

/B

But because the argument is an instance of a modus ponens, and thus valid, we can turn it into the logical truth at 3.8.11 or the logical truth at 3.8.12. 3.8.11 [(∼A ⊃ B) ∙ ∼A] ⊃ B 3.8.12 (∼A ⊃ B) ⊃ (∼A ⊃ B)

There are two options for constructing logical truths from any set of premises and a conclusion. On the first option, which I used at 3.8.11, conjoin all of the premises into one statement. Then write a conditional that takes the conjunction of the premises as the antecedent and the conclusion of the argument as the consequent. On the second option, which I used at 3.8.12, form a series of nested conditionals, using each premise as an antecedent and the conclusion as the final consequent. For a short argument, like 3.8.10, you can see the equivalence of the two methods by one use of exportation.

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The derivation of 3.8.10 is straightforward. The derivation of 3.8.11, as a logical truth with no premises, is just a bit more complicated, but it has the same technique at its core, a simple MP. The derivation is at 3.8.13. 3.8.13 1. (∼A ⊃ B) ∙ ∼A ACP 2. ∼A ⊃ B 1, Simp 3. ∼A ∙ (∼A ⊃ B) 1, Com 4. ∼A 3, Simp 5. B 2, 4, MP 6. [(∼A ⊃ B) ∙ ∼A] ⊃ B 1–5, CP QED

3.8.14 has more premises than 3.8.10, but can be converted to a logical truth in either of the ways just described. 3.8.14 1. P ⊃ (Q ∙ R) 2. R ⊃ S 3. T ∨ ∼S 4. ∼T / ∼(P ∨ S)

The first method for turning it into a logical truth, resulting in 3.8.15, is perhaps a little easier than the second, resulting in 3.8.16. It doesn’t matter how you group the four premises. Either way works. 3.8.15 {{[P ⊃ (Q ∙ R)] ∙ (R ⊃ S)} ∙ [(T ∨ ∼S) ∙ ∼T]} ⊃ ∼(P ∨ S) 3.8.16 [P ⊃ (Q ∙ R)] ⊃ {(R ⊃ S) ⊃ {(T ∨ ∼S) ⊃ [∼T ⊃ ∼(P ∨ S)]}}

The arguments we have been deriving so far, which include premises, are useful in applying logic to ordinary arguments. But the logical truths are the logician’s real interest, as they are the theorems of propositional logic. The transformations we have made at the object-language level can also be made at the metalinguistic level. Our rules of inference are written in a metalanguage. Any substitution instances of the premises in our rules of inference entail a substitution instance of the conclusion. We can similarly convert all of our rules of inference. 3.8.17 shows how modus ponens can be written as a single sentence of the metalanguage. 3.8.18 shows the same for constructive dilemma. 3.8.17

α⊃β α

/β

can be converted to: [(α ⊃ β) ∙ α] ⊃ β 3.8.18

α⊃β γ⊃δ α ∨ γ / β ∨ δ

can be converted to: {[(α ⊃ β) ∙ (γ ⊃ δ)] ∙ (α ∨ γ)} ⊃ (β ∨ δ)

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Any consistent substitution instance of these new forms, ones in which each metalinguistic variable is replaced by the same wffs of the object language throughout, will be a logical truth and provable in PL with no premises. All ten rules of equivalence we have been using can easily be turned into templates for constructing logical truths even more easily. We can just replace the metalinguis→ ’ with the object-language symbol ‘≡’, as I did for Impl and one version tic symbol ‘ ←  of DM in 3.8.19. 3.8.19 (α ⊃ β) ≡ (∼α ∨ β) ∼(α ∨ β) ≡ (∼α ∙ ∼β)

Again, any substitution instance of these forms will be a logical truth. These metalinguistic templates for logical truths are the kinds of rules one would adopt in an axiomatic system of logic. The templates are called axiom schemas. Such axiomatic theories can be constructed to derive the same logical theorems as our PL, to have the same strength as our system of logic, often with many fewer rules of inference or equivalence. Again, we are not using an axiomatic system, and we will retain all twenty-five rules, as well as the direct, conditional, and indirect derivation methods, the last of which is the subject of our next section.

Summary The primary goal of this section was to show you how to construct proofs of logical truths of PL, the theorems of propositional logic. Using conditional proof, we start by indenting and assuming the antecedents of a conditional logical truth and then derive the consequent. When we discharge our assumption, we have proven a formula of PL without any premises. The secondary goal of the section was to show the relation between our ordinary proofs so far, which contain premises, and the proofs of logical truths. Since every proof that assumes premises is convertible into a proof that does not, even the derivations that assume contingent premises can be seen as proofs of logical truths. KEEP IN MIND

The logical truths of PL are tautologies. Logical truths do not depend on any premises and can be proven with or without premises. Conditional proofs may be used to derive logical truths. We can construct logical truths from any set of premises and a conclusion in two ways: 1. Conjoin all premises and take the resulting conjunction as the antecedent of a complex conditional with the conclusion as the consequent. 2. Form a series of nested conditionals, using each premise as an antecedent and the conclusion as the final consequent.

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TELL ME MORE  • What is the role of logical truth in constructing logical theories? What is soundness for a logical system? See 6.4: Metalogic. • Of what do the laws of logic hold? See 6S.7: The Propositions of Propositional Logic. • How is logical truth related to necessity and other philosophical concepts? See 6S.9: Logical Truth, Analyticity, and Modality. • How do axiomatic inferential systems highlight the logical truths? See 6S.11: Axiomatic Systems. • Are there different views about logical truths? See 7S.11: Atomism and Color Incompatibility.

EXERCISES 3.8a Convert each of the following arguments to a logical truth, using either of the methods described above. 1. 1. ∼A ⊃ B 2. ∼B

/A

2. 1. ∼C ∨ D 2. C

/D

3. 1. E ∙ (F ∨ G) 2. ∼E /G 4. 1. ∼(H ∨ I) 2. J ⊃ I

/ ∼J

5. 1. K ∙ (∼L ∨ M) 2. L ⊃ ∼K / M 6. 1. N ⊃ (P ∙ Q  ) 2. ∼(O ∨ P) / ∼N

7. 1. R ⊃ S 2. S ⊃ T 3. ∼(T ∨ U)

/ ∼R

8. 1. V ⊃ W 2. ∼W ∨ X 3. V ∙ (Y ∙ Z) / X 9. 1. A ∨ (B ∙ C) 2. A ⊃ D 3. ∼(D ∨ E) /C 10. 1. F ⊃ G 2. H ⊃ F 3. H ∙ I

/ ∼G ⊃ I

EXERCISES 3.8b Use conditional proof to derive each of the following logical truths. 1. [A ∨ (B ∙ C)] ⊃ (A ∨ C) 2. [(A ⊃ B) ∙ C] ⊃ (∼B ⊃ ∼A)

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3. (O ∨ P) ⊃ [∼(P ∨ Q  ) ⊃ O] 4. [V ∙ (W ∨ X)] ⊃ (∼X ⊃ W) 5. [(P ∨ Q  ) ∨ (R ∨ S)] ⊃ [(R ∨ Q  ) ∨ (S ∨ P)] 6. [P ⊃ (Q ⊃ R)] ⊃ [(P ∙ ∼R) ⊃ ∼Q ] 7. [(P ∨ Q  ) ∨ R] ⊃ [∼P ⊃ (∼Q ⊃ R)] 8. (P ⊃ Q  ) ⊃ [(Q ⊃ S) ⊃ (∼S ⊃ ∼P)] 9. [(P ∨ Q  ) ∙ (P ∨ R)] ⊃ [∼P ⊃ (Q ∙ R)] 10. ∼[P ≡ (Q ∙ R)] ⊃ (∼P ⊃ Q  ) 11. ∼(P ≡ Q  ) ≡ [(P ∙ ∼Q  ) ∨ (Q ∙ ∼P)] 12. [(D ⊃ ∼E) ∙ (F ⊃ E)] ⊃ [D ⊃ (∼F ∨ G)] 13. [(H ⊃ I) ⊃ ∼(I ∨ ∼J)] ⊃ (∼H ⊃ J) 14. [(W ⊃ X) ∙ (Y ∨ ∼X)] ⊃ [∼(Z ∨ Y) ⊃ ∼W] 15. (P ≡ ∼Q  ) ⊃ ∼(P ∙ Q  ) 16. [P ⊃ (Q ⊃ R)] ⊃ [(Q ∙ ∼R) ⊃ ∼P] 17. [(P ≡ Q  ) ∙ ∼Q ] ⊃ (P ⊃ R) 18. [(P ∨ Q  ) ∙ ∼P] ⊃ [(Q ⊃ R) ⊃ R] 19. [P ⊃ (Q ∙ R)] ≡ [(∼P ∨ Q  ) ∙ (∼P ∨ R)] 20. [(P ⊃ Q  ) ∙ (P ⊃ R)] ⊃ {(S ⊃ P) ⊃ [S ⊃ (Q ∙ R)]} 21. [(R ∙ S) ⊃ U] ⊃ {∼U ⊃ [R ⊃ (S ⊃ T)]} 22. [(∼K ⊃ N) ∙ ∼(N ∨ L)] ⊃ [(K ⊃ L) ⊃ M] 23. [(D ∙ E) ⊃ (F ∨ G)] ≡ [(∼F ∙ ∼G) ⊃ (∼D ∨ ∼E)] 24. [(P ⊃ Q  ) ⊃ (R ⊃ P)] ⊃ [∼P ⊃ (P ≡ R)] 25. [(P ⊃ Q  ) ∙ (R ⊃ S)] ⊃ [(∼Q ∨ ∼S) ⊃ (∼P ∨ ∼R)]

3.9: INDIRECT PROOF We have seen two derivation methods, now, the direct and conditional. For ordinary derivations with assumptions, we can use either a direct or conditional proof. For logical truths, which need no assumptions, we use conditional proof. Our third and final method is called indirect proof. It is the formal version of what is commonly called a reductio ad absurdum, or just reductio, proof. Reductio ad absurdum is ‘reduction to the absurd’ in English. In reductio arguments, we assume a

Indirect proof, or reductio ad absurdum , is a third

method of derivation, along with the direct and conditional methods.

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premise, show that it leads to an unacceptable (or absurd) consequence, and then conclude the opposite of our assumption. Indirect proof, like conditional proof, is useful for proving logical truths. We can see the justification for indirect proof by considering the arguments 3.9.1, which we called explosion in section 3.5, and 3.9.2. 3.9.1 1. A ∙ ∼A 2. A 3. A ∨ B 4. ∼A ∙ A 5. ∼A 6. B QED

/B 1, Simp 2, Add 1, Com 4, Simp 3, 5, DS

3.9.2 1. B ⊃ (P ∙ ∼P) / ∼B 2. B ACP 3. P ∙ ∼P 1, 2, MP 4. P 3, Simp 5. P ∨ ∼B 4, Add 6. ∼P ∙ P 3, Com 7. ∼P 6, Simp 8. ∼B 5, 7, DS 9. B ⊃ ∼B 2–8, CP 10. ∼B ∨ ∼B 9, Impl 11. ∼B 10, Taut QED

The moral of 3.9.1 is that anything follows from a contradiction in PL. The moral of 3.9.2 is that if a statement entails a contradiction in PL, then its negation is provable. Indirect proof is based on these two morals, and it captures a natural style of inference: showing that some assumption leads to unacceptable consequences and then rejecting the assumption. To use an indirect proof, we assume the opposite of our desired conclusion and derive a contradiction. When we get the contradiction, then we can infer the negation of our assumption.

Method for Indirect Proof 1. Indent, assuming the opposite of what you want to conclude. 2. Derive a contradiction, using any wff. 3. Discharge the negation of your assumption.

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The last line of an indented sequence for indirect proof is always a contradiction. As in section 2.5, a contradiction, for the purposes of indirect proof, is any statement of the form α ∙ ∼α. The wffs listed in 3.9.3 are all contradictions. 3.9.3 P ∙ ∼P ∼ ∼P ∙ ∼ ∼ ∼P ∼(P ∨ ∼Q) ∙ ∼ ∼(P ∨ ∼Q)

We can assume any wff we want, for both CP and IP, by indenting and noting the assumption. But only certain assumptions will discharge in the desired way. For CP, we assume the antecedent of a desired conditional because when we discharge, the first line of the assumption becomes the antecedent of the resulting conditional. For IP, we always discharge the first line of the proof with one more tilde. Thus, if we wish to prove the negation of a formula, we can just assume the formula itself. 3.9.4 is a sample derivation using IP. At line 3, we are considering what would follow if the opposite of the conclusion is true. At line 6, we have found a contradiction, and so we discharge our assumption at line 7. 3.9.4 1. A ⊃ B 2. A ⊃ ∼B 3. A 4. B 5. ∼B 6. B ∙ ∼B 7. ∼A QED

/ ∼A AIP 1, 3, MP 2, 3, MP 4, 5, Conj 3–6, IP

Since the discharge step of an indirect proof requires an extra∼, we often need to use DN at the end of an indirect proof, as in 3.9.5. 3.9.5 1. F ⊃ ∼D 2. D 3. (D ∙ ∼E) ⊃ F /E 4. ∼E AIP 5. D ∙ ∼E 2, 4, Conj 6. F 3, 5, MP 7. ∼D 1, 6, MP 8. D ∙ ∼D 2, 7, Conj 9. ∼ ∼E 4–8, IP 10. E 9, DN QED

In addition to deriving simple statements and negations, the method of indirect proof is especially useful for proving disjunctions, as in 3.9.6. Assuming the negation of a disjunction leads quickly, by DM, to two conjuncts that you can simplify.

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3.9.6 1. ∼A ⊃ (B ⊃ C) 2. C ⊃ D 3. B 4. ∼(A ∨ D) 5. ∼A ∙ ∼D 6. ∼A 7. B ⊃ C 8. ∼D ∙ ∼A 9. ∼D 10. ∼C 11. C 12. C ∙ ∼C 13. ∼ ∼(A ∨ D) 14. A ∨ D QED

/A∨D AIP 4, DM 5, Simp 1, 6, MP 5, Com 8, Simp 2, 9, MT 7, 3, MP 11, 10, Conj 4–12, IP 13, DN

Indirect proof is compatible with conditional proof. Indeed, the structure of many mathematical proofs involves making a conditional assumption, and then assuming the opposite of a desired conclusion to get a contradiction. 3.9.7 is a formal example of exactly this procedure, nesting an IP within a CP. 3.9.7 1. E ⊃ (A ∙ D) 2. B ⊃ E / (E ∨ B) ⊃ A 3. E ∨ B ACP 4. ∼A AIP 5. ∼A ∨ ∼D 4, Add 6. ∼(A ∙ D) 5, DM 7. ∼E 1, 6, MT 8. B 3, 7, DS 9. ∼B 2, 7, MT 10. B ∙ ∼B 8, 9, Conj 11. ∼ ∼A 4–10, IP 12. A 11, DN 13. (E ∨ B) ⊃ A 3–12, CP QED

Essentially the same proof structure could have been used with a single assumption of the negation of the whole desired conclusion, as a single IP without using CP. I begin that alternative at 3.9.8.

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3.9.8 1. E ⊃ (A ∙ D) 2. B ⊃ E / (E ∨ B) ⊃ A 3. ∼[(E ∨ B) ⊃ A] AIP 4. ∼[∼(E ∨ B) ∨ A] 3, Impl 5. ∼ ∼(E ∨ B) ∙ ∼A 4, DM 6. (E ∨ B) ∙ ∼A 5, DN

Now the proof can proceed as it did from line 5 in 3.9.7. Either method is acceptable, though some find the nested structure of 3.9.7 both clearer and more conceptually useful. You can even nest indirect proofs within one another, though such measures are rarely warranted. When first learning to use IP, it is typical to try to invoke it as if it were magic, turning statements into their negations. Be very careful with your negations and with the structure of indirect proofs. DN always adds or subtracts pairs of consecutive tildes. IP always places a single tilde in front of the formula you assumed in the first line of your indented sequence after that sequence ends in a contradiction. Like conditional proof, the method of indirect proof is easily adapted to proving logical truths. To prove that ‘∼[(X ≡ Y) ∙ ∼(X ∨ ∼Y)]’ is a logical truth, as in 3.9.9, we again start with an assumption, the opposite of the theorem we wish to prove. 3.9.9 1. (X ≡ Y) ∙ ∼(X ∨ ∼Y) AIP 2. X ≡ Y 1, Simp 3. (X ⊃ Y) ∙ (Y ⊃ X) 2, Equiv 4. ∼(X ∨ ∼Y) ∙ (X ≡ Y) 1, Com ` 5. ∼(X ∨ ∼Y) 4, Simp 6. ∼X ∙ ∼ ∼Y 5, DM 7. ∼X ∙ Y 6, DN 8. (Y ⊃ X) ∙ (X ⊃ Y) 3, Com 9. Y ⊃ X 8, Simp 10. ∼X 6, Simp 11. ∼Y 9, 10, MT 12. Y ∙ ∼X 7, Com 13. Y 12, Simp 14. Y ∙ ∼Y 13, 11, Conj 15. ∼[(X ≡ Y) ∙ ∼(X ∨ ∼Y)] 1–14, IP QED

3.9.10 is another example of using IP to derive a logical truth, ‘(P ⊃ Q  ) ∨ (∼Q ⊃ P)’. Since our desired formula this time is a disjunction, an indirect proof quickly yields, by a use of DM, two simpler formulas with which to work. Since the assumption is a formula with a negation, though, we have to use DN at the end (line 17) to get our desired formula.

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3.9.10 1. ∼[(P ⊃ Q) ∨ (∼Q ⊃ P)] 2. ∼(P ⊃ Q) ∙ ∼(∼Q ⊃ P) 3. ∼(P ⊃ Q) 4. ∼(∼P ∨ Q) 5. ∼ ∼P ∙ ∼Q 6. P ∙ ∼Q 7. ∼(∼Q ⊃ P) ∙ ∼(P ⊃ Q) 8. ∼(∼Q ⊃ P) 9. ∼(∼ ∼Q ∨ P) 10. ∼(Q ∨ P) 11. ∼Q ∙ ∼P 12. ∼P ∙ ∼Q 13. ∼P 14. P 15. P ∙ ∼P 16. ∼ ∼[(P ⊃ Q) ∨ (∼Q ⊃ P)] 17. (P ⊃ Q) ∨ (∼Q ⊃ P) QED

AIP 1, DM 2, Simp 3, Impl 4, DM 5, DN 2, Com 7, Simp 8, Impl 9, DN 10, DM 11, Com 12, Simp 6, Simp 14, 13, Conj 1–15, IP 16, DN

We can nest proofs of logical truths inside a larger proof, as intermediate steps, as in 3.9.11. Notice that the antecedents of the conditionals on lines 4 and 8 are logical truths. 3.9.11 1. B ⊃ [(D ⊃ D) ⊃ E] 2. E ⊃ {[F ⊃ (G ⊃ F)] ⊃ (H ∙ ∼H)} / ∼B 3. B AIP 4. (D ⊃ D) ⊃ E 1, 3, MP 5. D ACP 6. D ⊃ D 5, CP 7. E 4, 6, MP 8. [F ⊃ (G ⊃ F)] ⊃ (H ∙ ∼H) 2, 7, MP 9. F ACP 10. F ∨ ∼G 9, Add 11. ∼G ∨ F 10, Com 12. G ⊃ F 11, Impl 13. F ⊃ (G ⊃ F) 9–12, CP 14. H ∙ ∼H 8, 13, MP 15. ∼B 3–14, IP QED

As with CP (see example 3.8.4 and lines 5–6 in 3.9.11, above), there is a trivial form of IP, at 3.9.12. The result is, perhaps unsurprisingly, the same. 3.9.12 1. P ∙ ∼P AIP 2. ∼(P ∙ ∼P) 1, IP 3. ∼P ∨ ∼ ∼P 2, DM 4. ∼ ∼P ∨ ∼P 3, Com 5. P ∨ ∼P 4, DN

Voila: the law of the excluded middle!

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Three Derivation Methods With the introduction of the methods of conditional and indirect proof, our proof system for PL is now complete: any tautology or valid argument of PL is provable. In chapters 4 and 5, we will explore a series of more refined logical languages. We will introduce new rules governing these refinements. These languages will contain all the vocabulary of PL, and the systems of inference will include all of the rules we have studied in this chapter. A small change to our definition of proof is worth noting here. In section 3.1, I wrote that a derivation, or proof, is a sequence of wffs, every member of which is an assumed premise or follows from earlier wffs in the sequence according to specified rules. Given the uses of CP and IP, we should expand that characterization. A derivation or proof is a sequence of wffs, every member of which is: a premise; or follows from earlier lines in the derivation using the rules; or is an (indented) assumption for CP or IP; or is in the scope of an assumption for CP or IP and follows from earlier lines of the derivation (but not from earlier closed indented sequences).

Summary We now have three derivation methods: direct, conditional, and indirect. Indirect proof is both a useful, legitimate tool of inference in classical systems like ours and the last hope of the desperate. If you are stuck in a proof and cannot see how to get your conclusion, it is often very useful just to assume the opposite of what you want and derive whatever you can, looking for a contradiction. The result might not be the most efficient derivation, but as long as you do not misuse any rules, the derivation will be legitimate. We now have two different kinds of assumptions: assumptions for conditional proof and assumptions for indirect proof. These assumptions are really no different. Indeed, you might think of indirect proof as a conditional proof of a formula whose consequent is a contradiction. Since the antecedent entails a contradiction, we know that the first line of the indented sequence is false, and we can, given bivalence, conclude its opposite. It is natural, especially at first, to wonder about which derivation method to use in any particular derivation. Some guidelines are generally useful, though they should not be taken as inviolable rules. Which Derivation Method Should I Use? If the main operator is a conditional or a biconditional, generally use conditional proof. If the main operator is a disjunction or a negation, generally use indirect proof. If the main operator is a conjunction, look to the main operators of each conjunct to determine the best derivation method.

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KEEP IN MIND

For indirect proof, assume (in the first indented line) the opposite of your desired conclusion. The last line of an indented sequence for IP should always be a contradiction. A contradiction is any statement of the form α ∙ ∼α. For IP, always discharge the first line of the proof with one more tilde. Logical truths may be proven using either CP or IP. You may use indirect proof whenever you are stuck in a derivation.

EXERCISES 3.9a Derive the conclusions of the following arguments using conditional proof and/or indirect proof where appropriate. 1. 1. U ⊃ (V ∨ W) 2. ∼(W ∨ V)

/ ∼U

2. 1. Y ∨ ∼Z 2. ∼X ∨ Z

/X⊃Y

3. 1. A ⊃ B 2. ∼(C ∨ ∼A)

/B

4. 1. L ⊃ M 2. L ∨ O

/M∨O

5. 1. A ∨ ∼B 2. (B ∨ C) ⊃ ∼A / ∼B 6. 1. F ⊃ (E ∨ D) 2. ∼E ∙ (∼D ∨ ∼F) / ∼F 7. 1. M ⊃ L 2. ∼(K ∙ N) ⊃ (M ∨ L) / K ∨ L 8. 1. H ⊃ G 2. H ∨ J 3. ∼(J ∨ ∼I)

/G∙I

9. 1. X ⊃ Y 2. ∼(Z ⊃ W)

/ X ⊃ (Y ∙ Z)

10. 1. ∼(G ⊃ H) ⊃ ∼F 2. G ∙ (F ∨ H)

/H

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11. 1. B ≡ (A ∙ D) 2. ∼A ⊃ (∼B ⊃ C)

/A∨C

12. 1. P ≡ (Q ∨ ∼R) 2. T ∙ ∼(Q ∙ P)

/ ∼(P ∙ R)

13. 1. (C ∨ ∼B) ⊃ (∼D ⊃ A) 2. (A ∨ B) ≡ D /D 14. 1. X ⊃ T 2. Y ⊃ T 3. T ⊃ Z

/ (X ∨ Y) ⊃ Z

15. 1. S ⊃ T 2. S ∨ (∼R ∙ U)

/R⊃T

16. 1. A ≡ (B ∙ D) 2. C ⊃ (E ∨ F) 3. A ∨ ∼E 4. A ∨ ∼F

/C⊃B

17. 1. M ⊃ (L ∙ ∼P) 2. K ⊃ ∼(O ∙ ∼P) 3. N ⊃ O

/ (K ∙ M) ⊃ ∼N

18. 1. A ⊃ B 2. ∼C ⊃ ∼(A ∨ ∼D) 3. ∼D ∨ (B ∙ C)

/ A ⊃ (B ∙ C)

19. 1. Z ⊃ Y 2. Z ∨ W 3. Y ⊃ ∼W 4. W ≡ ∼X

/X≡Y

20. 1. W ≡ (X ∙ Z) 2. ∼(∼X ∙ ∼W)

/Z⊃W

21. 1. ∼[J ∨ (F ∙ ∼H)] 2. ∼G ⊃ ∼H 3. G ∨ [∼F ⊃ (J ∙ K)] / E ∨ G 22. 1. (G ∙ ∼H) ⊃ F 2. G

/ (H ∨ F) ∙ G

23. 1. Y ≡ ∼(V ∙ X) 2. ∼W ⊃ ∼V 3. ∼(Y ⊃ ∼V) / ∼(W ⊃ X) 24. 1. ∼(I ⊃ J) ⊃ ∼F 2. (F ∨ H) ∙ (G ∨ I) 3. ∼H ⊃ ∼J

/H∨G

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25. 1. K ⊃ (L ∙ I) 2. ∼(J ⊃ M) 3. L ⊃ (∼K ∨ ∼I) / ∼[ J ⊃ (M ∨ K)] 26. 1. ∼(∼E ∙ ∼H) ∨ I 2. (E ∙ ∼I) ⊃ (H ∙ G)

/H∨I

27. 1. (T ⊃ U) ∙ (S ⊃ V) 2. [V ⊃ (∼T ⊃ W)] ⊃ ∼U 3. S

/ ∼T ∙ V

28. 1. M ⊃ (O ⊃ L) 2. ∼[(∼O ∙ ∼K) ≡ (L ∨ M)] / L ∨ ∼O 29. 1. P ⊃ (Q ∙ R) 2. ∼Q ⊃ R 3. (∼R ≡ ∼Q  ) ∨ P / ∼(Q ⊃ ∼R) 30. 1. A ≡ ∼(B ∨ C) 2. (D ∨ E) ⊃ ∼C 3. ∼(A ∙ D)

/D⊃B

31. 1. U ⊃ (P ∙ ∼Q  ) 2. T ⊃ (S ∨ U) 3. ∼T ⊃ ∼R

/ (P ⊃ Q  ) ⊃ (R ⊃ S)

32. 1. B ⊃ C 2. E ≡ ∼(B ∨ A) 3. D ⊃ ∼E

/ D ⊃ (A ∨ C)

33. 1. F ⊃ (K ≡ M) 2. ∼F ⊃ [L ⊃ (F ≡ H)] 3. ∼(M ∨ ∼L) 4. ∼H ⊃ ∼(∼K ∙ L)

/F≡H

34. 1. ∼P ∨ R 2. ∼P ⊃ ∼(N ⊃ ∼Q  ) 3. ∼R ≡ (P ∨ O)

/Q∙N

35. 1. ∼(R ∙ U) ⊃ T 2. [R ⊃ ∼(S ∙ ∼Q  )] ⊃ ∼T

/ R ∙ (S ∨ U)

36. 1. ∼L ⊃ ∼K 2. N ∙ ∼(K ∙ L)

/ ∼[K ∨ ∼(J ⊃ N)]

37. 1. (L ⊃ ∼J) ∨ (K ∙ M) 2. (∼M ⊃ K) ⊃ ( J ∙ L)

/K≡M

38. 1. (E ⊃ ∼A) ⊃ B 2. [(A ∙ D) ⊃ ∼C] ⊃ ∼B

/ A ∙ (C ∨ E)

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39. 1. ∼E ⊃ ∼(A ⊃ C) 2. (∼D ∙ A) ⊃ (B ∙ ∼B) 3. ∼(∼A ∙ E)

/D

40. 1. V ⊃ (T ∙ ∼W) 2. (T ⊃ W) ⊃ (∼X ∨ ∼Y) 3. ∼[∼(V ∨ Y) ∨ ∼(V ∨ X)] / ∼(T ⊃ W)

EXERCISES 3.9b Translate each of the following paragraphs into arguments written in PL. Then, derive the conclusions of the arguments. 1. If Lorena makes quiche, then she’ll make potatoes. She either doesn’t make potatoes or doesn’t make quiche. So, she doesn’t make quiche. 2. Stephanie either plays miniature golf and not netball, or she goes to the ocean. She doesn’t play miniature golf. So, she goes to the ocean. 3. If Grady eats quickly, then he’ll get hiccups. If he gets hiccups, then he’ll suck on an ice cube and will not eat quickly. So, Grady doesn’t eat quickly. 4. If either Xander or Yael go to the water park, then Vivian will go. Winston going to the water park is sufficient for Vivian not to go. So, if  Winston goes to the water park, then Xander will not. 5. If Esme grows olives, then she grows mangoes. She grows either olives or nectarines. So, she grows either mangoes or nectarines. 6. Having gorillas at the circus entails that there are elephants. There are either gorillas or hippos. Having fancy ponies means that there are no hippos. Thus, either there are elephants or there are no fancy ponies. 7. If the house is painted ivory and not green, then it will appear friendly. The neighbors are either happy or jealous. If the neighbors are jealous, then the house will be painted ivory. So, if it is not the case that either the house appears friendly or it is painted green, then the neighbors will be happy. 8. If tanks tops are worn in school, then the rules are not enforced. It is not the case that either short skirts or very high heels are in the dress code. Tank tops are worn in school, and either uniforms are taken into consideration or the rules are not enforced. So, it is not the case that either the rules are enforced or short skirts are in the dress code.

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9. If we are just, we help our friends. If we are unjust, we harm our enemies. So, we either help our friends or harm our enemies. 10. If beauty does not increase with familiarity, then it either is intellectual perfection or a manifestation of secret natural laws. But beauty is not intellectual perfection. If it’s a manifestation of secret natural laws, then it is intellection perfection. So, beauty increases with familiarity. 11. If I am my body, then I am constantly changing. If I am my conscious awareness, then I am sometimes changing. If I am either constantly or sometimes changing, then I do not have to repay my debts. But I do have to repay my debts. So, I am not my body and I am not my conscious awareness. 12. If there are no atoms, then multiplicity is an illusion. If there are no atoms, we can’t explain physical phenomena. Either we can explain physical phenomena or there is a physical world. Either there is no physical world or multiplicity is not an illusion. So, there are atoms. 13. If everything is either simple or real, then either causation is observable or time is an illusion. But time is no illusion. So, if everything is simple, then causation is observable. 14. Truth is not both correspondence of words to reality and consistency. If truth is not consistency, then we do not know whether our sentences are true and we are threatened with solipsism. If we have a good semantic theory, then we know whether our sentences are true. So, if truth is correspondence of words to reality, then we don’t have a good semantic theory. 15. If life is not all suffering, then we can be compassionate. If we can be compassionate or have empathy, then we are emotionally vulnerable. It is not the case that our sentience entails that we are emotionally vulnerable. So, life is all suffering. 16. If morality is relative, then it is either subjective or culturally conditioned. If morality is absolute, then either it is intuitive or not culturally conditioned. If morality is not intuitive, then it is not subjective. So, if morality is relative and not intuitive, then it is not absolute.

EXERCISES 3.9c Use conditional or indirect proof to derive each of the following logical truths. 1. ∼(∼P ∨ ∼Q  ) ⊃ P 2. [∼P ∨ (Q ∙ R)] ⊃ (Q ∨ ∼P)

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3. ∼(P ≡ ∼P) 4. (P ∨ Q  ) ∨ (∼P ∙ ∼Q  ) 5. A ∨ (B ∨ ∼A) 6. C ∨ (C ⊃ D) 7. ∼(P ∙ Q  ) ∨ P 8. ∼P ∨ (P ∨ Q  ) 9. ∼[(I ⊃ ∼I) ∙ (∼I ⊃ I)] 10. J ≡ [ J ∨ ( J ∙ K)] 11. (∼P ≡ Q  ) ≡ [(∼P ∙ Q  ) ∨ (P ∙ ∼Q  )] 12. [(∼P ∨ Q  ) ∙ (∼P ∨ R)] ∨ [P ∨ (∼Q ∙ ∼R)] 13. (P ∨ ∼Q  ) ∨ (∼P ∨ R) 14. (E ⊃ F) ∨ (F ⊃ E) 15. (G ⊃ H) ∨ (∼G ⊃ H) 16. (L ≡ ∼M) ≡ ∼(L ≡ M) 17. (P ⊃ Q  ) ≡ (Q ∨ ∼P) 18. (∼P ≡ Q  ) ∨ (∼P ∨ Q  ) 19. [(P ∙ Q  ) ∙ ∼R] ∨ [(P ∙ ∼Q  ) ∨ (∼P ∨ R)] 20. [(P ∙ ∼Q  ) ∨ (R ∙ ∼S)] ∨ [(Q ∙ S) ∨ (∼P ∨ ∼R)]

3.10: CHAPTER REVIEW We have come to the end of our study of proof theory for propositional logic. We have eleven rules of inference (sections 3.1, 3.2, and 3.6); fourteen rules of equivalence (sections 3.3, 3.4, and 3.6); and two alternatives to direct derivations (conditional proof in section 3.7 and indirect proof in section 3.9). In chapters 4 and 5, all of these tools continue to be used in constructing derivations in predicate logic. Practice with the rules and methods of proof can make them both intuitive and useful beyond pure logic. The main goal of the technical work of this book is a formal characterization of logical consequence: what follows from what. Our characterization in terms of our system of inference, by the equivalences between valid arguments and logical truths we saw in section 3.8, applies equally to logical truths and valid arguments. Our system allows us multiple ways of deriving the conclusion of an argument or proving a logical truth. By the soundness and completeness of our system, we can prove all and only the logical truths and we can derive the conclusions of all and only valid arguments.

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Proof Strategies Sometimes, when faced with the challenge of deriving the conclusion of an argument or proving a logical truth, we can quickly see our way through to the end. Other times, we get stuck. At such times, it is useful to work off to the side of the proof, or on scratch paper, trying different strategies. In particular, it can often be useful to work backward from our desired conclusions. Here are some useful strategies worth keeping in mind, for various kinds of conclusions. They can work for the main conclusions of an argument, or to get propositions that you see you need along the way. It is not a complete list, but it collects some of the most reliable strategies. If your desired conclusion is a simple propositional letter or a negation of one, it is useful to see where that letter exists in the premises. If it is in the consequent of a conditional, try to derive the antecedent of that conditional, so you can use MP. If it is in the antecedent of a conditional, try to derive the negation of the consequent, so you can use MT. If it is part of a disjunction, try to get the negation of the other disjunct, so you can use DS. You might also try an indirect proof, starting with the negation of your desired conclusion. Sometimes, though much less frequently, you can use Taut on statements of the form α ∨ α. If your desired conclusion is a conjunction, it is typical to derive each conjunct separately. Remember that conjunctions are the negations of disjunctions, by DM, so that statements of the form ∼(α ∨ β) turn into statements of the form ∼α ∙ ∼β. You can sometimes derive a disjunction merely by deriving one of the disjuncts and using Add for the other. If that fails, CD can be useful. Since Impl allows us to turn statements of the form α ∨ β into statements of the form ∼α ⊃ β, conditional proof can be effective with disjunctions, too. And an indirect proof of a disjunction allows you quickly to get two simpler statements. One use of DM on the negation of a statement of the form α ∨ β yields ∼α ∙ ∼β; you can simplify either side. Conditional proof is often effective in proving conditionals, especially for logical truths. Don’t forget HS, especially when you are given a few conditionals in the premises. Cont can help you set up HS properly. DM can turn disjunctions into conditionals on which you can use HS too. Lastly, while there are many rules for deriving biconditionals in section 3.6, it remains typical to derive each of the two component conditionals and then conjoin them. CP can help with each side, though you should try first to see if you really need CP; sometimes derivations are quicker without it.

Logical Truth or Not? If we combine our deductive system for proving logical truths and valid arguments with the semantic tools for constructing valuations and counterexamples in chapter 2, we have the ability, given any statement or argument of the language PL, to determine and demonstrate its validity. For example, we might be given a proposition like 3.10.1, without being told whether it is a logical truth or not. 3.10.1 [P ⊃ (R ⊃ Q)] ⊃ (P ⊃ Q)

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To determine whether it is a logical truth, we can just construct a truth table and see whether it comes out false in any row, as we did in section 2.5. Perhaps more easily, we can attempt to construct a derivation, as I begin to do at 3.10.2. 3.10.2 1. P ⊃ (R ⊃ Q) 2. P 3. R ⊃ Q

ACP (to prove P ⊃ Q) ACP (to prove Q) 1, 2, MP

At this point, I don’t see any further helpful inferences and I begin to worry that I might have a contingent (or even contradictory) proposition on my hands. I turn to my semantic techniques: Can I construct a valuation that makes 3.10.1 false? P

Q

R

[P

⊃

(R

⊃

Q)]

⊃

(P

⊃

Q)

To make the proposition false, I have to make the antecedent true and the consequent false. To make the consequent false, I must make P true and Q false. I can carry these values through the formula. P

Q

1

0

R

[P

⊃

(R

1

⊃

Q)]

⊃

0

(P

⊃

Q)

1

0

0

If 3.10.1 were a logical truth, I would not be able to make the antecedent true. But if I take R to be false, the antecedent comes out true and the whole formula comes out false. We have a valuation that shows that 3.10.2 is not a logical truth.

Valid or Invalid? We can use a similar combination of the methods of chapters 2 and 3 when given an argument that we do not know is valid or invalid, like 3.10.3. 3.10.3 1. P ≡ Q 2. ∼P ∨ R 3. R ⊃ S

/ ∼Q ∙ S

We might try to derive the conclusion, as I do at 3.10.4. 3.10.4 1. P ≡ Q 2. ∼P ∨ R 3. R ⊃ S 4. (P ⊃ Q) ∙ (Q ⊃ P) 5. (Q ⊃ P) ∙ (P ⊃ Q) 6. Q ⊃ P 7. ∼P ⊃ ∼Q 8. ∼Q ∨ S

/ ∼Q ∙ S 1, Equiv 4, Com 5, Simp 6, Cont 7, 3, 2, CD

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At this point, despite my ingenuity in using CD, I begin to suspect that the argument is invalid. I could try an indirect proof, but with the conjunction in the conclusion, it doesn’t seem promising. If the argument is invalid, I should be able to construct a counterexample. I turn to that task next. There’s no obvious place to start, so I’ll start with the first premise, which is true either when P and Q are both true or when they are both false. P

≡

Q

1

1

1

0

0

P

Q

1 0

R

S

∼

P

1

0

1

1

0

1

0

R

⊃

//

/

S

∨

R

/

∼

Q

∙

S

0

1

0

1

0

In the first row, our conclusion is already false, so we just need to make the second and third premises true. If we take R to be false, we make the third premise true, but the second premise is false. But if we take R to be true, we can make both premises true by taking S to be true. We have a counterexample when all atomic formulas are true. P

Q

R

S

P

≡

Q

∼

P

∨

R

1

1

1

1

1

1

1

0

1

1

1

0

0

0

1

0

1

0

R

⊃

S

//

∼

Q

∙

1

1

1

0

1

0

1

0

/

/

S

Since the argument has a counterexample, it is invalid. Since all and only valid arguments are provable in our system of deduction, the attempted derivation at 3.10.4 was indeed quixotic.

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To complete chapter 3, then, and our study of PL, use the tools from both chapters 2 and 3 on the exercises below, which give you arguments and propositions without telling you whether they are valid or invalid, logical truths or not.

TELL ME MORE  • How do we choose a set of rules for a logical theory? See 6.4: Metalogic. • What further rules can we use for logics of possibility, necessity, and other sentential operators? See 6.5: Modal Logics. • How do the deductive rules of this chapter contrast with other kinds of inferences? See 7.1: Deduction and Induction. • Are there alternatives to the system of derivation in this chapter? See 6S.11: Axiomatic Systems. • How can systems of deduction be used in science? See 7S.8: Logic and Science.

EXERCISES 3.10a Determine whether each of the following arguments is valid or invalid. If it is valid, provide a derivation of the conclusion. If it is invalid, provide a counterexample. 1. 1. A ≡ C 2. C ⊃ (D ∨ B) 3. D

/A⊃B

2. 1. E 2. (E ∨ G) ⊃ H 3. H ⊃ F 4. (F ∙ E) ⊃ ∼G / ∼G 3. 1. L ⊃ I 2. I ⊃ (K ⊃ J) 3. K ⊃ L

/J⊃L

4. 1. M ⊃ N 2. N ≡ ∼O 3. ∼N ⊃ (M ∙ O) / ∼N 5. 1. (Q ∨ R) ≡ ∼P 2. Q ∨ S 3. P

/S∙R

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6. 1. X ⊃ W 2. W ⊃ X 3. Y 4. (Z ∙ Y) ⊃ ∼X / ∼X 7. 1. (A ∙ B) ⊃ (C ∙ D) 2. ∼C 3. B 4. A ∨ (∼D ∙ ∼B) / ∼D 8. 1. E ∨ F 2. ∼F ∨ G 3. E ≡ G 4. F ⊃ (G ∨ E)

/F

9. 1. P ⊃ Q 2. R ∨ S 3. ∼R 4. Q ⊃ S

/ ∼P

10. 1. Z ≡ ∼X 2. ∼X ∨ Y 3. W ∙ ∼Y

/Z∙W

11. 1. A ≡ B 2. ∼B 3. C 4. (D ∙ C) ⊃ ∼(A ∨ D) / ∼A ∙ ∼D 12. 1. F ≡ (H ∙ I) 2. ∼H ∨ ∼I 3. ∼F ⊃ G 4. G ⊃ E

/E

13. 1. ∼P ⊃ R 2. Q ⊃ ∼R 3. (∼P ∙ Q  ) ∨ S 4. S ≡ T 5. T ⊃ ∼Q /∼Q 14. 1. (W ∙ X) ⊃ Y 2. Y ⊃ (Z ∨ ∼X) 3. ∼Z / ∼(W ∙ X) 15. 1. ∼A ⊃ ∼B 2. A ⊃ (C ∙ D) 3. (C ∙ D) ≡ A

/A

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16. 1. ∼(E ∨ F) 2. H ≡ F 3. (H ∙ G) ∨ (H ∙ I) / ∼(G ⊃ E) 17. 1. J ≡ K 2. ∼J ∙ L 3. M ⊃ J 4. N ⊃ (K ∨ M)

/ L ∙ ∼N

18. 1. P ⊃ Q 2. P ∨ R 3. Q ⊃ ∼R 4. R ≡ ∼S

/S≡Q

19. 1. ∼W ∨ X 2. Y ⊃ X 3. Y ⊃ ∼(Z ∙ X)

/ ∼Z ∨ ∼X

20. 1. (∼K ⊃ L) ∙ (∼M ⊃ N) 2. ∼(K ∙ M) 3. ∼L 4. N ≡ (K ∨ L) /M≡N 21. 1. P ≡ (∼Q ∙ R) 2. (R ⊃ Q  ) ⊃ S 3. S ⊃ T 4. S ⊃ ∼T 5. P ⊃ (T ≡ ∼X) / ∼(X ≡ T) 22. 1. A ⊃ [∼B ∨ (C ∙ ∼D)] 2. B ⊃ D / B ⊃ ∼A 23. 1. (E ∙ F) ⊃ (G ∙ H) 2. ∼G ∨ ∼H 3. F 4. I ⊃ ( J ⊃ E) / ∼I ∨ ∼J 24. 1. K ⊃ (∼L ⊃ M) 2. N ∨ K 3. L ⊃ ∼N

/M∨L

25. 1. ∼Z ⊃ Y 2. Z ⊃ ∼X 3. X ∨ ∼Z 4. Y ⊃ A 5. X ⊃ ∼A / ∼X

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EXERCISES 3.10b Determine whether each of the following propositions is a logical truth. If it is a logical truth, provide a proof using our system of natural deduction. If it is not a logical truth, provide a valuation that makes the statement false. 1. (G ∨ G) ⊃ G 2. (T ∨ ∼T) ⊃ T 3. (P ∙ Q  ) ⊃ (P ∨ Q  ) 4. (R ∨ S) ⊃ (R ∙ S) 5. [(A ∨ ∼B) ∙ ∼A] ⊃ B 6. [(C ∨ ∼D) ∙ ∼C] ⊃ (∼D ∨ E) 7. [(A ⊃ B) ∙ (B ⊃ C)] ⊃ (∼C ⊃ ∼A) 8. [E ⊃ (F ⊃ G)] ⊃ [F ⊃ (E ⊃ G)] 9. [(H ∨ I) ⊃ K] ⊃ [(H ∙ I) ⊃ K] 10. [(J ∙ L) ⊃ M] ⊃ [( J ∨ L) ⊃ M] 11. ∼(R ⊃ S) ≡ (T ⊃ R) 12. ∼(P ⊃ Q  ) ≡ (P ∨ ∼Q  ) 13. ∼(X ⊃ Y) ⊃ (Y ⊃ Z) 14. [(S ∨ T) ∙ ∼T] ⊃ (S ⊃ R) 15. [(P ∨ Q  ) ∙ ∼P] ⊃ [(Q ⊃ R) ⊃ R] 16. [P ⊃ (Q ∨ S)] ⊃ (∼Q ⊃ ∼P) 17. [J ≡ (K ∙ L)] ⊃ [(J ⊃ K) ∙ (K ⊃ J)] 18. ∼(A ∨ ∼B) ⊃ [(A ⊃ C) ∙ (C ⊃ B)] 19. [G ≡ (H ∨ I)] ⊃ [(H ⊃ G) ∙ (I ⊃ G)] 20. [A ≡ (B ∙ C)] ⊃ [(A ≡ B) ∙ (A ≡ C)] 21. (E ∨ F) ⊃ {(E ⊃ H) ⊃ [(F ⊃ H) ⊃ H]} 22. [D ≡ (E ∨ F)] ⊃ [(D ⊃ E) ∙ (D ⊃ F)] 23. [(W ≡ X) ⊃ (Y ≡ Z)] ⊃ [(Y ≡ ∼Z) ⊃ (∼W ≡ X)] 24. [(P ∨ Q  ) ⊃ (R ∙ S)] ⊃ [(P ⊃ R) ∙ (Q ⊃ S)] 25. [(W ∙ X) ⊃ (Y ∙ Z)] ⊃ [(W ⊃ X) ⊃ Y]

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KEY TERMS addition (Add), 3.2 association (Assoc), 3.3 commutativity (Com), 3.3 complete system of inference, 3.1 conditional proof, 3.7 conjunction (Conj), 3.2 constructive dilemma (CD), 3.2 contradiction, 3.5 contraposition (Cont), 3.4 De Morgan’s laws (DM), 3.3 derivation, 3.1, 3.7 direct proof, 3.7 disjunctive syllogism (DS), 3.1 distribution (Dist), 3.3 double negation (DN), 3.3 explosion, 3.5 exportation (Exp), 3.4 formal theory, 3.8 hypothetical syllogism (HS), 3.1 indented sequence, 3.7 indirect proof, 3.9

justification, 3.1 law of the excluded middle, 3.8 → ), 3.3 logically equivalent ( ←  material equivalence (Equiv), 3.4 material implication (Impl), 3.4 modus ponens (MP), 3.1 modus tollens (MT), 3.1 nested sequence, 3.7 PL, 3.1 proof, 3.1, 3.9 QED, 3.1 reductio ad absurdum, 3.9 rule of equivalence, 3.3 rule of inference, 3.1 simplification (Simp), 3.2 sound system of inference, 3.1 substitution instance, 3.1 system of inference, 3.1 tautology (Taut), 3.4 theorem, 3.8 theory, 3.8

Chapter 4 Monadic Predicate Logic

4.1: INTRODUCING PREDICATE LOGIC We started our study of logic with a casual understanding of what follows from what. Intuitively, a valid argument is one in which the truth of the premises ensures the truth of the conclusion. Then, we explored a semantic definition of validity, in chapter 2, and a proof system based on that semantic definition, in chapter 3. Our formal notion of validity for propositional logic captures many intuitively valid inferences. But it does not capture all of them. For example, argument 4.1.1 is an intuitively valid inference. 4.1.1

All philosophers are happy. Emily is a philosopher. So, Emily is happy.

But our tests for logical validity in propositional logic are of no help in showing the validity of the argument. 4.1.2 P Q

/R

The conclusion does not follow from the premises using our system of inference for PL. The truth tables show a counterexample, when P and Q are true and R is false. The rules for validity for propositional logic are thus insufficient as a general characterization of logical consequence. PL captures entailments among propositions. The entailments in 4.1.1 are within, rather than among, the simple propositions. We need a logic that explores logical relations inside propositions. Quantificational, or predicate, logic does that. In PL, we use the following vocabulary: Capital English letters for simple statements Five propositional operators Punctuation (brackets)

206

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In predicate logic, we extend the vocabulary. We retain the same propositional operators and punctuation. But the terms are more complex, revealing some subpropositional logical relations: Complex statements made of singular terms and predicates Quantifiers Five propositional operators Punctuation Our study of predicate logic starts with a simple language, which I will call M, for monadic predicate logic.

A predicate logic i ncludes singular

terms, predicates, and quantifiers.

M is monadic predicate

logic.

Singular Terms and Predicates In all predicate logic, we represent particular things using lower-case letters which we call singular terms. In monadic predicate logic, M, we have two kinds of singular terms: constants and variables. a, b, c, . . . u v, w, x, y, z

stand for specific objects and are called constants. are used as variables.

We might use ‘a’ to stand for a person (perhaps Alycia or Andres); a city (Abidjan or Athens); a work of art (Van Gogh’s Arles: View from the Wheat Fields or the movie The Amazing Spider-Man); a mountain (Annapurna); or any other object to which we give a name. The constant need not be the first letter of the object named; indeed, for objects with names beginning with the letters v . . . z we cannot use the first letter. But it is convenient to pick an obvious letter. We represent properties of objects using any of the twenty-six capital letters of English. Used this way, we call them predicates. Predicates are placed in front of singular terms so that ‘Pa’ is used to say that object a has property P. A predicate of M followed by a constant is called a closed sentence and expresses a proposition. 4.1.3 shows some closed sentences. 4.1.3

Amaya is clever. Baruch plays chess. Carlos is tall.

Ca Pb Tc

A predicate followed by a variable is called an open sentence. 4.1.4 shows some open sentences. Notice that closed sentences express what we might call a complete proposition, whereas open sentences do not. Indeed, they are not easily expressed in English. 4.1.4

v is admirable w is bold x is courteous

Av Bw Cx

Singular terms a re lower-

case letters which follow predicates. They may be constants or variables .

A predicate i s a capital letter that precedes a singular term.

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We call M monadic because the predicates

take exactly one singular term.

The predicates used in 4.1.3 and 4.1.4, and generally in M, are called one-place predicates since they are followed by only one singular term. In section 5.1, we will extend our uses of predicates, using capital letters followed by any number of singular terms to stand for relations among various objects. Returning to 4.1.1, we can now regiment the second premise and the conclusion. Emily is a philosopher. Emily is happy.

Pe He

To finish translating the argument in M, we must deal with the first premise, which is not about a single thing and so cannot be translated using a constant. We can use a variable, but variables are themselves insufficient to complete a proposition. ‘Px’ just means that x is a philosopher and ‘Hx’ just means that x is happy. Those claims are, by themselves, ambiguous among claims that a something is a philosopher, nothing is a philosopher, or everything is a philosopher; and among claims that something is happy, nothing is happy, or everything is happy. We need to disambiguate. Frege thought of predicates as functions from singular terms to complete propositions. He put the singular terms after the predicates in imitation of the mathematical practice of putting a function in front of its argument: f(3) or g(x). (See section 5.6 for more on functions.) We follow Frege, writing ‘Pe’ for ‘Emily is a philosopher’ instead of ‘eP’, or ‘Ep’, either of which might be a bit more natural. Just as a function needs an argument, a proposition expressed by a predicate has a hole in it, which must be filled with a singular term. When the singular term is a constant, we have a complete proposition, as at 4.1.3. But when the singular term is a variable, as at 4.1.4, we have to complete the proposition by indicating more about the variable, disambiguating among something, nothing, and everything. We do that with quantifiers.

Quantifiers

Quantifiers a re operators

that work with variables to stand for terms like ‘something’, ‘everything’, ‘nothing’, and ‘anything’. They may be existential or universal .

The subject of ‘All philosophers are happy’ is not a specific philosopher. No specific object is mentioned. Similarly, in ‘Something is made in the USA’, there is no specific thing to which the sentence refers. For sentences like these, we use quantifiers to bind and modify our singular terms. There are two quantifiers: existential and universal, which always appear with a variable. (∃x), (∃y), (∃z), (∃w), (∃v) (∀x), (∀y), (∀z), (∀w), (∀v)

Existential quantifiers are used to represent expressions like the following: There exists a thing such that For some thing There is a thing For at least one thing Something

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Universal quantifiers are used to represent expressions like the following: For all x Everything

Some terms, like ‘anything’, can indicate either an existential or a universal quantifier, depending on the context. 4.1.5 4.1.6

If anything is missing, you’ll be sorry. Anything goes.

In 4.1.5, we use an existential quantifier. ‘Anything’ in that case indicates ‘something’: if something is missing, then you’ll be sorry. In 4.1.6, we use a universal quantifier, since that sentence expresses that everything is acceptable. To know whether to use an existential or universal quantifier in cases where a quantifier is called for, you will have to judge from the context of the use. Similar remarks hold for the indefinite articles ‘a’ and ‘an’. 4.1.7 is universal, whereas 4.1.8 is existential. 4.1.7 4.1.8

A whale is a mammal. Ahab sought a whale.

4.1.9–4.1.11 are examples of simple translations using quantifiers. 4.1.9 4.1.10 4.1.11

Something is made in the USA. Everything is made in the USA. Nothing is made in the USA.

(∃x)Ux (∀x)Ux (∀x)∼Ux or ∼(∃x)Ux

Notice that the variables following the predicate match the quantifier variable. It doesn’t matter which variables you use in a translation, but it does matter that they match the appropriate quantifier. So, 4.1.9 could be written ‘(∃y)Uy’ or ‘(∃w)Uw’, but it could not be written ‘(∃x)Uy’. In this chapter, we generally work with one variable at a time. Some formulas will have more than one quantifier, but they will usually not overlap with each other. Notice also, in 4.1.11, that statements with quantifiers and negations can be translated in at least two different ways: everything lacks a property or it is not the case that something has the property. In the above examples, the quantifiers appear in the subject of the sentence. They can appear elsewhere, too, as in 4.1.12 and 4.1.13. 4.1.12 4.1.13

Kwame did everything. I wish that something would happen.

(∀x)Kx (∃x)Wx

Quantifiers are operators, like the unary propositional operator negation or the four binary propositional operators. The main operator of 4.1.9 is the existential quantifier. The main operator of 4.1.10 is the universal quantifier. The main operator of the first version at 4.1.11 is the universal quantifier; the second version is a negation.

‘Anything’ can indicate either an existential or a universal quantifier.

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Quantified Sentences with Two Predicates

The subject o f a sentence is what is discussed. The attribute of a sentence is what is said about the subject. Both may contain multiple logical predicates.

Most English sentences are best translated using at least two predicates. We can roughly divide most sentences into grammatical subjects and grammatical predicates. The grammatical subject (or just ‘subject’) is what the sentence is about. The grammatical predicate is what the sentence says about its grammatical subject. To avoid confusion between grammatical predicates and logical predicates, I’ll use the term ‘attribute’ for ‘grammatical predicate’. For example, in ‘Dinesh loves apples’, ‘Dinesh’ is the subject and ‘loves apples’ is the attribute. In ‘Mind-body materialists are chauvinists’, the subject is ‘mind-body materialists’ and the attribute is ‘are chauvinists’. When regimenting sentences such as the latter, it is typical to use one or more predicates for the subject of the sentence and another one or more predicates for the attribute of the sentence. Between the subject and attribute, there will be a propositional operator. 4.1.14 has the basic form of a universally quantified proposition, and 4.1.15 has the basic form of an existentially quantified sentence. 4.1.14 4.1.15

All persons are mortal. Some actors are vain.

(∀x)(Px ⊃ Mx) (∃x)(Ax ∙ Vx)

Notice that the propositional operator in the universally quantified 4.1.14 is a horseshoe: take anything you like; if it’s a person, then it is mortal. The existentially quantified proposition, 4.1.15, uses a conjunction: there are some things that are both actors and are vain. This is a useful lesson. Universally quantified propositions tend to use conditionals between the subject and the attribute. Existentially quantified propositions usually use conjunctions. These are not absolute rules, but are generally useful guidelines. Be careful not to confuse the two. A conjunction in the universally quantified expression 4.1.14 would assert that everything is a mortal person, not the meaning of the original sentence. Using a conjunction entails that each conjunct is asserted of everything; there are very few properties that hold of everything. A conditional in the existential claims 4.1.15 would weaken the force of the claim: there are some things such that if they are actors, then they are vain. My chair is such that if it is an actor, then it is vain; since my chair is not an actor, the claim is vacuously true. But the original English is best seen as asserting of one or more actors that they are in fact vain. As with simpler propositions, there are different ways of regimenting complex quantified propositions with negations, some with the negations in front and some with the negations embedded. 4.1.16 Some gods aren’t mortal. (∃x)(Gx ∙ ∼Mx) or ∼(∀x)(Gx ⊃ Mx) 4.1.17 No frogs are people. (∀x)(Fx ⊃ ∼Px) or ∼(∃x)(Fx ∙ Px)

The alternatives at 4.1.16 and 4.1.17 show that negations and quantifiers can combine differently. In 4.1.16, the first option says that there is something that is a God

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and is not mortal; the second option says that it is not the case that all gods are mortal, which would be the case only if some god is not mortal. The two forms are logically equivalent. In parallel, the first version at 4.1.17 says that everything that is a frog is not a person, whereas the second says, equivalently, that it is not the case that there is something that is a frog and a person. Notice that even with the negation, the universal statement is a conditional and the existential statement is a conjunction. Later, in section 4.5, we will move between these equivalent translations.

Languages of Predicate Logic We are starting our study of predicate logic by considering a simplified version of firstorder logical language: monadic predicate logic, or M. Predicate logic is monadic if the predicates take only one singular term. When predicates take more than one singular term, we call them relational and we call the resulting language full first-order predicate logic, or F. Chapter 4 focuses nearly exclusively on M. We use F (and some further extensions of F) in chapter 5. In constructing a formal language, we first specify the language, and then rules for wffs. Each time we extend our predicate logic, we will generate a slightly new language, with slightly new formation rules. From M, we proceed to F, and then to FF, full first-order predicate logic with functors. For PL, in chapters 2 and 3, we studied one language and one system of inference. But we can use the same language in different deductive systems and we can use the same deductive system with different languages. In the chapters on predicate logic, I use M and F with the same deductive system. Then, I add new inference rules covering a special identity predicate. It is typical to name both the deductive systems and the languages, but we need not do so and I will name only the different languages.

Summary The goal of this section is to start you translating between English and monadic predicate logic. When faced with a sentence of English, you first have to ask whether it uses constants (if it names particular objects) or quantifiers and variables (if it uses the quantifier terms like ‘all’, ‘some’, ‘none’, ‘any’, or ‘only’). Some sentences will use both constants and variables. The main subformulas of universally quantified sentences (after their quantifiers) are ordinarily conditionals, with subjects as their antecedents and attributes as their consequents. The main subformulas of existentially quantified sentences are ordinarily conjunctions; the order of the subject and attribute does not matter. Remember that sentences containing ‘nothing’ and related quantifiers can be translated either using a universal quantifier, with a negation embedded inside the formula, or using the negation of an existentially quantified sentence.

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KEEP IN MIND

Predicate logic extends propositional logic with predicates, singular terms, and quantifiers. Singular terms may be constants, standing for particular things, or variables, which must be modified by quantifiers to form a closed sentence that expresses a complete proposition. Quantifiers may be existential or universal. Statements with quantifiers and negations can be translated in at least two different ways. Start translating into M by asking whether the sentence is universal or existential. Think of English sentences in terms of the ordinary rules of subject-predicate grammar. The subject of the proposition is what we are talking about. The attribute of the proposition is what we are saying about it. The subject of a sentence is the antecedent in a universally quantified statement or the first conjunct in an existentially quantified statement. The attribute of a sentence is the consequent of the conditional in a universally quantified statement or the second conjunct of an existentially quantified statement. Quantifiers are logical operators and may be the main operators of a proposition.

TELL ME MORE  • How does using conditionals with universally quantified formulas raise a controversial question about their interpretation? See 6.6: Notes on Translation with M.

EXERCISES 4.1a Translate each sentence into predicate logic using constants in each. 1. Andre is tall. 2. Belinda sings well. 3. Deanna drives to New York City. 4. The Getty Museum is located in Los Angeles. 5. Snowy is called Milou in Belgium. 6. Cortez and Guillermo go to the gym after school. 7. Either Hilda makes dinner or Ian does. 8. Jenna doesn’t run for class president. 9. Ken doesn’t walk to school when it rains. 10. Either Lauren or Megan buys lunch. 11. Nate and Orlando play in the college orchestra.

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12. Paco will play football only if he’s not injured. 13. Ramona plays volleyball if, and only if, she sets up the net. 14. If Salvador invests all his money in the stock market, then he takes a second job. 15. Hamilton College is closed if, and only if, President Wippman invokes the closure policy.

EXERCISES 4.1b Translate each sentence into predicate logic. Do not use constants. 1. All computers are difficult to program. (Cx, Dx) 2. Some trees are green. (Tx, Gx) 3. Some flowers do not bloom. (Fx, Bx) 4. Every fruit has seeds. (Fx, Sx) 5. A few people walk fast. (Px, Wx) 6. Not all buses are yellow. (Bx, Yx) 7. A cloud is not fluffy. (Cx, Fx) 8. Every mistake is a lesson. (Mx, Lx) 9. Nothing worthwhile is easy. (Wx, Ex) 10. Most planes are safe. (Px, Sx) 11. Some mountains are not difficult to climb. (Mx, Dx) 12. Not all snakes are poisonous. (Sx, Px) 13. Some spiders are not harmful. (Sx, Hx) 14. No dog has antennae. (Dx, Ax) 15. No lions are not carnivorous. (Lx, Cx)

4.2: TRANSLATION USING M In section 4.1, we saw how to use singular terms (constants and variables), predicates, and quantifiers to translate some simple sentences into monadic predicate logic, M. In this section, we see how to use M to regiment more complex English sentences.

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Quantified Sentences with More than Two Predicates Most quantified sentences in M have a subject and an attribute separated by either a conjunction, if they are existential, or a conditional, if they are universal. But many subjects and attributes will themselves be complex. 4.2.1 and 4.2.2 have more than one predicate in the subject portion of the proposition. 4.2.1 Some wooden desks are uncomfortable.   (∃x)[(Wx ∙ Dx) ∙ ∼Cx] 4.2.2 All wooden desks are uncomfortable.   (∀x)[(Wx ∙ Dx) ⊃ ∼Cx]

4.2.3 and 4.2.4 have more than one predicate in the attribute part of the proposition. 4.2.3 Many applicants are untrained or inexperienced.   (∃x)[Ax ∙ (∼Tx ∨ ∼Ex)] 4.2.4 All applicants are untrained or inexperienced.   (∀x)[Ax ⊃ (∼Tx ∨ ∼Ex)]

As we saw in 4.1, when regimenting into predicate logic, start by asking whether the sentence is universal or existential. Then, think of the sentence in terms of the ordinary rules of subject-predicate grammar. What are we talking about? That’s the subject of the proposition. What are we saying about it? That’s the predicate, or attribute; I’ll use the latter term to avoid confusion with logical predicates. The subject is the antecedent in a universally quantified statement or the first conjunct in an existentially quantified statement. The attribute goes as the consequent or as the second conjunct. Subjects and attributes may be simple (as ‘philosophers’ and ‘are happy’ in ‘some philosophers are happy’) and be regimented as single predicates. But they may both be complex (as ‘green lemons’ in ‘green lemons are unripe’ or ‘is a big, strong, blue ox’ in ‘Babe is a big, strong, blue ox’) and regimented using multiple predicates.

Things and People The parallel sentences 4.2.5 and 4.2.6 each contain a quantifier but have different meanings. 4.2.5 4.2.6

‘Someone’, ‘anyone’, ‘everyone’, and ‘no one’ all

indicate both a quantifier and a predicate for ‘is a person’.

Something is making noise in the basement. Someone is making noise in the basement.

If 4.2.5 is true, anything could be making noise in the basement. But the scope of the claim in 4.2.6 is narrower. Unlike uses of 4.2.5, using 4.2.6 rules out mice and ghosts and wind through the broken window. We mark the difference by saying that a ‘one’ is a person, and we add a predicate for personhood to regimentations of sentences using ‘someone’. Thus, 4.2.5 is represented in M by 4.2.7, taking ‘Mx’ for ‘x is making noise’ and ‘Ix’ for ‘x is in the basement’. 4.2.7 (∃x)(Mx ∙ Ix)

But 4.2.6 is represented by the more complex 4.2.8, adding ‘Px’ for ‘x is a person’.

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4.2.8 (∃x)[Px ∙ (Mx ∙ Ix)]

I grouped the latter two terms in 4.2.8, but really, we could group any pairs, since the only operators (besides the existential quantifier) are the conjunctions. The same kind of adjustment can be made for ‘everyone’ (instead of ‘everything’), ‘anyone’ (‘anything’) and ‘no one’ (‘nothing’). 4.2.9 Everyone who takes logic works on derivations.   (∀x)[(Px ∙ Tx) ⊃ Wx] 4.2.10 Anyone who runs for office is corrupt.   (∀x)[(Px ∙ Rx) ⊃ Cx] 4.2.11 No one who reads Berkeley converts to idealism.   (∀x)[(Px ∙ Bx) ⊃ ∼Ix] or ∼(∃x)[(Px ∙ Bx) ∙ Ix]

Only Like ‘all’ and ‘some’, ‘only’ can modify an open sentence and so indicate the presence of a quantifier. But such translations can be tricky. ‘Only’ usually indicates a universal quantifier, as at 4.2.12. 4.2.12

Only men have been presidents.

4.2.12 claims that if something has been a president, it has been a man; all presidents have been men. Thus, it is equivalent to 4.2.13. 4.2.13

All presidents have been men.

In propositions with just two predicates, ‘only Ps are Qs’ is logically equivalent to ‘all Qs are Ps’. Thus, in simple cases, we can just invert the antecedent and consequent of a parallel sentence that uses ‘all’. Start with a related ‘all’ sentence, like 4.2.14 or 4.2.16. Then take the converse to find the ‘only’ sentence. 4.2.14 4.2.15 4.2.16 4.2.17

All men have been presidents. Only men have been presidents. All cats are animals. Only cats are animals.

(∀x)(Mx ⊃ Px) (∀x)(Px ⊃ Mx) (∀x)(Cx ⊃ Ax) (∀x)(Ax ⊃ Cx)

In more complex sentences, the rule of just switching antecedent and consequent between an ‘all’ sentence and its correlated ‘only’ sentence must be adjusted. 4.2.18 is standardly regimented as 4.2.19. 4.2.18 All intelligent students understand Kant. 4.2.19 (∀x)[(Ix ∙ Sx) ⊃ Ux]

If we regiment 4.2.20 merely by taking the converse of the conditional in 4.2.19, we get 4.2.21. 4.2.20 Only intelligent students understand Kant. 4.2.21 (∀x)[Ux ⊃ (Ix ∙ Sx)]

‘Only’ usually indicates

a universal quantifier. Sentences using ‘only’ must be carefully distinguished from their related ‘all’ sentences.

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4.2.21 says that anything that understands Kant must be an intelligent student. It follows from that regimentation that I don’t understand Kant, since I am no longer a student. I am not sure whether I understand Kant, but that I do not is not a logical consequence of 4.2.20. A preferred regimentation of 4.2.20 is 4.2.22, which says that any student who understands Kant is intelligent. 4.2.22 (∀x)[(Ux ∙ Sx) ⊃ Ix]

4.2.22 is a reasonable representation of 4.2.20. When regimenting, we need not assume that everything that is said is reasonable; that’s surely a false assumption. But it is customary and charitable to presume reasonableness unless we have good reason not to. Just above, I said that to regiment sentences into predicate logic, we think of them as divided into a subject and an attribute. Universally quantified sentences ordinarily have a horseshoe between the subject portion of the proposition and the attribute portion. In existential sentences, we use a conjunction between the subject and attribute. In sentences like 4.2.18, the subject portion of the sentence has both a subordinate subject (‘x is a student’) and a subordinate attribute (‘x is intelligent’); there is a single grammatical attribute (‘x understands Kant’). The relation between the only-­ quantified sentence and its corresponding all-quantified sentence is that the subordinate attribute is switched with the main attribute, but the subordinate subject remains where it is, in the antecedent. Thus, an amended rule could be that if an only-quantified sentence uses only two predicates, you can just switch the antecedent and consequent from the related ‘all’ sentence, the one that results from replacing ‘only’ with ‘all’; but if the grammatical subject contains two predicates (a subordinate subject and an attribute), then you should just switch the two subordinate attributes (‘x is intelligent’ and ‘x understands Kant’), leaving the subordinate subject alone. Let’s summarize this new guideline for ‘only’ as 4.2.23. 4.2.23

‘Only PQs are R’

is ordinarily the same as

‘All RQs are P’

4.2.23 is a good general rule, often applicable. But there are exceptions, and some sentences may be ambiguous. It is not especially clear whether 4.2.24 is best regimented as 4.2.25 or as 4.2.26. 4.2.24 Only famous men have been presidents. 4.2.25 (∀x)[Px ⊃ (Mx ∙ Fx)] 4.2.26 (∀x)[(Px ∙ Mx) ⊃ Fx]

4.2.25 and 4.2.26 are not logically equivalent. 4.2.25 says that if something is a president, then it is a famous man. 4.2.26 says that if something is a male president, then it is famous. If we take ‘president’ to refer to presidents of the United States, say, the former regimentation seems better. But imagine a place in which there have been

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both men and women presidents (like Switzerland). Of the women presidents, let’s imagine, some have been famous, and some have been obscure. But, all of the men who have been president have been famous. In such a case, we would favor the second regimentation, using an inflection on ‘men’ when we utter the original 4.2.24 to say that of the male presidents, all of them have been famous, but of the women, some have been famous and some have not. 4.2.27 is a good exception to the rule at 4.2.23. 4.2.27

Only probability-challenged ticket holders win the lottery.

Since one must hold a ticket to win the lottery, ‘winners of the lottery who are ticket holders’, at 4.2.28, which the rule at 4.2.23 would recommend, is redundant. The better regimentation is 4.2.29. 4.2.28 (∀x)[(Wx ∙ Tx) ⊃ Px] 4.2.29 (∀x)[Wx ⊃ (Px ∙ Tx)]

When translating ‘only’ sentences, then, you have to decide from the context whether to use the simple converse rule (as at 4.2.14–4.2.17) or the more complex rule at 4.2.23.

Propositions with More than One Quantifier The main operator of the quantified sentences we have seen so far has been the quantifier, whether existential or universal. But some propositions in M will contain more than one quantifier. The main operator of such sentences can be any of the propositional operators. Look for multiple quantifier indicators in the sentence, or a leading term that indicates that the main operator is one of the propositional operators. 4.2.30 If anything is damaged, then everyone in the house complains.   (∃x)Dx ⊃ (∀x)[(Ix ∙ Px) ⊃ Cx] 4.2.31 The gears are all broken if, and only if, a cylinder is missing.   (∀x)(Gx ⊃ Bx) ≡ (∃x)(Cx ∙ Mx) 4.2.32

Some philosophers are realists, but some philosophers are fiction   alists and some are modalists.   (∃x)(Px ∙ Rx) ∙ [(∃x)(Px ∙ Fx) ∙ (∃ x)(Px ∙ Mx)] 4.2.33

It’s not the case that either all conventionalists are logical empiricists    or some holists are conventionalists.   ∼[(∀x)(Cx ⊃ Lx) ∨ (∃x)(Hx ∙ Cx)]

Adjectives Adjectives are a main source of increasing complexity in our sentences and their regimentations. For example, in 4.2.1 we represented ‘wooden desks’ as ‘(Wx ∙ Dx)’, something that has the properties both of being wooden and of being a desk. 4.2.34 has a selection of similar examples.

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4.2.34

green book beautiful painting hungry puppy confused teenager

Gx ∙ Bx Bx ∙ Px Hx ∙ Px Cx ∙ Tx

A green book is something that is both green and a book; a confused teenager is something that is both confused and a teenager. But not all adjectives are properly regimented using an additional predicate, as in the items in the list at 4.2.35. 4.2.35

large baby smart bee old fruit fly

A large baby is not something that is large and a baby; it is something that is large for a baby. Such adjectives are context sensitive and cannot be ascribed to something in the way that ‘green’ or ‘hungry’ can. Nothing could be said to be large or smart or old by itself; things have these properties only relative to other things of their types. When faced with a sentence containing such context-sensitive adjectives, it is best to use one predicate for the modified noun. In 4.2.36, I use ‘Sx’ for ‘x is a jumbo shrimp’ and ‘Px’ for ‘x is on the plate’. 4.2.36 There are jumbo shrimp on the plate.   (∃x)(Sx ∙ Px)

In the exercises that follow, I provide predicates and specify what they are to represent, so you won’t find yourself challenged to make the distinction. But if you are regimenting completely on your own, it is worth keeping this phenomenon in mind.

Summary In section 4.1, we started translating between English and monadic predicate logic. In this section, we explored the subtleties of M. As sentences become more complicated, they have increasing numbers of predicates. A rough division of our natural-language sentences into subjects and attributes can be useful. While the main subformulas of universally quantified sentences are ordinarily conditionals, the antecedents and consequents of those conditionals may be complex formulas, often conjunctions, especially in the antecedents. The main subformulas of existentially quantified sentences are ordinarily conjunctions; again, the first and second conjuncts may be complex formulas themselves. There are lots of translation exercises in this section and the following sections that explore derivations in M. In section 5.1, we expand beyond monadic predicate logic into full first-order predicate logic. Even if you have mastered translation in M, the new translations there, and in section 5.4 where we look at identity theory, will be challenging. Practice! The translations to English from logic in exercises 4.2b can also be useful in learning how to translate from logic to English.

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KEEP IN MIND

Simple quantified English sentences often have two predicates, separated by a conditional, for universal sentences, or a conjunction, for existential sentences. More complex universal sentences may have complex antecedents (often conjunctions) or consequents. More complex existential sentences may have multiple predicates either before or after the main conjunction. Be careful to distinguish sentences with ‘someone’, ‘everyone’, ‘anyone’, and ‘no one’ from the simpler, more universal sentences that contain ‘something’, ‘everything’, ‘anything’, and ‘nothing’. To formalize sentences that use ‘only’ as a quantifier, there are two options: For two-predicate sentences, and some more complex sentences, just use the converse of the related ‘all’ sentence. For more-complex sentences, ‘Only PQs are R’ is often best rendered as ‘All RQs are P’. The meaning of the sentence, in context, will help you decide between the two alternatives. Sentences with multiple quantifiers often have propositional operators as their main operator.

TELL ME MORE  • When is an ‘and’ an ‘or’? How can I tell existential statements from universal ones? See 6.6: Notes on Translation with M. • How are existential quantifications related to debates about existence? See 7S.10: Quantification and Ontological Commitment.

EXERCISES 4.2a Translate each sentence into predicate logic using the given translation keys. For exercises 1–8, use: Fx: x is a flower Ox: x is orange Px: x is pink Sx: x is fragrant 1. Some pink flowers are fragrant. 2. Some pink flowers are not fragrant. 3. All orange flowers are fragrant.

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4. No orange flowers are fragrant. 5. No flowers are both orange and pink. 6. Some flowers are both pink and fragrant. 7. Some flowers are neither orange nor pink. 8. All fragrant flowers are pink, if they are not orange. For exercises 9–16, use: Cx: x is hypercritical Fx: x is friendly Ix: x is intelligent Px: x is a person Sx: x succeeds 9. Some people are friendly. 10. Some people are intelligent, but not friendly. 11. Everyone friendly succeeds. 12. No one friendly is hypercritical. 13. All friendly and intelligent people succeed. 14. Someone intelligent succeeds if s/he is friendly. 15. Hypercritical people who are not friendly do not succeed. 16. If some friendly people are intelligent, then no hypercritical people succeed. For exercises 17–24, use: Cx: x is a cat Dx: x is a dog Ex: x has pointed ears Lx: x likes humans Wx: x has whiskers 17. Some cats have whiskers, but not pointed ears. 18. No cats are dogs. 19. Some cats and all dogs have pointed ears. 20. All cats like humans if, and only if, some dogs do not have whiskers. 21. All cats and dogs have whiskers. 22. Not all dogs and cats like humans.

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23. It is not the case that both some dogs with pointed ears like humans and no cats with whiskers like humans. 24. All cats have whiskers if, and only if, they have pointed ears. For exercises 25–32, use: t: Theodore Roosevelt Ax: x is American Cx: x went to an Ivy League college Gx: x is a good communicator Lx: x is a politician Px: x is a president of the United States 25. Some American politicians are good communicators. 26. A few American politicians went to Ivy League colleges, and Theodore Roose­ velt is one. 27. All presidents of the United States are American politicians. 28. Only American politicians are presidents of the United States. 29. Theodore Roosevelt is not a good communicator if, and only if, some presidents of the United States who went to Ivy League colleges are not good communicators. 30. Only good communicators are politicians. 31. Only American politicians who went to Ivy League colleges are good communicators. 32. American politicians went to Ivy League colleges if, and only if, they are good communicators. For exercises 33–42, use: t: two Ex: x is even Nx: x is a number Ox: x is odd Px: x is prime 33. Two is an even prime number. 34. Some prime numbers are even. 35. Not all prime numbers are odd. 36. If all prime numbers are odd, then two is not even.

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37. No number is neither odd nor even. 38. If all prime numbers are odd, then no prime numbers are even. 39. Some odd numbers are not prime if, and only if, not all even numbers are not prime. 40. It is not the case that only prime numbers are odd, but it is the case that no even numbers are odd. 41. Either some prime numbers are not even or no even numbers are not prime. 42. Even prime numbers are not odd just in case not all prime numbers are odd. For exercises 43–50, use: Ax: x is an animal Cx: x is a cow Gx: x is a goat Hx: x has horns Mx: x (is a kind of animal that) produces milk Wx: x is a whale 43. All animals with horns produce milk. 44. Not all animals that produce milk have horns. 45. Whales produce milk but don’t have horns. 46. All goats and some cows have horns. 47. Some goats and cows produce milk. 48. Some goats, some cows, and some whales produce milk. 49. No whales have horns if, and only if, it is not the case that all animals that produce milk have horns. 50. Goats and cows are animals that produce milk. For exercises 51–58, use: Cx: x is creative Hx: x is hard-working Ix: x is imaginative Mx: x is a poem Px: x is a poet Rx: x rhymes Sx: x is successful 51. Only short poems rhyme.

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52. All successful poets are either creative or hard-working. 53. Some successful poets are creative but not imaginative. 54. Not all poets are creative and hard-working. 55. Some successful poems are creative but do not rhyme. 56. Some unsuccessful poems are imaginative. 57. Some hard-working poets are unsuccessful just in case not all imaginative poems are creative. 58. If every creative, imaginative poem is successful just in case some hard-­working poets are unsuccessful, then no poem that doesn’t rhyme is successful. For exercises 59–68, use: Ax: x is absolutist Ex: x is an ethicist Ox: x is objective Px: x is a person Rx: x is a relativist 59. All absolutist ethicists are objective. 60. Only absolutist ethicists are objective. 61. Some people are neither absolutists nor relativists. 62. No one who is a relativist is an objective ethicist. 63. Ethicists are relativists only if they are not absolutists. 64. If an ethicist is not an absolutist, then s/he is a relativist. 65. No ethicist is an absolutist without being objective. 66. Non-relativist ethicists are objective if, and only if, they are absolutists. 67. If someone is an objective relativist, then everyone who is an ethicist is absolutist. 68. Someone who is absolutist is not an objective ethicist. For exercises 69–76, use: Ex: x is an existentialist Hx: x is a humanist Nx: x is a nihilist Px: x is a phenomenologist 69. Some existential phenomenologists are humanists.

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70. Some existential humanists are not phenomenologists. 71. Not all phenomenologists are existential humanists. 72. No nihilists are humanists, if some phenomenologists are not existentialists. 73. Either some humanist phenomenologist is a nihilist or no nihilist is a humanist. 74. All humanist phenomenologists are either existentialists or nihilists. 75. Humanist existentialists are phenomenologists if they are not nihilists. 76. Some existentialist is not a humanist just in case it is false that no humanist is a phenomenologist. For exercises 77–84, use: h: Hume s: Spinoza Bx: x is British Ex: x is an empiricist Rx: x is a rationalist Sx: x is a skeptic 77. Hume is a British empiricist, but Spinoza is neither. 78. Some, but not all, British empiricists are skeptics. 79. It is not the case that skeptic empiricists are all British. 80. Hume is a skeptic empiricist just in case no rationalist is a skeptic. 81. If Spinoza is a British empiricist, then Hume is not a skeptic and all rationalists are British. 82. There are no empiricist rationalists unless Hume is not a skeptic. 83. Any rationalist skeptic is either not a rationalist or not a skeptic. 84. All empiricists are skeptics if, and only if, either Spinoza is not a rationalist or some British skeptics are rationalists. For exercises 85–92, use: Cx: x is a compatibilist Dx: x is a determinist Lx: x is a libertarian

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Mx: x is a materialist Ox: x is a monist Px: x is a philosopher 85. No libertarian philosophers are determinists. 86. Monists are compatibilists if they are materialists. 87. Monists are compatibilists only if they are materialists. 88. If you’re a compatibilist, then you’re a determinist, but not a libertarian. 89. Either every material monist is a compatibilist or some material monists are determinists. 90. Some materialists are compatibilists, if some philosophers are monists, but not determinists. 91. No determinist who is not a materialist is a compatibilist. 92. If all materialist monists are compatibilists, if they are philosophers, then some libertarian philosophers are actually determinists. For exercises 93–100, use: k: Kant m: Mill Cx: x is a consequentialist Dx: x is a deontologist Kx: x is a Kantian Px: x is a philosopher Ux: x is a utilitarian 93. Kant is a deontologist and a Kantian, but Mill is neither. 94. Some philosophers are deontologists without being Kantian. 95. Not all consequentialists are utilitarian philosophers, but Mill is both. 96. Deontologists are Kantians, just in case only consequentialists are utilitarians. 97. If all utilitarians are consequentialists, then all Kantians are deontologists. 98. Kantians are not utilitarians only if they are deontologists and not consequentialists. 99. No deontologist is a Kantian unless she is not a utilitarian. 100. Some philosophers are Kantian deontologists if, and only if, they are neither consequentialists nor utilitarians.

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EXERCISES 4.2b Use the given interpretations to translate the following arguments written in predicate logic into natural, English sentences. Ax: x is an athlete Bx: x is brawny Cx: x is a champion m: Malik g: Gita n: Ned 1. 1. (∀x)(Ax ⊃ Bx) 2. Am ∙ An

/ Bm ∙ Bn

2. 1. (∀x)(Ax ⊃ Bx) 2. (∀x)(Bx ⊃ Cx) / (∀x)(Ax ⊃ Cx) 3. 1. (∀x)(Bx ⊃ Cx) 2. (∃x)(Ax ∙ Bx)

/ (∃x)(Ax ∙ Cx)

4. 1. (∀x)(Ax ⊃ Bx) 2. ∼Bm

/ (∃x)∼Ax

5. 1. (∀x)[Ax ⊃ (Bx ∨ Cx)] 2. Ag ∙ ∼Bg / Cg 6. 1. (∀x)[(Ax ∙ Bx) ⊃ Cx] 2. (∃x)(Bx ∙ ∼Cx) / (∃x)∼Ax 7. 1. (∃x)Ax ⊃ (∀x)(Cx ⊃ Bx) 2. (∃x)(Ax ∨ Bx) 3. (∀x)(Bx ⊃ Ax) / (∀x)(Cx ⊃ Ax) 8. 1. (∀x)[Bx ∨ (Cx ∙ Ax)] 2. ~Bg / ~(∀x)(Cx ⊃~Ax) 9. 1. Cg ∙ (∃x)Bx 2. ∼Am ⊃ (∀x)∼Cx / ∼[(∃x)Ax ⊃ ∼(∃x)Bx] 10. 1. (∀x)[Bx ∙ (Ax ∨ Cx)] 2. Cn ⊃ (∀x)∼(Ax ∨ Bx) 3. ∼(∃ x)Cx / ∼Cn

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4.3: SYNTA X FOR M This section presents the syntax of M more formally than the previous two sections do, emphasizing the technical vocabulary and the concept of the scope of a quantifier, essential for constructing derivations in M and understanding the formation rules. We’ll start, as is customary, with the vocabulary of our language. Vocabulary of M Capital letters A . . . Z used as one-place predicates Lower-case letters used as singular terms a, b, c, . . . u are used as constants. v, w, x, y, z are used as variables. Five operators: ∼, ∙, ∨, ⊃, ≡ Quantifier symbols: ∃, ∀ Punctuation: (), [], {} The next step is to specify formation rules for formulas (wffs) of M. In order to explain the formation rules and use quantifiers properly, one has to be sensitive to their scope. The quantifiers in 4.3.1 and 4.3.2 have different scope. 4.3.1 (∀x)(Px ⊃ Qx) 4.3.2 (∀x)Px ⊃ Qx

Every P is Q. If everything is P, then x is Q.

We have already tacitly seen the notion of scope in using negations. If what follows the tilde is a single propositional variable, then the scope of the negation is just that propositional variable. If what follows the tilde is another tilde, then the scope of the first (outside) negation is the scope of the second (inside) negation plus that inside tilde. If what follows the tilde is a bracket, then the entire formula that occurs between the opening and closing of that bracket is in the scope of the negation. 4.3.3

The scope o f an operator is its range of application.

The scope of a negation is whatever directly follows the tilde.

∼{(P ∙ Q) ⊃ [∼R ∨ ∼ ∼(S ≡ T)]}

There are four tildes in 4.3.3. The first one has the broadest scope. Since what follows it is a bracket, the rest of the formula, everything enclosed in the squiggly brackets, is in the scope of the leading negation. The second tilde in the formula, which occurs just in front of the ‘R’, has narrow scope. It applies only to the ‘R’. The third tilde in the formula has ‘∼ (S ≡ T)’ in its scope. The fourth tilde has ‘(S ≡ T)’ in its scope. Similarly, the scope of a quantifier is whatever formula immediately follows the quantifier. If what follows the quantifier is a bracket, then any formulas that occur until that bracket is closed are in the scope of the quantifier.

The scope of a quantifier is whatever formula immediately follows the quantifier.

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If what follows the quantifier is a tilde, then the tilde and every formula in its scope is in the scope of the quantifier. If what follows the quantifier is another quantifier, then the inside quantifier and every formula in the scope of the inside quantifier is in the scope of the outside quantifier. 4.3.4 (∀w){Pw ⊃ (∃x)(∀y)[(Px ∙ Py) ⊃ (∃z)∼(Qz ∨ Rz)]}

The scope of a quantifier may be wider or narrower. We can increase the scope by using punctuation. There are four quantifiers in the formula at 4.3.4. Their scopes are as follows. Quantifier

A bound variable is attached, or related, to a quantifier. A variable is bound by a quantifier when it is in the scope of the quantifier and they share a variable. A free variable is not bound by a quantifier.

A closed sentence h as no free variables. An open sentence has at least one free variable.

Scope

(∀w)

{Pw ⊃ (∃x)(∀y)[(Px ∙ Py) ⊃ (∃z)∼(Qz ∨ Rz)]}

(∃x)

(∀y)[(Px ∙ Py) ⊃ (∃z)∼(Qz ∨ Rz)]

(∀y)

[(Px ∙ Py) ⊃ (∃z)∼(Qz ∨ Rz)]

(∃z)

∼(Qz ∨ Rz)

Scope is important for quantifiers because it affects which variables are bound by the quantifier. When we construct derivations in predicate logic, we will often remove quantifiers from formulas. When we do so, the variables bound by those quantifiers will become unbound. Similarly, we will add quantifiers to the fronts of formulas, binding variables that are in their scopes. We will see some rules for removing and replacing quantifiers, unbinding and binding variables, in the next section, with a few further restrictions to follow. If we are not careful in using these rules, observant about binding and unbinding variables, invalid inferences can result. Quantifiers bind every instance of their variable in their scope. A bound variable is connected to the quantifier that binds it. In 4.3.1, the ‘x’ in ‘Qx’ is bound, as is the ‘x’ in ‘Px’. In 4.3.2, the ‘x’ in ‘Qx’ is not bound, though the ‘x’ in ‘Px’ is bound. An unbound variable is called a free variable. Wffs that contain at least one unbound variable are open sentences, as we saw in section 4.1. Examples 4.3.5–4.3.8 are all open sentences. 4.3.5 Ax 4.3.6 (∀x)Px ∨ Qx 4.3.7 (∃x)(Px ∨ Qy) 4.3.8 (∀x)(Px ⊃ Qx) ⊃ Rz

4.3.6, 4.3.7, and 4.3.8 contain both bound and free variables. In 4.3.6, ‘Qx’ is not in the scope of the quantifier, so is unbound. In 4.3.7, ‘Q y’ is in the scope of the quantifier, but ‘y’ is not the quantifier variable, so is unbound. In 4.3.8, ‘Rz’ is neither in the scope of the quantifier, nor does it contain the quantifier variable. If a wff has no free variables, it is a closed sentence, and expresses a proposition. 4.3.9 and 4.3.10 are closed sentences. Translations from English into M should ordinarily yield closed sentences.

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4.3.9 (∀y)[(Py ∙ Qy) ⊃ (Ra ∨ Sa)] 4.3.10 (∃x)(Px ∙ Qx) ∨ (∀y)(Ay ⊃ By)

We are ready for the formation rules. They are fairly straightforward, and you have probably already gleaned most of the important points from working with translation in sections 4.1 and 4.2. Formation Rules for Wffs of M M1. A predicate (capital letter) followed by a singular term (lower-case letter) is a wff. M2. For any variable β, if α is a wff that does not contain either ‘(∃β)’ or ‘(∀β)’, then ‘(∃β)α’ and ‘(∀β)α’ are wffs. M3. If α is a wff, so is ∼α. M4. If α and β are wffs, then so are: (α ∙ β) (α ∨ β) (α ⊃ β) (α ≡ β) M5. These are the only ways to make wffs. A few observations concerning the formation rules are in order. As we saw in section 4.1, quantifiers are operators like the five propositional operators. As with PL, the last operator added according to the formation rules is called the main operator. By convention, we continue to drop the outermost brackets which are required by rule M4. Again, those brackets are implicit and replaced if we augment the formula. A wff constructed using only rule M1 is called an atomic formula; atomic formulas lack operators. 4.3.11–4.3.13 are atomic formulas. Notice that an atomic formula can be closed (as in 4.3.11 and 4.3.12) or open (as in 4.3.13).

An atomic formula i n M is formed by a predicate followed by a singular term.

4.3.11 Pa 4.3.12 Qt 4.3.13 Ax

A wff that is part of another wff is called a subformula. The proposition in the first line of 4.3.14 has all of the formulas to its right as subformulas. 4.3.14 (Pa ∙ Qb) ⊃ (∃x)Rx Subformulas: Pa Qb Rx (∃x)Rx Pa ∙ Qb

Lastly on the formation rules, rule M2 contains a clause used to prevent overlapping quantifiers of the same type (i.e., using the same variable). This clause prevents us from constructing propositions like the ill-formed 4.3.15. 4.3.15 (∃x)[Px ∙ (∀x)(Qx ⊃ Rx)]

A subformula is a formula that is part of another formula.

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The terms ‘Qx’ and ‘Rx’ contain variables that appear to be bound by both the leading existential quantifier and the universal quantifier inside the proposition. In the first few sections of chapter 4, we won’t normally be tempted to construct such sentences. But after we introduce relational predicates, we will have to be very careful to avoid such overlapping.

How to Expand Our Vocabulary We are using only a small, finite stock of singular terms and quantifiers. It is customary to use a larger stock—in fact, an infinite stock. To generate an indefinite number of singular terms and quantifiers, we could use the indexing functions of subscripts and superscripts. We could introduce arabic numerals, say, into the language. Then, we could index each constant and variable so that we have indefinitely many of them. a1, a2, a3 . . . x1, x2, x3 . . .

Similarly, we can create an indefinite number of quantifiers by using the indexed variables. (∃x1), (∃x2), (∃x3) . . . (∀x1), (∀x2), (∀x3) . . .

More austere languages avoid introducing numbers by using different numbers of prime symbols to indicate different variables. a′, a″, a‴, a⁗ . . . x′, x″, x‴, x⁗ . . . (∃x′), (∃x″), (∃x‴), (∃x⁗) . . . (∀x′), (∀x″), (∀x‴), (∀x⁗) . . .

Both of these techniques quickly become unwieldy as it becomes difficult to discern the different terms. Since we are going to need only a few variables and constants, we can use a cleaner, if more limited, syntax, remembering that there is a technique to extend our vocabulary if we were to need it.

Summary As we explore the languages of predicate language, we will focus mainly on two central tasks: translation and derivation. Each time we extend our language, I will show the changes to the vocabulary and formation rules. It will be important, especially as we learn the derivation rules for predicate logic, to understand scope and binding, the central concepts in proofs for predicate logic. As with PL, it will also be important to quickly determine the main operator of a wff. The few exercises in this section are aimed at helping you master these important concepts in order to make the derivations easier.

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In 4.7 and 5.2, we will look at the semantics for our languages of predicate logic, which are distinctly more complicated than the mere truth tables of PL. The semantics will allow us also to demonstrate the invalidity of arguments in predicate logic. KEEP IN MIND

Atomic wffs of M are predicates together with a singular term. Atomic wffs may be modified by any of the logical operators. They may be preceded by a quantifier. They may be negated. Pairs of atomic wffs may be joined by any of the binary operators. The scope of a quantifier is whatever formula immediately follows the quantifier. Quantifiers bind variables of their type within their scope. A formula with a free variable is open. Translations should ordinarily yield closed formulas and express propositions. Quantifiers are the main operators of a proposition when they are the last elements added to a wff according to the formation rules, at the front of the wff. We can expand our vocabulary if we want more quantifiers or variables, but we won’t do so in this book.

TELL ME MORE  • Is truth a predicate? See 7.5: Truth and Liars. • Are there other ways to represent quantification? Can a logic have zero-place predicates? See 6S.10: Alternative Notations.

EXERCISES 4.3 For each of the following wffs of M, answer each of the following questions: A. For each quantifier in the sentence, which subformulas are in its scope? (List them all.) B. For each quantifier in the sentence, which variables are bound by the quantifier? C. Which variables in the sentence are free? D. Is the sentence open or closed? E. What is the main operator of the sentence? 1. (∃x)(Px ∙ Qx) 2. (∀x)[(Px ∙ Qx) ⊃ ∼Ra]

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3. (∀x)(Px ∙ Qx) ⊃ (∃x)[(Px ∨ Q y) ∨ Rx] 4. (∃x)Py 5. (∀x)Px ⊃ (Qx ∙ Ra) 6. ∼(∀x)[Px ∨ (∼Q y ∙ Rx)] 7. (∀y)(Pa ⊃ Qb) 8. (∃x)(Ry ∙ Qx) ∙ Pa 9. (∃x)(Rx ∙ ∼Qx) ≡ (∀x)(Px ⊃ Qa) 10. (Pa ∨ Qb) ⊃ Rc 11. (∀x)(Px ∨ Qx) ⊃ (∀y)(∼Q y ⊃ ∼Py) 12. (∃x){[(Px ∨ Rx) ∙ Q y] ⊃ (∀y)[(Rx ⊃ Q y) ∙ Pb]} 13. ∼(∀x)[(Px ≡ Rx) ⊃ Qa] 14. ∼(∃y)(Qx ∨ Px) 15. (∀x){(Px ∙ Q y) ⊃ (∃y)[(Ry ⊃ Sy) ∙ Tx]}

4.4: DERIVATIONS IN M In this section, we start to construct derivations in M. All of the twenty-five rules we used with PL continue to hold, governing the uses of the propositional operators. There are four new rules governing removing and adding quantifiers, the subjects of this section. In the next section, we will add a rule for exchanging the universal and existential quantifiers. Then, in section 4.6, we will look at how the methods of conditional and indirect proof must be modified for M. The general structure of most of the derivations of this section is first to take off quantifiers; second, to use the rules we already saw for PL; and last, to put quantifiers on. So, we need four rules, two for taking off each of the quantifiers and two for putting on each of the quantifiers.

Taking Off the Universal Quantifier Recall the valid argument at 4.1.1, which we can now fully regiment. 4.1.1

All philosophers are happy. Emily is a philosopher. So, Emily is happy.

(∀x)(Px ⊃ Hx) Pe He

In order to derive the conclusion, we need a rule that will allow us to remove the quantifier and to show that the conclusion follows as a simple matter of modus ponens.

4 . 4 : Der i v at i ons i n M   2 3 3

Rule #1: Universal Instantiation (UI) (∀α)Fα Fβ

Universal instantiation (UI) is the rule of

for any variable α, any formula F, and any singular term β

To use UI, we remove the leading universal quantifier, as long as it is the main operator. Then, we replace all occurrences of variables bound by that quantifier with either a variable (v, w, x, y, z) or a constant (a, b, c, . . . u). When instantiating, you must change all the bound variables in the same way. Thus, 4.4.1 can be instantiated as any formula in the list 4.4.2. 4.4.1 (∀x)[Sx ∨ (Pa ∙ Tx)] 4.4.2 Sa ∨ (Pa ∙ Ta) Sb ∨ (Pa ∙ Tb) Sx ∨ (Pa ∙ Tx) Sy ∨ (Pa ∙ Ty)

But 4.4.1 cannot be instantiated as 4.4.3 or as 4.4.4. 4.4.3 Sa ∨ (Pa ∙ Tb) 4.4.4 Sx ∨ (Pa ∙ Ta)

Let’s see how we use UI in the derivation of our original argument, at 4.4.5. 4.4.5 QED

1. (∀x)(Px ⊃ Hx) 2. Pe 3. Pe ⊃ He 4. He

/ He 1, UI 3, 2, MP

Putting on the Universal Quantifier All of the propositions in 4.4.6, both premises and the conclusion, contain quantifiers as main operators. To derive the conclusion, we will remove the quantifiers from each premise, make some inferences, and put on a quantifier at the end. 4.4.6

1. Everything happy is content. 2. No miser is content. So, no miser is happy.



1. (∀x)(Hx ⊃ Cx) 2. (∀x)(Mx ⊃ ∼Cx)

/ (∀x)(Mx ⊃ ∼Hx)

We have UI to guide our removal of the quantifiers. We just need a rule allowing us to put a universal quantifier on the front of a formula in a derivation. We might be tempted to introduce a rule such as 4.4.7. 4.4.7

Bad Universal Generalization Rule

Fa (∀x)F x

inference in predicate logic that allows us to take off a universal quantifer.

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To see why 4.4.7 is a bad generalization rule, consider the instance of it at 4.4.8. 4.4.8

1. Pa 2. (∀x)Px

Inferring a universal claim from an existential one commits the fallacy of hasty generalization.

Now, interpret ‘P’ as ‘is a professor’ and ‘a’ as ‘Asha’. 4.4.7 thus licenses the conclusion that everything is a professor from just the premise that Asha is a professor. Such an inference is called the fallacy of hasty generalization. Most of the restrictions on the instantiation and generalization rules are constructed precisely to avoid confusing our existential assertions with our universal ones, to prevent our making a strong universal conclusion on the bases of weak existential assumptions. To avoid hasty generalization, we never universally generalize (or quantify) over a constant. In other words, we may not replace a constant with a variable bound by a universal quantifier. This restriction keeps us from ever universally quantifying over individual cases. While we do not universally quantify over constants, we may do so over variables. Indeed, the point of introducing variables, and distinguishing them from constants, is to mark where universal generalization is permitted. Variables, except in circumstances we will introduce in section 4.6, retain universal character, even when they are unbound. Generalizing over them (i.e., binding them with a universal quantifier) does not commit a fallacy because the variable can stand for anything and everything.

Universal generalization (UG) is the rule of

Rule #2: Universal Generalization (UG)

inference in predicate logic that allows us to put a universal quantifier onto a formula.

Fβ (∀α)Fα for any variable β, any formula F not containing α, and any variable α

UG, like all of the instantiation and generalization rules, works only on whole lines: we place the universal quantifier in front of a statement so that the scope of the quantifier is the entire rest of the proposition. Further, we replace all occurrences of the variable over which we are quantifying with the variable in the quantifier: we bind all instances of the variable. You must replace all occurrences! 4.4.9 contains a proper use of UG. 4.4.9 1. (∀x)(Hx ⊃ Cx) 2. (∀x)(Mx ⊃ ∼Cx) 3. Hy ⊃ Cy 4. My ⊃ ∼Cy 5. ∼Cy ⊃ ∼Hy 6. My ⊃ ∼Hy 7. (∀x)(Mx ⊃ ∼Hx) QED

/ (∀x)(Mx ⊃ ∼Hx) 1, UI 2, UI 3, Cont 4, 5, HS 6, UG

Notice that I changed all the ‘x’s to ‘y’s when instantiating at lines 3 and 4. I could have kept the variables as ‘x’s or used any other variable. Notice also that I replaced the ‘y’s with ‘x’s at the end. I could have kept the ‘y’s, adding a universal quantifier using a ‘y’ at line 7, yielding ‘(∀y)(My ⊃ ∼Hy)’. Strictly

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speaking, this would not be a proof of the stated conclusion. But since the statements are equivalent, such a derivation would suffice. UI would have allowed us to instantiate either premise to constants. Indeed, the derivation could have proceeded through line 6 with all of the ‘y’s changed to ‘a’s or ‘b’s. But line 7 would not have been permitted by UG had the ‘y’s been constants.

Putting on the Existential Quantifier We now have rules for removing and putting on the universal quantifier. There are parallel rules for the existential quantifier. We will use the rule for existentially generalizing to facilitate the inference 4.4.10. 4.4.10

Oscar is a Costa Rican. So, there are Costa Ricans.

Co (∃x)Cx

Rule #3: Existential Generalization (EG)

Fβ (∃α)Fα  for any singular term β, any formula F not containing α, and for any variable α To use EG, place an existential quantifier in front of any proposition and change all occurrences of the singular term (constant or variable) over which you are quantifying with the quantifier letter. Unlike UG, which results in a strong, universal claim, EG is a weak inference and so can be made from any claim, whether concerning constants or variables. Quantifying over a variable allows us to infer an existential claim from a universal one. In an empty universe, such an inference would be invalid. But we ordinarily make the very weak assumption that the universe is not completely empty. Again, the resulting formula will have the quantifier you just added as the main operator. The derivation of the argument at 4.4.10 is trivial. 4.4.11 QED

1. Co 2. (∃x)Cx

/ (∃x)Cx 1, EG

Although it is rarely useful, you need not bind all instances of a singular term in the scope of the quantifier when you use EG, as at 4.4.12. 4.4.12

1. Pa ∙ Qa 2. (∃x)(Px ∙ Qa) 3. (∃y)(∃x)(Px ∙ Qy)

1, EG 2, EG

The third line does not imply the existence of two different things, though it might seem to. Instantiating the third line, we must use two different constants. But they might, for all we know, refer to the same thing. The parallel inference for UG is not valid.

Taking off the Existential Quantifier Our fourth rule for managing quantifiers allows us to remove an existential quantifier. As with UG, we need a restriction.

Existential generalization (EG) is the rule of

inference in predicate logic that allows us to put an existential quantifier onto a formula.

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4.4.13

All New Yorkers are Americans. Some New Yorkers are bald. So, some Americans are bald.

1. (∀x)(Nx ⊃ Ax) 2. (∃x)(Nx ∙ Bx) / (∃x)(Ax ∙ Bx)

In order to derive 4.4.13, we have to take off the ‘∃x’ in the second premise. The existential quantifier commits us to the existence of only one thing. So, when we take it off, we have to put on a constant. Moreover, we cannot have said anything earlier in the derivation about that constant; it has to be a new thing. If a constant appears in the premises, in a prior derived line, or even in the stated conclusion of the argument, you may not instantiate an existentially quantified statement to that constant. Existential instantiation (EI) is the rule of

inference in predicate logic that allows us to remove an existential quantifier from a formula.

A new constant is one that does not appear in either any earlier line of the argument or the desired conclusion.

Rule #4: Existential Instantiation (EI) (∃α)Fα Fβ for any variable α, any formula F, and any

new constant β

As with all four of the quantifier management rules, EI must be used only on whole lines. We remove the leading existential quantifier and replace all occurrences that were bound by the quantifier with the same, new constant, one that does not appear in either the premises or the desired conclusion. An existentially quantified sentence commits you only to the existence of some thing that has the property ascribed to it in the formula, and not to any particular thing that might have other properties inconsistent with those in the formula. To see further why a new constant is required, consider what would happen without that restriction, in the fallacious inference at 4.4.14. 4.4.14 Uh-oh!

1. (∃x)(Ax ∙ Cx) 2. (∃x)(Ax ∙ Dx) 3. Aa ∙ Ca 4. Aa ∙ Da 5. Ca ∙ Aa 6. Ca 7. Da ∙ Aa 8. Da 9. Ca ∙ Da 10. (∃x)(Cx ∙ Dx)

1, EI 2, EI: but wrong! 3, Com 5, Simp 4, Com 7, Simp 6, 8, Conj 9, EG

To see that 4.4.14 contains a fallacious inference, let’s interpret ‘Ax’ as ‘x is an animal’; ‘Cx’ as ‘x is a cat’ and ‘Dx’ as ‘x is a dog’. The first two premises are perfectly reasonable: there are cats, and there are dogs. The conclusion indicates the existence of a cat-dog. Whatever the advances in biogenetic engineering may be, we cannot infer the existence of a cat-dog from the existence of cats and the existence of dogs.

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Since EI contains a restriction whereas UI does not, in the common case in which you have to instantiate both universally quantified and existentially quantified propositions, EI before you UI. 4.4.15 contains an acceptable use of EI. 4.4.15 QED

1. (∀x)(Nx ⊃ Ax) 2. (∃x)(Nx ∙ Bx) 3. Na ∙ Ba 4. Na ⊃ Aa 5. Na 6. Aa 7. Ba ∙ Na 8. Ba 9. Aa ∙ Ba 10. (∃x)(Ax ∙ Bx)

/ (∃x)(Ax ∙ Bx) 2, EI 1, UI 3, Simp 4, 5, MP 3, Com 7, Simp 6, 8, Conj 9, EG

Which Singular Term Should I Use? When instantiating, you must decide whether to replace your bound variables with either constants or variables. Here is a chart to help you understand how to choose.

I’m taking off an ∃ (Using EI)

I’m taking off an ∀ (Using UI)

Use a constant. Make sure that your constant does not appear earlier in the proof: in the premises, in the desired conclusion, or in any earlier row.

It depends . . .

Remember: You can EI the same formula repeatedly as long as you use a new constant each time.

Remember: You can UI to any singular term at any time. So if you UI to the wrong singular term, you can just UI the same formula again.

Will you want to UG the terms of this wff later? Use a variable.

Do you want to connect the terms of this wff with those of an existentially quantified wff? Use a constant, and EI the other formula first. (EI before you UI.)

Instantiation and Generalization Rules and Whole Lines Like the rules of inference in chapter 3, all four rules of instantiation and generalization may be used only on whole lines. We can use them only when the main operator of a formula is a quantifier. For example, in 4.4.16, line 2 cannot be instantiated; to use MP, we have to get the antecedent of line 2 on a line by itself.

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4.4.16 QED

1. (∀x)(Dx ∙ Ex) 2. (∀x)Dx ⊃ Fa 3. Dx ∙ Ex 4. Dx 5. (∀x)Dx 6. Fa 7. (∃x)Fx

/ (∃x)Fx 1, UI 3, Simp 4, UG 2, 5, MP 6, EG

Similarly, we cannot take off either quantifier in line 1 of 4.4.17. 4.4.17 1. (∀x)(Jx ∨ Kx) ⊃ (∃y)Ly 2. (∀x)(Jx ∨ Lx) 3. (∀x)(∼Lx ∨ Kx) 4. Jx ∨ Lx 5. ∼ ∼Jx ∨ Lx 6. ∼Jx ⊃ Lx 7. ∼Lx ∨ Kx 8. Lx ⊃ Kx 9. ∼Jx ⊃ Kx 10. ∼ ∼Jx ∨ Kx 11. Jx ∨ Kx 12. (∀x)(Jx ∨ Kx) 13. (∃y)Ly 14. La 15. (∃x)Lx QED

/ (∃x)Lx 2, UI 4, DN 5, Impl 3, UI 7, Impl 6, 8, HS 9, Impl 10, DN 11, UG 1, 12, MP 13, EI 14, EG

Instantiating the Same Quantifier Twice You may instantiate the same quantifier twice, including the existential quantifier. When the quantifier is universal, as in 4.4.18, there are no restrictions on instantiating it. 4.4.18 QED

1. (∀x)(Mx ⊃ Nx) 2. (∀x)(Nx ⊃ Ox) 3. Ma ∙ Mb 4. Ma ⊃ Na 5. Ma 6. Na 7. Mb ⊃ Nb 8. Mb ∙ Ma 9. Mb 10. Nb 11. Nb ⊃ Ob 12. Ob 13. Na ∙ Ob

/ Na ∙ Ob 1, UI 3, Simp 4, 5, MP 1, UI 3, Com 8, Simp 7, 9, MP 2, UI 11, 10, MP 6, 12, Conj

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When a quantifier is existential, the second instantiation must go to a new constant, as in line 9 of 4.4.19. 4.4.19 QED

1. (∃x)(Px ∙ Qx) 2. (∀x)(Px ⊃ Rx) 3. (∀x)(Qx ⊃ Sx) 4. Pa ∙ Qa 5. Pa 6. Pa ⊃ Ra 7. Ra 8. (∃x)Rx 9. Pb ∙ Qb 10. Qb ∙ Pb 11. Qb 12. Qb ⊃ Sb 13. Sb 14. (∃x)Sx 15. (∃x)Rx ∙ (∃x)Sx

/ (∃x)Rx ∙ (∃x)Sx 1, EI 4, Simp 2, UI 6, 5, MP 7, EG 1, EI 9, Com 10, Simp 3, UI 12, 11, MP 13, EG 8, 14, Conj

It may seem odd that we can instantiate an existential quantifier twice when the use of an existential quantifier only commits you to a single thing having a given property. That odd feeling should be removed by remembering that objects may have more than one name. We do not often instantiate an existential sentence more than once in M, but we do use this ability in full predicate logic (5.3), especially using identity (5.5).

Summary The four rules of inference in this section allow you to take off quantifiers and put them back on. Once the quantifiers are off, proofs generally proceed according to the rules of PL. Great care must be paid in order not to misuse the instantiation and generalization rules. Students first using these rules are sometimes not as sensitive as they should be to the differences between constants and variables. A proof can look perfectly fine, and use all of the PL rules well, and yet make serious errors of using constants when one must use variables, or vice versa. Be careful not to instantiate parts of lines. The exercises for this section mainly illustrate the instantiation and generalization rules and so mainly contain premises and conclusions that have quantifiers as the main operators. In the next section, we will work with propositions whose main operators are not quantifiers, but the propositional operators, and you will have to take care not to instantiate (or generalize) errantly.

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KEEP IN MIND

Pay close attention to the application conditions for each of the four rules, whether they hold for just constants, just variables, or for any singular term. EG and UI are anytime, anywhere rules; they have no restrictions. EI and UG require care: Never EI to a variable; always use a new constant. We need new constants when using EI in order not to confuse our claims about particular objects. A new constant is one that appears nowhere earlier in the derivation, not even in the stated conclusion. Never UG from (i.e., over) a constant. The restrictions on EI and UG are grounded mainly in avoiding hasty generalization. Constants may be replaced only by existentially quantified variables. Unbound variables are available for universal generalization. If you want to make inferences that connect existential and universal claims, EI before you UI.

Rules Introduced Universal Instantiation (UI) (∀α)Fα Fβ for any variable α, any formula F, and any

singular term β

Universal Generalization (UG) Fβ (∀α)Fα for any variable β, any formula F not

containing α, and any variable α Never UG over a constant.

Existential Instantiation (EI) (∃α)Fα Fβ for any variable α, any formula F, and any

new constant β Never EI to a variable.

Existential Generalization (EG) Fβ (∃α)Fα  for any singular term β, any formula F not containing α, and for any variable α

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TELL ME MORE  • How might inferences in predicate logic be used in science? See 7S.8: Logic and Science.

EXERCISES 4.4a Derive the conclusions of the following arguments. 1. 1. (∀x)(Ax ⊃ Bx) 2. (∀x)(Cx ⊃ ∼Bx) 3. Aa

/ ∼Ca

2. 1. (∀x)(Ax ⊃ Bx) 2. (∀x)(Cx ⊃ ∼Bx)

/ (∀x)(Cx ⊃ ∼Ax)

3. 1. (∃x)(Dx ∙ ∼Ex) 2. (∀x)(Ex ∨ Fx)

/ (∃x)Fx

4. 1. (∃x)(Ax ∙ ∼Bx) 2. (∀x)(Cx ⊃ Bx)

/ (∃x)(Ax ∙ ∼Cx)

5. 1. (∀x)Hx ∨ Ja 2. (∀x)[(∼ Jx ∙ Ix) ∨ (∼ Jx ∙ Kx)] / (∀x)Hx 6. 1. (∀x)(Jx ∙ Kx)

/ (∃x)Jx ∙ (∃x)Kx

7. 1. (∃x)(Px ∙ Qx) 2. (∃x)(Rx ∙ Sx)

/ (∃x)Px ∙ (∃x)Rx

8. 1. (∀x)(Fx ∨ Hx) ⊃ (∃x)Ex 2. (∀x)[Fx ∨ (Gx ∙ Hx)]

/ (∃x)Ex

9. 1. (∀x)(Ix ⊃ Kx) 2. (∀x)(Jx ⊃ Lx) 3. (∃x)(Jx ∨ Ix)

/ (∃x)(Kx ∨ Lx)

10. 1. (∀x)[Gx ⊃ (Hx ∨ Ix)] 2. (∃x)(Gx ∙ ∼Ix)

/ (∃x)(Gx ∙ Hx)

11. 1. (∀x)(Dx ⊃ Ex) 2. (∀x)(Ex ⊃ ∼Gx) 3. (∃x)Gx

/ (∃x) ∼Dx

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12. 1. (∀x)(Ox ⊃ Qx) 2. (∀x)(Ox ∨ Px) 3. (∃x)(Nx ∙ ∼Qx)

/ (∃x)(Nx ∙ Px)

13. 1. (∀x)[Ax ⊃ (Bx ∨ Cx)] 2. (∃x)∼(Bx ∨ ∼Ax)

/ (∃x)Cx

14. 1. (∃x)(Tx ∙ Ux) ⊃ (∀x)Vx 2. (∃x)[(Wx ∙ Tx) ∙ Ux]

/ (∀x)Vx

15. 1. (∃x)(Fx ∙ Hx) ≡ Gb 2. Gb

/ Fa

16. 1. (∀x)(Fx ≡ Gx)

/ (∀x)(Fx ⊃ Gx) ∙ (∀x)(Gx ⊃ Fx)

17. 1. (∀x)Ax ⊃ Ba 2. (∀x)∼(Ax ⊃ Cx)

/ (∃x)Bx

18. 1. (∃x)Lx ≡ Nb 2. (∃x)[(Lx ∙ Mx) ∙ Ox]

/ (∃x)Nx

19. 1. (∀x)(Mx ⊃ Nx) 2. (∀x)(Ox ⊃ Px) 3. (∀x)[Mx ∨ (Ox ∙ Qx)]

/ (∀x)(Nx ∨ Px)

20. 1. (∀x)(Lx ≡ Nx) 2. (∀x)(Nx ⊃ Mx) 3. (∀x)∼(Mx ∨ Ox)

/ (∃x)∼Lx

21. 1. (∃x)(Dx ∙ Fx) 2. (∃x)(Gx ⊃ Ex) 3. (∀x)∼(Hx ∨ Ex)

/ (∃x)Fx ∙ (∃x)∼Gx

22. 1. (∃x)[(Sx ∨ Tx) ∙ Ux] 2. (∀x)(Ux ⊃ ∼Sx)

/ (∃x)∼Sx ∙ (∃y)(Uy ∙ Ty)

23. 1. (∃x)(∼Tx ∙ Ux) ≡ (∀x)Wx 2. (∀x)(Tx ⊃ Vx) 3. (∃x)(Ux ∙ ∼Vx)

/ (∀x)Wx

24. 1. (∀x)[Lx ⊃ (Bx ∨ Ux)] 2. (∃x)(∼Bx ∙ ∼Ux) 3. (∀x)(Lx ≡ Ax)

/ (∃x)(~Ax ∨ Sx)

25. 1. (∀x)(Bx ≡ Fx) 2. (∃x)∼(∼Gx ∨ Cx) 3. (∀x)(∼Bx ⊃ Cx)

/ (∃x)Fx

26. 1. ∼(∀x)Mx 2. (∃x)Sx ∨ (∀x)(Qx ⊃ Tx) 3. (∀x)(Qx ⊃ Tx) ≡ (∀x)Mx / Sa

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27. 1. (∀x)(Mx ⊃ Nx) 2. (∃x)(∼Nx ∙ Ox) 3. (∃x)∼Mx ⊃ (∃x) ∼Ox

/ (∃x)Ox ∙ (∃x)∼Ox

28. 1. (∀x)(Px ∨ Qx) ≡ Rc 2. (∀x) ∼(Sx ∨ ∼Qx)

/ (∃x)Rx

29. 1. (∃x)Qx ≡ (∃x)Sx 2. (∀x)(Rx ∨ Sx) 3. (∃x)∼(Rx ∨ Qx)

/ Qb

30. 1. (∃x)Ax ⊃ (∀x)Cx 2. (∀x)(∼Bx ⊃ Dx) 3. (∀x)(Bx ⊃ Ax) 4. (∃x)∼(Dx ∨ ∼Cx)

/ (∀x)Cx

31. 1. (∃x)Kx ⊃ (∀x)(Lx ⊃ Mx) 2. (∀x)∼(Kx ⊃ ∼Lx) 3. (∀x)∼Mx

/ (∃x)∼Lx

32. 1. (∃x)[Ix ∨ (Hx ∨ Jx)] 2. (∀x) ∼(∼Ix ⊃ Jx) 3. (∀x) ∼(Hx ∙ Kx)

/ (∃x)∼Kx

33. 1. (∀x)(Ox ⊃ Mx) ⊃ (∃x)Nx 2. (∀x)Mx

/ (∃x)∼(∼Nx ∨ ∼Mx)

34. 1. (∀x)∼[∼(∼Px ∙ Mx) ⊃ (Px ∙ ∼Mx)] /(∀x)(Mx ≡ Px) 35. 1. (∀x)(Lx ⊃ ∼Nx) ∙ (∀x)(∼Mx ⊃ ∼Ox) 2. (∀x)∼(∼Nx ∙ ∼Ox) / (∀x)(Lx ⊃ Mx) 36. 1. (∀x)(Dx ∙ Ex) 2. (∃x)(∼Fx ∨ Gx)

/ (∃x)[(Dx ≡ Ex) ∙ (Fx ⊃ Gx)]

37. 1. (∀x)(Rx ≡ Tx) 2. (∃x)(Tx ∙ ∼Sx) 3. (∀x) [Sx ∨ (Rx ⊃ Ux)]

/ (∃x)Ux

38. 1. (∃x)( Jx ≡ Kx) ⊃(∀x)(Ix ∙ Lx) 2. (∀x)[(Ix ∙ Jx) ⊃ Kx] 3. (∃x)∼(Ix ⊃ Kx)

/ (∀y)Ly

39. 1. (∀x)(Kx ⊃ ~Lx) 2. (∃x)Jx ⊃ Ib 3. (∃x)[ Jx ∨ (Kx ∙ Lx)]

/ (∃x)(Hx ∨ Ix)

40. 1. (∀x)Tx ⊃ [(∀x)(Qx ∨ Sx) ⊃ (∀x)Rx] 2. (∀x)~(Tx ⊃ ~Sx) / (∃x)Rx

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EXERCISES 4.4b Translate each of the following paragraphs into arguments written in M, using the given translation key. Then, derive the conclusions of the arguments using the four quantifier rules, plus the rules of inference and equivalence for PL. 1. Some students are teenagers. Everything is either not a teenager or not a child. So, some students are not children. (Cx: x is a child; Sx: x is a student; Tx: x is a teenager) 2. If there are black holes, then there are star clusters. S5 0014+81 is a black hole.1 All star clusters are gravitationally bound. So, something is gravitationally bound. (a: S5 0014+81; Bx: x is a black hole; Gx: x is gravitationally bound; Sx: x is a star cluster) 3. Someone is either an elephant or a badger. No one is a badger. If there are elephants, then there are tusks. So, there are tusks. (Bx: x is a badger; Ex: x is an elephant; Px: x is a person; Tx: x is a tusk) 4. Some prime numbers are either Mersenne primes or semiprimes. Things are prime if, and only if, they are not composite. Semiprimes are composite. So, some prime numbers are Mersenne primes. (Cx: x is composite; Mx: x is a Mersenne prime; Px: x is a prime number; Sx: x is a semiprime) 5. Things are cats just in case they are feline. No feline is canine. There are cats. So, something is not canine. (Cx: x is canine; Fx: x is feline; Mx: x is a cat) 6. All trains run on tracks. Trains that run on tracks lack steering wheels. No cars lack steering wheels. Some trains are purple. So, some trains aren’t cars. (Lx: x lacks a steering wheel; Px: x is purple; Rx: x runs on tracks; Tx: x is a train) 7. If Shangri-La and the Shire exist, then so does Sodor. Anything that’s Sodor has tank engines. Nothing with tank engines has real people. But Utopia is Shangri-La and i Drann is the Shire. So, Sodor exists and does not have real people. (i: i Drann; u: Utopia; Lx: x is Shangri-La; Px: x has real people; Rx: x is the Shire; Sx: x is Sodor; Tx: x has tank engines)

1

S5 0014+81 is actually the name of a “blazar, in fact an FSRQ quasar, the most energetic subclass of objects known as active galactic nuclei, produced by the rapid accretion of matter by a central supermassive black hole,” according to its Wikipedia entry,  June 9, 2016. But let’s take it as the name of the black hole itself here.

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8. Someone is a composer but does not get paid. Others are composers and work in Hollywood. Anyone who works in Hollywood gets paid. So, some people get paid and some don’t. (Cx: x is a composer; Gx: x gets paid; Px: x is a person; Wx: x works in Hollywood) 9. All treatises are books. No journal article is a book. So, everything is either not a treatise or not a journal article. (Bx: x is a book; Jx: x is a journal article; Tx: x is a treatise) 10. All fallacies seem valid, if they resemble formal inferences. But nothing that seems valid is valid. So, everything valid, if it resembles a formal inference, is not a fallacy. (Fx: x is a fallacy; Rx: x resembles a formal inference; Sx: x seems valid; Vx: x is valid) 11. Some intuitions are reliable. Nothing reliable is obviously false. If some intuition is not obviously false, then there are useful epistemologies. So, there are useful epistemologies. (Ix: x is an intuition; Ox: x is obviously false; Rx: x is reliable; Ux: x is a useful epistemology) 12. There is a thing that is either a utilitarian or a Kantian. Any utilitarian is a consequentialist. Any Kantian is a deontologist. If something is either a consequentialist or a deontologist, then something is a moral theorist. So, something is a moral theorist. (Cx: x is a consequentialist; Dx: x is a deontologist; Kx: x is a Kantian; Mx is a moral theorist; Ux: x is a utilitarian) 13. All empiricists make sense experience primary. No rationalist does. And everything is either an empiricist or a rationalist. So, everything is an empiricist just in case it is not a rationalist. (Ex: x is an empiricist; Rx: x is a rationalist; Sx: x makes sense experience primary) 14. Everything good is beautiful and hard work. If something is hard work or rewarding, then it is worth pursuing. So, the good is worth pursuing. (Bx: x is beautiful; Gx: x is good; Hx: x is hard work; Rx: x is rewarding; Wx: x is worth pursuing) 15. Everything good is beautiful and hard work. If something is hard work, then either you do it yourself or you ask someone else to do it for you. Nothing you do yourself is beautiful. So, anything good you ask someone else to do for you. (Ax: you ask someone else to do x for you; Bx: x is beautiful; Gx: x is good; Hx: x is hard work; Yx: you do x yourself) 16. If some philosophers are existentialists, then some are nihilists. There are hermeneuticist philosophers. All philosophers are hermeneuticists just in case they are existentialists. All nihilist philosophers are empowering. So, something is empowering. (Ex: x is an existentialist; Hx: x is a hermeneuticist; Nx: x is a nihilist; Px: x is a philosopher; Sx: x is empowering)

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EXERCISES 4.4c Find the errors in each of the following illicit inferences. Some of the arguments are valid; some are not. All derivations contain errors. (We’ll show the invalid ones to be invalid in Exercises 4.8b.)   1. 1. (∀x)(Px ⊃ Qx) 2. (∃x)(Px ∙ Rx) 3. Px ∙ Rx 4. Px ⊃ Qx 5. Px 6. Qx 7. Rx ∙ Px 8. Rx 9. Qx ∙ Rx 10. (∃x)(Qx ∙ Rx) QED—Oops!  ​2. 1. (∃x)(Px ∙ Rx) 2. Pa ∙ Qa 3. Pa ∙ Ra 4. Qa ∙ Pa 5. Qa 6. Ra ∙ Pa 7. Ra 8. Qa ∙ Ra 9. (∃x)(Qx ∙ Rx) QED—Oops!  ​3. 1. (∃x)Px ⊃ (∃x)Qx 2. Pa 3. Pa ⊃ (∃x)Qx 4. (∃x)Qx QED—Oops!  ​4. 1. (∀x)(Px ⊃ Qx) 2. Pa ⊃ Qa 3. (∃x)(Px ⊃ Qx) QED—Oops!

/ (∃x)(Qx ∙ Rx) 2, EI 1, UI 3, Simp 4, 5, MP 3, Com 7, Simp 6, 8, Conj 9, EG

/ (∃x)(Qx ∙ Rx) 1, EI 2, Com 4, Simp 3, Com 6, Simp 5, 7, Conj 8, EG

/ (∃x)Qx 1, EI 3, 2, MP / (∃x)(Px ∙ Qx) 1, UI 2, EG

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  5. 1. (∃x)(Px ∙ Qx) 2. (∀x)(Px ⊃ Rx) 3. Pa ∙ Qa 4. Pa ⊃ Ra 5. Pa 6. Ra 7. Ra ∨ ∼Qa 8. ∼Qa ∨ Ra 9. Qa ⊃ Ra 10. (∀x)(Qx ⊃ Rx) QED—Oops!  ​6. 1. (∀x)(Px ⊃ Qx) 2. (∃x)(Qx ⊃ Rx) 3. Qx ⊃ Rx 4. Px ⊃ Qx 5. Px ⊃ Rx 6. (∀x)(Px ⊃ Rx) QED—Oops!

/ (∀x)(Qx ⊃ Rx) 1, EI 2, UI 3, Simp 4, 5, MP 6, Add 7, Com 8, Impl 9, UG

/ (∀x)(Px ⊃ Rx) 2, EI 1, UI 4, 3, HS 5, UG

 ​7. 1. (∀x)[Px ⊃ (Qx ≡ Rx)] 2. (∃x)(Px ∙ ∼Qx) 3. Pa ⊃ (Qa ≡ Ra) 4. Pa ∙ ∼Qa 5. Pa 6. Qa ≡ Ra 7. ∼Qa ∙ Pa 8. ∼Qa 9. ∼Ra 10. Pa ∙ ∼Ra 11. (∃x)(Px ∙ ∼Rx) QED—Oops!

/ (∃x)(Px ∙ Rx) 1, UI 2, EI 4, Simp 3, 5, MP 4, Com 7, Simp 6, 8, BMT 5, 9, Conj 10, EG

 ​8. 1. (∀x)(Px ⊃ Qx) ⊃ [(∃ x)Px ⊃ (∃x)Qx] 2. (∀x)(Px ⊃ Qx) 3. (∃x)Px 4. (∀x)(Px ⊃ Qx) ⊃ (∃ x)Qx 5. (∃x)Qx QED—Oops!

/ (∃x)Qx 1, 3, MP 4, 2, MP

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 ​9. 1. (∀x)Px ⊃ (∀x)Qx 2. (∃x)Px 3. Pa 4. Pa ⊃ Qa 5. Qa 6. (∃x)Qx QED—Oops! 10. 1. (∃x)(∼Px ∨ Qx) 2. (∀x)(∼Px ⊃ Qx) 3. ∼Pa ∨ Qa 4. Qa ∨ ∼Pa 5. ∼ ∼ Qa ∨ ∼Pa 6. ∼Qa ⊃ ∼Pa 7. ∼Pa ⊃ Qa 8. ∼Qa ⊃ Qa 9. ∼ ∼Qa ∨ Qa 10. Qa ∨ Qa 11. Qa QED—Oops!

/ (∃x)Qx 2, EI 1, UI 4, 3, MP 5, EG

/ Qa 1, EI 3, Com 4, DN 5, Impl 2, UI 6, 7, HS 8, Impl 9, DN 10, Taut

4.5: QUANTIFIER EXCHANGE The rules for removing and replacing quantifiers that we saw in the last section allow us to make many inferences in predicate logic. But some inferences need more machinery. Consider the argument at 4.5.1 and a natural expression of it in M. 4.5.1

All successful football players are hard-working. But, not all football players are hard working. So, not everything is successful.

1. (∀x)[(Fx ∙ Sx) ⊃ Hx] 2. ∼(∀x)(Fx ⊃ Hx)

/ ∼(∀x)Sx

We must remove the quantifier in the second premise of 4.5.1 to derive the conclusion. But the quantifier is not the main operator of that proposition, and so we cannot instantiate the premise as it stands. Further, we will want to put a quantifier on some proposition near the end of the derivation. But it’s unclear how we are going to sneak the quantifier in between the tilde and the ‘Sx’ in the conclusion. We need some rules for managing the interactions between quantifiers and negations. We already saw, in section 4.1, that there were alternative ways of translating sentences with quantifiers and negations. For example, 4.5.2 can be translated naturally as either 4.5.3 or 4.5.4.

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4.5.2 No apples are blueberries. 4.5.3 ∼(∃x)(Ax ∙ Bx) 4.5.4 (∀x)(Ax ⊃ ∼Bx)

As we will see in this section, every proposition that has a negation in front of a quantifier, like 4.5.3, is equivalent to another proposition in which the quantifier is the main operator, like 4.5.4. The two different quantifiers in predicate logic, the existential and the universal, are inter-definable. Indeed, some systems of logic take only one quantifier as fundamental and introduce the other by definition. We can see the relationship between the existential and universal quantifiers in natural language by considering the following four pairs of equivalent claims. 4.5.5 4.5.5′

Everything is made of atoms. It’s not the case that something is not made of atoms.

4.5.6 4.5.6′

Something is made of atoms. It’s wrong to claim that nothing is made of atoms.

4.5.7 4.5.7′

Nothing is made of atoms. It’s false that something is made of atoms.

4.5.8 4.5.8′

At least one thing isn’t made of atoms. Not everything is made of atoms.

Take your time to recognize that each pair above contains two different ways of saying the same thing. We can represent the equivalence of each pair in predicate logic, as I do at 4.5.9. 4.5.9 (∀x)Ax (∃x)Ax (∀x)∼Ax (∃x)∼Ax

is equivalent to is equivalent to is equivalent to is equivalent to

∼(∃x)∼Ax ∼(∀x)∼Ax ∼(∃x)Ax ∼(∀x)Ax

I’ll generalize those equivalences with the rule of quantifier exchange (QE). QE allows us to replace any expression of one of the above forms with its logical equivalent. Like rules of equivalence, QE is based on logical equivalence, rather than validity, and thus may be used on part of a line. Quantifier Exchange (QE) (∀α)Fα (∃α)Fα (∀α)∼Fα (∃α)∼Fα

→ ← → ← → ← → ←

∼(∃α)∼Fα ∼(∀α)∼Fα ∼(∃α)Fα ∼(∀α)Fα

QE appears as four rules. But we can really think of them as one more general rule. Consider the following three spaces: 1. The space directly before the quantifier 2. The quantifier itself 3. The space directly following the quantifier

Quantifier exchange (QE) is a rule of equivalence in

predicate logic.

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QE says that to change a quantifier, you change each of the three spaces. Add or remove a tilde directly before the quantifier. Switch quantifiers: existential to universal or vice versa. Add or remove a tilde directly after the quantifier. For example, in 4.5.10, we have a negation in front of a universal quantifier, but no negation directly after it. 4.5.10

∼(∀x)(Px ⊃ Qx)

Using quantifier exchange, we can transform 4.5.10 into 4.5.11, removing the leading quantifier (the main operator of 4.5.10), changing the universal quantifier to an existential quantifier, and adding a negation immediately following the existential quantifier. 4.5.11 (∃x)∼(Px ⊃ Qx)

We can also transform 4.5.12 into 4.5.13 by adding a negation in front, where there is none, changing the existential quantifier to a universal quantifier, and adding a negation directly after the quantifier where again there is none. 4.5.12 (∃x)(Px ∙ Qx) 4.5.13 ∼(∀x)∼(Px ∙ Qx)

Wffs like 4.5.11 and 4.5.13 may seem unnatural; we would rarely translate a sentence of English into forms like either of those. But a few uses of propositional rules of equivalence within those formulas can transform them into wffs that would be the obvious results of translations from natural language, as we will see in the next subsection.

Some Transformations Permitted by QE Understanding the relation between the existential and universal quantifiers facilitates some natural transformations, like between 4.5.14 and 4.5.15, as the following derivation shows. 4.5.14 4.5.15

It’s not the case that every P is Q. Something is P and not Q.

1. ∼(∀x)(Px ⊃ Qx) 2. (∃x)∼(Px ⊃ Qx) 3. (∃x)∼(∼Px ∨ Qx) 4. (∃x)(∼ ∼Px ∙ ∼Qx) 5. (∃x)(Px ∙ ∼Qx)

∼(∀x)(Px ⊃ Qx) (∃x)(Px ∙ ∼Qx)

Premise 1, QE 2, Impl 3, DM 4, DN

Similarly, 4.5.16, 4.5.17, and 4.5.18 are all equivalent, as the derivation given below them shows. 4.5.16 4.5.17 4.5.18

It’s not the case that something is both P and Q. Everything that’s P is not Q (or, no Ps are Qs). Everything that’s Q is not P (or, no Qs are Ps).

∼(∃x)(Px ∙ Qx) (∀x)(Px ⊃ ∼Qx) (∀x)(Qx ⊃ ∼Px)

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1. ∼(∃x)(Px ∙ Qx) 2. (∀x)∼(Px ∙ Qx) 3. (∀x)(∼Px ∨ ∼Qx) 4. (∀x)(Px ⊃ ∼Qx) 5. (∀x)(∼ ∼Qx ⊃ ∼Px) 6. (∀x)(Qx ⊃ ∼Px)

Premise 1, QE 2, DM 3, Impl 4, Cont 5, DN

Most of the proofs we have been doing require some instantiation and/or generalization. Now that we have QE available, we can derive arguments that require no removal or replacement of quantifiers, like 4.5.19. 4.5.19 1. (∃x)Lx ⊃ (∃y)My 2. (∀y)∼My / ∼La 3. ∼(∃y)My 2, QE 4. ∼(∃x)Lx 1, 3, MT 5. (∀x)∼Lx 4, QE 6. ∼La 5, UI QED

Note that in 4.5.19 you cannot existentially instantiate line 4. You may use EI only when it is the main operator on a line. On line 4, the main operator is the tilde. Thus, you must use QE before instantiating. On line 4, the quantifier is existential. But on line 5, it is universal; the rule for instantiating that claim is UI, not EI. Let’s return to 4.5.1, the conclusion of which I’ll derive at 4.5.20. The argument does not appear, at first glance, to have an existential premise. But since the main operator at line 2 is a tilde in front of a quantifier, in order to instantiate, we first must use QE on that formula. Using QE yields, after some quick transformations, the existential sentence at line 6. Then, we EI (line 7) before we UI (line 10). 4.5.20 1. (∀x)[(Fx ∙ Sx) ⊃ Hx] 2. ∼(∀x)(Fx ⊃ Hx) 3. (∃x)∼(Fx ⊃ Hx) 4. (∃x)∼(∼Fx ∨ Hx) 5. (∃x)(∼ ∼Fx ∙ ∼Hx) 6. (∃x)(Fx ∙ ∼Hx) 7. Fa ∙ ∼Ha 8. ∼Ha ∙ Fa 9. ∼Ha 10. (Fa ∙ Sa) ⊃ Ha 11. ∼(Fa ∙ Sa) 12. ∼Fa ∨ ∼Sa 13. Fa 14. ∼ ∼Fa 15. ∼Sa 16. (∃x)∼Sx 17. ∼(∀x)Sx QED

/ ∼(∀x)Sx 2, QE 3, Impl 4, DM 5, DN 6, EI 7, Com 8, Simp 1, UI 10, 9, MT 11, DM 7, Simp 13, DN 12, 14, DS 15, EG 16, QE

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Summary The rules of quantifier exchange (QE) allow us to manage the interactions between negations and quantifiers. They allow us to instantiate some wffs in which the main operator is not the quantifier but the negation; just make sure to use QE to change the wff so that the quantifier is the main operator before instantiating. QE also allows us to use the rules governing the propositional operators to make inferences with propositions whose main operators are neither negations nor quantifiers, especially in propositions with multiple quantifiers.

KEEP IN MIND

Never instantiate a quantifier if it is not the main operator in a wff. If the main operator is a negation followed by a quantifier, use QE before instantiating. Whether to use UI or EI to instantiate depends on the quantifier once it is the main operator. All four QE rules can be summarized in one procedure. Just change each of three spaces: Add or remove a tilde directly before the quantifier. Switch quantifiers: existential to universal or vice versa. Add or remove a tilde directly after the quantifier.

Rules Introduced Quantifier Exchange (QE) (∀α)Fα (∃α)Fα (∀α)∼Fα (∃α)∼Fα

→ ← → ← → ← → ←

∼(∃α)∼Fα ∼(∀α)∼Fα ∼(∃α)Fα ∼(∀α)Fα

EXERCISES 4.5a Derive the conclusions of each of the following arguments. Do not use CP or IP. 1. 1. (∀x)Ax ⊃ (∃x)Bx 2. (∀x)∼Bx

/ (∃x)∼Ax

2. 1. (∃x)[Qx ∙ (Rx ∙ ∼Sx)] / ∼(∀x)Sx 3. 1. (∀x)Xx ⊃ (∀x)Yx 2. (∃x)∼Yx

/ (∃x)∼Xx

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4. 1. (∃x)(Px ∙ Qx) 2. ∼(∃ x)(Px ∙ Rx)

/ (∃x)(Px ∙ ∼Rx)

5. 1. (∀x)(Dx ⊃ Ex) 2. ∼(∀x)(Dx ⊃ Fx)

/ (∃x)(Ex ∙ ∼Fx)

6. 1. (∃x)[(Gx ∙ Hx) ∙ Ix] 2. ∼(∃x)(Ix ∙ Jx)

/ (∃x)(Hx ∙ ∼Jx)

7. 1. (∀x)(Px ⊃ Qx) 2. (∀x)(Rx ⊃ ∼Qx) / ∼(∃x)(Px ∙ Rx) 8. 1. (∃x)Sx ⊃ (∃x)Tx 2. (∀x)∼Tx

/ (∀x)∼Sx

9. 1. (∃x)(Xx ∙ Yx) ⊃ (∃x)(Xx ∙ Zx) 2. (∀x)(Xx ⊃ ∼Zx) / ∼(∃x)(Xx ∙ Yx) 10. 1. (∀x)(Ax ⊃ Bx) ⊃ (∀x)(Ax ⊃ Cx) 2. (∃x)(Ax ∙ ∼Cx) / (∃x)(Ax ∙ ∼Bx) 11. 1. (∃x)(Tx ∙ ∼Vx) 2. (∃x)(Tx ∙ Vx)

/ ∼(∀x)(Tx ⊃ Vx) ∙ ∼(∀x)(Tx ⊃ ∼Vx)

12. 1. (∃x)∼Fx ∨ (∀x)(Gx ∙ Hx) 2. (∀x)[(Fx ∙ Gx) ∨ (Fx ∙ Hx)] / (∃x)(Gx ∙ Hx) 13. 1. ∼(∀x)(Qx ⊃ Rx) 2. (∀x)(∼Rx ⊃ Tx)

/ ∼(∀x)∼Tx

14. 1. (∀x)[Lx ∨ (Mx ∙ ∼Nx)] 2. ∼(∃x)Lx / ∼(∃x)(Lx ∨ Nx) 15. 1. (∀x)(Ax ∨ Bx) 2. (∀x)(Ax ⊃ Dx) 3. ∼(∀x)(Bx ∙ ∼Cx)

/ (∃y)(Dy ∨ Cy)

16. 1. ∼(∃x)(Ox ≡ Px) 2. Pa

/ ∼(∀x)Ox

17. 1. (∃x)(Px ∙ Qx) ⊃ (∃x)(Px ∙ Rx) 2. (∀x)(Px ⊃ ∼Rx) / (∀x)(Qx ⊃ ∼Px) 18. 1. ∼(∃x)(Lx ∙ ∼Mx) 2. ∼(∃x)(Mx ∙ Nx)

/ ∼(∃x)(Lx ∙ Nx)

19. 1. ∼(∃x)[(Px ∙ Qx) ∙ ∼Rx] 2. ∼(∃x)(Rx ∙ ∼Sx) / ∼(∃x)[(Px ∙ Qx) ∙ ∼Sx]

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20. 1. (∀x)(Tx ≡ ∼Vx) 2. (∃x)Vx 3. ∼(∀x)Tx ⊃ (∃x)Wx

/ (∃x)Wx

21. 1. ∼(∃x)(Rx ∨ Sx) ∨ (∀x)(Tx ⊃ ∼Rx) 2. Ra

/ ∼(∀x)Tx

22. 1. (∀x)[(Tx ∙ Ux) ⊃ Vx] 2. ∼(∀x)∼Tx / ∼(∀x)(Ux ∙ ∼Vx) 23. 1. (∃x)[Px ∙ (Rx ∙ ∼Sx)] 2. (∀x)[Qx ⊃ (Rx ⊃ Sx)]

/ ∼(∀x)(∼Px ∨ Qx)

24. 1. ∼(∀x)[Kx ⊃ (Lx ⊃ Mx)] 2. (∀x)[(Nx ∙ Ox) ≡ Mx] / ∼(∀x)(Nx ∙ Ox) 25. 1. ∼(∃x)[Ex ∙ (Fx ∨ Gx)] 2. (∀x)[Hx ⊃ (Ex ∙ Gx)] 3. (∃x)[∼Hx ⊃ (Ix ∨ Jx)]

/ (∃x)(∼Ix ⊃ Jx)

26. 1. ∼(∀x)[(Jx ∙ Kx) ∙ Lx] 2. (∀x)(Mx ⊃ Jx) 3. (∀x)(∼Nx ∙ Mx)

/ ∼(∀x)(Kx ∙ Lx)

27. 1. (∃x)(Nx ∨ ∼Ox) 2. ∼(∀x)(Px ∙ Qx) ∙ ∼(∃x)(Nx ∨ ∼Qx) / ∼[(∀x)Px ∨ (∀x)Ox] 28. 1. (∃x)[Ax ∙ (Bx ∨ Cx)] ⊃ (∀x)Dx 2. ∼(∀x)(Ax ⊃ Dx)

/ ∼(∀x)Cx

29. 1. ∼[(∃x)(Ax ∨ Bx) ∙ (∀x)(Cx ⊃ Dx)] 2. ∼(∀x)(∼Ax ∨ Ex)

/ (∃x)Cx

30. 1. (∃x)Px ≡ (∀x)(Qx ⊃ ∼Rx) 2. (∃x)[Qx ∙ (Rx ∨ Sx)] 3. ∼(∃x)Sx

/ (∀x)∼Px

31. 1. (∀x)[(Ax ∨ Bx) ⊃ ∼Cx] 2. (∀x)(Dx ⊃ Cx)

/ ∼(∃x)(Ax ∙ Dx)

32. 1. (∀x)(Kx ≡ ∼Lx) 2. ∼(∃x)(Lx ≡ ∼Mx) / ∼(∃x)(Mx ≡ Kx) 33. 1. (∃x)(Sx ∙ ∼Tx) ⊃ ∼(∀x)[Px ⊃ ∼(Qx ∨ Rx)] 2. (∀x)(Vx ⊃ Sx) 3. ∼(∀x)(Vx ⊃ Tx) 4. ∼(∃x)(Px ∙ Qx) / (∃x)(Px ∙ Rx) 34. 1. (∃x)[(Ax ∨ Cx) ⊃ Bx] 2. ∼(∃x)(Bx ∨ Ex) 3. (∃x)(Dx ⊃ Ex) ⊃ (∀x)(Ax ∨ Cx)

/ (∃y)Dy

4 . 5 : Q u ant i f i er E x c h ange   2 5 5

35. 1. (∃x)(Mx ∙ ∼Nx) ⊃ (∀x)(Ox ∨ Px) 2. ∼(∀x)(∼Nx ⊃ Ox) 3. ∼(∃x)Px / ∼(∀y)My 36. 1. (∀x)(Ex ∙ Fx) ∨ ∼(∀x)[Gx ⊃ (Hx ⊃ Ix)] 2. ∼(∀x)(Jx ⊃ Ex) / ∼(∀y) Iy 37. 1.∼(∃x)( Jx ∙ ∼Kx) 2. ∼(∃x)[Kx ∙ (∼Jx ∨ ∼Lx)]

/ (∀x)(Jx ≡ Kx)

38. 1. (∀x)(Fx ⊃ Hx) ∨ ∼(∃x)(Gx ≡ Ix) 2. (∃x)[Fx ∙ (∼Hx ∙ Ix)] / ∼(∀x)Gx 39. 1. ∼(∃x)[Px ∙ (Qx ∙ Rx)] 2. ∼(∀x)[∼Rx ∨ (Sx ∙ Tx)] 3. (∀x)(Px ∙ Qx) ∨ (∀x)(Tx ⊃ Rx) / ∼(∃x)(Tx ∙ ∼Rx) 40. 1. (∀x)[Ex ⊃ (Fx ∨ Gx)] 2. ∼(∃x)[Ex ∙ (Fx ∙ Gx)] 3. (∀x)(Hx ⊃ Ex)

/ (∀x){Hx ⊃ [(Fx ∨ Gx) ∙ ∼(Fx ∙ Gx)]}

EXERCISES 4.5b Translate each of the following arguments into propositions of M. Then, derive the conclusions of the arguments. 1. Everyone is weird. But not everyone is nice. So, some weird things aren’t nice. (Nx: x is nice; Px: x is a person; Wx: x is weird) 2. If there are gods, then everything is determined. But something is not determined. So, everything is not a god. (Dx: x is determined; Gx: x is a god) 3. Nothing blue is edible. This Sour Patch Kid is blue food. So, not all food is edible. (s: this Sour Patch Kid; Bx: x is blue; Ex: x is edible; Fx: x is food) 4. Someone in the class doesn’t keep up with the reading. Anyone who doesn’t keep up with the reading has trouble understanding the classwork. It is not the case that someone who has trouble understanding the classwork doesn’t struggle with the final. So, someone in the class struggles with the final. (Cx: x is in the class; Fx: x struggles with the final; Kx: x keeps up with the reading; Px: x is a person; Ux: x has trouble understanding the classwork) 5. All new phones have lots of memory and large screens. Not every new phone lacks a screen protector. So, not everything with a large screen lacks a screen protector. (Lx: x has a large screen; Mx: x has lots of memory; Px: x is a new phone; Sx: x has a screen protector)

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6. Any fruit on the table is either a strawberry or has a pit. Some fruits on the tables are apples. It is not the case that some apples are strawberries. So, it is not the case that no apples have a pit. (Ax: x is an apple; Fx: x is a fruit on the table; Px: x has a pit; Sx: x is a strawberry) 7. Everything is an Earthling just in case it is not an alien. It is false that some politicians are aliens from Mars. So, all politicians from Mars are Earthlings. (Ax: x is an alien; Ex: x is an Earthling; Mx: x is from Mars; Px: x is a politician) 8. No rock stars have bad hair. It is not the case that some rock stars lack amplifiers. Not every rock star has either devoted fans or a functioning website. So, not everything with amplifiers but not bad hair has either devoted fans or a functioning website. (Ax: x has amplifiers; Fx: x has devoted fans; Hx: x has bad hair; Rx: x is a rock star; Wx: x has a functioning website) 9. Some philosophers are A-theorists. It is not the case that some A-theorist doesn’t overvalue the present. So, some philosophers overvalue the present. (Ax: x is an A-theorist; Ox: x overvalues the present; Px: x is a philosopher) 10. Every Hegelian idealist believes in the transcendent. Not everything believes in the transcendent. So, not everything is a Hegelian idealist. (Hx: x is a Hegelian; Ix: x is an idealist; Tx: x believes in the transcendent) 11. All ethicists are utilitarians if, and only if, they are consequentialists. Not every ethicist is a utilitarian. So, not everything is a consequentialist. (Cx: x is a consequentialist; Ex: x is an ethicist; Ux: x is a utilitarian) 12. If all beliefs are grounded in sense experience, then some beliefs are abstract. All beliefs are mental states. It is not the case that some mental states are not grounded in sense experience. And it is not the case that something abstract is not ineffable. So, some beliefs are ineffable. (Ax: x is abstract; Bx: x is a belief; Ix: x is ineffable; Mx: x is a mental state; Sx: x is grounded in sense experience) 13. All existentialists are either nihilists or theists. All theists have faith. Not all existentialists have faith. So, it is not the case that no existentialists are nihilists. (Ex: x is an existentialist; Fx: x has faith; Nx: x is a nihilist; Tx: x is a theist) 14. Neither everything is material nor some people are zombies. It’s not the case that something is both not a zombie and not material. So, not everything is a person. (Mx: x is material; Px: x is person; Zx: x is a zombie) 15. All philosophers are determinists if, and only if, they are not libertarians. Not all philosophers are either determinists or nihilists. It is not the case that some libertarians are pessimists and not nihilists. So, not everything is either a determinist or a pessimist. (Dx: x is a determinist; Lx: x is a libertarian; Nx: x is a nihilist; Px: x is a philosopher; Sx: x is a pessimist)

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16. All empiricists either believe in abstract ideas or do not believe that we have mathematical knowledge. It is not the case that some empiricists who believe in abstract ideas are fictionalists. It is not the case that some empiricists who do not believe that we have mathematical knowledge approve of the calculus. So, it is not the case that some empiricists both are fictionalists and approve of the calculus. (Ax: x believes in abstract ideas; Cx: x approves of the calculus; Ex: x is an empiricist; Fx: x is a fictionalist; Mx: x believes that we have mathematical knowledge)

4.6: CONDITIONAL AND INDIRECT PROOF IN M The rules for instantiating and generalizing and the rules of quantifier equivalence are the main rules for predicate logic, whether monadic, in this chapter, or full, in chapter 5. I’ll add a few rules governing the identity predicate in section 5.5. There are some important restrictions on the rules as we refine our techniques and extend our language. But we’ve already seen most of the rules. For PL we had, in addition to our rules of inference and equivalence, three different derivation methods: direct, conditional, and indirect. To this point, we have used only direct proof with predicate logic. But the conditional and indirect derivation methods work just as well in M as they did in PL, with one small restriction. The restriction arises from considering the unrestricted and fallacious derivation 4.6.1. 4.6.1 1. (∀x)Rx ⊃ (∀x)Bx Premise 2. Rx ACP 3. (∀x)Rx 2, UG: but wrong! 4. (∀x)Bx 1, 3, MP 5. Bx 4, UI 6. Rx ⊃ Bx 2–5, CP 7. (∀x)(Rx ⊃ Bx) 6, UG Uh-oh!

Allowing line 7 to follow from the premise at line 1 would be wrong. We can show that the inference is invalid by interpreting the predicates. Let’s take ‘Rx’ to stand for ‘x is red’ and ‘Bx’ to stand for ‘x is blue’. 4.6.1 would allow the inference of ‘Everything red is blue’ (the conclusion) from ‘If everything is red, then everything is blue’ (the premise). But that premise can be true while the conclusion is false. Indeed, since it is not the case that everything is red, the first premise is vacuously true; it is a conditional with a false antecedent. But the conclusion is clearly false: it is not the case that all red things are blue. So, the derivation should be invalid. We must restrict conditional proof.

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All lines of an indented sequence are within the

scope of an assumptionin

the first line.

The problem with 4.6.1 can be seen at step 3. The assumption for conditional proof at line 2 just means that a random thing has the property denoted by ‘R’, not that everything has that property. While variables ordinarily retain their universal character in a proof, when they are used within an assumption (for CP or IP), they lose that universal character. It is as if we are saying, “Imagine that some (particular) thing has the property ascribed in the assumption.” If it follows that the object in the assumption also has other properties, we may universally generalize after we’ve discharged, as in line 7, for we have not made any specific claims about the thing outside of the assumption. Using conditional proof in this way should be familiar to mathematics students. Often in mathematics we will show that some property holds of a particular example. Then we claim, without loss of generality, that since our example was chosen arbitrarily, whatever we derived using our assumption holds universally. Within the assumption, we have a particular example and we treat it existentially. Once we are done with that portion of the proof, we can treat our object universally. Consider an indirect proof of some universally quantified formula, ‘(∀x)α’. To begin the proof, we assume its opposite: ‘∼(∀x)α’. We can then change that assumption, using QE, to ‘(∃x)∼α’. In other words, we start an indirect proof of a universal claim with an existential assertion: let’s say that something is not α. Another way to do such an indirect proof would be to assume ‘∼α’ immediately. We could do this by making the free variables in α constants or variables. Either way, they have to act as constants within the assumption, so we must not use UG within the assumption on those singular terms. Whenever we use CP or IP, we start by indenting, drawing a vertical line, and making an assumption. All lines of the proof until we discharge the assumption are also indented, indicating that they are within the scope of the assumption in the first line of the indented sequence. To summarize the restriction, we may not UG on a variable within the scope of an assumption in which that variable is free. Once the assumption is discharged, the restriction is dismissed and you may UG on the variable. This restriction holds on both CP and IP, though it would be unusual to use IP with a free variable in the first line. Addendum to the rule of inference UG: Within the scope of an assumption for conditional or indirect proof, never UG on a variable that is free in the assumption.

Derivations in Predicate Logic with CP There are two typical uses of CP in predicate logic. The first way, using CP when you want to derive a wff whose main operator is a ⊃, is rather obvious and entails no complications or restrictions. When you desire such a proposition, just assume the whole antecedent to prove the whole consequent, as in 4.6.2.

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4.6.2 1. (∀x)[Px ⊃ (Qx ∙ Rx)] 2. (∀x)(Rx ⊃ Sx) / (∃x)Px ⊃ (∃x)Sx 3. (∃x)Px ACP 4. Pa 3, EI 5. Pa ⊃ (Qa ∙ Ra) 1, UI 6. Qa ∙ Ra 5, 4, MP 7. Ra ∙ Qa 6, Com 8. Ra 7, Simp 9. Ra ⊃ Sa 2, UI 10. Sa 9, 8, MP 11. (∃x)Sx 10, EG 12. (∃x)Px ⊃ (∃x)Sx 3–11, CP QED

The other typical use of CP within predicate logic involves assuming the antecedent of the conditional we ordinarily find inside a universally quantified formula. Since universally quantified propositions ordinarily have conditional subformulas, CP can be useful. But in such cases, the typical assumption will have a free variable in the first line of an indented sequence, so we must be aware of the restriction on UG within the scope of an assumption, as in 4.6.3. 4.6.3 1. (∀x)[Ax ⊃ (Bx ∨ Dx)] 2. (∀x)∼Bx 3. Ay 4. Ay ⊃ (By ∨ Dy) 5. By ∨ Dy 6. ∼By 7. Dy 8. Ay ⊃ Dy 9. (∀x)(Ax ⊃ Dx) QED

/ (∀x)(Ax ⊃ Dx) ACP 1, UI 4, 3, MP 2, UI 5, 6, DS 3–7, CP 8, UG

In 4.6.3, at line 3, we pick a random object that has property A. From lines 3 to 7, we show that given any object, if it has A, then it has D; we make that claim at step 8. Then, at line 9, since we are no longer within the scope of the assumption, we may use UG. Thus, to prove statements of the form (∀x)(αx ⊃ βx), we use the method sketched at 4.6.4. 4.6.4 Assume αx Derive βx Discharge (αx ⊃ βx) UG to get your desired conclusion: (∀x)(αx ⊃ βx)

Derivations in Predicate Logic with IP Indirect proof ordinarily works just as it did in propositional logic, as you can see in 4.6.5.

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4.6.5 1. (∀x)[(Ax ∨ Bx) ⊃ Ex] 2. (∀x)[(Ex ∨ Dx) ⊃ ∼Ax] / (∀x)∼Ax 3. ∼(∀x)∼Ax AIP 4. (∃x)Ax 3, QE 5. Aa 4, EI 6. ∼ ∼Aa 5, DN 7. (Ea ∨ Da) ⊃ ∼Aa 2, UI 8. ∼(Ea ∨ Da) 7, 6, MT 9. ∼Ea ∙ ∼Da 8, DM 10. ∼Ea 9, Simp 11. (Aa ∨ Ba) ⊃ Ea 1, UI 12. ∼(Aa ∨ Ba) 11, 10, MT 13. ∼Aa ∙ ∼Ba 12, DM 14. ∼Aa 13, Simp 15. Aa ∙ ∼Aa 5, 14, Conj 16. ∼ ∼(∀x)∼Ax 3–15, IP 17. (∀x)∼Ax 16, DN QED

With CP, we sometimes assume only part of a line and then generalize outside the assumption. With IP, we almost always assume the negation of the whole conclusion, as in line 3 of 4.6.5. Remember, after you make your assumption, you’re looking for any contradiction. A contradiction may be an atomic formula and its negation, or it may be a more complex formula and its negation. It can contain quantifiers, or not. But be sure to maintain our definition of a contradiction as any statement of the form α ∙ ∼α. In particular, do not make the mistake of thinking that statements of the form at 4.6.6 are contradictions; they are not. 4.6.6 (∃x)Px ∙ (∃x)∼Px

There is nothing contradictory about something having a property and something not having that property. Some things are red and other things are not; some things have wings and others do not. Statements like 4.6.7 are not, strictly speaking, contradictions, though contradictions may easily be derived from them. 4.6.7 (∀x)Px ∙ (∀x)∼Px

4.6.8 is a list of typical contradictions in predicate logic. 4.6.8 (∃x)(Px ∙ Qx) ∙ ∼(∃x)(Px ∙ Qx) (∀x)(Px ⊃ Qx) ∙ ∼(∀x)(Px ⊃ Qx) (∃x)Px ∙ ∼(∃x)Px (∀x)Px ∙ ∼(∀x)Px Pa ∙ ∼Pa Px ∙ ∼Px

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Logical Truths of M Just as CP and IP allowed us to use our proof theory to prove that some formulas of PL were logical truths, these methods allow us to prove that some formulas of M, like 4.6.9, are logical truths. 4.6.9 (∀x)(Px ∨ ∼Px) 1. ∼(∀x)(Px ∨ ∼Px) AIP 2. (∃x)∼(Px ∨ ∼Px) 1, QE 3. ∼(Pa ∨ ∼Pa) 2, EI 4. ∼Pa ∙ ∼ ∼Pa 3, DM 5. ∼ ∼(∀x)(Px ∨ ∼Px) 1–4, IP 6. (∀x)(Px ∨ ∼Px) 5, DN QED

4.6.10–4.6.13 are further logical truths of M. Note that each one has a similarity to one of the four rules for removing or replacing quantifiers. 4.6.10 (∀y)[(∀x)Fx ⊃ Fy] 4.6.11 (∀y)[Fy ⊃ (∃x)Fx] 4.6.12 (∃y)[Fy ⊃ (∀x)Fx] 4.6.13 (∃y)[(∃x)Fx ⊃ Fy]

I’ll prove the first, at 4.6.14, leaving the others for Exercises 4.6c. 4.6.14 1. ∼(∀y)[(∀x)Fx ⊃ Fy] 2. (∃y)∼[(∀x)Fx ⊃ Fy] 3. (∃y)∼[∼(∀x)Fx ∨ Fy] 4. (∃y)[∼ ∼(∀x)Fx ∙ ∼Fy] 5. (∃y)[(∀x)Fx ∙ ∼Fy] 6. (∀x)Fx ∙ ∼Fa 7. (∀x)Fx 8. Fa 9. ∼Fa ∙ (∀x)Fx 10. ∼Fa 11. Fa ∙ ∼Fa 12. (∀y)[(∀x)Fx ⊃ Fy] QED

AIP 1, QE 2, Impl 3, DM 4, DN 5, EI 6, Simp 7, UI 6, Com 9, Simp 8, 10, Conj 1–11, IP

Summary In PL, we first showed that statements were logical truths semantically by using the truth tables to show that they were tautologies. We can show that statements are logical truths of M semantically, too, though the semantics for predicate logic are more complicated; we’ll deal with them in the next two sections, after which we’ll be able to show that arguments are invalid, too.

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KEEP IN MIND

The conditional and indirect derivation methods are useful in predicate logic, though there is an important restriction on UG within any indented sequence. Within the scope of an assumption for conditional or indirect proof, never UG on a variable that is free in the assumption. You may UG on a variable that is free in an assumption after the assumption is discharged. Conditional proof is especially useful for deriving universally quantified conclusions or for deriving conditional conclusions. Indirect proof is often used just as in PL, by assuming the opposite of your desired conclusion. Be sure to maintain our strict sense of ‘contradiction’ for the last line of an indirect proof. Either CP or IP is useful in proving logical truths of M.

TELL ME MORE  • How can we use conditional proofs to show that different translations are logically equivalent? See 6.6: Notes on Translation with M. • How are the logical truths of this section related to the tautologies of PL? See 6S.9: Logical Truth, Analyticity, and Modality.

EXERCISES 4.6a Derive the conclusions of the following arguments. 1. 1. (∀x)(Ax ⊃ Bx) 2. (∀x)∼(Bx ∙ ∼Cx)

/ (∀x)(Ax ⊃ Cx)

2. 1. (∀x)(Dx ∨ Ex) 2. (∀x)(Fx ⊃ ∼Ex)

/ (∀x)(∼Dx ⊃ ∼Fx)

3. 1. (∀x)(Gx ≡ ∼Hx) 2. (∀x)(Ix ⊃ Hx)

/ (∃x)Ix ⊃ (∃x)∼Gx

4. 1. (∀x)[Mx ⊃ (Nx ∙ Ox)] 2. (∃x)∼Nx / (∃x)∼Mx 5. 1. (∀x)[Px ⊃ (Qx ∙ Rx)] 2. (∀x)(Qx ⊃ Sx)

/ (∀x)(Px ⊃ Sx)

6. 1. (∀x)(Tx ≡ ∼Vx) 2. (∀x)[Vx ⊃ (Wx ∙ Xx)] / (∀x)(∼Tx ⊃ Xx) 7. 1. (∀x)(Ax ⊃ Cx) 2. ∼(∃x)(Bx ∙ ∼Cx)

/ (∀x)[(Ax ∨ Bx) ⊃ Cx]

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8. 1. (∃x)[(Dx ∙ Ex) ∙ ∼Fx] 2. (∀x)(Gx ⊃ Fx)

/ ∼(∀x)(Ex ⊃ Gx)

9. 1. (∀x)[Hx ≡ (Ix ∨ Jx)] 2. ∼(∃x) Jx

/ (∀x)(Hx ≡ Ix)

10. 1. (∀x)(Gx ⊃ Hx) 2. ∼(∃x)(Ix ∙ ∼Gx) 3. (∀x)(∼Hx ⊃ Ix)

/ (∀x)Hx

11. 1. (∀x)(Rx ⊃ Ux) 2. ∼(∃x)(Ux ∙ Sx)

/ (∃x)Rx ⊃ (∃x)∼Sx

12. 1. (∀x)[Ax ⊃ (Dx ∨ Ex)] 2. (∀x)[(∼Dx ⊃ Ex) ⊃ (∼Cx ⊃ Bx)] / (∀x)[Ax ⊃ (Bx ∨ Cx)] 13. 1. (∀x)[∼Nx ∨ (Qx ∙ Rx)] 2. (∀x)(Px ≡ Qx)

/ (∃x)Nx ⊃ (∃x)Px

14. 1. (∀x)(Ox ⊃ Nx) 2. (∀x)(Nx ⊃ Px) 3. ∼(∃x)(Px ∨ Qx)

/ (∀x)∼Ox

15. 1. (∀x)[(Fx ∨ Gx) ⊃ Ix] 2. (∀x)[(Ix ∙ Ex) ⊃ Gx]

/ (∀x)[Ex ⊃ (Fx ⊃ Gx)]

16. 1. (∀x)[Sx ⊃ (∼Tx ∨ ∼Rx)] 2. (∀x)(Ux ⊃ Sx)

/ (∃x)(Rx ∙ Tx) ⊃ (∃x)(∼Sx ∙ ∼Ux)

17. 1. (∀x)(Ex ≡ Hx) 2. (∀x)(Hx ⊃ ∼Fx)

/ (∀x)Ex ⊃ ∼(∃x)Fx

18. 1. (∀x)(Cx ⊃ Ax) 2. (∃x)∼Bx ⊃ (∀x)Cx

/ (∃x)(Ax ∨ Bx)

19. 1. (∀x)[ Jx ⊃ (∼Kx ⊃ ∼Lx)] 2. (∃x)(Jx ∙ ∼Kx) / ∼(∀x)Lx 20. 1. (∃x)Ax ⊃ ∼(∀x)Bx 2. (∃x)Cx ⊃ (∀x)Bx 3. (∀x)Ax ∨ (∀x)∼Cx / ∼(∃x)Cx 21. 1. (∀x)[(Px ∨ Qx) ≡ Rx] 2. (∀x)(Rx ⊃ Sx) 3. ∼(∃x)(Sx ∙ ∼Px)

/ (∀x)(Px ≡ Rx)

22. 1. (∀x)[Ax ⊃ (Cx ∙ Dx)] 2. (∃x)(Bx ∙ ∼Cx) / ∼(∀x)(Ax ≡ Bx) 23. 1. (∀x)[ Jx ⊃ (Mx ∙ Lx)] 2. (∀x)[(∼Kx ∨ Nx) ∙ (∼Kx ∨ Lx)]

/ (∀x)[( Jx ∨ Kx) ⊃ Lx]

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24. 1. (∀x)(Ix ⊃ Kx) 2. (∀x)(Lx ⊃ Jx) 3. ∼(∃x)(∼Kx ⊃ Jx)

/ ∼(∃x)[Ix ∨ (Lx ∙ Mx)]

25. 1. (∀x)[Fx ⊃ (Dx ∙ ∼Ex)] 2. (∀x)(Fx ⊃ Hx) 3. (∃x)Fx / ∼(∀x)(Dx ⊃ Ex) ∨ (∃x)[Fx ∙ (Gx ∙ Hx)] 26. 1. (∀x)(Ax ⊃ Bx) 2. (∀x)(Dx ⊃ Cx) 3. (∃x)(Ax ∨ Dx)

/ (∃x)Bx ∨ (∃x)Cx

27. 1. (∃x)Xx ⊃ (∀x)(Yx ≡ Zx) 2. (∃x)Yx

/ (∀x)[Xx ⊃ (∃y)Zy]

28. 1. (∃x)(Sx ∨ Tx) 2. (∃x)(Ux ⊃ ∼Vx) 3. (∃x)Tx ⊃ (∀x)Ux / ∼(∀x)(∼Sx ∙ Vx) 29. 1. (∀x)[(Lx ∙ Ix) ⊃ ∼Kx] 2. (∀x)[Mx ∨ ( Jx ∙ Nx)] 3. (∀x)(Kx ⊃ ∼Mx) 4. (∃x)(Ix ∙ Kx)

/ ∼(∀x)(Jx ⊃ Lx)

30. 1. ∼(∃x)(Dx ∙ ∼Ex) 2. (∀x)(Fx ⊃ Gx) 3. ∼(∃x)(Gx ∙ Ex)

/ ∼(∃x)(Dx ∙ Fx)

31. 1. (∀x)[Dx ≡ (∼Ex ∙ ∼Fx)] 2. (∀x)(Gx ⊃ Ex) 3. (∀x)[∼(Gx ∨ Fx) ⊃ Hx]

/ (∀x)(Dx ⊃ Hx)

32. 1. (∃x)(Px ∙ ∼Qx) ⊃ (∀x)(∼Rx ⊃ Sx) 2. (∃x)(∼Qx ∙ ∼Rx) / ∼(∃x)Sx ⊃ ∼(∀x)Px 33. 1. (∃x)[Fx ∨ (Gx ∨ Hx)] 2. (∀x)[∼Jx ⊃ (∼Fx ∙ ∼Hx)] 3. (∀x)(∼Gx ⊃ ∼Jx)

/ (∃x)Gx

34. 1. (∃x)Ax ⊃ ∼(∀x)Cx 2. (∃x)Bx ⊃ ∼(∀x)Dx 3. (∃x)∼Cx ⊃ (∀x)(Ex ⊃ Fx) 4. (∃x)∼Dx ⊃ (∀x)(Fx ⊃ Gx)

/ (∃x)(Ax ∙ Bx) ⊃ ∼(∃x)(Ex ∙ ∼Gx)

35. 1. (∀x)[Px ⊃ (Qx ⊃ ∼Rx)] 2. (∀x)[Px ⊃ (Sx ⊃ ∼Rx)] 3. (∀x)(Qx ∨ Sx)

/ (∀x)(Px ⊃ ∼Rx)

36. 1. ∼(∃x)[Rx ≡ (Tx ∙ Ux)] 2. (∀x){(Tx ⊃ ∼Ux) ⊃ [Sx ≡ (Rx ∨ Wx)]} /(∀x)[Rx ⊃ (Sx ∨ Vx)]

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37. 1. (∀x)(Ax ≡ Dx) 2. (∀x)[(∼Bx ⊃ Cx) ⊃ Dx] 3. (∀x)[(Ex ⊃ Bx) ∙ (Dx ⊃ Cx)] / (∀x)[Ax ≡ (Bx ∨ Cx)] 38. 1. (∃x)[Kx ∙ (Lx ∙ Mx)] 2. (∀x)[Ox ⊃ ∼(Lx ∙ Mx)] 3. (∃x)(Kx ∙ Nx) 4. (∀x)(Nx ⊃ Ox)

/ (∃x)(Kx ∙ Ox) ∙ ∼(∀x)(Kx ⊃ Ox)

39. 1. (∃x)(Px ∙ Qx) ≡ (∃x)(Rx ∙ Qx) 2. (∃x)(Px ∙ Qx) ≡ (∃x)(Rx ∙ ∼Qx) 3. (∀x)(Px ⊃ Qx) 4. (∃x)Px / ∼[(∀x)(Rx ⊃ Qx) ∨ ∼(∃x)(Rx ∨ Qx)] 40. 1. ∼(∃x)[(Kx ∙ Lx) ∙ (Mx ≡ Nx)] 2. (∀x){Kx ⊃ [Ox ∨ (Px ⊃ Qx)]} 3. (∀x)[(Lx ∙ Mx) ⊃ Px] 4. (∀x)[Nx ∨ (Kx ∙ ∼Qx)] / (∀x)[Lx ⊃ (Nx ∨ Ox)]

EXERCISES 4.6b Translate each of the following arguments into propositions of M. Then, derive the conclusions of the arguments. 1. All gibbons are apes. It’s not the case that there are apes that are not primates. So, if there are gibbons, there are primates. (Ax: x is an ape; Gx: x is a gibbon; Px: x is a primate) 2. All living things are carbon-based. Things that aren’t living are eternal. So, anything not eternal is carbon-based. (Cx: x is carbon-based; Ex: x is eternal; Lx: x is living) 3. Anything corrupt is not happy if it’s real. There are real dinosaurs. So, if everything is corrupt, then there are unhappy dinosaurs. (Cx: x is corrupt; Dx: x is a dinosaur; Hx: x is happy; Rx: x is real) 4. All plays are either comedies or tragedies. Everything is not a tragedy if, and only if, it ends well. So, if some play is not a comedy, then something doesn’t end well. (Cx: x is a comedy; Ex: x ends well; Px: x is a play; Tx: x is a tragedy) 5. No violent thunderstorms are safe. There are safe thunderstorms. So, not everything is violent. (Sx: x is safe; Tx: x is a thunderstorm; Vx: x is violent) 6. All restaurants have chefs. It’s not the case that there are lazy chefs. There are restaurants. So, something isn’t lazy. (Cx: x is a chef; Lx: x is lazy; Rx: x is a restaurant)

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7. All deserts are arid and cool at night. Anything arid or semi-arid has lizards. So, it is not the case that some deserts lack lizards. (Ax: x is arid; Cx: x is cool at night; Dx: x is a desert; Lx: x has lizards; Sx: x is semi-arid) 8. Good parents are either not too busy or don’t fail to make time for their children. So, if all good parents are too busy, then if something is a good parent, then not everything fails to make time for its children. (Bx: x is too busy; Px: x is a good parent; Tx: x fails to make time for its children) 9. Platonists believe that forms are causes. Aristotelians believe that forms are material. So, if there are Platonists or Aristotelians, then something believes either that forms are causes or that they are material. (Ax: x is an Aristotelian; Cx: x believes that forms are causes; Mx: x believes that forms are material; Px: x is a Platonist) 10. All art is either expressive or representational. All art is either expressive or formal. Art exists. So, either something is expressive or something is both representational and formal. (Ax: x is art; Ex: x is expressive; Fx: x is formal; Rx: x is representational) 11. Everything is a human if, and only if, it is rational. Everything is an animal if, and only if, it is either human or not rational. So, there are animals. (Ax: x is an animal; Hx: x is human; Rx: x is rational) 12. Everything is either a substance or an accident. Something is not a substance, but a shape. So, something is an accident and a shape. (Ax: x is an accident; Fx: x is a shape; Sx: x is a substance) 13. All desire is self-destructive. It is not the case that something is both not desire and not self-destructive. So, something is self-destructive. (Dx: x is a desire; Sx: x is self-destructive) 14. It is not the case that some historians are not both broadly trained and learned. Some philosophers are not broadly trained. So, it’s not the case that everything is an historian if, and only if, it is a philosopher. (Hx: x is an historian; Lx: x is learned; Px: x is a philosopher; Tx: x is broadly trained) 15. If no morality is objective, then all morality is relative. Some morality is not relative. If something is objective, then something lacks perspective. So, not everything has perspective. (Mx: x is morality; Ox: x is objective; Px: x has perspective; Rx: x is relative) 16. All idealists are either empirical or transcendental. Some idealist is not empirical. All transcendentalists are empirical if they haven’t read Kant. So, not everything hasn’t read Kant. (Ex: x is empirical; Ix: x is an idealist; Kx: x has read Kant; Tx: x is transcendental)

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EXERCISES 4.6c Derive the following logical truths of M. 1. (∀y)[Fy ⊃ (∃x)Fx] 2. (∃y)[Fy ⊃ (∀x)Fx] 3. (∃y)[(∃x)Fx ⊃ Fy] 4. (∃x)Ax ∨ (∀x)∼Ax 5. (∀x)Bx ⊃ (∃x)Bx 6. (∀x)(Cx ⊃ Dx) ⊃ [(∀x)Cx ⊃ (∀x)Dx] 7. [(∀x)(Gx ⊃ Hx) ∙ (∃x)Gx] ⊃ (∃x)Hx 8. (∀x)(Ix ⊃ Jx) ∨ (∃x)(Ix ∙ ∼Jx) 9. Fa ∨ [(∀x)Fx ⊃ Ga] 10. (∃x)(Px ∙ Qx) ⊃ [(∀x)(Qx ⊃ Rx) ⊃ (∃x)(Px ∙ Rx)] 11. (∃x)(Ax ∙ Bx) ⊃ [(∃x)Ax ∙ (∃x)Bx] 12. [(∀x)Dx ∨ (∀x)Ex] ⊃ (∀x)(Dx ∨ Ex) 13. (∃x)Ix ∨ (∀x)(Ix ⊃ Jx) 14. [(∀x)(Px ⊃ Qx) ∙ (∀x)(Qx ⊃ Rx)] ⊃ (∀x)(∼Rx ⊃ ∼Px) 15. [(∀x)(Mx ⊃ Nx) ∙ ∼(∃x)(Ox ∙ Nx)] ⊃ ∼(∃x)(Mx ∙ Ox) 16. ∼(∃x)Kx ≡ [(∀x)(Kx ⊃ Lx) ∙ (∀x)(Kx ⊃ ∼Lx)] 17. [(∃x)Ax ⊃ Ba] ≡ (∀x)(Ax ⊃ Ba) 18. [∼(∃x)Cx ∙ ∼(∃x)Dx] ⊃ (∀x)(Cx ≡ Dx) 19. {[(∃x)Fx ∨ (∃x)Gx] ∙ (∀x)(Gx ⊃ Hx)} ⊃ [(∃x)Fx ∨ (∃x)Hx] 20. (∃x)(Ka ∙ Lx) ≡ [Ka ∙ (∃x)Lx]

4.7: SEMANTICS FOR M We have been constructing and using formal theories of logic. Some theories have just finitely many theorems. Many interesting formal theories are infinite. For the theories we are using with the languages PL and M, the theorems are the logical truths. Those theories are infinite since there are infinitely many logical truths. To construct a formal theory, we first specify a language and its syntax: vocabulary and rules for well-formed formulas. We have looked carefully at the syntax of both

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Proof theory is the study

of axioms (if any) and rules for a formal theory.

An interpretation of a formal language describes the meanings or truth conditions of its components.

PL (in section 2.2) and M (in section 4.3). Once we have specified the wffs of a language, we can use that language in a theory. But until we specify a semantics or a proof theory, a language can be used in a variety of theories. We could have, for instance, adopted a three-valued semantics for PL, which would have generated different logical truths and thus a different logical theory. There are different ways to specify a theory. We can just list some theorems. Listing the theorems of infinite theories like those we use with PL or M would be an arduous task. More promisingly, we can describe some limited ways of generating theorems. For example, we can adopt some axioms and rules of inference. In geometry, the Euclidean axioms, along with a background logic, characterize what we call Euclidean geometry. We can also axiomatize physical theories, like quantum mechanics, and purely logical systems too. The logical systems in this book do not include any axioms. Instead, to characterize the theories we are using, we have two options. The first option involves what we call proof theory, the subject of chapter 3. Proof theory studies the axioms, for theories that include axioms, and rules of a formal theory. Our proof theory included both rules of inference and rules of equivalence. To generate the theorems of the theory we used with the language PL, we just stated our rules of inference, including the methods of conditional and indirect proof that allow us to derive the logical truths. By adding the inference rules of the previous few sections to those of chapter 3, we have been developing a proof theory for monadic predicate logic. The second option, which is independent of proof theory, is to provide a semantics for our language, a pursuit more generally called model theory. Our semantics for propositional logic consists of assigning truth values to the simple sentences and using the basic truth tables to compute truth conditions for complex sentences. We simply interpret formulas by assigning 1 or 0 to each atomic sentence. We compute truth values of complex propositions by combining the truth values of the atomic sentences according to the truth table definitions. Since we have only twenty-six simple terms, there are only 226 = ∼6.7 million possible interpretations, a large, but finite, number. The semantics for PL was thus pretty easy, using the truth tables. For M, and the other languages of predicate logic, the semantics is more complicated. We have to deal with logical particles, singular terms, predicates, and quantifiers. That is the goal of this section, and we’ll use the framework described here to show the invalidity of arguments in M in the next section.

Interpretations, Satisfaction, and Models The first step in formal semantics for predicate logic is to show how to provide an interpretation of a language. Then, we can determine its logical truths. The logical truths will be the wffs that come out as true under every interpretation. To define an

4 . 7 : S e m ant i cs f or M   2 6 9

interpretation of a theory written in our language M, we specify, in a metalanguage, how to handle constants, predicates, quantifiers, and the propositional operators. To interpret predicates and quantifiers, we use some set theory in our metalanguage as a tool for talking about the terms and formulas of M. Set theory is an important mathematical theory that can get sophisticated and technical. But our uses of sets will be elementary. Indeed, for this section, you need to know only two facts about set theory. First, a set is an imagined collection and the order of objects in that collection is unimportant. Second, a subset of a set is a set all of whose members are in the larger set; a subset can be empty, and it can have all of the members of the original set. (A set that contains strictly fewer members is called a proper subset.) We ordinarily write the members of a set in curly braces. Some sets are small and finite, like the set of current U.S. Supreme Court justices. Other sets are infinite, like the set of natural numbers. Some sets are empty, like the set of trees growing on the moon. We can describe sets in various ways, including listing all of their members or describing a rule for generating members.

A set is an unordered collection of objects.

A subset o f a set is a collection, all of whose members are in the larger set.

S1: {Bill Clinton, George W. Bush, Barack Obama} S2: {Winners of the Academy Award for Best Actress} S3: {1, 2, 3 . . . } S4: {}

There are three members of S1; seventy-four members of S2 (as of 2017); and an infinite number of members of S3. There are no members of S4. There are eight different subsets of S1. {Bill Clinton, George W. Bush, Barack Obama} {Bill Clinton, George W. Bush} {Bill Clinton, Barack Obama} {George W. Bush, Barack Obama} {Bill Clinton} {George W. Bush} {Barack Obama} {}

Moving on to our main work, we interpret a first-order theory in four steps. Step 1. Specify a set to serve as a domain of interpretation. The domain of interpretation (sometimes called a domain of quantification) is the universe of the theory, the objects to which we apply the theory. We can consider small finite domains, sets of even just one or two objects. Or we can consider larger domains, like the whole universe or all of the real numbers. In showing arguments to be invalid, in the next section, we’ll use small finite domains. But often we implicitly think of our domain of interpretation as much larger. Step 2. Assign a member of the domain to each constant. We introduced constants to be used as names of particular things. In giving an interpretation of our language, we pick one thing out of the domain for each constant.

A domain of

interpretation, or domain of quantification , is a set

of objects to which we apply a theory.

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Different constants may correspond to the same object, just as an individual person or thing can have multiple names. For example, if we are using M and working with a small domain of interpretation {1, 2, 3}, we can assign the number 1 to ‘a’, the number 2 to ‘b’, and the number 3 to all of the remaining nineteen constants (‘c’, . . .‘u’). Just as not every object in our world has a name, not every object in a domain of interpretation needs to have a name in a theory. So we can pick a universe of many objects and name only some of them. Also, since one object can have multiple names, a theory with many different constants can be interpreted with a domain of fewer objects. But we ordinarily use a different name for each object. Step 3. Assign some set of objects in the domain to each predicate. We interpret predicates as subsets of the domain of interpretation, the objects of which that predicate holds. We can interpret predicates by providing a list of members of the domain or by providing a rule. If we use a predicate ‘Dx’ to stand for ‘x is a Democrat who has been elected president of the United States’, then the interpretation of that predicate will be the set of things in the domain of interpretation that were elected president as Democrats. Using a domain of S1, the interpretation of ‘Dx’ will be {Bill Clinton, Barack Obama}. Using a domain of S2 , it will be empty. In the domain of natural numbers, S3, we might define a predicate of even numbers, ‘Ex’, as the set of all objects that are multiples of two: {2, 4, 6 . . . }. Step 4. Use the customary truth tables to interpret the propositional operators. We are familiar with step 4 of the semantics from our work with PL, and we naturally assume the truth table definitions for all the propositional operators when interpreting theories written in M. Let’s take, for an example, the interpretation of a small set of sentences that I’ll call Theory TM1, with a small domain. Theory TM1 1. Pa ∙ Pb 2. ∼Ib 3. (∃x)Px 4. (∀x)Px 5. (∀x)(Ix ⊃ Px) 6. (∀x)(Px ⊃ Ix) An Interpretation of TM1 Domain: {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} a: Venus b: Mars c: Neptune

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Px: {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} Ix: {Mercury, Venus, Earth, Mars}

We can think of ‘Px’ as meaning that x is a planet in our solar system. We can think of ‘Ix’ as meaning that x is an inner planet. But the interpretation, speaking strictly, is made only extensionally, by the members of the sets listed. Sentence 1 is true on our interpretation because the objects assigned to ‘a’ and ‘b’ are in the set assigned to ‘Px’. We say that the given objects satisfy that predicate. Sentence 2 is not true on our interpretation, since the object assigned to ‘b’ is in the set assigned to ‘Ix’ and sentence 2 denies that it is. Sentences 3–6 of  TM1 require interpreting quantified sentences. Sentence 3 is true because there is an object in the domain that is in the set which interprets ‘Px’. Sentence 4 is true because all objects in the domain are in the set which interprets ‘Px’. We can define satisfaction for quantified sentences too. An existentially quantified sentence is satisfied if, and only if, it is satisfied by some object in the domain; a universally quantified sentence is satisfied if, and only if, it is satisfied by all objects in the domain. Sentence 5 is true since every object in the domain that satisfies ‘Ix’ also satisfies ‘Px’. But sentence 6 is false since there are some objects in the domain that satisfy ‘Px’, but do not satisfy ‘Ix’. Not all Ps are Is, on our interpretation. Let’s interpret a new theory, TM 2 , using the same domain and assignments as above. Theory TM2

An object satisfies a predicate if it is in the set that interprets that predicate.

An existentially

quantified sentence is satisfiedwhen it

is satisfied by some object in the domain. A universally quantified sentence is satisfied

when it is satisfied by all objects in the domain.

1. Pa ∙ Pb 2. Ib ∙ ∼Ic 3. (∃x)(Px ∙ Ix) ∙ (∃x)(Px ∙ ∼Ix) 4. (∀x)(Ix ⊃ Px)

Notice that all of the sentences of TM 2 are true on our interpretation. We thus call our interpretation a model of TM 2 . To construct a model for a given set of sentences, we specify an interpretation, using the four steps above. The exercises at the end of this section contain some theories for which you are to construct models. You may pick any domain of interpretation and any assignment of objects of that domain for your constants and predicates.

A model o f a theory is an interpretation on which all of the sentences of the theory are true.

Logical Truth: Semantic Arguments The logical truths of PL are the tautologies. We can show that a formula of PL is a logical truth either semantically, by the truth table method, or proof-theoretically, using either conditional or indirect proof. In section 4.6, we saw how to prove that a wff of  M is a logical truth with our proof theory, using conditional or indirect proof. We can show that a wff is a logical truth semantically, too. To show that a formula of  M is a logical truth, semantically, we have to show that it is true for every interpretation.

A wff of M is a logical truth when it is true for every interpretation.

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To show that a formula is true for every interpretation, we have to think about various domains, with various assignments of objects to constants and sets of objects to predicates. I will show semantically that 4.7.1 is a logical truth. 4.7.1 Pa ∨ [(∀x)Px ⊃ Qa]

QED

Suppose that ‘Pa ∨ [(∀x)Px ⊃ Qa]’ is not a logical truth. Then there is an interpretation on which it is false. On that interpretation, the object assigned to ‘a’ will not be in the set assigned to ‘Px’, and there is some counterexample to ‘(∀x)Px ⊃ Qa’. Any counterexample to a conditional statement has to have a true antecedent. So, every object in the domain of our interpretation will be in the set assigned to ‘Px’. That contradicts the claim that the object assigned to ‘a’ will not be in the set assigned to ‘Px’. So, our assumption must be false: no interpretation will make that sentence false. So, ‘Pa ∨ [(∀x)Px ⊃ Qa]’ is logically true.

Semantic proofs of the logical truth of wffs of M are essentially metalogical, and very different in feel from the semantic proofs for PL. The truth tables are also metalogical, not part of the object language, but they are more mechanical. Semantics proofs for logical truths of M are often structured as reductio arguments: suppose that the given proposition is not a logical truth. Then there will be an interpretation that makes it false. If the statement is a logical truth, a contradiction should follow. We’ll spend more time on proof theory for M and F than we will on the semantics for logical truth. Still, there is a nice, simple, and agreeable method for showing that an argument is invalid using the semantics for M, one that we will examine in our next section.

Summary We are near the end of our studies of M. We have translated between natural language and predicate logic. We have a proof system to show that arguments are valid and which can be used to show that formulas are logical truths. And we have a semantic method for interpreting our theories of M, constructing models and showing that formulas are logical truths. When we introduced our system of inference for PL, we already had a way of distinguishing the valid from the invalid arguments, using truth tables. In M, we need a corresponding method for showing that an argument is invalid. In the next section, we will explore a formal, semantic method for showing that an argument is invalid in M. Then, we will proceed to a new language, of relational predicate logic.

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KEEP IN MIND

We interpret a theory of M in four steps: Step 1. Specify a set to serve as a domain of interpretation. Step 2. Assign a member of the domain to each constant. Step 3. Assign some set of objects in the domain to each predicate. Step 4. Use the customary truth tables to interpret the propositional operators. We can pick any domain to interpret a theory. Not every member of a domain of interpretation must have a name (be assigned to a constant). We can interpret predicates using any subjects of objects of the domain. We can show semantically that wffs of M are logical truths, by showing that they are true on any interpretation.

TELL ME MORE  • What is the relationship between models and consistency? See 6.4: Metalogic. • How can understanding the semantics for M help us use the language more precisely and effectively? See 6.6: Notes on Translation with M. • How does the difference between syntax and semantics underlie important claims about consciousness? See 7.3: Logic and Philosophy of Mind. • Is there a domain of everything? See 7.7: Logicism. • Why is quantification so important for dealing with questions about existence? See 7S.10: Quantification and Ontological Commitment. • How do the semantics for M help us understand the limits of predicate logic? See 7S.11: Atomism and Color Incompatibility.

EXERCISES 4.7a Construct models for each of the following theories by specifying a domain of interpretation (make one up) and interpreting the constants and predicates. Translate each of the sentences of the theory to English, given your interpretation. 1. Pa ∙ ∼Pb Qa ∙ Qb (∀x)(Px ⊃ Qx) (∃x)(∼Px ∙ ∼Qx)

2. Mb ∙ ∼Md ∼La ∙ ∼Wa Wc ∙ Wd (∃x)(Mx ∙ Lx) (∃x)(Mx ∙ ∼Wx) (∀x)(Lx ⊃ ∼Wx)

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3. Eb ∙ Ec Kd ∙ ∼Ka ∼Ea ∙ Pa (∀x)(Ex ⊃ ∼Kx) (∃x)(Px ∙ Kx) (Eb ∨ Ed) ⊃ ∼Ka

5. (Pa ∙ Pb) ∙ Pc (Qa ∙ Qb) ∙ ∼Qc (∀x)[(Px ∙ Qx) ⊃ Rx] (∃x)[(Px ∙ Qx) ∙ Sx] (∀x)[(Px ∙ ∼Qx) ⊃ (∼Rx ∙ ∼Sx)]

4. Oa ∙ ∼Ob Ra ∙ ∼Ea Rd ∙ Od ∙ ∼Ed (∃x)(Rx ∙ Ox) ∼(∃x)(Ex ∙ Ox) (∃x)(Ex ∙ Rx) ⊃ ∼Oc

EXERCISES 4.7b Show, semantically, that the following propositions selected from Exercises 4.6c are logical truths. 1. (∃x)Ax ∨ (∀x)∼Ax

(4.6c.4)

2. (∀x)(Cx ⊃ Dx) ⊃ [(∀x)Cx ⊃ (∀x)Dx] (4.6c.6) 3. (∃x)(Px ∙ Qx) ⊃ [(∀x)(Qx ⊃ Rx) ⊃ (∃x)(Px ∙ Rx)]

(4.6c.10)

4. [(∀x)Dx ∨ (∀x)Ex] ⊃ (∀x)(Dx ∨ Ex)

(4.6c.12)

5. [(∃x)Ax ⊃ Ba] ≡ (∀x)(Ax ⊃ Ba)

(4.6c.17)

4.8: INVALIDITY IN M We studied proof-theoretic methods for showing that an argument in M is valid in sections 4.4–4.6. In this section, I demonstrate a semantic method for showing that an argument in M is invalid. A valid argument is one that is valid under any interpretation, using any domain. An invalid argument will have counterexamples, interpretations on which the premises come out true and the conclusion comes out false. Understanding how we interpret theories in the language of predicate language, the subject of section 4.7, will help us here to formulate a method for showing that an argument in predicate logic is invalid.

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Recall how we proved that an argument in PL, such as 4.8.1, is invalid. 4.8.1 1. A ⊃ B 2. ∼(B ∙ A)

/A ≡ B

We lined up the propositional variables on the left side of the table, and the premises and conclusion on the right. Then we assigned truth values to the component sentences to form a counterexample, a valuation that makes the premises true and the conclusion false. 4.8.2 A

B

A

⊃

B

0

1

0

1

1

/

∼

(B

∙

A)

1

1

0

0

//

A

≡

B

0

0

1

The table at 4.8.2 shows that the argument is invalid since there is a counterexample when A is false and B is true. We will adapt this method for first-order logic. Just as logical truths are true for all interpretations, if an argument is valid, then it is valid no matter what we choose as our domain of interpretation. Even if our domain has only one member, or two or three or a million, valid arguments have no counterexamples. Inversely, if an argument is invalid, then there will be a counterexample in some finite domain, though there may be no counterexample in any particular finite domain. As in PL, we will show that arguments of M are invalid by constructing counterexamples. Our approach is sometimes called the method of finite universes. Of course, the counterexamples for M will be more complex. Assigning truth values to closed atomic propositions is easy enough. It’s the quantifiers that create complexity. So, to construct a counterexample, we transform propositions with quantifiers into unquantified equivalents in finite domains. Then we will have propositions whose operators are just the operators of PL and we’ll be able to use our old methods. We’ll start with some examples in domains of one member and then move to more complex examples that require larger domains.

Domains of One Member Argument 4.8.3 is invalid. 4.8.3 (∀x)(Wx ⊃ Mx) (∀x)(Px ⊃ Mx)

/ (∀x)(Wx ⊃ Px)

We can see intuitively that 4.8.3 is invalid by interpreting the predicates. For example, we can take ‘Wx’ to stand for ‘x is a whale’, ‘Mx’ to stand for ‘x is a mammal’, and ‘Px’ to stand for ‘x is a polar bear’. All whales are mammals, all polar bears are mammals, but it’s not the case that all whales are polar bears. True premises; false conclusion.

The method of finite universesis a semantic

method that can produce counterexamples to arguments in predicate logic.

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To show that 4.8.3 is invalid more formally, I will start by choosing a domain of one object. We will call it ‘a’. Since there is only one object in the domain, the universally quantified formulas are equivalent to statements about that one object. 4.8.4 (∀x)(Wx ⊃ Mx) (∀x)(Px ⊃ Mx) (∀x)(Wx ⊃ Px)

is equivalent to is equivalent to is equivalent to

Wa ⊃ Ma Pa ⊃ Ma Wa ⊃ Pa

We can thus eliminate the quantifiers and use the same method we used for arguments in PL. We assign truth values to make the premises true and the conclusion false, as in 4.8.5. 4.8.5 Wa

Ma

Pa

Wa

⊃

Ma

1

1

0

1

1

1

/

Pa

⊃

Ma

0

1

1

//

Wa

⊃

Pa

1

0

0

The argument 4.8.3 is thus shown invalid because there is a counterexample in a one-member domain, where Wa is true, Ma is true, and Pa is false. In other words, there is a counterexample in a domain of one object, when that object is a whale, and a mammal, but not a polar bear. Again, a specification of the assignments of truth values to the atomic sentences of the theory, as in the previous sentence, is called a counterexample. The method of finite domains works with existential quantifiers as well, as in argument 4.8.6. 4.8.6

1. (∀x)[Ux ⊃ (Tx ⊃ Wx)] 2. (∀x)[Tx ⊃ (Ux ⊃ ∼Wx)] 3. (∃x)(Ux ∙ Wx)

/ (∃x)(Ux ∙ Tx)

Expanding an existentially quantified formula to a one-member domain, as in 4.8.7, works exactly like it does for universally quantified formulas. In a world with just one thing, ‘everything’ is the same as ‘something’. 4.8.7 (∀x)[Ux ⊃ (Tx ⊃ Wx)] (∀x)[Tx ⊃ (Ux ⊃ ∼Wx)] (∃x)(Ux ∙ Wx) (∃x)(Ux ∙ Tx)

is equivalent to is equivalent to is equivalent to is equivalent to

Ua ⊃ (Ta ⊃ Wa) Ta ⊃ (Ua ⊃ ∼Wa) Ua ∙ Wa Ua ∙ Ta

The construction of a counterexample proceeds in the same way, too. The table at 4.8.8 shows that there is a counterexample in a one-member domain, where Ua is true, Ta is false, and Wa is true.

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4.8.8 Ua

Ta

Wa

Ua

⊃

(Ta

⊃

Wa)

1

0

1

1

1

0

1

1

Wa)

/

Ua 1

1

Ta

⊃

(Ua

⊃

∼

0

1

1

0

0

∙

Wa

//

Ua

∙

Ta

1

1

1

0

0

/

Be careful not to confuse expansions into finite domains with instantiation in natural deductions. In each case, we remove quantifiers. But the restrictions on EI play no role in expansions. To show that an argument is invalid, we need only one counterexample. For many simple arguments, we can construct a counterexample in a domain of one member. But not all invalid arguments have counterexamples in a one-member domain. To construct a counterexample, we often must use a larger domain.

Domains of Two Members Argument 4.8.9 is invalid, but has no counterexample in a one-member domain. 4.8.9 (∀x)(Wx ⊃ Hx) (∃x)(Ex ∙ Hx)

/ (∀x)(Wx ⊃ Ex)

To make the conclusion false, we have to make ‘Wa’ true and ‘Ea’ false. Then the second premise is false no matter what value we assign to ‘Ha’. Wa

Ha

Ea

Wa

⊃

Ha

/

Ea 0

∙

Ha

//

Wa

⊃

Ea

1

0

0

Thus, to show that 4.8.9 is invalid, we have to consider a larger domain. If there are two objects in a domain, a and b, then the expansions of quantified formulas become more complex. Universally quantified formulas become conjunctions: a universally quantified proposition states that every object in the domain has some property. Existentially quantified formulas become disjunctions: at least one object in the domain has the property ascribed by an existential formula. 4.8.10 shows the rules for expanding quantified formulas into two- and three-­ member domains.

2 7 8    C h apter 4  Mona d i c P re d i cate L og i c

4.8.10 In a two-member domain: (∀x)Fx becomes Fa ∙ F b (∃x)Fx becomes Fa ∨ F b In a three-member domain: (∀x)Fx becomes Fa ∙ F b ∙ Fc (∃x)Fx becomes Fa ∨ F b ∨ Fc

Returning to argument 4.8.9, let’s expand the argument into a domain of two members, as in 4.8.11, to look for a counterexample. 4.8.11 (Wa ⊃ Ha) ∙ (Wb ⊃ Hb) (Ea ∙ Ha) ∨ (Eb ∙ Hb)

/ (Wa ⊃ Ea) ∙ (Wb ⊃ Eb)

Let’s assign values to each of the terms to construct a counterexample. Wa

Wb

Ha

Hb

Ea

Eb

(Wa

⊃

Ha)

∙

(Wb

⊃

Hb)

1

0

1

1

0

1

1

1

1

1

0

1

1

/

(Ea

∙

Ha)

∨

(Eb

∙

Hb)

0

0

1

1

1

1

1

(Wa

⊃

Ea)

∙

(Wb

⊃

Eb)

1

0

0

0

0

1

1

//

Note that in a two-membered domain, each quantified wff has two instances, one for each object in the domain.

Constants When expanding formulas into finite domains, constants remain themselves; there is no need to expand a term with a constant when moving to a larger domain. If an argument contains more than one constant, then it will require a domain larger than one object. Remember that expanding formulas into finite domains is not the same as instantiating. In particular, the restriction on EI that we must instantiate to a new constant does not apply. If an argument contains both an existential quantifier and a constant, you may expand the quantifier into a single-member domain using the constant already present in the argument. It need not be a new constant. 4.8.12 cannot be shown invalid in a one-member domain.

4 . 8 : In v al i d i t y i n M   2 7 9

4.8.12 (∃x)(Ax ∙ Bx) Ac Ac

Bc

/Bc

Ac

∙

Bc

0

0

/

Ac

//

Bc 0

We can generate a counterexample in a two-member domain, though, as at 4.8.13. 4.8.13 Ac

Aa

Bc

Ba

(Ac

∙

Bc)

∨

(Aa

∙

Ba)

1

1

0

1

1

0

0

1

1

1

1

/

Ac

//

1

The counterexample is in a two-member domain, when Aa, Ac, and Ba are true and Bc is false. Some arguments require three-member, four-member, or larger domains to be shown invalid. The pattern apparent at 4.8.10 can be extended for larger domains, adding further conjunctions for universal quantifiers and further disjunctions for existential quantifiers.

Domains of Three or More Members The argument at 4.8.14 is invalid, but it has no counterexamples in domains of fewer than three members. (Check it!) Let’s see how to expand it to a domain of three objects. 4.8.14 Pa ∙ Qa (∃x)(Px ∙ ∼Qx) (∃x)(Qx ∙ Rx) (∀x)(Px ⊃ ∼Rx)

/ (∃x)(Rx ∙ ∼Qx)

I’ll unpack each proposition on a separate line, starting with a row for the atomic formulas. Pa

Qa

Ra

Pb

Qb

Rb

Pa

∙

Qa

Pc

Qc

Rc

Bc 0

2 8 0    C h apter 4  Mona d i c P re d i cate L og i c

(Pa

∙

∼

Qa)

∨

(Pb

∙

∼

Qb)

∨

(Pc

∙

∼

Qc)

(Qa

∙

Ra)

∨

(Qb

∙

Rb)

∨

(Qc

∙

Rc)

(Pa

⊃

∼

Ra)

∙

(Pb

⊃

∼

Rb)

∙

(Pc

⊃

∼

Rc)

//

(Ra

∙

∼

Qa)

∨

(Rb

∙

∼

Qb)

∨

(Rc

∙

∼

Qc)

Note that the first premise does not get expanded to other objects; only quantified sentences expand. No matter how large a domain you choose, a statement without quantifiers remains the same. Also notice that I do not group the three disjuncts in the second premise, the third premise, or the conclusion, and that I do not group the three conjuncts in the fourth premise. Technically, according to our formation rules, each pair of disjuncts or conjuncts should be grouped. But since conjunction and disjunction are both associative and commutative, the grouping really doesn’t matter. For a disjunction to be true, only one of however many disjuncts appear must be true; it doesn’t matter which. For a disjunction to be false, every one of the disjuncts must be false. For a conjunction to be true, every one of the conjuncts must be true. For a conjunction to be false, just one of the conjuncts has to be false. The extra punctuation as you reach three- or four-membered domains is less helpful than it is cluttering, so I relax the need for groupings of pairs when unpacking quantifiers into larger domains. I’ll still use it for derivations until section 5.5, when similar considerations lead me again to relax punctuation in long conjunctions or disjunctions. Returning to our work at hand, the counterexample is relatively easy to construct. I’ll describe my process of constructing a counterexample and provide a completed table. I started by assigning values for Pa and Qa in the first premise, both true. The conclusion has three disjuncts that each have to be false, and the truth of Qa means that the first disjunct is false.

4 . 8 : In v al i d i t y i n M   2 8 1

The fourth premise includes three conjuncts, each of which must be true, and the truth of Pa entails that Ra must be false in order for the first conjunct to be true. The second and third premises are both series of disjuncts. The values so far assigned entail that the first disjunct in each expanded premise is false, but we have two other disjuncts that we can make true for each, and only one of the disjuncts has to be true. I assigned true to Pb and false to Qb to take care of the second premise. The truth of Pb, carried to the fourth premise, entails that Rb must be false. And the falsity of Rb makes the second disjunct in the conclusion false, which was needed there given the falsity of Qb. Still, the third premise now had two false disjuncts, so I had to make Qc and Rc both true. Then all that remained was making the last conjunct of the fourth premise true and the last disjunct of the conclusion false. The truth of Qc already accomplished the latter task, and making Pc false accomplishes the former. The counterexample is constructed. Pa

Qa

Ra

Pb

Qb

Rb

Pc

Qc

Rc

1

1

0

1

0

0

0

1

1

Pa

∙

Qa

1

1

1

(Pa

∙

∼

Qa)

∨

(Pb

∙

∼

Qb)

1

0

0

1

1

1

1

0

(Qa

∙

Ra)

∨

(Qb

∙

Rb)

∨

(Qc

∙

Rc)

1

0

0

0

0

0

1

1

1

(Pa

⊃

∼

Ra)

∙

(Pb

⊃

∼

Rb)

∙

1

1

1

0

1

1

1

0

∨

(Pc

∙

∼

Qc)

0

0

0

1

(Pc

⊃

∼

Rc)

0

1

0

1

2 8 2    C h apter 4  Mona d i c P re d i cate L og i c

//

(Ra

∙

∼

Qa)

0

0

1

∨

(Rb

∙

∼

Qb)

0

0

1

0

∨

(Rc

∙

∼

Qc)

1

0

0

1

There is no easy rule for determining how large a domain is required for a counter­ example for a given argument. The standard approach is just to start with a one-­ membered domain and work upward as needed. But students often ask for guidelines, and a rough one is that the size of the required domain increases with the number of existential premises. Universal premises are easily satisfied trivially, with false antecedents of their conditionals. But existentials often require conflicting assignments of truth values and so can increase the size of the required domain. It is useful and elegant to find a counterexample in the smallest domain possible. But whatever the minimum size of the domain required to construct a counterexample for a particular argument, there will be counterexamples in all larger domains. So, if you mistakenly miss a counterexample in, say, a two-membered domain, there will be one in a three-membered domain, and in larger ones.

Propositions Whose Main Operator Is Not a Quantifier The main operator of the second premise of 4.8.15 is a ⊃, not a quantifier. On each side of the conditional, there is a quantifier. There is no counterexample to the argument in a one-member domain, though the expansion is straightforward. 4.8.15 (∃x)(Px ∙ Qx) (∀x)Px ⊃ (∃x)Rx (∀x)(Rx ⊃ Qx) Pa

Qa 0

Ra

Pa

∙

Qa

0

0

/

Pa

/ (∀x)Qx ⊃

Ra

/

Ra

⊃

Qa

//

Qa 0

In a two-member domain, each quantifier in the second premise is unpacked independently, as in 4.8.16. Notice that the main operator of the premise remains the conditional. 4.8.16 (∀x)Px ⊃ (∃x)Rx becomes (Pa ∙ Pb) ⊃ (Ra ∨ Rb)

We can clearly see here the difference between instantiation and expansion into a finite domain. In a derivation, we could not instantiate the second premise, since the main operator is not a quantifier. But interpreting the argument in a finite domain, we can expand each quantifier. We can construct a counterexample for the argument 4.8.15 in a two-member domain.

4 . 8 : In v al i d i t y i n M   2 8 3

4.8.16 Pa

Qa

Ra

Pb

Qb

Rb

1 or 0

0

0

1

1

1

/

/

(Pa

(Pa

∙

Qa)

∨

(Pb

∙

Qb)

0

0

1

1

1

1

∙

Pb)

⊃

(Ra

∨

Rb)

1

1

0

1

1

(Ra

⊃

Qa)

∙

(Rb

⊃

Qb)

0

1

0

1

1

1

1

//

Qa

∙

Qb

0

0

1

Logical Truths The method of finite domains can easily be adapted to show that individual propositions are not logical truths. If a proposition is a logical truth, it will be true on any valuation, in any domain. So, if we can find a valuation that makes it false in a domain of any size, we have a counterexample to the claim that the proposition is a logical truth. 4.8.17 is not a logical truth. 4.8.17 (∀x)(Px ⊃ Qx) ∨ (∀x)(Qx ⊃ Px)

Let’s start by translating it into a domain of one object, at 4.8.18. 4.8.18 (Pa ⊃ Qa) ∨ (Qa ⊃ Pa)

In a one-object domain, no false valuation is possible. Making either disjunct false makes the other disjunct true. We’ll have to expand it into a domain of two objects, at 4.8.19. 4.8.19 [(Pa ⊃ Qa) ∙ (Pb ⊃ Qb)] ∨ [(Qa ⊃ Pa) ∙ (Qb ⊃ Pb)]

The expansion into a two-object domain, 4.8.20, is more promising for a false valuation.

2 8 4    C h apter 4  Mona d i c P re d i cate L og i c

4.8.20 Pa

Qa

Pb

Qb

[(Pa

⊃

Qa)

∙

1

0

0

1

1

0

0

0

∨

[(Qa

⊃

Pa)

0

(Pb

⊃

Qb)]

∙

(Qb

⊃

Pb)]

0

1

0

0

We can make each disjunct false, so that the whole proposition is false. Thus, we have a valuation that shows that 4.8.17 is not a logical truth.

Overlapping Quantifiers Sometimes, two quantifiers of M overlap. Unpacking propositions with overlapping quantifiers requires some care. Consider a logical truth such as 4.8.21. (We saw the derivation of this proposition at example 4.6.14.) 4.8.21 (∀y)[(∀x)Fx ⊃ Fy]

To expand 4.8.21 into a finite domain, we have to manage the overlapping quantifiers. For a one-membered domain, the expansion is simple, as at 4.8.22. 4.8.22 Fa ⊃ Fa

For larger domains, just work in stages, starting with the outside quantifiers, as I do at 4.8.23, in a two-object domain, and at 4.8.24, in a three-object domain. 4.8.23 [(∀x)Fx ⊃ Fa] ∙ [(∀x)Fx ⊃ Fb] [(Fa ∙ Fb) ⊃ Fa] ∙ [(Fa ∙ Fb) ⊃ Fb] 4.8.24 [(∀x)Fx ⊃ Fa] ∙ [(∀x)Fx ⊃ Fb] ∙ [(∀x)Fx ⊃ Fc] [(Fa ∙ Fb ∙ Fc) ⊃ Fa] ∙ [(Fa ∙ Fb ∙ Fc) ⊃ Fb] ∙ [(Fa ∙ Fb ∙ Fc) ⊃ Fc]

As you should be able to see, no matter how large a domain we consider, we will not be able to construct a counterexample. Now consider a related claim that is not a logical truth, 4.8.25. 4.8.25 (∀y)[(∃x)Fx ⊃ Fy]

To show that it is not a logical truth, we just need a valuation that makes the statement false. There is no counterexample in a one-membered domain, which looks exactly like 4.8.22. For a two-membered domain, once again start with the outside quantifier, as at 4.8.26. 4.8.26 [(∃x)Fx ⊃ Fa] ∙ [(∃x)Fx ⊃ Fb] [(Fa ∨ Fb) ⊃ Fa] ∙ [(Fa ∨ Fb) ⊃ Fb]

4 . 8 : In v al i d i t y i n M   2 8 5

Now we can assign truth values to show the proposition to be false.

Fa

Fb

[(Fa

∨

Fb)

⊃

Fa]

∙

0

1

0

1

1

0

0

0

[(Fa

∨

Fb)

⊃

Fb]

There is a false valuation of 4.8.25 in a domain of two objects, when Fa is false and Fb is true.

Negations of Quantified Formulas You may notice that none of the arguments we’ve examined in this section, and none of those in the exercises below, contain negations of quantified formulas. You can expand the negation of a quantified formula by merely leaving the negation alone, expanding the quantified formula, and then negating the entire result. But it is simpler just to use the rules of quantifier exchange to turn a negated formula into its unnegated equivalent before expanding it. So, given an invalid argument like 4.8.27, we can construct a countermodel for the equivalent argument at 4.8.28. 4.8.27

∼(∃x)(Px ∙ ∼Qx) ∼(∀x)(Qx ⊃ Rx)

4.8.28 (∀x)(Px ⊃ Qx) (∃x)(Qx ∙ ∼Rx)

/ ∼(∀x)(Px ⊃ Rx) / (∃x)(Px ∙ ∼Rx)

Given the ready availability of such equivalents, we won’t bother with the expansions of quantified formulas with leading negations.

Summary The method of constructing counterexamples to arguments by considering interpretations in finite domains draws on both our semantics for PL, in the uses of truth tables, and our semantics for M, in translating quantified sentences into unquantified claims in finite domains. We now have a semantic method for proving arguments of M invalid and a proof-theoretic method of proving arguments valid. We can also adapt our method of expansion into finite domains to provide a semantic method for showing that a statement is not a logical truth. We can prove logical truths of M using the methods of conditional or indirect proof, or the semantic method sketched at the end of section 4.7. We can now show that a wff is not a logical truth by providing a valuation that makes it false in a finite domain.

2 8 6    C h apter 4  Mona d i c P re d i cate L og i c

KEEP IN MIND

To show that an argument of M is invalid, we translate quantified sentences into unquantified equivalents in finite domains and then construct a counterexample. As for PL, a counterexample is a valuation that makes the premises true and the conclusion false. Existential statements are equivalent to series of disjunctions. Universal statements are equivalent to series of conjunctions. Series of disjunctions are true if at least one is true; they are false if every one is false. Series of conjunctions are true if every one is true; they are false if at least one is false. Only quantified formulas expand in finite domains; propositions with constants and no variables do not expand. In propositions with more than one quantifier, expand each quantifier independently. When expanding overlapping quantifiers, work in stages, from the outside quantifier in. There is no rule about the size of the domain needed to construct a counterexample, though the size tends to increase with the number of existential premises. If there is a counterexample in a domain of a certain size, there will be counterexamples in domains of all larger sizes; still, you should seek the smallest domain possible. To show that a wff is not a logical truth, construct a false valuation in a finite domain.

TELL ME MORE  • How does invalidity in finite domains relate to broader questions about validity? See 7.2: Fallacies and Argumentation.

EXERCISES 4.8a Show that each of the following arguments is invalid by generating a counterexample. 1. 1. (∃x)(Ax ∨ Bx) 2. (∀x)Ax

/ (∀x)Bx

2. 1. (∀x)(Cx ⊃ Dx) 2. Da

/ Ca

3. 1. (∀x)(Kx ≡ Lx) 2. (∃x)(Mx ∙ Lx)

/ (∃x)(Nx ∙ Kx)

4. 1. (∀x)[(Gx ∙ Hx) ∨ Ix] 2. (∼Hc ⊃ Jc) ⊃ ∼Ic

/ (∃x)(Gx ∙ ∼Jx)

4 . 8 : In v al i d i t y i n M   2 8 7

5. 1. (∀x)(Px ≡ Rx) 2. (∃x)(Qx ∙ ∼Sx)

/ (∀x)(Qx ⊃ ∼Rx)

6. 1. (∃x)(Ex ∙ Fx) ⊃ (∀x)(Gx ⊃ Hx) 2. ∼(∀x)(Fx ⊃ Ex) / (∀x)(∼Hx ⊃ ∼Gx) 7. 1. (∃x)(Ix ∙ Jx) ≡ (∀x)(Lx ⊃ Kx) 2. (∃x)( Jx ∙ ∼Kx) ≡ (∀x)(Lx ⊃ ∼Kx) 3. (∃x)(Ix ∙ ∼Kx) / (∃x)(Lx ∙ Kx) 8. 1. (∃x)[(Ax ∙ Bx) ∙ Cx] 2. (∃x)[(Ax ∙ Bx) ∙ ∼Cx] 3. (∃x)(Bx ∙ Dx) 4. ∼Da / (∀x)(Cx ⊃ Dx) 9. 1. (∃x)(Ex ∙ Fx) 2. Fb

/ Eb

10. 1. (∃x)Dx ⊃ (∃x)Gx 2. (∃x)(Dx ∙ Ex)

/ (∃x)(Ex ∙ Gx)

11. 1. (∃x)(Sx ∙ Tx) 2. (∃x)(Tx ∙ Vx)

/ (∃x)(Sx ∙ Vx)

12. 1. (∃x)(Xx ∙ Yx) 2. (∀x)(Yx ⊃ Zx) 3. (∃x)(Zx ∙ ∼Yx) / ∼(∀x)(Xx ⊃ Yx) 13. 1. Pa ∙ Qb 2. (∃x)(Rx ∙ Sx) 3. (∃x)(Rx ∙ ∼Sx) 4. (∀x)(Sx ⊃ Qx)

/ (∀x)(Rx ⊃ Px)

14. 1. (∃x)(Lx ∙ Nx) 2. (∃x)(Mx ∙ ∼Nx) 3. (∀x)(Lx ⊃ Ox)

/ (∀x)(Mx ⊃ Ox)

15. 1. (∃x)(Rx ∨ ∼Tx) 2. (∃x)(∼Rx ∙ Tx) 3. (∀x)(Sx ≡ Tx)

/ (∀x)(Sx ⊃ Rx)

16. 1. (∃x)(Ax ∙ Bx) 2. (∃x)(Cx ∙ ∼Bx) 3. (∀x)[(Ax ∙ Cx) ⊃ Dx] / (∀x)(Bx ⊃ Dx) 17. 1. (∃x)(Ex ∙ Fx) 2. ∼(∀x)(Ex ⊃ Fx) 3. (∀x)(Fx ⊃ Ex)

/ (∀x)(∼Fx ⊃ ∼Ex)

18. 1. (∃x)( Jx ∨ Kx) ⊃ (∃x)(Lx ∙ ∼Jx) 2. (∃x)(Lx ∙ Jx) / (∃x)(Kx ∙ ∼Jx)

2 8 8    C h apter 4  Mona d i c P re d i cate L og i c

19. 1. (∀x)Ax ⊃ (∀x)Bx 2. (∃x)(Ax ∙ ∼Bx) 3. (∀x)(Cx ⊃ Bx)

/ (∀x)(Cx ⊃ Ax)

20. 1. (∃x)[Ox ∙ (Px ≡ Qx)] 2. (∃x)[∼Ox ∙ (Px ⊃ Qx)] 3. (∀x)(Rx ⊃ Ox) / (∀x)(Rx ⊃ Qx) 21. 1. (∃x)Ex ⊃ (∃x)Fx 2. (∃x)(Ex ∙ ∼Fx) 3. (∀x)[(Gx ∨ Hx) ⊃ Fx] / (∀x)(Hx ⊃ Ex) 22. 1. (∀x)( Jx ≡ Ix) ∙ (∃x)Kx 2. (∃x)(Ix ∙ ∼Kx) 3. (∀x)(Lx ⊃ Kx) 4. ∼Ja ∙ Jb / (∀x)(Lx ⊃ Ix) 23. 1. (∃x)[∼Wx ∙ (Xx ∙ Yx)] 2. (∀x)(Xx ≡ Yx) 3. (∃x)(Yx ∙ Zx) / (∀x)(Wx ⊃ Zx) 24. 1. (∀x)[Ax ⊃ (Bx ∙ Cx)] 2. (∃x)[(Bx ∙ Cx) ∙ Ax]

/ (∀x)[Ax ≡ (Bx ∙ Cx)]

25. 1. (∃x)Dx ⊃ (∃x)Ex 2. (∃x)(Ex ∙ Fx) ⊃ (∀x)(Dx ⊃ Gx) 3. (∃x)(Dx ∙ Fx) / ∼(∃x)(Dx ∙ ∼Gx) 26. 1. (∃x)(∼Hx ∙ Ix) 2. (∃x)(Hx ∙ ∼Ix) 3. (∀x)( Jx ≡ Ix)

/ (∀x)(Hx ⊃ Jx)

27. 1. (∀x)(Mx ⊃ Nx) 2. (∃x)(Mx ∙ Ox) 3. Oa

/ Oa ∙ Na

28. 1. (∃x)[Dx ∙ (Ex ∨ Fx)] 2. ∼(∃x)(Ex ∙ Fx) 3. (∃x)Ex 4. Fa

/ (∃x)(Dx ∙ Fx)

29. 1. (∃x)[(Ix ∙ Jx) ∙ ∼Kx] 2. (∃x)[(Ix ∙ Jx) ∙ Kx] 3. (∀x)(Kx ≡ Lx)

/ (∃x)(Lx ∙ ∼Ix)

30. 1. (∃x)(Ex ∙ Fx) 2. (∃x)(Ex ∙ ∼Fx) 3. (∀x)(Fx ≡ Gx)

/ (∀x)Ex

4 . 8 : In v al i d i t y i n M   2 8 9

31. 1. (∃x)(Kx ∙ Mx) 2. La ∙ Lb

/ (∃x)(Lx ≡ Mx)

32. 1. (Ha ∙ ∼Ia) ∙ Ja 2. (∃x)[Ix ∙ (Jx ≡ ∼Kx)] 3. (∃x)(∼Jx ∨ Kx)

/ (∃x)Kx

33. 1. (∃x)(Kx ∙ ∼Lx) 2. (∃x)(Kx ∙ Lx) 3. (∀x)[(Mx ∨ Nx) ⊃ Lx]

/ (∀x)(Mx ⊃ Kx)

34. 1. (∃x)(Lx ∙ Mx) 2. (∃x)(∼Lx ∙ Mx) 3. (∀x)(Mx ⊃ Nx)

/ (∀x)(Lx ⊃ Nx)

35. 1. (∃x)(Fx ∙ Gx) 2. (∃x)(∼Fx ∙ Gx) 3. (∀x)[Gx ⊃ (Fx ≡ Hx)]

/ (∀x)(Fx ≡ Hx)

36. 1. (∃x)(Ax ∙ Bx) 2. (∃x)[(Ax ∙ ∼Bx) ∙ Cx] 3. ∼(∃ x)(∼Ax ∙ ∼Dx) 4. (∀x)(Dx ⊃ Cx)

/ Ca ∨ Cb

37. 1. (La ∙ ∼Lb) ∙ (∼Mc ∙ Md) 2. (∃x)(Lx ∙ Nx) 3. (∃x)(Mx ∙ Ox) 4. (∀x)[(Lx ∨ Mx) ⊃ Ox] / (∀x)(Nx ⊃ Ox) 38. 1. (∀x)(Mx ⊃ Nx) 2. ∼(∀x)(Ox ⊃ Nx) 3. (∀x)(Px ⊃ ∼Ox) 4. Ma ∙ Mb

/ (∀x)(Px ⊃ Nx)

39. 1. (∀x)(Px ⊃ Qx) 2. (∀x)(∼Px ≡ Rx) 3. (∃x)(Qx ∙ Rx) 4. (Pa ∙ Pb) ∙ (∼Pc ∙ ∼Pd) / ∼Qb 40. 1. (∃x)[(Ex ∙ Fx) ∙ Gx] 2. (∃x)[(Ex ∙ ∼Fx) ∙ Gx] 3. (∃x)(∼Ex ∙ Gx) 4. (∀x)(Gx ⊃ Hx) 5. (∀x)(∼Gx ⊃ ∼Ex)

/ Ha ∨ Fa

2 9 0    C h apter 4  Mona d i c P re d i cate L og i c

EXERCISES 4.8b Show that each of the invalid arguments from Exercises 4.4c, listed here, is invalid. 1. (4.4c: 2)

1. (∃x)(Px ∙ Rx) 2. Pa ∙ Qa

/ (∃x)(Qx ∙ Rx)

2. (4.4c: 4)

1. (∀x)(Px ⊃ Qx)

/ (∃x)(Px ∙ Qx)

3. (4.4c: 5)

1. (∃x)(Px ∙ Qx) 2. (∀x)(Px ⊃ Rx)

/ (∀x)(Qx ⊃ Rx)

4. (4.4c: 6)

1. (∀x)(Px ⊃ Qx) 2. (∃x)(Qx ⊃ Rx)

/ (∀x)(Px ⊃ Rx)

5. (4.4c: 9)

1. (∀x)Px ⊃ (∀x)Qx 2. (∃x)Px / (∃x)Qx

6. (4.4c: 10)

1. (∃x)(∼Px ∨ Qx) 2. (∀x)(∼Px ⊃ Qx) / Qa

EXERCISES 4.8c For each argument, determine whether it is valid or invalid. If it is valid, derive the conclusion using our rules of inference and equivalence. If it is invalid, provide a counterexample. 1. 1. (∀x)[Ax ⊃ (Bx ∙ Cx)] 2. (∀x)(Bx ⊃ Ax)

/ (∀x)(Ax ≡ Cx)

2. 1. (∀x)[Ax ⊃ (Bx ∙ Cx)] 2. (∀x)[(Bx ∨ Cx) ⊃ Ax] / (∀x)(Ax ≡ Cx) 3. 1. (∀x)[(Dx ∨ Ex) ⊃ Fx] 2. ∼(∃x)(Fx ∙ Gx)

/ (∀x)(Ex ⊃ ∼Gx)

4. 1. (∀x)[(Dx ∙ Ex) ⊃ Fx] 2. ∼(∃x)(Fx ∙ Gx)

/ (∀x)(Ex ⊃ ∼Gx)

5. 1. (∀x)(Hx ⊃ Ix) 2. (∀x)( Jx ⊃ Ix) 3. ∼(∃x)(Hx ∙ Jx)

/ (∀x)(Ix ⊃ Jx)

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6. 1. (∀x)[Xx ⊃ (Yx ≡ Zx)] 2. (∃x)(Xx ∙ ∼Yx) 3. (∀x)(Zx ∨ Wx)

/ (∃x)(Xx ∙ Wx)

7. 1. (∃x)(Xx ∙ Yx) ⊃ (∀x)[Xx ⊃ (Yx ∙ Zx)] 2. ∼(∃x)(∼Xx ∙ ∼Yx) 3. (∃x)(Yx ∙ Zx) / (∃x)(Xx ∙ Zx) 8. 1. (∀x)[Px ≡ (Qx ∨ Rx)] 2. (∀x)(Rx ≡ Sx) 3. (∃x)(Sx ∙ Tx)

/ (∃x)(Px ∙ Tx)

9. 1. (∀x)[Px ≡ (Qx ∨ Rx)] 2. (∀x)(Rx ≡ Sx) 3. (∃x)(Sx ∙ Tx)

/ ∼(∃x)(Px ∙ ∼Tx)

10. 1. (∀x)[Xx ⊃ (Yx ≡ Zx)] 2. (∃x)(Xx ∙ ∼Yx) 3. (∃x)(Zx ∙ Yx)

/ (∃x)(Xx ∙ Yx)

11. 1. (∀x)(Kx ⊃ Lx) 2. ∼(∃x)(∼Mx ∙ Lx) 3. (∀x)(∼Mx ∨ Nx) 4. ∼(∃x)∼(∼Nx ∨ ∼Kx) / ∼(∃x)Kx 12. 1. (∃x)(Ax ∙ ∼Bx) ⊃ (∃x)(Ax ∙ Bx) / (∃x)Ax ⊃ ∼(∀x)∼Bx 13. 1. (∀x)[(Cx ∨ Dx) ⊃ ∼Ex] 2. ∼(∃x)(∼Fx ∙ ∼Ex)

/ (∀x)(Fx ∨ ∼Dx)

14. 1. (∃x)(Gx ∙ ∼Hx) 2. (∃x)(Hx ∙ ∼Gx) 3. (∀x)[Ix ⊃ (Gx ∨ Hx)]

/ (∃x)[Ix ∙ (∼Gx ∨ ∼Hx)]

15. 1. (∃x)[( Jx ∙ Kx) ∙ ∼Lx] 2. (∀x)[Lx ⊃ ( Jx ∨ Kx)] 3. (∃x)(Mx ∙ Lx)

/ (∃x)(Mx ∙ ∼Lx)

16. 1. (∀x)(Px ⊃ Qx) ≡ (∀x)(Qx ⊃ Rx) 2. (∃x)(Px ∙ ∼Qx) 3. (∀x)(∼Sx ⊃ Rx) / (∃x)(Qx ∙ Sx) 17. 1. (∀x)[Gx ⊃ (Hx ∨ Ix)] 2. (∀x)(Gx ⊃ Hx) ⊃ (∀x)(Jx ∙ ∼Hx) 3. ∼(∃x)Ix / (∃x)Jx 18. 1. (∀x)(Px ≡ ∼Qx) 2. (∀x)[(Rx ∨ Sx) ⊃ Qx] 3. (∀x)(Tx ⊃ Rx) 4. ∼(∃x)(∼Ux ∙ ∼Tx)

/ (∀x)(Px ⊃ Ux)

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19. 1. (∀x)[Gx ⊃ (Hx ∙ ∼Ix)] 2. (∃x)(Gx ∙ Hx) ⊃ (∀x)(Jx ⊃ ∼Hx) 3. ∼(∃x)Ix / (∃x)Gx ⊃ (∀x)Jx 20. 1. (∃x)[Ax ∙ (Bx ∙ ∼Cx)] 2. (∃x)[∼Ax ∙ (Bx ∙ ∼Cx)] 3. (∀x)(Bx ⊃ ∼Dx) / (∀x)(∼ Cx ⊃ ∼Dx)

EXERCISES 4.8d For each proposition, determine if it is a logical truth. If it is a logical truth, provide a derivation. If it is not, provide a valuation that shows it false in some finite domain. 1. (∀x)Ax ⊃ (∃x)Ax 2. (∀x)(Bx ⊃ ∼Bx) 3. (∃x)Cx ∨ (∃x)∼Cx 4. (∀x)Dx ∨ (∀x)∼Dx 5. (∀x)(Ex ⊃ Fx) ⊃ (∃x)(Ex ∙ Fx) 6. [(∀x)(Gx ⊃ Hx) ∙ (∃x)∼Hx] ⊃ (∃x)∼Gx 7. ∼(∃x)∼(Kx ∙ Lx) ⊃ (∀x)(Kx ∙ Lx) 8. (∃x)(Ix ∙ ∼Jx) ≡ ∼(∀x)(Ix ⊃ Jx) 9. (∀x)[(Mx ∨ Nx) ⊃ Ox] ⊃ (∀x)(Mx ⊃ Ox) 10. (∀x)[(Px ∙ Qx) ⊃ Rx] ⊃ (∀x)(Px ⊃ Rx) 11. [(∃x)(Sx ∙ ∼Tx) ∙ (∃x)Tx] ⊃ (∃x)∼Sx 12. (∀x)[Xx ⊃ ∼(Yx ∨ Zx)] ⊃ ∼(∃x)(Xx ∙ Yx) 13. (∀x)[Ax ⊃ ∼(Bx ∙ Cx)] ⊃ (∃x)(∼Bx ∨ ∼Cx) 14. (∀x)(Dx ⊃ ∼Ex) ∨ (∃x)(Dx ∙ Ex) 15. (∀x)[(Fx ∨ Gx) ∨ Hx] ⊃ [(∃x)(∼Fx ∙ ∼Gx) ⊃ (∃x)Hx] 16. (∀x)[(Ix ∨ Jx) ∨ Kx] ⊃ (∀x)[∼(Ix ∙ Jx) ⊃ Kx] 17. (∃x)(Lx ∨ Mx) ≡ ∼(∀x)(Lx ∙ Mx) 18. (∃x)(Nx ∨ Ox) ∨ (∃x)(∼Nx ∨ ∼Ox) 19. (∃x)(Px ∙ Qx) ∨ (∃x)(∼Px ∙ ∼Qx) 20. [(∀x)(Rx ∙ Sx) ∨ (∃x)(Rx ∙ ∼Sx)] ∨ [(∃x)(∼Rx ∙ Sx) ∨ (∃x)(∼Rx ∙ ∼Sx)]

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KEY TERMS anyone, 4.2 anything, 4.1 atomic formula, 4.3 attribute, 4.1 bound variable, 4.3 closed sentence, 4.3 constant, 4.1 domain of interpretation, 4.7 domain of quantification, 4.7 everyone, 4.2 existential generalization (EG), 4.4 existential instantiation (EI), 4.4 existential quantifier, 4.1 free variable, 4.3 hasty generalization, 4.4 interpretation, 4.7 logical truth for M, 4.7 M, 4.1 method of finite universes , 4.8 model, 4.7 monadic predicate logic, 4.1 new constant, 4.4 no one, 4.2

only, 4.2 open sentence, 4.3 predicate, 4.1 predicate logic, 4.1 proof theory, 4.7 quantifier, 4.1 quantifier exchange (QE), 4.5 satisfaction, 4.7 scope, 4.3 scope of an assumption, 4.6 scope of a negation, 4.3 scope of a quantifier, 4.3 set, 4.7 singular terms, 4.1 someone, 4.2 subformula, 4.3 subject, 4.1 subset, 4.7 universal generalization (UG), 4.4 universal instantiation (UI), 4.4 universal quantifier, 4.1 variable, 4.1

Chapter 5 Full First-Order Logic

5.1: TRANSLATION USING RELATIONAL PREDICATES Argument 4.1.1 showed that some intuitively valid inferences were not valid in PL; we explored M in response. Argument 5.1.1 shows that some intuitively valid inferences are not valid in M and that we should examine a further refinement of our logic. 5.1.1

Alyssa is taller than Bhavin. Bhavin is taller than Carlos. Given any three things, if one is taller than another, and the latter is    taller than the third, then the first is also taller than the third. So, Alyssa is taller than Carlos.

In M, with only monadic predicates, we translate the two first sentences with different predicates. The first sentence ascribes to Alyssa the property of being taller than Bhavin. The second sentence ascribes to Bhavin the property of being taller than Carlos. Being taller than Carlos is a different property from being taller than Bhavin. So, if I use ‘Tx’ for ‘x is taller than Bhavin’, I need a different predicate, say, ‘Ux’ for ‘x is taller than Carlos’. Relational predicates, or polyadic predicates,

are followed by more than one singular term. Dyadic predicates are followed by two singular terms. Triadic predicates are followed by three singular terms.

5.1.2

Alyssa is taller than Bhavin. Bhavin is taller than Carlos.

Ta Ub

But what we really want is a more general predicate, being taller than, that relates two singular terms. Such a predicate is called dyadic. 5.1.3 contains examples of various dyadic predicates. 5.1.3

Txy: x is taller than y Kxy: x knows y Pxy: x precedes y

We can construct three-place predicates too, called triadic predicates, as at 5.1.4. 5.1.4

294

Gxyz: x gives y to z Bxyz: x is between y and z Kxyz: x kisses y at z

We can construct four-place and higher-place predicates, as well. All predicates that take more than one singular term are called relational, or polyadic. With relational predicates, we now have a choice of how to regiment sentences like 5.1.5.

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5.1.5

Andrés loves Beatriz.

We could regiment 5.1.5 in monadic predicate logic as ‘La’. In that case, ‘L’ stands for the property of loving Beatriz. But if we want to use ‘L’ to stand for the general, relational property of loving, it will take two singular terms: one for the lover and one for the beloved. We can introduce a two-place predicate, ‘Lxy’, which means that x loves y. Then, we regiment 5.1.5 as 5.1.6. 5.1.6 Lab

A similar translation, using a three-place relation for giving, can help us avoid using an overly simple monadic predicate for 5.1.7. 5.1.7

Camila gave David the earring.

Instead of using ‘Gx’ for ‘x gives David the earring,’ we can invoke ‘Gxyz’ for ‘x gives y to z’. Then, 5.1.7 is regimented as 5.1.8, using constants ‘c’, ‘d’, and ‘e’ for Camila, David, and the earring. (Using a constant to stand for the object picked out by a definite description—‘the earring’—is somewhat contentious; we’ll see a more standard way to represent definite descriptions in section 5.4.) 5.1.8 Gced

By using relational predicates, we reveal more logical structure. The more logical structure we reveal, the more we can facilitate inferences. We will rarely, in this text, use relational predicates of more than three places. But more-complex relations can be useful. For example, 5.1.9, couched in a serious scientific theory, might be regimented using a five-place relation. 5.1.9

There is something blue over there now.

We need a predicate for blueness and one place for the object. To indicate the spatial position of the object, we could use three places: one for each position on a threedimensional coordinate axis. And we can add one more place for a temporal location, resulting in the formula at 5.1.10. 5.1.10 (∃v)Bvabct

In other words, there is a thing that is blue at spatial location a,b,c at time t. 5.1.10 thus uses constants for spatial locations (a, b, and c) and temporal location (t), but we could of course quantify over them as well, as at 5.1.11. 5.1.11 (∃v)(∃x)(∃y)(∃z)(∃w)Bvxyzw

The utility of a language with more variables (and thus more quantifier variables), which we discussed in section 4.3 (‘How to Expand Our Vocabulary’), should be apparent. The order of the singular terms that follow a predicate is important. The property of loving is distinct from the property of being loved. And, as many of us sadly know, the loving relation is not always symmetric: things we love don’t always love us back. For these reasons, we have to be careful to be clear about the precise nature of our relational predicates. I’ll be pedantically explicit as we proceed.

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By introducing relational predicates, we have extended our language. We are now using a language I call F, for full first-order predicate logic, rather than M. The differences here between F and M are minor. The two languages use the same vocabulary except for two small additions introduced in section 5.4 and used merely as shorthands and nearly the same formation rules. But beyond this text, the differences between M and F are significant.

Quantifiers with Relational Predicates We can now translate the first two premises of 5.1.1 and its conclusion, using ‘Txy’ for ‘x is taller than y’. 5.1.1

Alyssa is taller than Bhavin. Bhavin is taller than Carlos. Alyssa is taller than Carlos.

Tab Tbc Tac

To regiment the third premise of 5.1.1, we need multiple, overlapping quantifiers. Let’s see how to use quantifiers with relational predicates in steps. We’ll start with sentences with just one quantifier. The sentences at 5.1.12 use ‘Bxy’ for ‘x is bigger than y’. (Ignore for now the inconsistency of the last two sentences, which entail that something [Joe] is bigger than itself. We’ll introduce a device to eliminate that inconsistency in section 5.4.) 5.1.12

Joe is bigger than something. Something is bigger than Joe. Joe is bigger than everything. Everything is bigger than Joe.

(∃x)Bjx (∃x)Bxj (∀x)Bjx (∀x)Bxj

Next, we can introduce overlapping quantifiers. 5.1.13 uses ‘Lxy’ for ‘x loves y’. 5.1.13

Everything loves something.

(∀x)(∃y)Lxy

Note the different quantifier letters: overlapping quantifiers must use different variables in order not to violate the formation rules, which we’ll see in detail in the next section. Also, the order of quantifiers matters. 5.1.14 differs from 5.1.13, but only in the order of the quantifiers. 5.1.14

Something loves everything.

(∃x)(∀y)Lxy

Switching the order of the quantifiers in front of a formula thus changes its meaning. Note that the leading quantifier in each sentence of F corresponds to the first word of the corresponding English sentence. Changing the order of the singular terms changes the meaning as well, as we can see at 5.1.15 and 5.1.16. 5.1.15 5.1.16

Everything is loved by something. Something is loved by everything.

(∀x)(∃y)Lyx (∃x)(∀y)Lyx

We now can regiment 5.1.1 completely. The remaining third premise is ‘given any three things, if one is taller than another, and the latter is taller than the third, then the

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first is also taller than the third’. We translate that claim, also known as the transitive property of ‘taller than’ with three universal quantifiers, as in the third premise of the argument 5.1.17. 5.1.17

1. Tab 2. Tbc 3. (∀x)(∀y)(∀z)[(Txy ∙ Tyz) ⊃ Txz]

/ Tac

We will return to deriving the conclusion of this argument in section 5.3. For the remainder of this section, and in the next section as well, we will look at some more complicated translations.

People and Things and Using Relational Predicates Instead of Monadic Ones In full first-order logic, we often use the terms ‘someone’ and ‘everyone’ in addition to ‘something’ and ‘everything’. We represent this difference by invoking a predicate ‘Px’ for ‘x is a person’. Sometimes we’ll use ‘some people’ or ‘all people’ to mean the same thing. The difference should be clear in 5.1.18–5.1.22, which also use ‘Txy’ for ‘x teaches y’ and the constant ‘p’ for ‘Plato’. 5.1.18 5.1.19 5.1.20 5.1.21 5.1.22

Something teaches Plato. Someone teaches Plato. Plato teaches everyone. Everyone teaches something. Some people teach themselves.

(∃x)Txp (∃x)(Px ∙ Txp) (∀x)(Px ⊃ Tpx) (∀x)[Px ⊃ (∃y)Txy] (∃x)(Px ∙ Txx)

We can also use relational predicates to reduce our dependence on monadic predicates. For example, we can take ‘teacher’ to refer to someone who teaches something and ‘student’ to refer to someone who is taught (by something). Then, we can use ‘Txy’, for ‘x teaches y’ to characterize both teachers and students, eliminating any need for the monadic predicates ‘x is a teacher’ and ‘x is a student’. 5.1.23 5.1.24

There are teachers. There are students.

(∃x)(∃y)Txy (∃x)(∃y)Tyx

In 5.1.25, we say that anything that is such that there is something that it teaches (i.e., any teacher) is interesting, if it is skilled. 5.1.25

Skilled teachers are interesting.

(∀x)[(∃y)Txy ⊃ (Sx ⊃ Ix)]

5.1.26 expands this practice, using two different predicates for teachers: the skilled ones and the unskilled ones. 5.1.26 Skilled teachers are better than unskilled teachers. (∀x){[(∃y)Txy ∙ Sx] ⊃ {(∀z)[(∃y)Tzy ∙ ∼Sz] ⊃ Bxz}}

Notice that we can use ‘(∃y)’ in both the first antecedent and the second, since the scope of the first quantifier has ended before we need the second one.

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A quantifier’s scope is wider t he more subformulas it contains; it is narrower the fewer subformulas it contains.

Wide and Narrow Scope The scope of a quantifier may be wider (have more subformulas in its scope) or narrower (having fewer). When you have multiple quantifiers in a proposition, they can take wide scope by standing in front of the proposition, as in 5.1.27. Or one can take a narrower scope by being located inside the proposition, as in 5.1.28. 5.1.27 (∃x)(∃y)[(Px ∙ Py) ∙ Lxy] 5.1.28 (∃x)[Px ∙ (∃y)(Py ∙ Lxy)]

5.1.27 and 5.1.28 are equivalent. But sometimes changing the scope of a quantifier changes the meaning of the sentence. 5.1.29 and 5.1.30, for example, are not logically equivalent. 5.1.29 (∀x)[Px ⊃ (∃y)(Py ∙ Qxy)] 5.1.30 (∃y)(∀x)[Px ⊃ (Py ∙ Qxy)]

5.1.29 could be used for ‘all people love someone’. Using the same interpretation of the predicates, 5.1.30 would stand for ‘there is someone everyone loves’. 5.1.29 is plausible. 5.1.30 is not. In general, the scope of the quantifiers doesn’t much matter when the quantifiers are all existential or all universal. Just as 5.1.27 and 5.1.28 are equivalent, 5.1.31, in which we introduce all of the quantifiers we need at the beginning of our formula, is equivalent to 5.1.32, in which we alter the formula by a simple use of exportation and wait to introduce ‘(∀y)’. 5.1.31 (∀x)(∀y)[(Mx ∙ My) ⊃ Pxy] 5.1.32 (∀x)[Mx ⊃ (∀y)(My ⊃ Pxy)]

When some quantifiers are existential and some are universal, changes of scope can alter meaning. When translating, it is best form to introduce quantifiers only when needed, giving them as narrow a scope as possible. On occasion, we will put all quantifiers in front of a formula, using wide scope. But moving quantifiers around is not always simple, and we must be careful.

More Translations For 5.1.33–5.1.38, I use Px: x is a person, and Kxy: x knows y. 5.1.33 Someone knows everything. (∃x)[Px ∙ (∀y)Kxy] 5.1.34 Someone knows everyone. (∃x)[Px ∙ (∀y)(Py ⊃ Kxy)] 5.1.35 Everyone knows someone. (∀x)[Px ⊃ (∃y)(Py ∙ Kxy)] 5.1.36 Everyone knows everyone. (∀x)[Px ⊃ (∀y)(Py ⊃ Kxy)] or (∀x)(∀y)[(Px ∙ Py) ⊃ Kxy)] 5.1.37 No one knows everything. (∀x)[Px ⊃ (∃y)∼Kxy] or ∼(∃x)[Px ∙ (∀y)Kxy] 5.1.38 No one knows everyone. (∀x)[Px ⊃ (∃y)(Py ∙ ∼Kxy)] or ∼(∃x)[Px ∙ (∀y)(Py ⊃ Kxy)]

Notice the structural similarities among many of these propositions, especially with the quantifiers having narrow scopes. The leading quantifier matches the first

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word in the English sentence. The main operator of the subformula that follows depends on the quantifier (conditionals for universals; conjunctions for existentials). And similarly, when the second quantifier is introduced, the operator in the ensuing subformula will match the leading quantifier. 5.1.39 Every child is stronger than some adult.    (Cx: x is a child; Ax: x is an adult; Sxy: x is stronger than y) (∀x)[Cx ⊃ (∃y)(Ay ∙ Sxy)] 5.1.40 No cat is smarter than any horse.    (Cx: x is a cat; Hx: x is a horse; Sxy: x is smarter than y) (∀x)[Cx ⊃ (∀y)(Hy ⊃ ∼Sxy)] ∼(∃x)[Cx ∙ (∃y)(Hy ∙ Sxy)] (∀x)(∀y)[(Cx ∙ Hy) ⊃ ∼Sxy]

5.1.40 uses a ‘no’ as a quantifier, but the first and second version I provide maintains the structure. The third one uses a wide scope, which is acceptable since there are two universal quantifiers. Sometimes it is useful, before translating, to try to put a proposition into a semiformal form. Take the adage ‘dead men tell no tales’, which has only a slightly more complex structure. To translate the adage into F, we can first think about how we would add the quantifiers: for all x, if x is a dead man, then for all y, if y is a tale, then x does not tell y. 5.1.41 Dead men tell no tales.    (Dx: x is dead; Mx: x is a man; Tx: x is a tale; Txy: x tells y) (∀x)[(Dx ∙ Mx) ⊃ (∀y)(Ty ⊃ ∼Txy)]

Also worth noting, at 5.1.41 is that the same predicate letter can be used twice, as I did with ‘T’. You can distinguish a monadic predicate from a dyadic or triadic predicate by looking at how they are used. In 5.1.41, you can see which is which by just looking at the number of singular terms that follow the predicate letter. The structure of our wffs really remains the same even with three-place predicates, as at 5.1.42–5.1.44. 5.1.42 There is a city between New York and Washington.    (n: New York; w: Washington; Cx: x is a city; Bxyz: y is    between x and z) (∃x)(Cx ∙ Bnxw) 5.1.43 Everyone gives some gift to someone.    (Gx: x is a gift; Px: x is a person; Gxyz: y gives x to z) (∀x){Px ⊃ (∃y)[Gy ∙ (∃z)(Pz ∙ Gxyz)]} 5.1.44 Everyone gives something to someone.    (Px: x is a person; Gxyz: y gives x to z) (∀x)[Px ⊃ (∃y)(∃z)(Pz ∙ Gxyz)]

When punctuating, make sure never to leave variables unbound. It is often useful to punctuate after the translation is done, rather than along the way, or at least to check your punctuation once you have completed a translation. Leading quantifiers

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generally have the whole statement in their scope. Other quantifiers tend to have smaller scopes. In 5.1.26, we saw two quantifiers with very narrow scopes. The second quantifier in 5.1.45, as in many of the earlier examples, has the remainder of the formula in its scope since it has to bind a variable in the last term of the wff. 5.1.45 A dead lion is more dangerous than a live dog.    (Ax: x is alive; Dx: x is a dog; Lx: x is a lion;    Dxy: x is more dangerous than y) (∀x){(Lx ∙ ∼Ax) ⊃ (∀y)[(Dy ∙ Ay) ⊃ Dxy]}

The Power of F F allows us to translate some neat subtleties and facilitate the understanding of many aspects of our language. Using F can be pretty amusing, too. For example, check out the formalization of William Carlos Williams’s “The Red Wheelbarrow,” at 5.1.46, using: Bx: x is a wheelbarrow; Bxy: x is beside y; Cx: x is a chicken; Dxy: x depends on y; Gxy: x glazes y; Rx: x is red; Sx: x is rainwater; Wx: x is white. 5.1.46 so much depends upon a red wheel barrow glazed with rain water beside the white chickens. (∃x){(Bx ∙ Rx) ∙ (∃y)Dyx ∙ (∃z)(Sz ∙ Gzx) ∙ (∃w)(Cw ∙ Ww ∙ Bxw)}

An interesting exercise would be to discuss the virtues and weaknesses of this formalization. Another interesting exercise would be to translate other work. There is a translation of Williams’s “This Is Just to Say” at the end of the exercises; look it up and give it a try before peeking!

Summary F is a powerful language, nearly the strongest of the formal languages we will study. It allows us to represent, in logical language, a wide range of propositions and inferences of English, without the ambiguity of natural languages. Exercises 5.1b, which I adapted from the logic textbook I used as an undergraduate, asks you to translate into English some well-known sentences that have been rendered in F; you’ll be able to see how much subtlety and expression you can pack into wffs of F there especially. Once again, the best way to get comfortable with the difficulties and subtleties of F is to practice your translations as much as possible. Remember to check your punctuation. In the next section, I’ll lay out the formal syntax and semantics of F. You might find that looking at that material will help you with the translation exercises of this

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section. Then we’ll look at derivations in section 5.3. There’s one more major topic, identity theory, that we will study in sections 5.4 and 5.5. Identity theory actually doesn’t change our language, but it introduces a special predicate and some rules governing inferences with it. KEEP IN MIND

Relational predicates can be followed by any number of singular terms, though most of our work in this text will use one- to three-place predicates. The order of the singular terms matters. The order of quantifiers also matters. Try to keep the scope of your quantifiers as narrow as possible. When all quantifiers are existential or all are universal, putting them all in front, with wide scope, is acceptable. Be careful to distinguish “someone” from “something” and “everyone” from “everything.” It is important to punctuate correctly, never leaving an unbound variable.

TELL ME MORE  Why do we differentiate between M and F? See 6.4: Metalogic. Are there rules for moving quantifiers through a formula? See 6S.12: Rules of Passage.

EXERCISES 5.1a Translate each of the following into predicate logic using relational predicates. For exercises 1–10, use: b: Ben Gx: x is gray Mx: x is a mouse Rx: x is a rat Lxy: x is larger than y 1. All rats are larger than Ben. 2. Ben is larger than all rats. 3. Some rats are larger than Ben. 4. No rats are larger than Ben.

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5. All gray rats are larger than Ben. 6. All rats are larger than some mice. 7. No rats are larger than some mice. 8. Some gray rats are larger than all mice. 9. No gray mice are larger than some gray rats. 10. If some gray mouse is larger than all rats, then Ben is not larger than some gray mouse. For exercises 11–20, use: Gx: x is a god Px: x is a pen Sx: x is a sword Mxy: x is mightier than y 11. All pens are mightier than all swords. 12. All gods are mightier than all pens. 13. No pens are mightier than some gods. 14. All gods are mightier than all pens and all swords. 15. No sword is mightier than any pen. 16. No god is mightier than herself. 17. Any sword mightier than some god is mightier than all pens. 18. Some pens are mightier than some swords, but some swords are mightier than some pens. 19. No swords are mightier than all gods, but some swords are mightier than some gods. 20. If some pens are mightier than all swords, then some gods are not mightier than some pens. For exercises 21–30, use: Dx: x is a dancer Px: x is a person Rx: x is a runner Hxy: x is healthier than y Sxy: x is stronger than y

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21. All runners are healthier than some people. 22. Some dancers are healthier than some people. 23. All dancers are healthier than some people. 24. No dancer is healthier than some runner. 25. No runner is stronger than every person. 26. Some runners are healthier than no dancers. 27. Some people who are dancers are stronger than some people who are not runners. 28. If someone is stronger than someone, then s/he is healthier than someone. 29. If some dancer is stronger than all people, then no person is healthier than some dancer. 30. Either some runner is stronger than all dancers or some dancer is healthier than all runners. For exercises 31–40, use: l: literature m: mathematics p: philosophy Sx: x is a student Mxy: x majors in y Cxy: x is a course in y Txy: x takes y 31. Every student majors in something. 32. All math majors take a philosophy class. 33. Some math majors do not take a literature class. 34. Some math majors take a literature class and a philosophy class. 35. No literature majors take a mathematics course. 36. All literature majors take a course in philosophy or a course in mathematics. 37. Some students major in philosophy and take courses in mathematics. 38. No student majors in philosophy and mathematics without taking a course in literature. 39. Every student who majors in literature or philosophy takes a course in mathematics.

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40. If some students take courses in philosophy and mathematics, then all literature majors take courses in philosophy or mathematics. For exercises 41–50, use: c: Chiara m: Marina o: Orsola Px: x is a person Ixyz: x introduces y to z 41. Orsola introduces Chiara to Marina. 42. Someone introduces Chiara to Marina. 43. Someone introduces Chiara to everyone. 44. No one introduces Chiara to Orsola. 45. Orsola introduces Chiara to everyone. 46. Orsola introduces someone to Marina. 47. Orsola introduces someone to everyone. 48. Marina does not introduce Chiara to Orsola. 49. No one introduces Marina to everyone. 50. No one introduces someone to everyone. For exercises 51–60, use: Bx: x is big Hx: x is a home Ox: x is an office Px: x is a person Dxyz: x drives from y to z 51. Some people drive from an office to a home. 52. Some people do not drive from an office to a home. 53. Some people drive from a home to an office. 54. No one drives from an office to a home. 55. Someone drives from a big home to all offices. 56. Someone drives from an office to all big homes.

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57. A big person drives from a home to an office. 58. No one drives from a big home to an office that is not big. 59. Everyone who drives from a home to an office drives from the office back home. 60. If you don’t drive from some home to any office, then you don’t drive from some office to a home. For exercises 61–70, use: a: Asimov j: Jin Bx: x is a book Ix: x is intelligent Px: x is a person Sx: x is a scholar Rxy: x reads y Wxy: x writes y 61. Jin reads all books written by Asimov. 62. Jin is an intelligent person who reads some books by Asimov. 63. If Jin writes intelligent books, then he is a scholar. 64. Some people read all books written by Asimov. 65. Some people read all books written by someone. 66. Some scholars read all intelligent books written by Asimov. 67. No scholar reads any book written by Asimov unless s/he is intelligent. 68. If a scholar writes a book, then all intelligent people read it. 69. No intelligent person reads any book by any scholar. 70. All intelligent people read books written by some scholar. For exercises 71–80, use: Cx: x is a child Ex: x is elderly Hx: x is a home Jx: x is a jewel Px: x is a person Tx: x is a thief Txyz: x takes y from z

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71. Some thieves take jewels from elderly people. 72. Some children take jewels from thieves. 73. Some thieves take children from homes. 74. Every thief takes something from some home. 75. No thief takes jewels from elderly people. 76. No elderly thief takes children from a home. 77. No children take jewels from anything. 78. Some thieves take jewels from both children and elderly people. 79. No children who are thieves take anything from elderly people. 80. If some thieves take some jewels from some homes, then some thieves take all jewels from some homes.

For exercises 81–90, use: b: Judith Butler d: Simone de Beauvoir m: Mary Wollstonecraft Px: x is a philosopher Axy: x anticipates y Rxy: x respects y 81. Mary Wollstonecraft anticipates both Simone de Beauvoir and Judith Butler. 82. Some philosophers respect Butler, but some do not. 83. Butler respects both Wollstonecraft and de Beauvoir but anticipates neither. 84. Any philosopher who anticipates Butler respects de Beauvoir. 85. No philosopher anticipates either de Beauvoir or Wollstonecraft. 86. All philosophers respect Wollstonecraft, but some philosophers also anticipate Butler. 87. Any philosopher who anticipates de Beauvoir also respects her. 88. Any philosopher who does not respect Butler doesn’t respect Wollstonecraft. 89. A philosopher respects de Beauvoir if, and only if, she respects herself. 90. If some philosopher anticipates either Wollstonecraft or de Beauvoir, then no philosopher does not respect both de Beauvoir and Butler.

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For exercises 91–100, use: a: Aristotle c: that forms are causes p: Plato s: that forms are in a separate world Px: x is a person Bxy: x believes y Dxy: x denies y 91. Plato believes that forms are causes, and in a separate world, but Aristotle believes neither. 92. Plato believes something that Aristotle denies. 93. Aristotle believes everything that Plato denies. 94. Not only does Aristotle not believe that forms are causes, but he denies that they are. 95. Plato believes that forms are in a separate world just in case Aristotle denies it. 96. No one believes that forms are causes, even though Plato does not deny it. 97. Aristotle denies either that forms are causes or that they are separate, but not both. 98. If someone believes that forms are in a separate world, then Plato does. 99. No one denies that forms are separate even though not everyone believes it. 100. If Plato believes nothing that Aristotle believes, then no one believes that forms are causes. For exercises 101–110, use: c: Christina, Queen of Sweden d: Descartes e: Elisabeth, Princess of Bohemia Px: x is a person Cxy: x corresponded with y Ixy: x influenced y 101. Elisabeth, Princess of Bohemia, and Christina, Queen of Sweden, corresponded with Descartes. 102. Elisabeth did not correspond with Christina but did influence her.

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103. Everyone who corresponded with Descartes influenced Descartes. 104. No one who corresponded with Descartes was not influenced by him. 105. No one who influenced Descartes corresponded with both Elisabeth and Christina. 106. If Elisabeth corresponded with Descartes and influenced him, then so did Christina. 107. Everyone who corresponded with Descartes was influenced by someone who corresponded with Elisabeth. 108. Everyone who corresponded with Descartes corresponded with each other. 109. Elisabeth did not influence Descartes if, and only if, no one who corresponded with him influenced him. 110. Someone who was influenced by Elisabeth corresponded with someone who was influenced by Christina. For exercises 111–120, use: b: boastfulness c: courage m: mock modesty t: truthfulness Cx: x is a characteristic Ex: x is an extreme Vx: x is a virtue Bxyz: y is between x and z 111. Truthfulness is a virtue between the characteristics of boastfulness and mock modesty. 112. Boastfulness and mock modesty are not virtues, but extremes. 113. No extreme is a virtue, and no virtue is an extreme. 114. Mock modesty is an extreme characteristic, and not between anything. 115. No virtue is between boastfulness and itself. 116. Courage is a virtue, but it is not between boastfulness and mock modesty. 117. Some virtue is between boastfulness and some characteristic. 118. All virtues are between some extremes.

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119. If truthfulness is between two extremes, then boastfulness is not a virtue. 120. If mock modesty is a virtue, then anything is between any two characteristics.

For exercises 121–130, use: Bx: x is British Cx: x is continental Ex: x is an empiricist Rx: x is a rationalist Rxy: x is read more often than y Wxy: x wrote more than y 121. Some empiricist wrote more than some rationalist. 122. Some empiricist wrote more than all rationalists. 123. All rationalists wrote more than some empiricists. 124. No rationalist wrote more than some empiricist. 125. Some British empiricists wrote more than all continental rationalists. 126. No continental rationalist wrote more than all British empiricists. 127. Some continental rationalists wrote more than, but are not read more often than, some British empiricists. 128. All continental rationalists are read more than all British empiricists. 129. A British empiricist is read more than a continental rationalist. 130. If some British empiricist wrote more than all continental rationalists, then some continental rationalist is read more than all British empiricists.

For exercises 131–140, use: Ax: x is an act Gx: x is good Lx: x is laudable Px: x is punished Bxy: x produces better consequences than y Hxy: x is more heinous than y 131. All good acts are laudable.

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132. Some unpunished act is more heinous than any punished act. 133. For any act that is punished, there is some act more heinous. 134. Any act that produces better consequences than some act is laudable. 135. No laudable act is more heinous than all punished acts. 136. For any laudable act, some more heinous act produces better consequences. 137. Some laudable act does not produce better consequences than some act that is not laudable. 138. Some good acts are punished even though they produce better consequences than some acts that are not good. 139. No punished act is laudable if it doesn’t produce better consequences than some good act. 140. If no good acts are punished, then no acts which are not good produce better consequences than any laudable acts.

For exercises 141–150, use: Bx: x is a barber Fx: x has facial hair Mx: x is a man Tx: x is in town Sxy: x shaves y 141. Some men shave themselves. 142. Some men do not shave themselves. 143. All barbers shave some men. 144. A town barber shaves himself. 145. A town barber shaves everyone in town. 146. Some men with facial hair get shaved by a town barber. 147. No men with facial hair get shaved by a town barber. 148. Some barbers are not men and shave some barbers in town. 149. Some men in town are not shaved by any barber. 150. A barber in town shaves all men in town who do not shave themselves.

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EXERCISES 5.1b Use the translation key to translate the formulas into natural English sentences.1 Ax: x is silver Bxy: x belongs to y Cx: x is a cloud Cxy: x keeps company with y Dx: x is a dog Ex: x is smoke Fx: x is fire Fxy: x is fair for y g: God Gx: x is glass Gxy: x gathers y Hx: x is home Hxy: x helps y Ixy: x is in y Jxy: x is judged by y Kxy: x is a jack of y Lx: x is a lining Lxy: x is like y Mx: x is moss Mxy: x is master of y Px: x is a person Qx: x is a place Rx: x rolls Sx: x is a stone Tx: x is a trade Txy: x should throw y Ux: x is a house Uxy: x comes to y Vxy: x ventures y Wx: x waits Yx: x is a day

1. (∀x)[Dx ⊃ (∃y)(Yy ∙ Byx)] 2. (∀x)[(∃y)(Py ∙ Fxy) ⊃ (∀z)(Pz ⊃ Fxz)] 3. (∀x)[(Rx ∙ Sx) ⊃ (∀y)(My ⊃ ∼Gxy)] 4. (∀x)[(Px ∙ Wx) ⊃ (∀y)Uyx] 5. (∀x)[(Px ∙ Hxx) ⊃ Hgx] 6. (∀x)[Hx ⊃ (∀y)(Q y ⊃ ∼Lyx)] 7. (∀x){Cx ⊃ (∃y)[(Ay ∙ Ly) ∙ Byx]} 8. (∀x)[Px ⊃ (∀y)(Cxy ⊃ Jxy)] 9. (∀x){Qx ⊃ [(∃y)(Ey ∙ Iyx) ⊃ (∃z)(Fz ∙ Izx)]} 10. (∀x){[Px ∙ (∀y)(Ty ⊃ Kxy)] ⊃ (∀z)(Tz ⊃ ∼Mxz)} 11. (∀x){{Px ∙ (∃y)[(Gy ∙ Uy) ∙ Ixy]} ⊃ (∀z)(Sz ⊃ ∼Txz)} 12. (∀x){[Px ∙ (∀y)∼Vxy] ⊃ (∀z)∼Gxz}

1

Adapted from I. Copi, Symbolic Logic, 5th ed. (New York: Macmillan, 1979), 127–128.

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EXERCISE 5.1c, A WRITING ASSIGNMENT Consider the formalization in F of William Carlos Williams’s “The Red Wheelbarrow,” example 5.1.46, and this version of his “This Is Just to Say.” I have eaten the plums that were in the icebox and which you were probably saving for breakfast Forgive me they were delicious so sweet and so cold (∃x){Px ∙ Dx ∙ Sx ∙ Cx ∙ (∃y)(Iy ∙ Ixy) ∙ ◊Sux ∙ Eix ∙ Fiu} where: i: me; u: you; Cx: x is cold; Dx: x is delicious; Exy: x eats y; Fxy: x asks forgiveness from y; Ix: x is an icebox; Ixy: x is in y; Sx: x is sweet; and ◊ is to be taken (contentiously) as modal operator representing ‘probably’. See section 6.5 on modal logics.

What are the virtues and weaknesses of the regimentations?

5.2: SYNTA X, SEMANTICS, AND INVALIDITY IN F In our last section, we started translating arguments of F. In this section, I’ll lay out the syntax and semantics of F more carefully. Moving from M to F requires no change of vocabulary. The formation rules for F are almost the same, too. Formation Rules for Wffs of F F1. A predicate followed by any number of singular terms is a wff. F2. For any variable β, if α is a wff that does not contain either ‘(∃β)’ or ‘(∀β)’, then ‘(∃β)α’ and ‘(∀β)α’ are wffs. F3. If α is a wff, so is ∼α.

5 . 2 : S y nta x , S e m ant i cs , an d In v al i d i t y i n F   3 1 3

F4. If α and β are wffs, then so are: (α ∙ β) (α ∨ β) (α ⊃ β) (α ≡ β) F5. These are the only ways to make wffs. The only difference is to the first rule: predicates may be followed by any number of singular terms (constants and variables, for now), yielding monadic, dyadic, triadic, and other polyadic atomic formulas. The change in our formation rules thus multiplies the number of predicate letters we have available: twenty-six monadic predicate letters, twenty-six dyadic ones, and so on. Despite using the same predicate letter, ‘P’, the predicates at 5.2.1 may be distinguished by counting the number of singular terms that follow it.

A n atomic formula of F

is an n-placed predicate followed by n singular terms.

5.2.1 Pa Pab Pabc Pabcd

The semantics of M must also be adjusted to account for relational predicates. Recall that there were four steps for providing a standard formal semantics for M. Step 1. Specify a set to serve as a domain of interpretation, or domain of quantification. Step 2. Assign a member of the domain to each constant. Step 3. Assign some set of objects in the domain to each predicate. Step 4. U  se the customary truth tables for the interpretation of the propositional operators. The introduction of relational predicates requires adjustment to step 3. For an interpretation of F, we can also assign sets of ordered n-tuples to each relational predicate. Let’s take a moment to see the little bit more of set theory you need to understand ‘n-tuple’. An n-tuple is a set with structure used to describe an n-place relation. ‘N-tuple’ is a general term for pairs, triples, quadruples, and so on. Sets are unordered collections; n-tuples are sets in which order matters. The sets {1, 2} and {2, 1} are equivalent, since all that matters for the constitution of a set is its members. In contrast, the triple is distinct from the triple , which is distinct from the triple , even though they all have the same members. For the semantics of F, a two-place predicate is assigned sets of ordered pairs, a threeplace predicate is assigned sets of three-place relations, and so on. Given a domain of {1, 2, 3}, the relation ‘Gxy’, which could be understood as meaning ‘is greater than’, would be standardly interpreted by the set of ordered pairs: {, , }.

An ‘n-tuple’ is a general term for pairs, triples, quadruples, and so on.

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For relational predicates, our definitions of satisfaction and truth must be adjusted as well. Objects in the domain can satisfy predicates; that remains the case for oneplace predicates. Ordered n-tuples may satisfy relational predicates. A wff will be satisfiable if there are objects in the domain of quantification that stand in the relations indicated in the wff. A wff will be true for an interpretation if all objects in the domain of quantification stand in the relations indicated in the wff. The definition of logical truth remains the same: a wff is logically true if, and only if, it is true for every interpretation. For an example, let’s extend the interpretation we considered when originally discussing semantics of M, in section 4.7, to the theory TF1. Theory TF1:

1. Pa ∙ Pb 2. Ib ∙ ∼Ic 3. Nab 4. Nbc 5. (∃x)(Px ∙ Nxb) 6. (∃x)(Px ∙ Nbx) 7. (∀x)[Ix ⊃ (∃y)(Py ∙ Nxy)]

An Interpretation of TF1 Domain: {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} a: Venus b: Mars c: Neptune Px: Ix:

{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} {Mercury, Venus, Earth, Mars}

Nxy: {, , , , , , , , , , , , , , , , , , , , , , , , , , , }

Notice that our interpretation is a model of TF1; all of the statements of the theory come out true. Constructing an interpretation of a theory of F can be arduous, especially if the theory contains lots of relational predicates. I wrote out all of the ordered pairs for ‘Nxy’. But, as you probably observed, I could have just said that I was taking that relation to be interpreted as ‘x is nearer to the sun than y’, in which case I would have at least provided a rule that allows us to generate the list if we need it. For a three-place predicate, we use ordered triples. We can interpret the predicate ‘Bxyz’ as at 5.2.2, with a small domain.

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5.2.2

Domain: {1, 2, 3, 4, 5}



Bxyz: {, , , , , , , , , }

Here, the predicate can be understood as betweenness; “Bxyz” says that y is between x and z. But the interpretation of the relation is given purely extensionally, above, by the list of ordered triples.

Invalidity in F The method of finite domains of section 4.8 can be used in F just as well as it can be used in M, though the preponderance of overlapping quantifiers in many formulas of F can make the process more arduous. Let’s work with the invalid argument 5.2.3. 5.2.3 (∀x)[Px ⊃ (∃y)(Py ∙ Lxy)] (∃x)(Px ∙ Qx)

/ (∃x)[Qx ∙ (∃y)Lyx]

The argument is easily expanded into a domain of one object, though there is no counterexample there. Pa

Qa

Laa

Pa

⊃

(Pa

∙

Laa)

/

Pa

∙

Qa

//

Qa

To construct a counterexample, we have to make Pa and Qa true in the second premise. Then we have to make Laa false for the conclusion. But that makes the first premise false. We’ll have to move to a domain of two objects. In a two-membered domain, we unpack the second premise just as we did in M. The first premise and conclusion will take a little more work. I’ll work in stages, as we did with the overlapping quantifiers of section 4.8, starting with the outer quantifier and moving to the inner quantifier, showing the process at 5.2.4. 5.2.4 (∀x)[Px ⊃ (∃y)(Py ∙ Lxy)] [Pa ⊃ (∃y)(Py ∙ Lay)] ∙ [Pb ⊃ (∃y)(Py ∙ Lby)] {Pa ⊃ [(Pa ∙ Laa) ∨ (Pb ∙ Lab)]} ∙ {Pb ⊃ [(Pa ∙ Lba) ∨ (Pb ∙ Lbb)]}

In the first step I remove the outside quantifier and replace all the ‘x’s with ‘a’s, conjoining that formula with the same formula that replaces all the ‘x’s with ‘b’s. Then I take that long second formula and replace all the existentially quantified subformulas with disjunctions of the subformula with the ‘y’s replaced by ‘a’s and the subformula with the ‘y’s replaced by ‘b’s. The process of expanding the conclusion, at 5.2.5, is parallel, and just a bit simpler. 5.2.5 (∃x)[Qx ∙ (∃y)Lyx] [Qa ∙ (∃y)Lya] ∨ [Qb ∙ (∃y)Lyb] [Qa ∙ (Laa ∨ Lba)] ∨ [Qb ∙ (Lab ∨ Lbb)]

∙

Laa

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We’re ready to construct the counterexample, lining up the premises and a conclusion, after a list of all the atomic formulas. It has taken a little more work to get to the unquantified expansion, but the work from here is no more difficult than it was in M.

Pa

Qa

{Pa

Pb

Qb

[Qa

∙

Lab

Lba

Lbb

⊃

[(Pa

∙

Laa)

∨

(Pb

∙

Lab)]}

∙

{Pb

⊃

[(Pa

∙

Lba)

∨

(Pb

∙

Lbb)]}

/

//

Laa

(Laa

∨

∙

(Pa

Lba)]

∨

∨

Qa)

[Qb

∙

∙

(Pb

(Lab

∨

Qb)

Lbb)]

I’ll start with the second premise. One of the disjuncts has to be true, so I’ll arbitrarily choose the first one, making Pa and Qa true. Carrying those values into the conclusion, we see that Laa and Lba must be false. Then, on the left side of the first premise, we can see that Pb and Lab each must be true. Pa

Qa

Pb

1

1

1

Qb

Laa

Lab

Lba

0

1

0

Lbb

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{Pa

⊃

[(Pa

∙

Laa)

∨

(Pb

∙

Lab)]}

1

1

1

0

0

1

1

1

1

{Pb

⊃

[(Pa

∙

Lba)

∨

(Pb

∙

1

1

/

//

0

∙

Qa)

∨

1

1

1

1

∙

(Laa

∨

Lba)]

1

0

0

0

0

∨

Lbb)]}

1

(Pa

[Qa

∙

[Qb

∙

∙

(Pb

(Lab

∨

Qb)

Lbb)]

1

The second premise is done, but we still have to make the right conjunct of the first premise true and the right disjunct of the conclusion false. All we need to do to make the first premise true is make Lbb true. Then the disjunction is true, and so the conditional is also true, finishing our work with the premise. Only the conclusion remains, and that’s easily completed by making Qb false. Pa

Qa

Pb

Qb

Laa

Lab

Lba

Lbb

1

1

1

0

0

1

0

1

{Pa

⊃

[(Pa

∙

Laa)

∨

(Pb

∙

Lab)]}

∙

1

1

1

0

0

1

1

1

1

1

{Pb

⊃

[(Pa

∙

Lba)

∨

(Pb

∙

Lbb)]}

1

1

1

0

0

1

1

1

1

3 1 8    C h apter 5  F u ll F i rst - O r d er L og i c

/

//

(Pa

∙

Qa)

∨

(Pb

∙

Qb)

1

1

1

1

1

0

0

[Qa

∙

(Laa

∨

Lba)]

∨

[Qb

∙

(Lab

∨

Lbb)]

1

0

0

0

0

0

0

0

1

1

1

The counterexample is complete. When expanding formulas into finite domains, it is typical to find some redundancy, which you can eliminate before constructing a valuation. For example, consider 5.2.6. 5.2.6 (∀x)(∀y)[(Px ∙ Py) ⊃ (Lax ∙ Lay)]

I’ll expand 5.2.6 into a domain of two members, a and b. First, I’ll remove the outside quantifier, yielding 5.2.7. 5.2.7 (∀y)[(Pa ∙ Py) ⊃ (Laa ∙ Lay)] ∙ (∀y)[(Pb ∙ Py) ⊃ (Lab ∙ Lay)]

So far, so good; no redundancy. Now, let’s remove the remaining quantifier from each conjunct, yielding 5.2.8. 5.2.8 [(Pa ∙ Pa) ⊃ (Laa ∙ Laa)] ∙ [(Pa ∙ Pb) ⊃ (Laa ∙ Lab)] ∙ [(Pb ∙ Pa) ⊃    (Lab ∙ Laa)] ∙ [(Pb ∙ Pb) ⊃ (Lab ∙ Lab)]

Notice the redundancies. There are two in both the first and last conjuncts, and the second and third conjuncts are equivalent. It’s prudent to eliminate these before proceeding, as I do at 5.2.9. Make sure you understand how to convert a statement like 5.2.8 to one like 5.2.9 before taking on Exercises 5.2b. 5.2.9 (Pa ⊃ Laa) ∙ [(Pa ∙ Pb) ⊃ (Laa ∙ Lab)] ∙ (Pb ⊃ Lab)

Summary The semantics for F are not much different from the semantics for M, except for the interpretations of relational predicates by ordered n-tuples. The semantic definitions of validity and logical truth remain unaltered. We can also still use our method of finite domains, though its utility is limited. The expansions of formulas with three or more quantifiers can get unpleasantly long, even in a two-membered domain, let alone larger domains. But this method can generate counterexamples reliably for many invalid arguments. There are other methods for generating counterexamples for invalid arguments of F and the further extensions of logic in this book. Most notably, truth trees, sometimes called semantic tableaux, can be both amusing and effective. But we’ll stick with our work on natural deduction, moving to proof theory for F in the next section.

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KEEP IN MIND

In F, predicates may be followed by any number of singular terms. The same predicate letter may be used as a monadic predicate, dyadic predicate, and any other polyadic predicate in the same formula. Relational predicates are interpreted using sets of ordered n-tuples. A two-place predicate is interpreted by a set of ordered pairs. A three-place predicate is interpreted by a set of ordered triples. A four-place predicate is interpreted by a set of ordered quadruples. And so on. The method of finite domains can be effective in generating counterexamples in F. To expand formulas with overlapping quantifiers into finite domains, work in stages, from the outside quantifier inward.

TELL ME MORE  • How does logic relate to set theory? See 6S.13: Second-Order Logic and Set Theory. • How does quantification help us understand questions about existence? See 7S.10: Quantification and Ontological Commitment.

EXERCISES 5.2a Construct models for each of the given theories by specifying a domain of interpretation and interpreting the constants and predicates so that all sentences of the theory come out true. 1. 1. Aa ∙ Ab 2. Rab ∙ Rba 3. (∃x)∼Rax ∙ (∃x)∼Rbx 4. (∃x)∼Rxa ∙ (∃x)∼Rxb 2. 1. (Pa ∙ Pb) ∙ Pc 2. Babc ∙ ∼Bcba 3. (∀x)(∃y)(∃z)(Byxz ∨ Bzxy) 3. 1. Pa ∙ ∼Sa 2. Pb ∙ ∼Tb 3. (∃x)(∃y)[(Px ∙ Py) ∙ (Rxy ∙ ∼Ryx)] 4. (∀x)[Px ⊃ (Sx ∨ Tx)]

3 2 0    C h apter 5  F u ll F i rst - O r d er L og i c

4. 1. (∀x)(∀y)[(Px ∙ Py) ⊃ (∃z)(Rxyz ∨ Ryxz)] 2. (∃x)Px ∙ ∼(∀x)Px 3. (∀x)(∀y)(∀z){{[(Rxyz ∨ Rxzy) ∨ (Ryxz ∨ Ryzx)] ∨ (Rzxy ∨ Rzyx)} ⊃   [(Qx ∙ Q y) ∙ Qz]} 5. 1. (Pa ∙ Pb) ∙ Pc 2. Hc 3. Cab ∙ Cde 4. Cba 5. (∃x)(Px ∙ Cxe) 6. (∃x)(Px ∙ Cxm) 7. (∀x)[Hx ⊃ (∃y)(Py ∙ Cyx)]

EXERCISES 5.2b Show that each of the following arguments is invalid by generating a counterexample. 1. 1. Aa ∙ Ab 2. Bab ∙ ∼Bba

/ (∃x)(Ax ∙ Bxa)

2. 1. (∃x)Cax 2. (∃x)Cbx

/ (∃x)(Cax ∙ Cbx)

3. 1. Da ∙ (∃x)Eax 2. Db ∙ (∃x)Ebx

/ (∀x)[Dx ∙ (∃y)Exy]

4. 1. (∀x)(Fax ⊃ Gx) 2. (∀x)[Gx ⊃ (∃y)Fyx] 3. Faa / (∀x)(∃y)Fyx 5. 1. (∀x)(∀y)[( Jx ∙ Jy) ⊃ (Kxa ∙ Kya)] 2. Jb / Kba 6. 1. (∀x)[Lx ⊃ (∃y)Mxy] 2. ∼Mab / ∼La 7. 1. (∃x)[Px ∙ (∃y)(Py ∙ Qxy)] 2. (∀x)(Px ⊃ Rx) / (∀x)[Rx ⊃ (∃y)(Py ∙ Q yx)] 8. 1. (∀x)(∀y)[(Hx ∙ Hy) ⊃ Ixy] 2. Ha / (∀x)[Hx ⊃ (∀y)Ixy] 9. 1. Da ∙ Eab 2. (∃x)(∃y)(Eyx ∙ Fx) / (∃x)(Dx ∙ Fx)

5 . 3 : Der i v at i ons i n F   3 2 1

10. 1. (∀x)[Ax ⊃ (∃y)Bxy] 2. (∀x)[(∃y)Byx ⊃ Cx] 3. (∃x)Ax

/ (∀x)Cx

11. 1. (∀x)[Px ⊃ (∃y)(Q y ∙ Rxy)] ⊃ (∃x)[Px ∙ (∃y)(Sy ∙ Rxy)] 2. (∃x)[Px ∙ (∃y)(Q y ∙ ∼Rxy)] / (∃x)[Px ∙ (∃y)(Sy ∙ ∼Rxy)] 12. 1. (∃x)[Lx ∙ (∀y)(My ⊃ Nxy)] 2. (∃x)[Mx ∙ (∀y)(Ly ⊃ Oxy)] / (∃x)(∃y)(Nxy ∙ Oxy) 13. 1. (∃x)[Lx ∙ (∃y)(My ∙ Nxy)] 2. (∀x)[Lx ⊃ (∃y)(My ∙ Oxy)] / (∃x)(∃y)(Nxy ∙ Oyx) 14. 1. (∃x)(∃y)(Dxy ∙ ∼Dyx) 2. Dab

/ ∼Dba

15. 1. (∃x)[(∀y)(Dy ⊃ Fyx) ∙ (∀y)(Ey ⊃ Fyx)] 2. (∃x)(Dx ∨ Ex) / (∀x)Fxx 16. 1. (∃x)[(∀y)(Dy ⊃ Fyx) ∨ (∀y)(Ey ⊃ Fyx)] 2. (∀x)(Dx ∨ Ex) / (∃x)Fxx 17. 1. (∀x)[Px ⊃ (∀y)(Q yx ≡ Ryx)] 2. (∃x)(∀y)(Q yx ∙ ∼Ryx) / (∀x)∼Px 18. 1. (∀x)[Px ⊃ (∃y)Q yx] 2. (∃x)[Px ∙ (∀y)Ryx]

/ (∃x)[Px ∙ (∃y)(Qxy ∙ Rxy)]

19. 1. (∀x)[Px ⊃ (∃y)(Qx ∙ Rxy)] 2. (∃x)[Px ∙ (∀y)(Sy ⊃ Rxy)]

/ (∃x)(∃y)(Rxy ∙ Ryx)

20. 1. (∀x)[Gx ⊃ (∃y)(Gy ∙ Ixy)] 2. (∀x)[Hx ⊃ (∃y)(Hy ∙ Ixy)] 3. (∃x)(Gx ∙ Hx)

/ (∃x)(∃y)(Ixy ∙ Iyx)

5.3: DERIVATIONS IN F In section 5.1, I motivated extending our language M to a language F by introducing relational predicates to regiment argument 5.1.1. 5.1.1

Alyssa is taller than Bhavin. Bhavin is taller than Carlos. Given any three things, if one is taller than another, and the latter is    taller than the third, then the first is also taller than the third. So, Alyssa is taller than Carlos.



1. Tab 2. Tbc 3. (∀x)(∀y)(∀z)[(Txy ∙ Tyz) ⊃ Txz]

/ Tac

3 2 2    C h apter 5  F u ll F i rst - O r d er L og i c

To derive the conclusion, we use the same rules of inference we used with M. When instantiating, we remove quantifiers one at a time, taking care to make appropriate instantiations to variables or constants. We will need to make only one small adjustment to the rule UG, which I will note shortly. A derivation of our motivating argument is below, at 5.3.1. Notice that the removal of quantifiers from the third premise takes three steps. 5.3.1 QED

1. Tab 2. Tbc 3. (∀x)(∀y)(∀z)[(Txy ∙ Tyz) ⊃ Txz] 4. (∀y)(∀z)[(Tay ∙ Tyz) ⊃ Taz] 5. (∀z)[(Tab ∙ Tbz) ⊃ Taz] 6. (Tab ∙ Tbc) ⊃ Tac 7. (Tab ∙ Tbc) 8. Tac

/ Tac 3, UI 4, UI 5, UI 1, 2, Conj 6, 7, MP

Sometimes, as in 5.3.1, we start our derivations by removing all quantifiers. Sometimes we remove the quantifiers in the middle of the proof, rather than at the beginning, as in 5.3.2. 5.3.2 QED

1. (∃x)[Hx ∙ (∀y)(Hy ⊃ Lyx)] 2. Ha ∙ (∀y)(Hy ⊃ Lya) 3. Ha 4. (∀y)(Hy ⊃ Lya) ∙ Ha 5. (∀y)(Hy ⊃ Lya) 6. Ha ⊃ Laa 7. Laa 8. Ha ∙ Laa 9. (∃x)(Hx ∙ Lxx)

/ (∃x)(Hx ∙ Lxx) 1, EI 2, Simp 2, Com 4, Simp 5, UI 6, 3, MP 3, 7, Conj 8, EG

The Restriction on UG All of our rules for removing and replacing quantifiers work in F just as they did in M, with only one exception. Consider the problematic 5.3.3, beginning with a proposition that can be interpreted as ‘Everything loves something’. 5.3.3

1. (∀x)(∃y)Lxy 2. (∃y)Lxy 3. Lxa 4. (∀x)Lxa 5. (∃y)(∀x)Lxy

1, UI 2, EI 3, UG: but wrong! 4, EG

5 . 3 : Der i v at i ons i n F   3 2 3

Given our interpretation of line 1, line 5 reads, ‘There’s something that everything loves’. It does not follow from the proposition that everything loves something that there is one thing that everything loves. Imagine that we arranged all the things in a circle and everyone loved just the thing to its left. Line 1 would be true, but line 5 would be false. We should not be able to derive step 5 from step 1. We can locate the problem in step 4 of 5.3.3. In line 2 we universally instantiated to some random object x. So, ‘x’ could have stood for any object. It retains its universal character, even without a universal quantifier to bind it, and so we are free to UG over x. Then, in line 3, we existentially instantiated. In existentially instantiating, we gave a name, ‘a’ to the thing that bore relation L to it, to the thing that x loves. Once we gave a name to the thing that x loves, x lost its universal character. It could no longer be anything that loves something. It now is the thing that loves a. Thus ‘x’ became as particular an object as ‘a’ is. So, the generalization at line 4 must be blocked. In other words, variables lose their universal character if they are free when EI is used. We formulate the resultant restriction on UG as 5.3.4. 5.3.4

Never UG on a variable when there’s a constant present and the    variable was free when the constant was introduced.

A constant may be introduced as the result of EI or UI, and these are the cases you will have to keep your eye on. Constants may also be introduced in the premises, though there are ordinarily no free variables in premises, since premises should be closed formulas. The restriction on UG debars line 4 of 5.3.3 because ‘x’ was free in line 3 when ‘a’ was introduced. 5.3.5 contains an acceptable use of UG in F. 5.3.5 QED

1. (∃x)(∀y)[(∃z)Ayz ⊃ Ayx] 2. (∀y)(∃z)Ayz 3. (∀y)[(∃z)Ayz ⊃ Aya] 4. (∃z)Ayz ⊃ Aya 5. (∃z)Ayz 6. Aya 7. (∀y)Aya 8. (∃x)(∀y)Ayx

/ (∃x)(∀y)Ayx 1, EI 3, UI 2, UI 4, 5, MP 6, UG 7, EG

Note that at line 7, UG is acceptable because ‘y’ was not free when ‘a’ was introduced in line 3. The restriction 5.3.4 applies only to UG. All other rules are just as they are in monadic predicate logic.

Accidental Binding When using UG or EG, watch for illicit accidental binding. 5.3.6 contains an instance of accidental binding.

3 2 4    C h apter 5  F u ll F i rst - O r d er L og i c

5.3.6 (Pa ∙ Qa) ⊃ (Fx ∨ Gx) (∃x)[(Px ∙ Qx) ⊃ (Fx ∨ Gx)]

EG

The first proposition already contains two instances of the variable ‘x’. If you try to quantify over the ‘a’ using EG with the variable ‘x’, you illicitly bind the latter two singular terms with the same quantifier that binds the first two terms. 5.3.7 has an acceptable inference. 5.3.7 (Pa ∙ Qa) ⊃ (Fx ∨ Gx) (∃y)[(Py ∙ Qy) ⊃ (Fx ∨ Gx)]

In 5.3.7, the latter two singular terms, the ‘x’s, remain free. We can bind them with either a universal quantifier or an existential quantifier, later, as in either of the propositions at 5.3.8. 5.3.8 (∀x)(∃y)[(Py ∙ Qy) ⊃ (Fx ∨ Gx)] (∃x)(∃y)[(Py ∙ Qy) ⊃ (Fx ∨ Gx)]

More Derivations Derivations in F often involve propositions with overlapping quantifiers. Nevertheless, we must adhere to the rules and restrictions we had in M, as well as the new restriction on UG for F. UI and EG remain anytime-anywhere rules. The restrictions on EI can be trickier to manage, since quantifiers may be buried inside formulas. Still, remember always to EI to a new constant. Derivations with more than one existential quantifier in the premises are likely to need multiple constants, as in 5.3.9, where at line 4 I EI line 2 to ‘b’ because I had already EIed line 1 to ‘a’. 5.3.9 QED

1. (∃x)[Px ∙ (∀y)(Py ⊃ Qxy)] 2. (∃x)(Px ∙ Sx) 3. Pa ∙ (∀y)(Py ⊃ Qay) 4. Pb ∙ Sb 5. (∀y)(Py ⊃ Qay) ∙ Pa 6. (∀y)(Py ⊃ Qay) 7. Pb ⊃ Qab 8. Pb 9. Qab 10. (∃y)Qyb 11. Sb ∙ Pb 12. Sb 13. Sb ∙ (∃y)Qyb 14. (∃x)[Sx ∙ (∃y)Qyx]

/ (∃x)[Sx ∙ (∃y)Qyx] 1, EI 2, EI 3, Com 5, Simp 6, UI 4, Simp 7, 8, MP 9, EG 4, Com 11, Simp 12, 10, Conj 13, EG

It remains generally useful to EI before you UI. But sometimes an existential quantifier is buried in a line and we cannot instantiate its subformula until we have the quantifier as the main operator, as in 5.3.10, which uses conditional proof.

5 . 3 : Der i v at i ons i n F   3 2 5

5.3.10 1. (∀x)[Ax ⊃ (∀y)Bxy] 2. (∀x)[Ax ⊃ (∃y)Dyx] 3. Ax 4. Ax ⊃ (∀y)Bxy 5. Ax ⊃ (∃y)Dyx 6. (∀y)Bxy 7. (∃y)Dyx 8. Dax 9. Bxa 10. Bxa ∙ Dax 11. (∃y)(Bxy ∙ Dyx) 12. Ax ⊃ (∃y)(Bxy ∙ Dyx) 13. (∀x)[Ax ⊃ (∃y)(Bxy ∙ Dyx)] QED

/ (∀x)[Ax ⊃ (∃y)(Bxy ∙ Dyx)] ACP 1, UI 2, UI 4, 3, MP 5, 3, MP 7, EI 6, UI 9, 8, Conj 10, EG 3–11, CP 12, UG

We could not instantiate the existential quantifier in line 2 until we instantiated the leading universal quantifier and used modus ponens to get the existential quantifier as the main operator at line 7. I thus had to wait to UI the universal formula at line 6 until after line 8. The methods for indirect proofs in M carry over neatly to F, as at 5.3.11, in which I make a strategic assumption for IP. 5.3.11 1. (∀x)[Px ⊃ (∃y)(Ry ∙ Fxy)] 2. (∀x)[Qx ⊃ (∀y)(Ry ⊃ ∼Fxy)] 3. (∀x)(Px ∙ Qx) 4. Px ∙ Qx 5. Px 6. Px ⊃ (∃y)(Ry ∙ Fxy) 7. (∃y)(Ry ∙ Fxy) 8. Qx ∙ Px 9. Qx 10. Qx ⊃ (∀y)(Ry ⊃ ∼Fxy) 11. (∀y)(Ry ⊃ ∼Fxy) 12. Ra ∙ Fxa 13. Ra 14. Ra ⊃ ∼Fxa 15. Fxa ∙ Ra 16. Fxa 17. ∼Fxa 18. Fxa ∙ ∼Fxa 19. ∼(∀x)(Px ∙ Qx) 20. (∃x)∼(Px ∙ Qx) 21. (∃x)(∼Px ∨ ∼Qx) QED

/ (∃x)(∼Px ∨ ∼Qx) AIP 3, UI 4, Simp 1, UI 6, 5, MP 4, Com 8, Simp 2, UI 10, 9, MP 7, EI 12, Simp 11, UI 12, Com 15, Simp 14, 13, MP 16, 17, Conj 3–18, IP 19, QE 20, DM

Note that the proof would work just as well, and be one line shorter, had I assumed ‘Px ∙ Qx’ at line 3.

3 2 6    C h apter 5  F u ll F i rst - O r d er L og i c

5.3.12 is a more complex derivation using CP and illustrating the importance of remaining vigilant about the differences between constants and variables in F. 5.3.12 1. (∀x)(Wx ⊃ Xx) 2. (∀x)[(Yx ∙ Xx) ⊃ Zx] 3. (∀x)(∃y)(Yy ∙ Ayx) 4. (∀x)(∀y)[(Ayx ∙ Zy) ⊃ Zx] 5. (∀y)(Ayx ⊃ Wy) 6. (∃y)(Yy ∙ Ayx) 7. Ya ∙ Aax 8. Aax ⊃ Wa 9. Aax ∙ Ya 10. Aax 11. Wa 12. Wa ⊃ Xa 13. Xa 14. Ya 15. Ya ∙ Xa 16. (Ya ∙ Xa) ⊃ Za 17. Za 18. (∀y)[(Ayx ∙ Zy) ⊃ Zx] 19. (Aax ∙ Za) ⊃ Zx 20. Aax ∙ Za 21. Zx 22. (∀y)(Ayx ⊃ Wy) ⊃ Zx 23. (∀x)[(∀y)(Ayx ⊃ Wy) ⊃ Zx] QED

/ (∀x)[(∀y)(Ayx ⊃ Wy) ⊃ Zx] ACP 3, UI 6, EI 5, UI 7, Com 9, Simp 8, 10, MP 1, UI 12, 11, MP 7, Simp 14, 13, Conj 2, UI 16, 15, MP 4, UI 18, UI 10, 17, Conj 19, 20, MP 5–21, CP 22, UG

Notice that at line 17, you might be tempted to discharge your assumption and finish your CP. But you wouldn’t be able to UG over the ‘Za’. We have to UI at line 18, retaining a variable for the predicate ‘Z’.

Logical Truths We can use CP and IP to prove logical truths in F. 5.3.13 proves that ‘(∃x)(∀y)Pxy ⊃ (∀x)(∃y)Pyx’ is a logical truth by conditional proof. 5.3.13 1. (∃x)(∀y)Pxy ACP 2. (∀y)Pay 1, EI 3. Pax 2, UI 4. (∃y)Pyx 3, EG 5. (∀x)(∃y)Pyx 4, UG 6. (∃x)(∀y)Pxy ⊃ (∀x)(∃y)Pyx 1–5, CP QED

Notice that the use of UG at line 5 is legitimate since the constant at line 3 was bound at line 4; there’s no constant present on the line on which I used UG.

5 . 3 : Der i v at i ons i n F   3 2 7

We can prove that ‘(∃x)∼Pxx ∨ (∀x)(∃y)Pxy’ is a logical truth by indirect proof, as I do at 5.3.14. 5.3.14 1. ∼[(∃x)∼Pxx ∨ (∀x)(∃y)Pxy] AIP 2. ∼(∃x)∼Pxx ∙ ∼(∀x)(∃y)Pxy 1, DM 3. ∼(∃x)∼Pxx 2, Simp 4. (∀x)Pxx 3, QE 5. ∼(∀x)(∃y)Pxy ∙ ∼(∃x)∼Pxx 2, Com 6. ∼(∀x)(∃y)Pxy 5, Simp 7. (∃x)∼(∃y)Pxy 6, QE 8. (∃x)(∀y)∼Pxy 7, QE 9. (∀y)∼Pay 8, EI 10. ∼Paa 9, UI 11. Paa 4, UI 12. Paa ∙ ∼Paa 11, 10, Conj 13. ∼ ∼[(∃x)∼Pxx ∨ (∀x)(∃y)Pxy] 1–12, IP 14. (∃x)∼Pxx ∨ (∀x)(∃y)Pxy 13, DN QED

As with all other proofs in F, take your time with the quantifiers. Notice that the exchange of the consecutive quantifiers from lines 6–8 takes two separate steps. Be careful also to obey the restrictions on UG, and always EI to a new constant.

Summary Derivations in F look different from those in M, and they are generally more complex, but the rules are basically the same. The presence of multiple quantifiers tends to lengthen any derivation, since instantiation, generalization, and exchanging quantifiers has to be done one step at a time. Keep track of your variables and constants, make sure to obey the restrictions on UG and EI, and be patient. And, of course, practice. It is much better to do a little every day than to try to do a lot at once. KEEP IN MIND

All rules for M are the same for F, with one exception, a restriction on UG. Never UG on a variable when there’s a constant present and the variable was free when the constant was introduced. Remove quantifiers from formulas one at a time, and only when they are the main operators. Logical truths of F can be derived using conditional or indirect proof, just as for M.

TELL ME MORE  • How does F differ from M? See 6.4: Metalogic. • Are there ways to move quantifiers through formulas? See 6S.12: Rules of Passage.

3 2 8    C h apter 5  F u ll F i rst - O r d er L og i c

EXERCISES 5.3a Derive the conclusions of each of the following arguments. 1. 1. Bab 2. (∀x)(Bax ⊃ Ax)

/ (∃x)Ax

2. 1. Da ∙ (∃x)Eax 2. Db ∙ (∀x)Ebx

/ (∃x)(Eax ∙ Ebx)

3. 1. Fab 2. (∀x)(Fax ⊃ Gx) 3. (∀x)(Gx ⊃ Fxa)

/ Fba

4. 1. ∼(∃x)(Hx ∙ Ixa) 2. (∃x)Ixa

/ (∃x)∼Hx

5. 1. (∀x)[Lx ⊃ (∃y)Mxy] 2. (∀y)∼May / ∼La 6. 1. Aa ∙ (Ba ∙ ∼Cab) 2. (∀y)Cay ∨ (∀z)Dbz

/ (∃y)(∀z)Dyz

7. 1. (∀x)[(∃y)Bxy ⊃ (Ax ∨ Cx)] 2. (∃z)(∼Az ∙ ∼Cz)

/ (∃z)(∀y)∼Bzy

8. 1. Db ∙ Eab 2. (∀x)[(∃y)Eyx ⊃ Fx]

/ (∃x)(Dx ∙ Fx)

9. 1. (∃x)[Nx ∙ (∃y)(Ny ∙ Qxy)] 2. (∀x)(Nx ⊃ Px)

/ (∃x)[Px ∙ (∃y)(Py ∙ Q yx)]

10. 1. (∃x)[Qx ∨ (∃y)(Ry ∙ Pxy)] 2. ∼(∃x)(Sx ∨ Qx)

/ (∃z)(∃y)(Ry ∙ Pzy)

11. 1. (∀x)[(∀y)Uxy ⊃ (Tx ∙ Vx)] 2. ∼(∃x)Tx

/ (∃z)∼Uza

12. 1. (∀x)[Ax ⊃ (∃y)Bxy] 2. (∀x)[(∃y)Bxy ⊃ (Cx ∨ Dx)] 3. (∃x)(Ax ∙ ∼Cx)

/ (∃x)(Ax ∙ Dx)

13. 1. (∃x)[Mx ∙ (∃y)(Ny ∙ Lxy)] 2. (∀x)(∀y)[Lxy ⊃ (∃z)Oyz]

/ (∃x)(∃y)Oxy

14. 1. (∀x)[Ex ∙ (Fx ∨ Gx)] 2. (∃x){Hx ∙ (∀y)[(Fy ∨ Gy) ⊃ Ixy]} / (∃y)(∃x)Ixy 15. 1. (∀x)[Ax ⊃ (∃y)(Cy ∙ Dxy)] 2. (∀x)(∀y)(Dxy ⊃ By)

/ (∀x)Ax ⊃ (∃y)(By ∙ Cy)

5 . 3 : Der i v at i ons i n F   3 2 9

16. 1. (∀x)(∀y){Fxy ⊃ [(Gx ∙ Hy) ∨ (∼Gx ∙ ∼Hy)]} 2. Fmb 3. ∼Hb / ∼Gm 17. 1. (∀x)[Px ⊃ (∃y)Qxy] 2. (∃x)[Px ∙ (∀y)Rxy]

/ (∃x)[Px ∙ (∃y)(Qxy ∙ Rxy)]

18. 1. (∃x)[Lx ∙ (∀y)(My ⊃ Nxy)] 2. (∃x)[Mx ∙ (∀y)(Ly ⊃ Oxy)]

/ (∃x)(∃y)(Nxy ∙ Oyx)

19. 1. (∀x)(∀y)(∀z)[(Bxy ∙ Byz) ⊃ Bxz] 2. (∀x)(∀y)(Bxy ⊃ Byc) 3. (∃x)Bax / Bac 20. 1. (∀x)[(Fx ∙ Hx) ⊃ (∀y)(Gy ∙ Ixy)] 2. (∃x)[ Jx ∙ (∀y)(Gy ⊃ ∼Ixy)] / ∼(∀z)(Fz ∙ Hz) 21. 1. (∀x)[Ax ⊃ (∀y)(Dy ⊃ Byx)] 2. (∃x)[Dx ∙ (∀y)(Bxy ⊃ Cy)]

/ (∀x)Ax ⊃ (∃y)Cy

22. 1. (∃x)[(∀y)(Hy ⊃ Jyx) ∙ (∀y)(Iy ⊃ Jyx)] 2. (∀x)(Hx ∨ Ix) / (∃x)Jxx 23. 1. (∀x)[Lx ⊃ (∃y)(Ly ∙ Nxy)] 2. (∀x)[Mx ⊃ (∃y)(My ∙ Nxy)

/ (∃x)(Lx ∙ Mx) ⊃ (∃x)(∃y)(∃z)(Nzx ∙ Nzy)

24. 1. (∃x){Px ∙ (∀y)[Oy ⊃ (∀z)(Rz ⊃ Qxyz)]} 2. (∀x)[Px ≡ (Ox ∙ Rx)] / (∃x)Qxxx 25. 1. (∀x)(Mx ⊃ ∼Ox) ⊃ (∃y)Ny 2. (∀y)[Ny ⊃ (∃z)(Pz ∙ Q yz)] 3. ∼(∃x)(Mx ∙ Ox)

/ (∃x)[Nx ∙ (∃y)Qxy]

26. 1. (∀x)(Kx ≡ Lx) ∙ (∀x)Jx 2. (∀x)[ Jx ⊃ (∃y)(∼Ky ∙ Mxy)] / (∀x)(∃y)(∼Ly ∙ Mxy) 27. 1. (∀x)[Rx ⊃ (∀y)(Ty ⊃ Uxy)] 2. (∀y)[(∀x)(Uxy ⊃ Sy)]

/ (∀x)[(Rx ∙ Tx) ⊃ (∃y)Sy]

28. 1. (∀x)[Kx ⊃ (∃y)( Jy ∙ Ixy)] 2. (∀x)(∀y)(Ixy ⊃ Lx)

/ (∀x)(∼Kx ∨ Lx)

29. 1. (∀x)(Fx ≡ Hx) 2. (∀x)(Hx ⊃ ∼Ix) 3. (∃x)[Fx ∙ (∃y)(Iy ∙ ∼Gxy)]

/ (∃x)[(Fx ∙ ∼Ix) ∙ (∃y)(Iy ∙ ∼Gxy)]

30. 1. (∀x){Ax ⊃ (∃y)[By ∙ (∀z)(∼Cz ∙ Dzxy)]} 2. ∼(∀x)(Ax ⊃ Cx) / (∃x)(∃y)Dxxy 31. 1. (∀x)[Tx ⊃ (∀y)(Vy ⊃ Uxy)] 2. ∼(∃x)(Tx ∙ Sx) 3. Ta ∙ Vb

/ (∃x)[∼Sx ∙ (∃y)Uxy]

3 3 0    C h apter 5  F u ll F i rst - O r d er L og i c

32. 1. (∀x)[Ax ⊃ (∀y)(Dyx ≡ ∼Byx)] 2. (∃x)(∀x)(Dyx ∙ Byx) / (∃x)∼Ax 33. 1. (∀x)[Fx ⊃ (∃y)(Gy ∙ Hxy)] 2. (∃x)[Fx ∙ (∀y)(Iy ⊃ Hyx)] 3. (∀x)(Gx ⊃ Ix) / (∃x)(∃y)(Hxy ∙ Hyx) 34. 1. (∀x)[(Ox ⊃ Nx) ⊃ (∀y)(Q y ∙ ∼Rxy)] 2. (∀y)(∀x)(Pxy ⊃ Rxy) / (∀x)[(Nx ∨ Ox) ⊃ (∀y) ∼(Q y ⊃ Pxy)] 35. 1. (∀x)[(Bx ⊃ Ax) ⊃ (∃y)(Cy ∙ Dxy)] 2. (∀x)[(∀y)∼Dxy ∨ Ex] 3. (∃x)Ex ⊃ ∼(∃x)Cx / (∀x)Bx 36. 1. (∀x){(Tx ⊃ ∼Sx) ⊃ (∃y)[Uy ∨ (∀z)(Vz ⊃ Wxyz)]} 2. ∼(∃x)(Tx ≡ Sx) 3. ∼(∃x)(Vx ⊃ Ux) / (∃x)(∃y)Wxyy 37. 1. (∀x)[Fx ⊃ (∃y)(Hy ∙ Gxy)] 2. (∀x)[Hx ⊃ (∃y)(Ey ∙ Gxy)] 3. (∀x)[Ex ⊃ (∀y)Fy] / (∀x)Fx ≡ (∃x)Ex 38. 1. (∃x)(∃y)[(Px ∙ Py) ∙ (∀z)(∃w)Fxywz] 2. (∀x)(∀y){(Px ∙ Py) ⊃ [(∃w)(∃z)Fxywz ⊃ Rxy]} 3. (∀x)(∀y)(Rxy ≡ Ryx) 4. (∀x)(∀y){[(Rxy ∙ Ryx) ∙ (Px ∙ Py)] ⊃ (Qx ∙ Q y)} / (∃x)(Px ∙ Qx) 39. 1. (∀x)[(Dx ∨ Gx) ⊃ (∃y)(Ey ∙ Fxy)] 2. (∀x)[Dx ⊃ (∀y)(Hy ⊃ Fyx)] 3. (∀x)[Gx ⊃ (∀y)(Ey ⊃ Fyx)] 4. (∀y)(∃z)(Hz ∙ ∼Fzy) / (∀x)[(Dx ∨ Gx) ⊃ (∃y)(Fyx ∙ Fxy)] 40. 1. (∀x){ Jx ⊃ (∀y)[My ⊃ (∀z)(Lz ⊃ Kxyz)]} 2. (∃x)(∃y)[Mx ∙ ( Jy ∙ Nxy)] 3. ∼(∀x)(Lx ⊃ Ox) / (∃x){Mx ∙ (∃y)[Nxy ∙ (∃z)(∼Oz ∙ Kyxz)]}

EXERCISES 5.3b Translate each of the following arguments into propositions of F using the indicated formulas. Then, derive the conclusions of the arguments. 1. Some ballet dancers are shorter than some gymnasts. No gymnasts are clumsy. So, it is not the case that all things are clumsy. (Bx: x is a ballet dancer; Gx: x is a gymanst; Cx: x is clumsy; Sxy: x is shorter than y)

5 . 3 : Der i v at i ons i n F   3 3 1

2. Anyone who teaches a math class is intelligent. Professor Rosen is a person who teaches Calculus I. Calculus I is a math class. So, Professor Rosen is intelligent. (c: Calculus I; r: Professor Rosen; Px: x is a person; Ix: x is intelligent; Mx: x is a math class; Txy: x teaches y) 3. All cats love all dogs. It is not the case that everything loves Brendan; and all things are cats. So, it is not the case that everything is a dog. (b: Brendan; Cx: x is a cat; Dx: x is a dog; Lxy: x loves y) 4. Alice buys a baguette from some store. Baguettes are food. Alice is a resident of Clinton. So, some residents of Clinton buy some food from some store. (a: Alice; c: Clinton; Bx: x is a baguette; Fx: x is food; Sx: x is a store; Rxy: x is a resident of y; Bxyz: x buys y from z) 5. All philosophers have some mentor to whom they respond. Either something isn’t a philosopher or nothing is a mentor. So, not everything is a philosopher. (Mx: x is a mentor; Px: x is a philosopher; Rxy: x responds to y) 6. Some students read books written by professors. All books written by professors are well-researched. So, some professor wrote a well-researched book. (Bx: x is a book; Px: x is a professor; Sx: x is a student; Wx: x is well-researched; Rxy: x reads y; Wxy: x wrote y) 7. Sunflowers and roses are plants. Some sunflowers grow taller than all roses. Russell gave a rose to Emily. So, some plant is taller than some rose. (e: Emily; r: Russell; Px: x is a plant; Rx: x is a rose; Sx: x is a sunflower; Gxy: x grows taller than y; Gxyz: x gives y to z) 8. There is something trendier than everything that’s expensive or of good quality. Anything that’s meaningful or serves a purpose is either expensive, or there’s something more uninteresting than it. Not everything is expensive or not meaningful, but everything is of good quality. So, there is something trendier, and there is something more uninteresting, than something of good quality. (Ex: x is expensive; Mx: x is meaningful; Px: x serves a purpose; Qx: x is of good quality; Txy: x is trendier than y; Uxy: x is more uninteresting than y) 9. All philosophers are more skeptical than some physicists. All physicists are scientists. So, all philosophers are more skeptical than some scientists. (Px: x is a philosopher; Sx: x is a scientists; Yx: x is a physicist; Sxy: x is more skeptical than y) 10. Some sets include sets. If something includes all sets, then it is not a set. So, some set does not include some set. (Sx: x is a set; Ixy: x includes y) 11. All philosophers who influenced Mill influenced Quine. Bentham was a political theorist and a philosopher who influenced Mill. Any philosopher who influenced Quine was an empiricist. So, Bentham was an empiricist. (b: Bentham; m: Mill; q: Quine; Ex: x is an empiricist; Px: x is a philosopher; Tx: x is a political theorist; Ixy: x influenced y)

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12. Any act with better consequences than some act is more morally required than it. Pulling the lever in a trolley case is an act with better consequences than the act of ignoring it. If pulling the lever is more morally required than ignoring it, then the doctrine of acts and omissions is unsupportable. So, the doctrine of acts and omissions is unsupportable. (a: the doctrine of acts and omissions; i: ignoring the lever in a trolley case; p: pulling the lever in a trolley case; Ax: x is an act; Sx: x is supportable; Cxy: x has better consequences than y; Mxy: x is more morally required than y) 13. Any characteristic that is between extremes is a virtue. Cowardice and rashness are vices. Every vice is an extreme. Courage is a characteristic between cowardice and rashness. So, courage is a virtue. (c: courage; f: cowardice; r: rashness; Cx: x is a characteristic; Ex: x is an extreme; Gx: x is a virtue; Vx: x is a vice; Bxyz: y is between x and z) 14. All virtues are between some extremes. Any characteristic between any two things is not an extreme. Any characteristic that is not extreme has some benefit. Temperance is a characteristic that is a virtue. So, temperance has some benefit. (t: temperance; Cx: x is a characteristic; Ex: x is an extreme; Vx: x is a virtue; Bxy: x is a benefit of y; Bxyz: y is between x and z) 15. Philosophers who are read more widely than other philosophers have greater influence than them. No philosopher has greater influence than the philosopher Plato. So, no philosopher is read more widely than Plato. (p: Plato; Px: x is a philosopher; Ixy: x has greater influence than y; Rxy: x is read more widely than y) 16. Given any three works of philosophy, if the first has greater influence than the second, and the second has greater influence than the third, then the first has greater influence than the third. Gorgias, Republic, and Laws are all dialogues written by Plato. Everything written by Plato is a work of philosophy. Gorgias has more influence than Laws, but Republic has more influence than Gorgias. So, Republic has greater influence than Laws. (g: Gorgias; l: Laws; p: Plato; r: Republic; Dx: x is a dialogue; Wx: x is a work of philosophy; Ixy: x has greater influence than y; Wxy: x wrote y)

EXERCISES 5.3c Derive the following logical truths of F. 1. (∀x)(∀y)Axy ⊃ (∃x)(∃y)Axy 2. (∃x)(∀y)Dyx ⊃ (∃x)Dxx

5 . 3 : Der i v at i ons i n F   3 3 3

3. (∀x)Fmxn ⊃ (∃x)(∃y)Fxoy 4. (∀x)(∃y)(Gxy ∨ ∼Gxx) 5. (∃x)Exx ⊃ (∃x)(∃y)Exy 6. (∃x)∼Bxa ∨ (∃x)Bbx 7. (∃x)(∀y)Cxy ⊃ (∀y)(∃x)Cxy 8. (∀x)(∃y)Hxy ⊃ (∃x)(∃y)Hxy 9. (∀x)[Px ⊃ (∃y)(Q y ∙ Rxy)] ⊃ {(∃x)Px ⊃ (∃x)(∃y)[(Px ∙ Q y) ∙ Rxy]} 10. (∃x)(∀y)( Jxy ∙ ∼Jyx) ∨ (∀x)(∃y)(Jxy ⊃ Jyx) 11. (∃x)(∀y)(Kxy ∨ Kyx) ⊃ (∃x)[(∃y)∼Kxy ⊃ (∃y)Kyx] 12. (∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∀x)(∀y)∼Qxy ⊃ ∼(∃x)Px] 13. (∃x)[Px ∙ (∃y)(Q y ∙ Rxy)] ⊃ (∃x)[Qx ∙ (∃y)(Py ∙ Ryx)] 14. (∀x)[Px ⊃ (∀y)Qxy] ≡ (∀x)(∀y)(Py ⊃ Q yx) 15. (∀x)[Px ⊃ (∃y)(Q y ∙ Rxy)] ∨ (∃x)(∀y)[Px ∙ ∼(Q y ∙ Rxy)]

EXERCISES 5.3d For each argument, determine whether it is valid or invalid. If it is valid, derive the conclusion using our rules of inference and equivalence. If it is invalid, provide a counterexample. 1. 1. (∀x)(∀y)(Bxy ≡ Byx) 2. Bab ∙ Bbc / Bac 2. 1. (∀x)(∀y)(Pxy ≡ ∼Pyx) 2. ∼(∃x)Pxa / (∃x)Pax 3. 1. (∀x)(Px ⊃ Qxi) 2. (∃x)(Qix ∙ Px) 3. Pa

/ Qia

4. 1. (∀x)(∀y)(∃z)(Bxzy ≡ Byzx) 2. Babc / Bcba 5. 1. (∀x)[Px ⊃ (∃y)(Py ∙ Qxy)] 2. (∀x)(Px ⊃ ∼Rx) / (∀x)[Rx ⊃ (∀y)(Ry ⊃ ∼Qxy)] 6. 1. (∀x)[Px ⊃ (∃y)(Py ∙ Rxy)] 2. (∀x)(Px ⊃ Qx) / (∀x)[Px ⊃ (∃y)(Q y ∙ Rxy)]

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7. 1. (∀x)[Px ⊃ (∃y)Qxy] 2. (∃x)∼Qax

/ (∃x)∼Px

8. 1. (∀x)[Ux ⊃ (∃y)(Ty ∙ Vxy)] 2. (∃x)Vax ⊃ (∀x)Vax 3. Ua

/ (∃x)(∀y)Vxy

9. 1. (∃x)(∃y)[(Px ∙ Py) ∙ Rxy] 2. (∃x)(∃y)[(Px ∙ Py) ∙ Qxy]

/ (∃x)(∃y)(Qxy ∙ Rxy)

10. 1. (∀x)(∀y)(Pxy ⊃ Pyx) 2. (∃x)[Qx ∙ (∀y)Pxy]

/ (∃x)[Qx ∙ (∀y)Pyx]

11. 1. (∀x)[(Px ∙ Qx) ⊃ Rxx] 2. (∃x)(Px ∙ ∼Rxx) 3. (∀x)[Qx ⊃ (∃y)(Py ∙ Rxy)] / (∃x)(∃y)(Rxy ∙ ∼Rxx) 12. 1. (∀x)[(∃y)Pxy ⊃ (∃y)Qxy] 2. (∃x)(∀y)∼Qxy

/ (∃x)(∀y)∼Pxy

13. 1. (∀x)[(∃y)Pxy ⊃ (∃y)Qxy] 2. (∃x)(∃y)∼Qxy

/ (∃x)(∃y)∼Pxy

14. 1. (∃x)(∀y)[(Fx ∙ Dx) ∨ (Ey ⊃ Gxy)] 2. (∀x)[(∃y)Gxy ⊃ (∃z)Hxz] 3. ∼(∃x)Fx ∙ (∀z)Ez / (∃y)(∃z)Hyz 15. 1. (∀x)(∀y)(Pxy ⊃ Pyx) 2. Pab ∙ Pbc

/ Pac

16. 1. (∀x)(∀y)(∀z)[(Pxy ∙ Pyz) ⊃ Pxz] 2. Pab ∙ ∼Pac / Pbc 17. 1. (∀x)(∀y)(∀z)[(Pxy ∙ Pyz) ⊃ Pxz] 2. Pab ∙ Pba / (∃x)Pxx 18. 1. (∀x)(∀y)(∀z)[(Pxy ∙ Pyz) ⊃ Pxz] 2. (∀x)Pxx 3. Pac ∙ ∼Pba / ∼Pcb 19. 1. (∀x)(∀y)(∀z)(Bxzy ≡ ∼Byzx) 2. (∀x)(∀y)(∀z){[(Px ∙ Py) ∙ Pz] ⊃ Bxyz} 3. Pa ∙ Pb 4. Babc / ∼Pc 20. 1. (∀x)(∀y)(Pxy ⊃ Pyx) 2. (∀x)[Qx ⊃ (∃y)(Sy ∙ Rxy)] 3. (∀x)(Sx ⊃ Qx) 4. Qa ∙ Pba

/ (∃x)(Qx ∙ Pxb) ∙ (∃x)(Qx ∙ Rax)

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5.4: THE IDENTITY PREDICATE: TRANSLATION We’ve come to the last major topic in the formal work of this book. In this section and the next, we will explore an extension to the system of inference we have adopted for our language F. This extension concerns a special two-place relation, identity. In translation, identity allows us to use F to express a wide range of concepts including some fundamental mathematical concepts. In the next section, we will add some simple derivation rules governing the identity predicate. There is some debate about whether identity is strictly a logical relation. I start by explaining that debate, and then proceed, in the remainder of this section, to show how to use identity in translation.

Introducing Identity Theory Some claims, like 5.4.1, are paradigmatically logical. 5.4.1

If P, then P.

P⊃P

Other claims, like 5.4.2, are paradigmatically nonlogical. 5.4.2

It snows in winter in Quebec.

Still other claims fall somewhere in between. 5.4.3 is generally not considered a logical truth, even though it has something of the feel of a logical truth. 5.4.3

All bachelors are unmarried.

Philosophers generally characterize the truth of 5.4.3 as semantic, rather than logical, though not in the sense of ‘semantic’ that we have been using in this book. ‘Semantics’ in logic refers to interpretations of logical vocabulary. Semantics more broadly is the study of meanings. That bachelors are not married is not a logical entailment. It follows from the meaning of the word ‘bachelor’ that ‘unmarried’ follows from (most uses of ) it, not the logic of our language. The line between logical and nonlogical claims is not always clear. Other predicates have logical properties: ‘taller than’ is transitive and anti-reflexive, and ‘is married to’ is symmetric. But we don’t assume those logical properties as part of a logical system; we just add them as axioms or premises when using the terms. Still, entailments surrounding identity, like the inference at 5.4.4, are so thin and uncontroversial that they are generally considered logical. 5.4.4

1. Superman can fly. 2. Superman is Clark Kent. So, Clark Kent can fly.

If we write the second premise as ‘Isc’, as at 5.4.5, the conclusion of the argument does not follow in our inferential system. 5.4.5

1. Fs 2. Isc

/ Fc

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So, we introduce derivation rules that govern inferences like this one and give identity its own symbol, ‘=’.

Syntax and Rules for Identity Statements Identity sentences, like those at 5.4.6, look a little different from others with dyadic relations. 5.4.6

Clark Kent is Superman. Mary Ann Evans is George Eliot.

c=s m=g

We need not extend our language F by introducing the identity predicate. We only set aside a particular two-place predicate. But, for convenience, we add a new shorthand (=) for it. We need no new formation rules, though we should clarify how the shorthand works. Formulas like ‘a=b’ are really short for ‘Iab’, taking ‘Ixy’ as the identity relation. Just as we do not put brackets around ‘Iab,’ we do not punctuate ‘a=b’. As far as the logical language is concerned, identities are just special kinds of two-place relations. Negations of identity claims, strictly speaking, are written just like the negations of any other two-place relation, with a tilde in front, though there is another shorthand (≠). Both ways of writing negations are displayed at 5.4.7. 5.4.7 ∼a=b a≠b

Remember that negation applies to the identity predicate, and not to the objects related by that predicate. We cannot negate names. The statements at 5.4.7 say that it is not the case that the objects named by ‘a’ and ‘b’ are identical. By adopting new derivation rules governing uses of the identity predicate, we introduce a new deductive system using the same language F. There are three rules, based on three principles surrounding identity: that every object is identical to itself; that identity is symmetrical (if one thing is identical to another, then the second is also identical to the first); and a claim, called Leibniz’s law, that identical objects share all properties. Perhaps more clearly, this latter property of identity is just that any object with two different names has all the same characteristics whether we call it by one name or another. We’ll see the inference rules based on these three principles in the next section. For the rest of this section, we focus on translation.

Translation The identity predicate allows us to reveal inferential structure for a wide variety of propositions, making it extraordinarily powerful. It allows us to express propositions with ‘only’ and ‘except’; superlatives; and ‘at least’, ‘at most’, and ‘exactly’; and to manage a problem with names and definite descriptions. To start, note that, as a convention for the rest of the chapter, I will drop the requirement on wffs that series of conjunctions and series of disjunctions have

5 . 4 : T h e I d ent i t y P re d i cate : T ranslat i on   3 3 7

brackets for every two conjuncts or disjuncts. Propositions using identity can become long and complex. To reduce the amount of punctuation in our formulas, given that commutativity and association hold for both conjunction and disjunction, we allow such series, even if they have many terms, to be collected with one set of brackets. Thus, 5.4.8 can be written as 5.4.9, and 5.4.10 can be written as 5.4.11. 5.4.8 (∃x)(∃y){(Ax ∙ Bxj) ∙ [(Ay ∙ Iyj) ∙ x≠y)]} 5.4.9 (∃x)(∃y)(Ax ∙ Bxj ∙ Ay ∙ Iyj ∙ x≠y) 5.4.10 (∀x)(∀y)(∀z)(∀w){[(Px ∙ Py) ∙ (Pz ∙ Pw)] ⊃ {[(x=y ∨ x=z) ∨    (x=w ∨ y=z)] ∨ (y=w ∨ z=w)}} 5.4.11 (∀x)(∀y)(∀z)(∀w)[(Px ∙ Py ∙ Pz ∙ Pw) ⊃ (x=y ∨ x=z ∨ x=w ∨ y=z   ∨ y=w ∨ z=w)] SIMPLE IDENTIT Y CLAIMS

As we have seen, simple identity claims are easily written, as in 5.4.6. Ordinarily, we think of such claims as holding between two names of a single object. EXCEPT AND ONLY

Statements using terms like ‘except’ and ‘only’ can be regimented usefully using identity. To say that Julio loves only Maria, we add to the claim that Julio loves Maria, 5.4.12, the claim that anyone Julio loves is Maria, as at 5.4.13. 5.4.12 5.4.13

Julio loves Maria. Julio loves only Maria.

Ljm Ljm ∙ (∀x)(Ljx ⊃ x=m)

To say that only Julio loves Maria, we add to 5.4.12 the claim that anyone who loves Maria is Julio. 5.4.14

Only Julio loves Maria.

Ljm ∙ (∀x)(Lxm ⊃ x=j)

Notice that each ‘only’ statement contains two parts. ‘Julio loves only Maria’ means both that Julio loves Maria and Maria is the only love of Julio. ‘Only Julio loves Maria’ means again that Julio loves Maria, but this time also that he is the only lover of Maria. These two clauses are present in all ‘only’ sentences, as in the further examples 5.4.15 and 5.4.16. Note that the negation in 5.4.16 is present in both clauses, and that we need two leading clauses for 5.4.17. 5.4.15 Nietzsche respects only Spinoza. Rns ∙ (∀x)(Rnx ⊃ x=s) 5.4.16

Only Nietzsche doesn’t like Nietzsche. ∼Lnn ∙ (∀x)(∼Lxn ⊃ x=n)

5.4.17 Only Kant is read more widely than Descartes and Hume. Mkd ∙ Mkh ∙ (∀x)[(Mxd ∨ Mxh) ⊃ x=k]

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‘Only’ sentences can be even more complex, as at 5.4.18, in which both clauses contain quantification. 5.4.18

Only Locke plays billiards with some rationalist who is read more widely than Descartes.    (Rx: x is a rationalist; Mxy: x is read more widely than y;    Pxy: x plays billiards with y) (∃x){(Rx ∙ Mxd ∙ Plx) ∙ (∀y)[(Ry ∙ Myd) ⊃ (∀z)(Pzy ⊃ z=l)]}

Sentences with ‘except’ also contain universal claims and a preceding clause. As usual, universal claims have a conditional as the main propositional operator in their scope. But identity shows up in the consequent of the conditional for ‘only’ claims, while it shows up in the antecedent in ‘except’ claims, allowing us to omit the desired exception, as in 5.4.19. 5.4.19

Everyone except Julio loves Maria. ∼Ljm ∙ (∀x)[(Px ∙ x≠j) ⊃ Lxm]

Ordinarily, when we use ‘except’, not only do we exempt one individual from a universal claim, we also deny that whatever we are ascribing to everyone else holds of the exemption. Julio doesn’t love Maria, and every other person does. As with ‘only’ sentences, these denials are extra clauses that I put at the beginning. 5.4.20 and 5.4.21 have slightly more complex preceding clauses; you can see the role of negation in the latter. 5.4.20 Every philosopher except Berkeley respects Locke. Pb ∙ ∼Rbl ∙ (∀x)[(Px ∙ x≠b) ⊃ Rxl] 5.4.21 Nietzsche does not respect any philosopher except Spinoza. Ps ∙ Rns ∙ (∀x)[(Px ∙ x≠s) ⊃ ∼Rnx]

The exception clause added to the antecedent of the conditional following the universal quantifier can also be longer, as when we except more than one thing, as at 5.4.22. 5.4.22 Some philosopher respects all philosophers except Plato and Aristotle. Pp ∙ Pa ∙ (∃x){Px ∙ ~Rxp ∙ ~Rxa ∙ (∀y)[(Py ∙ y≠p ∙ y≠a) ⊃ Rxy]}

Some uses of ‘but’ work just like ordinary uses of ‘except’, as at 5.4.23, which also has a quantified preceding clause. 5.4.23

Every philosopher but Socrates wrote a book.    (Bx: x is a book; Px: x is a philosopher; Wxy: x wrote y) Ps ∙ ∼(∃x)(Bx ∙ Wsx) ∙ (∀x)[(Px ∙ x≠s) ⊃ (∃y)(By ∙ Wxy)]

Socrates is a philosopher, and there is no book that he wrote, but for all philosophers except Socrates, there is a book that they wrote. Of course, 5.4.23 is false, though that’s no barrier to writing it.

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SUPERLATIVES

Relational predicates allow us to express comparisons: larger than, smaller than, older than, funnier than, and so on. The identity predicate allows us to express superlatives. We have a comparison at 5.4.24 where ‘Ix’ stands for ‘x is an impressionist’ and ‘Bxy’ stands for ‘x is a better impressionist than y’. 5.4.24

Degas is a better impressionist than Monet.

Id ∙ Im ∙ Bdm

We don’t really need the ‘Ix’ clauses for 5.4.24, and we don’t need identity. But what if you want to say that Degas is the best impressionist, or to say that you are the nicest person? If you are nicer than anyone, then you are nicer than yourself, which is impossible. We really need to say ‘nicer than anyone else’, ‘nicer than anyone except oneself ’. We thus add a universal quantifier with an identity clause to except the single, reflexive case: better (or more profound or nicer or whatever) than anyone else, as at 5.4.25. 5.4.25

Degas is the best impressionist.

Id ∙ (∀x)[(Ix ∙ x≠d) ⊃ Bdx]

Notice that we do need the ‘Ix’ clauses here: Degas is an impressionist, and no matter what other impressionist you pick, he’s a better impressionist. 5.4.26 is another standard superlative sentence. 5.4.27 adds a negation, which leads to two equivalent propositions (given QE). 5.4.26

Hume is the biggest philosopher.    (h: Hume; Px: x is a philosopher; Bxy: x is bigger than y) Ph ∙ (∀x)[(Px ∙ x≠h) ⊃ Bhx] 5.4.27

Hume is not the most difficult empiricist to read.    (h: Hume; Ex: x is an empiricist; Dxy: x is more difficult to    read than y) Eh ∙ ∼(∀x)[(Ex ∙ x≠h) ⊃ Dhx] Eh ∙ (∃x)[(Ex ∙ x≠h) ∙ ∼Dhx]

5.4.28 just complicates the sentence slightly, and 5.4.29 a bit more. 5.4.28

The Ethics is the most difficult book by Spinoza to read.   (e: The Ethics; Bx: x is a book; Wxy: x wrote y; Dxy:    x is more difficult to read than y) Be ∙ Wse ∙ (∀x)[(Bx ∙ Wsx ∙ x≠e) ⊃ Dex] 5.4.29 Either The Critique of Pure Reason or The Ethics is the most difficult book to read.   (c: The Critique of Pure Reason; e: The Ethics;    Bx: x is a book; Dxy: x is more difficult to read than y) Bc ∙ Be ∙ (∀x)[(Bx ∙ x≠c ∙ x≠e) ⊃ (Dcx ∨ Dex)]

The last few uses of identity that I will discuss are especially philosophically interesting. The next few (‘at least’, ‘at most’, and ‘exactly’) concern how much mathematics can be developed using just logic. The latter (‘definite descriptions’) concerns a puzzle in the philosophy of language, often called the problem of empty reference.

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AT LEAST AND AT MOST

Frege’s development of formal logic was intricately linked to his logicist project of trying to show that mathematics is just logic in complex form. Frege’s logicism, as he developed it, was a failure; he used an inconsistent logic. Subsequent logicist (or neologicist) projects rely on substantial set-theoretic principles that many philosophers believe are not strictly logical. Normally, we extend logical systems to mathematical ones by including one more element to the language, ‘∈’, standing for set inclusion, and axioms governing set theory. Mathematics is uncontroversially definable in terms of logic plus set theory. Part of the contemporary neo-logicist project is to see just how little set theory we need to add to logic in order to develop mathematics. It is edifying to see, then, how much mathematics can be generated by the logical machinery of just F, using the identity predicate. For example, we can express many adjectival uses of numbers in F. We have already seen how to say that there is at least one of something; that’s just using the existential quantifier. To say that there is exactly one of something, we can use ‘only’ as in 5.4.30. 5.4.30

There is only one aardvark.

(∃x)[Ax ∙ (∀y)(Ay ⊃ x=y)]

So, we have already seen how to translate sentences including ‘exactly one’ clauses. To regiment ‘exactly’ sentences for larger numbers, to say that there are exactly n of some object, for any n we need just a little more machinery, combining at-least sentences with at-most clauses. Let’s start with the at-least sentences, 5.4.31–5.4.34. Notice that there is a natural procedure for translating ‘at least’ for any number. You use as many quantifiers as the number you are trying to represent. The identity predicate is used to make sure that each of the quantifiers refers to a distinct individual. 5.4.31 There is at least one aardvark.   (∃x)Ax 5.4.32 There are at least two aardvarks.   (∃x)(∃y)(Ax ∙ Ay ∙ x≠y) 5.4.33 There are at least three aardvarks.   (∃x)(∃y)(∃z)(Ax ∙ Ay ∙ Az ∙ x≠y ∙ x≠z ∙ y≠z) 5.4.34 There are at least four aardvarks.   (∃x)(∃y)(∃z)(∃w)(Ax ∙ Ay ∙ Az ∙ Aw ∙ x≠y ∙ x≠z ∙ x≠w ∙   y≠z ∙ y≠w ∙ z≠w)

Note that with ‘at least one’, we don’t need an identity clause. With ‘at least two’, we need one identity clause. ‘At least three’ takes three clauses, and ‘at least four’ takes six. We won’t do ‘at least’ for numbers greater than four, but if you’re looking for a formula, I’ll put it in the Keep in Minds at the end of the section. The identity clauses at the end become increasingly long as the number we are expressing increases, but the algorithm is simple: just make sure to include one clause for each pair of variables.

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5.4.35–5.4.39 contain relational predicates as well as ‘at least’. The increasing complexity just follows the pattern. 5.4.35

At least one materialist respects Berkeley.    (b: Berkeley; Mx: x is a materialist; Rxy: x respects y) (∃x)(Mx ∙ Rxb) 5.4.36 At least two materialists respect Berkeley. (∃x)(∃y)(Mx ∙ Rxb ∙ My ∙ Ryb ∙ x≠y) 5.4.37 There are at least three materialists who respect Berkeley. (∃x)(∃y)(∃z)(Mx ∙ Rxb ∙ My ∙ Ryb ∙ Mz ∙ Rzb ∙ x≠y ∙ x≠z ∙ y≠z) 5.4.38

At least two idealist philosophers respect each other.    (Ix: x is an idealist; Px: x is a philosopher; Rxy: x respects y) (∃x)(∃y)(Ix ∙ Px ∙ Iy ∙ Py ∙ Rxy ∙ Ryx ∙ x≠y) 5.4.39 At least three idealist philosophers respect each other. (∃x)(∃y)(∃z)(Ix ∙ Px ∙ Iy ∙ Py ∙ Iz ∙ Pz ∙ Rxy ∙ Ryx ∙ Rxz ∙ Rzx ∙   Ryz ∙ Rzy ∙ x≠y ∙ x≠z ∙ y≠z)

The pairs of ‘respect’ clauses in 5.4.39 follow a pattern similar to that for the identity clauses. 5.4.40 requires a fourth quantifier to take care of ‘some book written by Descartes’. 5.4.40

At least three coherentists respect some book written by Descartes.    (d: Descartes; Bx: x is a book; Cx: x is a coherentist;    Wxy: x wrote y; Rxy: x respects y) (∃x)(∃y)(∃z){Cx ∙ Cy ∙ Cz ∙ (∃w)[(Bw ∙ Wdw) ∙ Rxw] ∙ (∃w)[(Bw ∙   Wdw) ∙ Ryw] ∙ (∃w)[(Bw ∙ Wdw) ∙ Rzw] ∙ x≠y ∙ x≠z ∙ y≠z}

Notice that we can use the same quantifier, (∃w), repeatedly: there is some book written by Descartes that x respects, and one that y respects, and one that z respects. We need all three clauses in case x, y, and z respect different books. Let’s move on to ‘at most’ sentences. At-most clauses use universal quantifiers. The core idea is that to say that one has at most n of something, we say that if we think we have one more than n of it, there must be some redundancy. Again, the complexity increases in a predictable way, as at 5.4.41–5.4.43. 5.4.41

There is at most one aardvark.    (Ax: x is an aardvark) (∀x)(∀y)[(Ax ∙ Ay) ⊃ x=y] 5.4.42 There are at most two aardvarks. (∀x)(∀y)(∀z)[(Ax ∙ Ay ∙ Az) ⊃ (x=y ∨ x=z ∨ y=z)] 5.4.43 There are at most three aardvarks. (∀x)(∀y)(∀z)(∀w)[(Ax ∙ Ay ∙ Az ∙ Aw) ⊃ (x=y ∨ x=z ∨ x=w ∨ y=z   ∨ y=w ∨ z=w)]

As with at-least sentences, we have identity clauses at the end. For at-most sentences, though, the identity clauses are affirmative and we disjoin them. Again, make

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sure to have one clause for each pair of variables. The complexity of relational predicates and quantified subformulas, which we see in 5.4.44–5.4.49, does not change the ‘at most’ pattern. 5.4.44

Nietzsche respects at most one philosopher.    (n: Nietzsche; Px: x is a philosopher; Rxy: x respects y) (∀x)(∀y)[(Px ∙ Rnx ∙ Py ∙ Rny) ⊃ x=y] 5.4.45 Nietzsche respects at most two philosophers. (∀x)(∀y)(∀z)[(Px ∙ Rnx ∙ Py ∙ Rny ∙ Pz ∙ Rnz) ⊃ (x=y ∨ x=z ∨ y=z)] 5.4.46 Nietzsche respects at most three philosophers. (∀x)(∀y)(∀z)(∀w)[(Px ∙ Rnx ∙ Py ∙ Rny ∙ Pz ∙ Rnz ∙ Pw ∙ Rnw) ⊃   (x=y ∨ x=z ∨ x=w ∨ y=z ∨ y=w ∨ z=w)] 5.4.47

Kant likes at most two empiricists better than Hume.    (h: Hume; k: Kant; Ex: x is an empiricist; Lxyz: x likes y better   than z) (∀x)(∀y)(∀z)[(Ex ∙ Lkxh ∙ Ey ∙ Lkyh ∙ Ez ∙ Lkzh) ⊃    (x=y ∨ x=z ∨ y=z)] 5.4.48

At most one idealist plays billiards with some rationalist.    (Ix: x is an idealist; Rx: x is a rationalist; Pxy: x plays    billiards with y) (∀x)(∀y){Ix ∙ (∃z)(Rz ∙ Pxz) ∙ Iy ∙ (∃z)(Rz ∙ Pyz)] ⊃ x=y} 5.4.49

At most two rationalists wrote a book more widely read than every book written by Hume.    (h: Hume; Bx: x is a book; Rx: x is a rationalist; Wxy: x wrote y;    Mxy x is read more widely than y) (∀x)(∀y)(∀z){{Rx ∙ (∃w)[Bw ∙ Wxw ∙ (∀v)(Bv ∙ Whv) ⊃ Mwv] ∙   Ry ∙ (∃w)[Bw ∙ Wyw ∙ (∀v)(Bv ∙ Whv) ⊃ Mwv] ∙ Rz ∙ (∃w)[Bw ∙   Wzw ∙ (∀v)(Bv ∙ Whv) ⊃ Mwv]} ⊃ (x=y ∨ x=z ∨ y=z)} EX ACTLY

To express “exactly,” we combine the at-least and at-most clauses. 5.4.30 says that there is exactly one aardvark. The first portion says that there is at least one. The second portion, starting with the universal quantifier, expresses the redundancy that follows from supposing that there are two aardvarks. We still need n+1 quantifiers in an ‘exactly’ sentence. The first n quantifiers are existential. Then we add the one further universal quantifier. The identity clauses at the end of the at-most portion of the proposition hold between only the variable bound by the universal quantifier and the other variables, not among the existentially bound variables: there are n things that have such and such a property; if you think that you have another one, an n+1 thing, it must be identical to

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one of the first n. As you can see at 5.4.50–5.4.52, the ‘at most’ clause always has just one universal quantifier. 5.4.50 There are exactly two aardvarks. (∃x)(∃y){Ax ∙ Ay ∙ x≠y ∙ (∀z)[Az ⊃ (z=x ∨ z=y)]} 5.4.51 There are exactly three aardvarks. (∃x)(∃y)(∃z){Ax ∙ Ay ∙ Az ∙ x≠y ∙ x≠z ∙ y≠z ∙   (∀w)[Aw ⊃ (w=x ∨ w=y ∨ w=z)]} 5.4.52 There are exactly four aardvarks. (∃x)(∃y)(∃z)(∃w){Ax ∙ Ay ∙ Az ∙ Aw ∙ x≠y ∙ x≠z ∙ x≠w ∙ y≠z ∙   y≠w ∙ z≠w ∙ (∀v)[Av ⊃ (v=x ∨ v=y ∨ v=z ∨ v=w)]}

These numerical sentences get very long very quickly. Indeed, our language of predicate logic, F, cannot express ‘exactly five’ or more, since we have run out of quantifiers. To abbreviate numerical sentences, logicians sometimes introduce special shorthand quantifiers like the ones at 5.4.53. 5.4.53 (∃1x), (∃2x), (∃3x) . . .

The quantifiers at 5.4.53 might be taken to indicate that there are at least the number indicated. To indicate exactly a number, ‘!’ is sometimes used. For exactly one thing, people sometimes write ‘(∃!x)’. For more things, we can insert the number and the ‘!’, as at 5.4.54. 5.4.54 (∃1!x), (∃2!x), (∃3!x) . . .

These abbreviations are useful for translation. But once we want to make inferences using the numbers, we have to unpack their longer forms. We will not extend our language F to include more variables, or to include numerals or ‘!’, but it is easy enough to do so. 5.4.55–5.4.58 contain further ‘exactly’ translations, with the same kinds of complications we saw above with ‘at least’ and ‘at most’ sentences. 5.4.55 There is exactly one even prime number. (∃x){(Ex ∙ Px ∙ Nx) ∙ (∀y)[(Ey ∙ Py ∙ Ny) ⊃ y=x]} 5.4.56 There are exactly two chipmunks in the yard. (∃x)(∃y){Cx ∙ Yx ∙ Cy ∙ Yy ∙ x≠y ∙ (∀z)[(Cz ∙ Yz) ⊃ (z=x ∨ z=y)]} 5.4.57 There are exactly three aardvarks on the log. (∃x)(∃y)(∃z){Ax ∙ Lx ∙ Ay ∙ Ly ∙ Az ∙ Lz ∙ x≠y ∙ x≠z ∙ y≠z ∙   (∀w)[(Aw ∙ Lw) ⊃ (w=x ∨ w=y ∨ w=z]} 5.4.58

Exactly three idealists play billiards with some rationalist. (∃x)(∃y)(∃z){[Ix ∙ (∃w)(Rw ∙ Pxw) ∙ Iy ∙ (∃w)(Rw ∙ Pyw) ∙ Iz ∙   (∃w)(Rw ∙ Pzw) ∙ x≠y ∙ x≠z ∙ y≠z] ∙ (∀v){[Iv ∙ (∃w)(Rw ∙   Pvw)] ⊃ (v=x ∨ v=y ∨ v=z)}}

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DEFINITE DESCRIPTIONS

Our last use of the identity predicate is in a solution to a problem in the philosophy of language. The problem can be seen in trying to interpret 5.4.59. 5.4.59

The king of America is bald.

We might regiment 5.4.59 as 5.4.60, taking ‘k’ for ‘the king of America’. 5.4.60 Bk

5.4.60 is false, since there is no king of America. Given our bivalent semantics, then, 5.4.61 should be true since it is the negation of a false statement. 5.4.61

∼Bk

5.4.61 seems to be a perfectly reasonable regimentation of 5.4.62. 5.4.62

The king of America is not bald.

5.4.62 has the same grammatical form as 5.4.63. 5.4.63

This happy man is not bald.

We take 5.4.63 to be true because the happy man has a lot of hair. So, 5.4.61 may reasonably be taken to say that the king of America has hair. But that’s not something we want to assert as true. In fact, we want both 5.4.60 and 5.4.61 to be false. The conjunction of their negations is the contradiction 5.4.64. 5.4.64

A definite description picks out an object by

using a descriptive phrase beginning with ‘the’.

∼Bk ∙ ∼ ∼Bk

And given what we saw about explosion in section 3.5, we certainly don’t want to assert that! We had better regiment our sentences differently. Bertrand Russell, facing just this problem, focused on the fact that ‘the king of America’ is a definite description that refers to no real thing. Like a name, a definite description is a way of referring to a specific object. A definite description picks out an object by using a descriptive phrase that begins with ‘the’, as in ‘the person who . . .’, or ‘the thing that . . .’. Both 5.4.59 and 5.4.62 use definite descriptions to refer to an object. They are both false due to a false presupposition in the description that there exists a king of America. Russell’s solution to the problem is to rewrite sentences that use definite descriptions. Definite descriptions, he says, are disguised complex propositions, and the grammatical form of sentences that contain definite descriptions are more complicated than they look. We have to unpack them to reveal their true logical form. So, according to Russell, 5.4.59, properly understood, consists of three simpler expressions. 5.4.59A 5.4.59B 5.4.59C

There is a king of America. There is only one king of America. That thing is bald.

(∃x)Kx (∀y)(Ky ⊃ y=x) Bx

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Putting them together, so that every term is within the scope of the original existential quantifier, we get 5.4.65, which Russell claims is the proper analysis of 5.4.59. 5.4.65 (∃x)[Kx ∙ (∀y)(Ky ⊃ y=x) ∙ Bx]

5.4.59 is false because clause A is false. 5.4.62 is also false, for the same reason, which we can see in its proper regimentation, 5.4.66. 5.4.66 (∃x)[Kx ∙ (∀y)(Ky ⊃ y=x) ∙ ∼Bx]

The tilde in 5.4.66 affects only the third clause. The first clause is the same in 5.4.65 and 5.4.66, and still false. Further, when we conjoin 5.4.65 and 5.4.66, we do not get a contradiction, as we did in 5.4.64. 5.4.67 (∃x)[Kx ∙ (∀y)(Ky ⊃ y=x) ∙ Bx] ∙ (∃x)[Kx ∙ (∀y)(Ky ⊃ y=x) ∙ ∼Bx]

5.4.67 is no more problematic than 5.4.68. 5.4.68 Some things are purple, and some things are not purple. (∃x)Px ∙ (∃x)∼Px

There is much more to say about Russell’s theory of definite descriptions. In particular, Russell’s distinction between grammatical form and logical form is both enormously influential and deeply contentious. Here, though, let’s put away the problem of empty reference for definite descriptions and see how Russell’s analysis guides translation generally. A definite description starts with ‘the’, rather than ‘a’ or ‘an’, indicating the definiteness of the description. Identity allows us to represent that definiteness. We regiment sentences of the form of 5.4.69 as sentences like 5.4.70; I separated the three clauses one more time. 5.4.69

The country called a subcontinent is India. 5.4.69A There is a country called a subcontinent. 5.4.69B There is only one such country. 5.4.69C That country is (identical with) India.

5.4.70 (∃x){(Cx ∙ Sx) ∙ (∀y)[(Cy ∙ Sy) ⊃ y=x] ∙ x=i}

Russell’s original example is at 5.4.71. 5.4.71 The author of Waverley was a genius. (∃x){Wx ∙ (∀y)[Wy ⊃ y=x] ∙ Gx}

Summary The identity symbol, =, is just an ordinary binary relation between two singular terms. But the logic of that relation is both simple and powerful in translation, allowing us to regiment sentences with ‘except’, ‘only’, superlatives, ‘at least’, ‘at most’, ‘exactly’, and definite descriptions. Each kind of translation follows a standard pattern that can be learned without too much effort, if you have mastered F.

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In our next section, we will construct derivations using the rules governing identity that I introduced in this section. Take your time to get comfortable with the translations before moving on to derivations.

KEEP IN MIND

The identity predicate is a two-place relation of singular terms. Instead of Ixy, we write ‘x=y’. For negations of identity statements, ∼Ixy, we write ∼x=y or x≠y. Do not punctuate identity sentences; reserve punctuation for wffs connected by dyadic propositional operators and for quantifiers. Identity is reflexive and symmetric. The identity predicate is especially useful in translating sentences containing ‘only’ and ‘except’; superlatives; ‘at least’, ‘at most’, and ‘exactly’; and definite descriptions. Propositions including ‘except’ and ‘only’ have two clauses, one of which is universal. Identity appears in the consequent of the conditional following the universal quantifier for ‘only’ claims and in the antecedent of that conditional in ‘except’ claims. For superlatives, use a relational predicate with an identity clause to except the reflexive case. For ‘at least n’ statements, use n existential quantifiers. ‘At least’ statements greater than ‘one’ require negative identity clauses to ensure that each quantifier refers to a distinct thing. The formula for the number of negative identity clauses in an ‘at least’ statement is n(n–1)/2. For ‘at most n’ statements, use n+1 universal quantifiers. The formula for the number of identity clauses in an ‘at most’ statement is n(n+1)/2. ‘Exactly’ sentences combine ‘at least’ sentences with ‘at most’ sentences. For ‘exactly n’ objects, use n existential quantifiers and one universal quantifier. Definite descriptions use ‘the’ and a descriptive phrase to pick out an object; identity is used to represent the uniqueness of the object that fits the description.

TELL ME MORE  • What is a name? Should our best logic include names? What questions surround Russell’s theory of definite descriptions? See 7.6: Names, Definite Descriptions, and Logical Form. • How do logicians understand numbers as sets? See 7.7: Logicism and 6S.13: SecondOrder Logic and Set Theory. • How do philosophers deal with numbers beyond natural numbers such as infinite numbers? See 7S.9: Infinity.

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EXERCISES 5.4 Translate into first-order logic, using the identity predicate where applicable. For exercises 1–8, use: b: Bob e: Emilia Dx: x attends the conference dinner Px: x presents original work Sx: x is a speaker at the conference 1. There are at least two speakers at the conference. 2. Exactly two speakers at the conference present original work. 3. Only Bob and Emilia present original work. 4. All speakers at the conference except Bob attend the conference dinner. 5. Exactly one speaker at the conference presents original work and attends the conference dinner. 6. The speaker at the conference presents original work. 7. All speakers at the conference that present original work attend the conference dinner, except Emilia and Bob. 8. If at least three speakers at the conference present original work then some of those presenting original work do not attend the conference dinner.

For exercises 9–16, use: e: Zoe l: Leah r: Riverdale High s: Sunnydale University Hx: x is a high school Sx: x is a student Tx: x is in our town Ux: x is a university

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Axy: x attends y Exy: x enrolls at y 9. At most two students who attend Riverdale High enroll at Sunnydale University. 10. At most three students who attend Riverdale High enroll at Sunnydale University. 11. All students who attend Riverdale High enroll at Sunnydale, except Leah. 12. All students who enroll in some university attend some high school, except Zoe and Leah. 13. Exactly three students who attend Riverdale High enroll at Sunnydale University. 14. Only Zoe attends high school without enrolling in some university. 15. The university in our town is Sunnydale. 16. If exactly one student attends Riverdale High and enrolls in Sunnydale University, then Zoe enrolls in a university in our town.

For exercises 17–24, use: c: Carla f: Fifi r: Ravi Dx: x is a dog Bxy: x is better trained than y Txy: x trains y 17. Only Carla trains dogs. 18. Carla only trains dogs. 19. If only Carla trains dogs, then Ravi does not train dogs. 20. Fifi is the best trained dog. 21. Fifi is the best dog that is trained by Carla. 22. Exactly two dogs trained by Carla are better trained than Fifi. 23. At least three dogs trained by Ravi are better trained than some dog trained by Carla. 24. Every dog trained by Carla is better trained than every dog trained by Ravi, except Fifi.

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For exercises 25–32, use: c: Canada f: Fernanda h: Honora Hx: x is heavy Px: x is a person Wx: x is a worker Cxy: x carries y Ixy: x is in y Lxy: x likes y Sxy: x is stronger than y 25. At least two workers like Honora. 26. Exactly two workers like Honora. 27. At least three workers are stronger than Fernanda. 28. The strongest worker is Fernanda. 29. Fernanda is the strongest worker in Canada. 30. Only Fernanda carries heavy things. 31. Everyone except Honora likes Fernanda. 32. No one who likes Honora is stronger than Honora, except Fernanda. For exercises 33–40, use: g: Grant’s Tomb j: Jalisa n: New York City o: One World Trade Center Ax: x is an apartment Bx: x is called the Big Apple Cx: x is a city Ex: x is a building Sx: x is a student Bxy: x is bigger than y Fxy: x is from y Ixy: x is in y Hxy: x has y

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33. The city called the Big Apple is New York. 34. Exactly one city is called the Big Apple. 35. At most two students are from New York. 36. At most one New Yorker has an apartment bigger than Grant’s Tomb. 37. At most two New Yorkers have apartments bigger than Grant’s Tomb. 38. One World Trade Center is bigger than Grant’s Tomb. 39. One World Trade Center is the biggest building in New York City. 40. No student has an apartment bigger than Grant’s Tomb except Jalisa. For exercises 41–48, use: c: Chemistry 200 j: Juan n: Nicola p: Physics 101 r: Rick Bx: x is a biology major Gx: x is a grade Hxy: x is higher than y Ixy: x is in y Rxy: x received y Txy: x takes y 41. There are at least two biology majors in Physics 101. 42. There are exactly three biology majors in Physics 101. 43. Only Nicola and Rick received a higher grade than Juan. 44. Either Nicola or Rick received the highest grade in Physics 101. 45. Every biology major except Nicola takes Physics 101. 46. The biology major who received the highest grade in Physics 101 takes Chemistry 200. 47. The biology major who received the highest grade in Chemistry 200 received the highest grade in Physics 101. 48. Every biology major who takes Physics 101, except Rick, received a higher grade in that class than any grade that some biology major received in Chemistry 200.

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For exercises 49–56, use: a: Andre j: Juliana l: Logic t: Tarski Px: x is a paper Sx: x is a student Bxy: x is busier than y Ixy: x is in y Rxy: x reads y Wxy: x wrote y 49. Andre is the busiest student in Logic. 50. At least two students in Logic read a paper written by Tarski. 51. At least three students in Logic are busier than Andre. 52. Exactly three students in Logic are busier than Juliana. 53. Every student in Logic reads a paper written by Tarski except Andre. 54. No student in Logic reads all papers by Tarski, except Juliana. 55. At most two papers by Tarski are read by all students in Logic. 56. If only Andre reads all papers by Tarski, then no student in Logic is busier than Andre.

For exercises 57–64, use: g: Gödel p: Principia Mathematica s: Schmidt Lx: x is a logician Bxy: x is a better logician than y Dxy: x discovered the incompleteness of y 57. At most one logician is better than Gödel. 58. No logician is better than Gödel, except Schmidt. 59. Schmidt is the best logician. 60. Exactly two logicians are better than Gödel. 61. At most two logicians discovered the incompleteness of Principia Mathe­ matica—­Gödel or Schmidt.

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62. Only Schmidt discovered the incompleteness of Principia Mathematica. 63. The logician who discovered the incompleteness of Principia Mathematica was either Gödel or Schmidt. 64. Schmidt discovered the incompleteness only of Principia Mathematica. For exercises 65–72, use: b: Berkeley d: Descartes k: Kant Ax: x is an atomist Ix: x is an idealist Mx: x is a materialist Tx: x is a transcendental idealist Fxy: x is more famous than y 65. Exactly three materialists are idealists. 66. At most two atomists are not materialists. 67. The most famous idealist is Berkeley. 68. The transcendental idealist is Kant. 69. No materialist except Kant is an idealist. 70. No idealist except Berkeley is more famous than Kant. 71. Only Descartes, among materialists, is not an atomist. 72. At least two materialists are atomists, if, and only if, at least one idealist is a transcendental idealist. For exercises 73–80, use: d: Descartes h: Hume s: Spinoza Cx: x is a compatibilist Dx: x is a determinist Fx: x believes in free will Lx: x is a libertarian Px: x is a philosopher Rx: x believes in moral responsibility

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73. At least one determinist believes both in free will and moral responsibility. 74. At least two determinists believe in moral responsibility, but not free will. 75. At most three compatibilists do not believe in moral responsibility. 76. All compatibilists who believe in moral responsibility are determinists, except Hume. 77. No philosopher is a libertarian except Descartes. 78. The libertarian is Descartes; the determinist is Spinoza; the compatibilist is Hume. 79. The only determinist who does not believe in free will but does believe in moral responsibility is Spinoza. 80. If exactly one compatibilist believes in free will, then only Hume believes in moral responsibility. For exercises 81–88, use: a: Aristotle b: Bentham j: Jones m: Mill Cx: x is a consequentialist Dx: x is a deontologist Gx: x teaches in the graduate school Kx: x is a Kantian Px: x is a philosopher Ux: x is a utilitarian 81. At most two deontologists are Kantians. 82. At least two consequentialists who teach in the graduate school are utilitarians. 83. Exactly one philosopher is both a consequentialist and a deontologist. 84. No consequentialist is also a deontologist except Aristotle. 85. The deontologist who teaches in the graduate school is a Kantian. 86. There are at least three consequentialists who teach in the graduate school, but the only utilitarians are Mill and Bentham. 87. Every philosopher who teaches in the graduate school is either a consequentialist or a deontologist, except Jones. 88. No philosophers who teach in the graduate school are Kantians if, and only if, exactly one philosopher who teaches in the graduate school is a deontologist.

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5.5: THE IDENTITY PREDICATE: DERIVATIONS In this section, we will use three rules governing the identity predicate to construct derivations using identity. The rules are generally simple to use, but the complexity of the propositions that identity allows us to express make some proofs quite long. Some can be intimidatingly complicated. But the core ideas are not particularly difficult, so patience and persistence can pay off. Rules Governing Identity For any singular terms α and β: IDr (reflexivity) IDs (symmetry) IDi (indiscernibility of identicals) Identity rules (IDi, IDr, IDs) are three rules governing the identity relation.

α=α → β=α α=β ← ℱα α=β / ℱ β

IDr is an axiom schema that says that any singular term stands for an object that is identical to itself. While we are not generally using an axiomatic system of inference, we will follow tradition and allow any instance of the schema IDr into any proof, with no line justification. IDr is not often useful in derivations, but it helps to characterize the relation. IDs is a rule of equivalence that says that identity is commutative: if one thing is identical to another, then the second is also identical to the first. As we have noticed, many relations, like loving or being taller than, are not symmetric. Identity, like the relations of being married to or being collinear with, is symmetric. We can use IDs on whole lines or on parts of lines, switching the order of the singular terms in the relation. IDs often assists us in managing the uses of identity statements with other rules. IDi is a rule of inference, and the most useful of the ID rules. IDi says that if you have α=β, then you may rewrite any formula containing α with another formula that has β in the place of α throughout. With IDi, we always rewrite a whole line, switching one singular term for another. To understand IDi, consider again Superman and Clark Kent. We know that the two people are the same, so anything true of one is true of the other. Since Clark Kent works at The Daily Planet, Superman works at The Daily Planet, too, even though his coworkers do not generally know this. Since Superman can fly, Clark Kent can fly, though characters in the Superman universe do not generally know this. The property captured by IDi is called Leibniz’s law, or the indiscernibility of identicals. Be careful not to confuse this simple logical property, written as a single schematic sentence at 5.5.1, with the related identity of indiscernibles, written at 5.5.2 and defended by the philosopher G. W. Leibniz. 5.5.1 (∀x)(∀y)[x=y ⊃ (ℱx ≡ ℱ y)] indiscernibility of identicals 5.5.2 (∀x)(∀y)[(ℱx ≡ ℱ y) ⊃ x=y] identity of indiscernibles The indiscernibility of identicals says that if two terms refer to the same object, then whatever we predicate of the one term can be predicated of the other. The contentious

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identity of indiscernibles says that no two things share all properties. Whether two things can share all of their properties is a vexing question that depends for its truth on a theory of properties, a topic well beyond our range. For examples of these rules in use, let’s start with 5.4.4, the inference with which I motivated identity theory. Superman can fly. Superman is Clark Kent. So, Clark Kent can fly.

To derive the conclusion, we need only a simple application of IDi, as at 5.5.3. 5.5.3 QED

1. Fs 2. s=c 3. Fc

/ Fc 1, 2, IDi

5.5.4 uses IDs and IDi. 5.5.4 QED

1. a=b ⊃ j=k 2. b=a 3. Fj 4. a=b 5. j=k 6. Fk

/ Fk 2, IDs 1, 4, MP 3, 5, IDi

To derive the negation of an identity statement, one ordinarily uses indirect proof as in 5.5.5. 5.5.5 1. Rm 2. ∼Rj / m≠j 3. m=j AIP 4. Rj 1, 3, IDi 5. Rj ∙ ∼Rj 4, 2, Conj 6. m≠j 3–5, IP QED

5.5.6 uses the reflexivity rule, at line 4, to produce a contradiction. Alternatively, one could use it to set up a modus tollens with line 3. 5.5.6 1. (∀x)(∼Gx ⊃ x≠d) / Gd 2. ∼Gd AIP 3. ∼Gd ⊃ d≠d 1, UI 4. d=d IDr 5. d≠d 3, 2, MP 6. d=d ∙ d≠d 4, 5, Conj 7. ~ ~Gd 2-6, IP 8. Gd 7, DN QED

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We can generalize over variables in identity statements, as in the use of EG at line 9 in 5.5.7. Notice also the use of IDs at line 5, working like commutativity for singular terms. 5.5.7 1. Rab 2. (∃x)∼Rxb / (∃x)∼x=a 3. ∼Rcb 2, EI 4. c=a AIP 5. a=c 4, IDs 6. Rcb 1, 5, IDi 7. Rcb ∙ ∼Rcb 6, 3, Conj 8. ∼c=a 4–7, IP 9. (∃x)∼x=a 8, EG QED

The derivations 5.5.3–5.5.7 have been quick. But many simple arguments using identity require long derivations. The argument 5.5.8 is valid. 5.5.8

The Joyce scholar at Hamilton College is erudite. Therefore, all    Joyce scholars at Hamilton College are erudite.

It may seem a little odd, since it derives a universal conclusion from an existential premise. But the universality of the conclusion is supported by the uniqueness clause in the definite description. Remember that a definite description is definite: there is only one thing that fits the description. The premise thus entails that there is only one Joyce scholar at Hamilton College. Anything we say of a Joyce scholar at Hamilton holds of all Joyce scholars at Hamilton (viz., only the one). Let’s translate 5.5.8 into F, at 5.5.9. As I noted in section 5.4, by convention we may drop brackets from series of conjunctions or disjunctions. 5.5.9 (∃x){Jx ∙ Hx ∙ (∀y)[(Jy ∙ Hy) ⊃ x=y] ∙ Ex}

/ (∀x)[(Jx ∙ Hx) ⊃ Ex]

Given our convention about dropping brackets among series of conjunctions and series of disjunctions, we should add corresponding conventions governing inferences. Conventions for Derivations with Dropped Brackets If a wff is a series of conjunctions, you may use Simp to infer, immediately, any of the conjuncts, including multiple conjuncts. If a wff is a series of disjunctions and you have the negation of one of the disjuncts on a separate line, you may eliminate it, using DS, from the series. You may use Conj to conjoin any number of propositions appearing on separate lines into a single proposition in a single step. If there is a negation in front of a bracket containing a series of conjunctions you may use DM to negate each of the conjuncts and change all the ∙ s to ∨ s. If there is a negation in front of a bracket containing a series of disjunctions you may use DM to negate each of the disjuncts and change all the ∨ s to ∙ s.

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You may use Dist to distribute a conjunction over any number of disjuncts and to distribute a disjunction over any number of conjuncts. You may use Com to re-order, in any way, any series of disjuncts or of conjuncts. In the proof of the argument 5.5.9, which is at 5.5.10 and uses a standard CP, I avail myself of the first of these conventions at lines 4 and 7. 5.5.10 1. (∃x){Jx ∙ Hx ∙ (∀y)[(Jy ∙ Hy) ⊃ x=y] ∙ Ex} / (∀x)[(Jx ∙ Hx) ⊃ Ex] 2. Jx ∙ Hx ACP 3. Ja ∙ Ha ∙ (∀y)[(Jy ∙ Hy) ⊃ a=y] ∙ Ea 1, EI 4. (∀y)[(Jy ∙ Hy) ⊃ a=y] 3, Simp 5. (Jx ∙ Hx) ⊃ a=x 4, UI 6. a=x 5, 2, MP 7. Ea 3, Simp 8. Ex 7, 6, IDi 9. (Jx ∙ Hx) ⊃ Ex 2–8, CP 10. (∀x)[(Jx ∙ Hx) ⊃ Ex] 9, UG QED

5.5.11 contains another substantial proof using propositions with identity, this time showing how ‘at least’ and ‘at most’ entail ‘exactly’ proof-theoretically. 5.5.11

There is at least one moon of Earth. There is at most one moon of Earth. / So, there is exactly one    moon of Earth.

1. (∃x)Mx 2. (∀x)(∀y)[(Mx ∙ My) ⊃ x=y] / (∃x)[Mx ∙ (∀y)(My ⊃ x=y)] 3. Ma 1, EI 4. My ACP 5. (∀y)[(Ma ∙ My) ⊃ a=y] 2, UI 6. (Ma ∙ My) ⊃ a=y 5, UI 7. Ma ∙ My 3, 4, Conj 8. a=y 6, 7, MP 9. My ⊃ a=y 4–8, CP 10. (∀y)(My ⊃ a=y) 9, UG 11. Ma ∙ (∀y)(My ⊃ a=y) 3, 10, Conj 12. (∃x)[Mx ∙ (∀y)(My ⊃ x=y)] 11, EG QED

5.5.12 has an even longer derivation, even with our new conventions (especially at lines 27, 33, and 40). Removing and replacing multiple quantifiers, moving negations across multiple quantifiers using QE, and just working with the complex statements that identity helps us represent all lengthen the proofs. When working with long proofs, be especially careful to keep track of your different singular terms, which ones are constants and which are variables. Look ahead to see whether you are going to need to UG, in which case you’ll need to work with variables. And, as always, indirect proof is the refuge of the desperate.

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5.5.12 There are at least two cars in the driveway. All the cars in the driveway belong to Jasmine. Jasmine has at most two cars. / So, there are exactly two cars in the driveway. 1. (∃x)(∃y)(Cx ∙ Dx ∙ Cy ∙ Dy ∙ x≠y) 2. (∀x)[(Cx ∙ Dx) ⊃ Bxj] 3. (∀x)(∀y)(∀z)[(Cx ∙ Bxj ∙ Cy ∙ Byj ∙ Cz ∙ Bzj) ⊃ (x=y ∨ x=z ∨ y=z)] / (∃x)(∃y){Cx ∙ Dx ∙ Cy ∙ Dy ∙ x≠y ∙ (∀z)[(Cz ∙ Dz) ⊃ (z=x ∨ z=y)]} 4. (∃y)(Ca ∙ Da ∙ Cy ∙ Dy ∙ a≠y) 1, EI 5. Ca ∙ Da ∙ Cb ∙ Db ∙ a≠b 4, EI 6. Ca ∙ Da 5, Simp 7. (Ca ∙ Da) ⊃ Baj 2, UI 8. Baj 7, 6, MP 9. Cb ∙ Db 5, Simp 10. (Cb ∙ Db) ⊃ Bbj 2, UI 11. Bbj 10, 9, MP 12. a≠b 5, Simp 13. ∼(∀z)[(Cz ∙ Dz) ⊃ (z=a ∨ z=b)] AIP 14. (∃z)∼[(Cz ∙ Dz) ⊃ (z=a ∨ z=b)] 13, QE 15. (∃z)∼[∼(Cz ∙ Dz) ∨ (z=a ∨ z=b)] 14, Impl 16. (∃z)[∼ ∼(Cz ∙ Dz) ∙ ∼(z=a ∨ z=b)] 15, DM 17. (∃z)[(Cz ∙ Dz) ∙ ∼(z=a ∨ z=b)] 16, DN 18. Cc ∙ Dc ∙ ∼(c=a ∨ c=b) 17, EI 19. Ca 6, Simp 20. Ca ∙ Baj 19, 8 Conj 21. Cb 9, Simp 22. Cb ∙ Bbj 21, 11, Conj 23. Cc ∙ Dc 18, Simp 24. (Cc ∙ Dc) ⊃ Bcj 2, UI 25. Bcj 24, 23, MP 26. Cc 23, Simp 27. Cc ∙ Bcj 26, 25, Conj 28. Ca ∙ Baj ∙ Cb ∙ Bbj ∙ Cc ∙ Bcj 20, 22, 27, Conj 29. (∀y)(∀z)[(Ca ∙ Baj ∙ Cy ∙ Byj ∙ Cz ∙ Bzj) ⊃ (a=y ∨ a=z ∨ y=z)] 3, UI 30. (∀z)[(Ca ∙ Baj ∙ Cb ∙ Bbj ∙ Cz ∙ Bzj) ⊃ (a=b ∨ a=z ∨ b=z)] 29, UI 31. (Ca ∙ Baj ∙ Cb ∙ Bbj ∙ Cc ∙ Bcj) ⊃ (a=b ∨ a=c ∨ b=c) 30, UI 32. a=b ∨ a=c ∨ b=c 31, 28, MP 33. a≠b 5, Simp 34. a=c ∨ b=c 32, 33, DS 35. ∼(c=a ∨ c=b) 18, Simp 36. ∼(c=a ∨ b=c) 35, IDs 37. ∼(a=c ∨ b=c) 36, IDs 38. (a=c ∨ b=c) ∙ ∼(a=c ∨ b=c) 34, 37, Conj

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39. ∼ ∼(∀z) (Cz ∙ Dz) ⊃ (z=a ∨ z=b)] 40. (∀z)[(Cz ∙ Dz) ⊃ (z=a ∨ z=b)] 41. Ca ∙ Da ∙ Cb ∙ Db ∙ a≠b ∙ (∀z)[(Cz ∙ Dz) ⊃ (z=a ∨ z=b)] 42. (∃y){Ca ∙ Da ∙ Cy ∙ Dy ∙ a≠y ∙ (∀z)[(Cz ∙ Dz) ⊃ (z=a ∨ z=y)]} 43. (∃x)(∃y){Cx ∙ Dx ∙ Cy ∙ Dy ∙ x≠y ∙ (∀z)[(Cz ∙ Dz) ⊃ (z=x ∨ z=y)]} QED

Summary The rules governing the identity predicate are fairly simple and easy to learn. The propositions that use identity, though, can be long and complex. Arguments that use such propositions tend to be consequently long, and sometimes difficult, mainly just because of the complexity of the propositions.

KEEP IN MIND

Singular terms of which identity holds may be exchanged in wffs; we call this property the indiscernibility of identicals, or Leibniz’s law. Do not confuse Leibniz’s law with its converse, the identity of indiscernibles. IDi allows us to rewrite a whole line, switching one singular term for another. IDs is a rule of equivalence, allowing us to commute the two singular terms flanking a ‘=’. IDr allows us to insert an identity sentence, of a singular with itself, with no line justification; it is rarely useful in derivations. Our conventions for dropping brackets in series of conjunctions or disjunctions lead to further conventions within derivations for some rules.

Rules Introduced For any singular terms α and β: IDr (reflexivity)

α=α

IDs (symmetry)

α=β

IDi (indiscernibility of identicals)

ℱα



α=β / ℱβ

→ β=α ←

13–38, IP 39, DN 6, 9, 12, 40, Conj 41, EG 42, EG

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EXERCISES 5.5a Derive the conclusions of each of the following arguments. 1. 1. (∀x)[(∃y)Pxy ⊃ (∃z)Pzx] 2. (∃x)(Pxb ∙ x=d) / (∃z)Pzd 2. 1. (∀x)(∀y)[Ax ⊃ (By ⊃ Cxy)] 2. Aa ∙ Ba 3. a=b / Cab 3. 1. (∃x)(Mx ∙ Px) 2. (∀x)[Mx ⊃ (∀y)(Ky ⊃ x=y)] 3. Kf / Mf ∙ Pf 4. 1. Pa ∙ (∀x)[(Px ∙ x≠a) ⊃ Qax] 2. Pb ∙ a≠b / Qab 5. 1. Dkm ∙ (∀x)(Dkx ⊃ x=m) 2. Dab 3. Fb ∙ ∼Fm / a≠k 6. 1. (∀x)[ Jx ∨ (Kx ∙ Lx)] 2. ∼(Ja ∨ Kb)

/ a≠b

7. 1. (∀x)[(Mx ∨ Nx) ⊃ Ox] 2. ∼Oc 3. Md / c≠d 8. 1. (∀x)(Qx ⊃ Sx) 2. (∀x)(Rx ⊃ Tx) 3. (∀x)[Qx ∨ (Rx ∙ Ux)] 4. a=b

/ Sb ∨ Ta

9. 1. (∀x)[Ax ∨ (Bx ∙ Cx)] 2. ∼(∀x)Bx 3. (∀x)(Ax ⊃ x=c)

/ (∃x)x=c

10. 1. (∃x){Px ∙ Qx ∙ (∀y)[(Py ∙ Q y ∙ x≠y) ⊃ Axy]} 2. (∀x)(∀y)(Axy ⊃ Byx) / (∃x){Px ∙ Qx ∙ (∀y)[(Py ∙ Q y ∙ x≠y) ⊃ Byx]} 11. 1. (∀x)[(Px ∙ Qx ∙ x≠a) ⊃ (∃y)Rxy] 2. ∼(∃y)Rby 3. Sa ∙ ∼Sb / ∼(Pb ∙ Qb) 12. 1. Pa ∙ Qab ∙ (∀x)[(Px ∙ Qxb ∙ x≠a) ⊃ Rax] 2. Pc ∙ Qcb ∙ ∼Rac / c=a 13. 1. Dp ∙ (∃x)(Ex ∙ ∼Fxp) 2. (∀x)[Gx ⊃ (∀y)Fyx]

/ (∃x)(Dx ∙ ∼Gx)

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14. 1. La ∙ Lb ∙ a≠b 2. (∀x)(∀y)(∀z)[(Lx ∙ Ly ∙ Lz) ⊃ (x=y ∨ y=z ∨ x=z)] / (∀x)[Lx ⊃ (x=a ∨ x=b)] 15. 1. (∃x){Px ∙ Qx ∙ (∀y)[(Py ∙ Q y) ⊃ y=x]} 2. (∃x){Rx ∙ Qx ∙ (∀y)[(Ry ∙ Q y) ⊃ y=x]} 3. (∀x)(Px ≡ ∼Rx)

/ (∃x)(∃y)(Qx ∙ Q y ∙ x≠y)

16. 1. (∀x)(∀y)[(Px ∙ Qx ∙ Py ∙ Q y) ⊃ x=y] 2. (∃x)(∃y)(Px ∙ Rx ∙ Py ∙ Ry ∙ x≠y)

/ (∃x)(Px ∙ ∼Qx)

17. 1. (∃x)[Px ∙ (∀y)(Py ⊃ y=x)] 2. (∀x){Px ⊃ (∃y)[Q y ∙ (∀z)(Qz ⊃ z=y) ∙ Rxy]} / (∃x)(∃y)[Px ∙ Q y ∙ Rxy ∙ (∀z)(Pz ⊃ z=x) ∙(∀z)(Qz ⊃ z=y)] 18. 1. (∀x)[(Px ∙ Qx) ⊃ x≠a] 2. (∃x){Px ∙ Rx ∙ (∀y)[(Py ∙ Ry) ⊃ y=x]} 3. (∀x)(Rx ⊃ Qx)

/ ∼(Pa ∙ Ra)

19. 1. (∃x)[Px ∙ (∀y)(Py ⊃ y=x) ∙ Qx] 2. (∀x)[Qx ⊃ (∃y)Rxy] 3. (∃x)(Px ∙ Sx)

/ (∃x)[Qx ∙ Sx ∙ (∃y)Rxy]

20. 1. (∃x)(∃y)(Px ∙ Qx ∙ Py ∙ Q y ∙ x≠y) 2. (∀x)(Px ⊃ Rx) 3. (∀x)(∀y)(∀z)[(Qx ∙ Rx ∙ Q y ∙ Ry ∙ Qz ∙ Rz) ⊃ (x=y ∨ x=z ∨ y=z)] / (∃x)(∃y){Px ∙ Qx ∙ Py ∙ Qy ∙ x≠y ∙ (∀z)[(Pz ∙ Qz) ⊃ (z=x ∨ z=y)]} 21. 1. (∃x)(∃y)(∃z)(Px ∙ Py ∙ Pz ∙ x≠y ∙ x≠z ∙ y≠z ∙ Qxyz ∙ Qzyx) 2. (∀x)(∀y)(∀z)(Qxyz ≡ Q yxz) 3. (∀x)(∀y)(∀z)(Qxyz ≡ Qxzy) / (∃x)(∃y)(∃z)(Px ∙ Py ∙ Pz ∙ x≠y ∙ x≠z ∙ y≠z ∙ Qxyz ∙ Qxzy ∙ Q yxz ∙ Q yzx ∙ Qzxy ∙ Qzyx) 22. 1. (∃x)(∃y)(Hx ∙ Ix ∙ Jx ∙ Hy ∙ Iy ∙ Jy ∙ x≠y) 2. (∀x)(∀y)(∀z)[(Hx ∙ Ix ∙ Jx ∙ Hy ∙ Iy ∙ Jy ∙ Hz ∙ Iz ∙ Jz) ⊃ (x=y ∨ x=z ∨ y=z)] / (∃x)(∃y){Hx ∙ Ix ∙ Jx ∙ Hy ∙ Iy ∙ Jy ∙ x≠y ∙ (∀z)[(Hz ∙ Iz ∙ Jz) ⊃ (z=x ∨ z=y)]} 23. 1. Na ∙ Oa ∙ Nb ∙ Ob ∙ a≠b ∙ (∀x)[(Nx ∙ Ox) ⊃ (x=a ∨ x=b)] 2. Na ∙ ∼Pa ∙ (∀x)[(Nx ∙ x≠a) ⊃ Px] / (∃x){Nx ∙ Ox ∙ Px ∙ (∀y)[(Ny ∙ Oy ∙ Py) ⊃ y=x]} 24. 1. (∃x)(∃y)(Kx ∙ Lx ∙ Ky ∙ Ly ∙ x≠y) 2. Ka ∙ La ∙ Ma ∙ (∀y)[(Ky ∙ Ly ∙ My) ⊃ y=a] / (∃x)(Kx ∙ Lx ∙ ∼Mx) 25. 1. (∃x)(∃y)(Ax ∙ Cx ∙ Ay ∙ Cy ∙ x≠y) 2. (∀x)(∀y)(∀z)[(Cx ∙ Cy ∙ Cz) ⊃ (x=y ∨ x=z ∨ y=z)] 3. (∃x)(Bx ∙ ∼Ax) / ∼(∀x)(Bx ⊃ Cx)

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26. 1. (∃x)(∃y)(Qx ∙ Rx ∙ Q y ∙ Ry ∙ x≠y) 2. (∀x)(∀y)(∀z)[(Rx ∙ Sx ∙ Ry ∙ Sy ∙ Rz ∙ Sz) ⊃ (x=y ∨ x=z ∨ y=z)] 3. (∀x)(∼Qx ∨ Sx) / (∃x)(∃y){Qx ∙ Rx ∙ Sx ∙ Q y ∙ Ry ∙ Sy ∙ x≠y ∙ (∀z)[(Rz ∙ Sz) ⊃ (z=x ∨ z=y)]} 27. 1. (∀x)(∀y)(∀z)[(Px ∙ Qxa ∙ Rxb ∙ Py ∙ Q ya ∙ Ryb ∙ Pz ∙ Qza ∙ Rzb) ⊃   (x=y ∨ x=z ∨ y=z)] 2. Pc ∙ Sc ∙ Qca ∙ Rcb 3. Pd ∙ ∼Sd ∙ Qda ∙ Rdb / (Pe ∙ Qea ∙ Reb) ⊃ (e=c ∨ e=d) 28. 1. (∀x)(∀y)(∀z)[(Px ∙ Qx ∙ Py ∙ Q y ∙ Pz ∙ Qz) ⊃ (x=y ∨ x=z ∨ y=z)] 2. (∃x)(∃y)(Rx ∙ Qx ∙ Ry ∙ Q y ∙ x≠y) 3. (∀x)(Px ≡ Rx) / (∃x)(∃y){Px ∙ Qx ∙ Py ∙ Q y ∙ x≠y ∙ (∀z)[(Pz ∙ Qz) ⊃ (z=x ∨ z=y)]} 29. 1. (∃x)(∃y){Px ∙ Qx ∙ Py ∙ Q y ∙ x≠y ∙ (∀z)[(Pz ∙ Qz) ⊃ (z=x ∨ z=y)]} 2. (∃x)(∃y)(∃z)(Sx ∙ Sy ∙ Sz ∙ Qx ∙ Q y ∙ Qz ∙ x≠y ∙ x≠z ∙ y≠z) 3. (∀x)(Px ⊃ Rx) 4. ∼(∃x)(Rx ∙ Sx) / (∃x)(∃y)(∃z)(∃w)(∃v)(Qx ∙ Q y ∙ Qz ∙ Qw ∙ Qv ∙ x≠y ∙ x≠z ∙ x≠w ∙ x≠v ∙ y≠z ∙ y≠w ∙ y≠v ∙ z≠w ∙ z≠v ∙ w≠v) 30. 1. Ma ∙ ∼Pa ∙ Mb ∙ ∼Pb ∙ (∀x)[(Mx ∙ x≠a ∙ x≠b) ⊃ Px] 2. Qb ∙ (∀x)[(Mx ∙ Qx) ⊃ x=b] 3. (∀x){Mx ⊃ [∼(Qx ∨ Px) ≡ Rx]} 4. a≠b / (∃x){Mx ∙ Rx ∙ (∀y)[(My ∙ Ry) ⊃ y=x]}

EXERCISES 5.5b Translate each of the following arguments into F, using the given terms and the identity predicate, where useful. Then, derive the conclusion using our rules of inference. 1. Polly flies. Olivia doesn’t. So, Polly is not Olivia. (o: Olivia; p: Polly; Fx: x flies) 2. If George is Dr. Martin, then Dr. Martin is married to Mrs. Wilson. Dr. Martin is George. Mrs. Wilson is Hilda. So, George is married to Hilda. (g: George; h: Hilda; m: Dr. Martin; w: Mrs. Wilson; Mxy: x is married to y) 3. If something is a not superhero, then everything is not Wonder Woman. So, Wonder Woman is a superhero. (w: Wonder woman; Sx: x is a superhero) 4. Katerina is the fastest runner on the team. Pedro is a runner on the team. Katerina is not Pedro. So, Katerina is faster than Pedro. (k: Katerina; p: Pedro; Rx: x is a runner; Tx: x is on the team; Fxy: x is faster than y)

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5. The author of Republic was a Greek philosopher. John Locke was a philosopher, but he was not Greek. Therefore, John Locke did not write Republic. (l: John Locke; r: Republic; Gx: x is Greek; Px: x is a philosopher; Wxy: x wrote y) 6. The only person who went skiing was James. The only person who caught a cold was Mr. Brown. Some person who went skiing also caught a cold. So, James is Mr. Brown. (b: Mr. Brown; j: James; Cx: x caught a cold; Px: x is a person; Sx: x went skiing) 7. Exactly one student in the class gives a presentation about Spinoza. At least two students in the class give a presentation about Leibniz. No student in the class gives a presentation about both Leibniz and Spinoza. So, there are at least three students in the class. (l: Leibniz; s: Spinoza; Sx: x is a student in the class; Gxy: x gives a presentation about y) 8. Every employee except Rupert got a promotion. The only employee to get a promotion was Jane. So, there are exactly two employees. (j: Jane; r: Rupert; Ex: x is an employee; Px: x gets a promotion) 9. No philosopher except Descartes is a dualist. Spinoza is a philosopher, distinct from Descartes. Every philosopher is either a dualist or a monist. So, Spinoza is a monist. (d: Descartes; s: Spinoza; Dx: x is a dualist; Mx: x is a monist; Px: x is a philosopher) 10. Kierkegaard and Sartre are both existentialists, but Kierkegaard is a theist and Sartre is not. If all existentialists are nihilists, then Kierkegaard and Sartre are identical. So, some existentialists are not nihilists. (k: Kierkegaard; s: Sartre; Ex: x is an existentialist; Nx: x is a nihilist; Tx: x is a theist) 11. No idealist is more renowned than Berkeley, except Kant. Russell, who is neither Berkeley nor Kant, is more renowned than Berkeley. So, Russell is not an idealist. (b: Berkeley; k: Kant; r: Russell; Ix: x is an idealist; Rxy: x is more renowned than y) 12. Every platonist except Plato believes in the existence of the material world. Every platonist believes in an abstract realm. Gödel is a platonist who is not Plato. So, something believes in both a material world and an abstract realm, and something does not. (g: Gödel; p: Plato; Ax: x believes in an abstract realm; Mx: x believes in the existence of a material world; Px: x is a platonist) 13. At least two philosophers are more prolific than the philosopher Hume. No philosopher is more insightful than Hume. Nothing is more prolific than itself. So, at least two philosophers are more prolific, without being more insightful, than a third philosopher. (h: Hume; Px: x is a philosopher; Ixy: x is more insightful than y; Pxy: x is more prolific than y) 14. At most one argument for consequentialism is not utilitarian. There are some non-utilitarian arguments for consequentialism. Any argument for consequentialism faces trolley-case objections. So, exactly one non-utilitarian argument

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for consequentialism faces trolley-case objections. (Ax: x is an argument for consequentialism; Fx: x faces trolley-case objections; Ux: x is utilitarian) 15. Exactly two students in the class are compatibilists. Exactly one student in the class is a hard determinist. No compatibilist or hard determinist is a libertarian. No compatibilist is a hard determinist, and vice versa. So, at least three students in the class are not libertarians. (Cx: x is a compatibilist; D: x is a hard determinist; Lx: x is a libertarian; Sx: x is a student in the class) 16. Any two distinct points determine exactly one line that contains both of those points. A and B are distinct points. Line L contains points A and B. Line M is distinct from Line L. So, Line M does not contain both points A and B. (a: Point A; b: Point B; l: Line L; m: Line M; Lx: x is a line; Px: x is a point; Cxy: x contains y)

EXERCISES 5.5c Derive the following logical truths of identity theory. 1. (∀x)(∀y)(x=y ≡ y=x) 2. (Fa ∙ a=b) ⊃ Fb 3. (∃x)x=a ∨ (∀x)x≠a 4. (∀x)(∀y)(∀z)[(x=y ∙ y=z) ⊃ x=z] 5. (∀x)(∀y)(∀z)[(x=y ∙ x=z) ⊃ y=z] 6. (∀x)(∀y)[(Fx ∙ ∼Fy) ⊃ x≠y] 7. (∀x)(∀y)[x=y ⊃ (Fx ≡ Fy)] 8. (∀x)(∀y)[x=y ⊃ (∀z)(Pxz ≡ Pyz)] 9. (∀x)(∀y){(x=a ∙ y=a) ⊃ [Rab ≡ (Rxb ∙ Ryb)]} 10. (∀x)(Pax ⊃ x=b) ⊃ [(∃y)Pay ⊃ Pab]

5.6: TRANSLATION WITH FUNCTIONS In the last two sections of this chapter, we will look at one final, formal topic: functions. This extension beyond F is contentious. Some philosophers consider functions to be mathematical, not purely logical. But their introduction gives us some efficient translations and facilitates some natural inferences. Consider, as a motivating example, the intuitively valid argument 5.6.1.

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5.6.1

All applicants will get a job. Jean is an applicant. Jean is the first child of Dominique and Henri. So, some first child will get a job.

The first two premises are easily regimented into F. 5.6.2

1. (∀x)(Ax ⊃ Gx) 2. Aj

We have several options for the third premise. We could take ‘first child of Dominique and Henri’ as a monadic predicate, as at 5.6.3. 5.6.3

3. Fj

Then we would need a different predicate for being the first child (of any parents) for the conclusion. Being the first child of Dominique and Henri is a different monadic property than being some first child. A second option is to regiment the third premise of 5.6.1 by using Russell’s theory of definite descriptions. We can use ‘Fxyz’ for ‘x is a first child of y and z’ and add a uniqueness clause. 5.6.4

3. (∃x)[Fxdh ∙ (∀y)(Fydh ⊃ y=x) ∙ x=j]

5.6.4 has the advantage of taking ‘first child of ’ to be a three-place relation. That option reveals more logical structure than 5.6.3, and so may be useful. Correspondingly, we can regiment the conclusion of 5.6.1 as 5.6.5. 5.6.5 (∃x){(∃y)(∃z)[Fxyz ∙ (∀w)(Fwyz ⊃ w=x)] ∙ Gx}

The conclusion 5.6.5 follows from the premises at 5.6.2 and 5.6.4, as we can see at 5.6.6. 5.6.6 QED

1. (∀x)(Ax ⊃ Gx) 2. Aj 3. (∃x)[Fxdh ∙ (∀y)(Fydh ⊃ y=x) ∙ x=j] 4. Fadh ∙ (∀y)(Fydh ⊃ y=a) ∙ a=j 5. a=j 6. j=a 7. Aa 8. Aa ⊃ Ga 9. Ga 10. (∀y)(Fydh ⊃ y=a) 11. Fwdh ⊃ w=a 12. (∀w)(Fwdh ⊃ w=a) 13. Fadh 14. Fadh ∙ (∀w)(Fwdh ⊃ w=a) 15. (∃z)[Fadz ∙ (∀w)(Fwdz ⊃ w=a)] 16. (∃y)(∃z)[Fayz ∙ (∀w)(Fwyz ⊃ w=a)] 17. (∃y)(∃z)[Fayz ∙ (∀w)(Fwyz ⊃ w=a)] ∙ Ga 18. (∃x){(∃y)(∃z)[Fxyz ∙ (∀w)(Fwyz ⊃ w=x)] ∙ Gx}

3, EI 4, Simp 5, IDs 2, 6, IDi 1, UI 8, 7, MP 4, Simp 10, UI 11, UG 4, Simp 13, 12, Conj 14, EG 15, EG 16, 9, Conj 17, EG

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When working with functions, an argument is an element or ordered n-tuple of elements of the domain paired with exactly one element of the range.

While the derivation at 5.6.6 is successful, there is a more efficient, and more fecund, option for regimenting ‘the first child of x and y’: we can take ‘the first child of x and y’ to be a function. Using a function allows us to regiment both the third premise and the conclusion more simply, and to construct tighter derivations. Let’s take a moment to explore functions before returning to 5.6.1. Consider terms like ‘the biological father of ’, ‘the successor of ’, ‘the sum of ’, and ‘the academic adviser of ’. Each takes one or more arguments, from their domain, and produces a single output, the range. We can tell that there is a single output by the use of the definite description. One-place functions take one argument, two-place functions take two arguments, and n-place functions take n arguments. With a small extension of F, adding functors like ‘f(x)’, we can express such functions neatly. 5.6.7 lists some functions and some possible logical representations. 5.6.7 f(x) g(x) f(x, y) f(a, b) g(x1 . . . xn)

the father of the successor of the sum of the truth value of the conjunction of A and B the teacher of

The last function can take as arguments, say, all the students in a class. An essential characteristic of functions is that they yield exactly one value no matter how many arguments they take. Thus, the expressions at 5.6.8 are not functions. 5.6.8

A functor is a symbol used to represent a function.

the biological parents of a the classes that a and b share the square root of x

These expressions are relations. Relations may be one-many, like ‘the square root of n’, which pairs a single number, say 4, with both its positive and negative square roots, +2 and −2. Relations may be many-many, like the classes that Johanna and Alexis share when they are both taking Logic, Organic Chemistry, and The Study of the Novel. Functions are special types of relations that always yield a single value. ‘The positive square root of x’ is a function, as is ‘the first class of the day for student x’. Functions play an important role in mathematics and science, as well as logic. We have seen that we can use the identity predicate to simulate adjectival uses of numbers: three apples, seven seas. With functions, we can express even more mathematics. A functor is a symbol used to represent a function, like any of the functions ubiquitous in mathematics and science. In mathematics, there are linear functions, exponential functions, periodic functions, quadratic functions, and trigonometric functions. In science, force is a function of mass and acceleration; momentum is a function of mass and velocity. The genetic code of a child is a function of the genetic codes of its biological parents. Functions are also essential for metalogic. Recall that the semantics for PL is presented in terms of truth functions. All the operators are truth functions, taking one argument (negation) or two arguments (the rest of the operators) and yielding a specific truth value.

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By adding functors to our language F, we adopt a new language, which I call FF, for full first-order predicate logic with functors. Vocabulary of FF Capital letters A . . . Z, used as predicates Lower-case letters a, b, c, d, e, i, j, k . . . u are used as constants. f, g, and h are used as functors. v, w, x, y, z are used as variables. Five propositional operators: ∼, ∙, ∨, ⊃, ≡ Quantifiers: ∃, ∀ Punctuation: (), [], {} In order to specify the formation rules for FF, we invoke n-tuples of singular terms, ordered series of singular terms: constants, variables, or functor terms. As we saw in section 5.2, n-tuples are like sets in that they are collections of objects but differ from sets in that the order of their objects (which we call arguments) matters. Often, n-tuples are represented using angle brackets: , , or . For FF, we will represent n-tuples of singular terms by listing the singular terms separated by commas, as at 5.6.9. 5.6.9

a, b a, a, f(a) x, y, b, d, f(x), f(a, b, f(x)) a

An n-tuple of singular terms is an ordered series of singular terms.

two arguments three arguments six arguments one argument

Now that we have characterized n-tuples, we can use them to define functor terms. Suppose α is an n-tuple of singular terms. Then a functor symbol, followed by an n-tuple of singular terms in brackets, is a functor term. The expressions at 5.6.10 are all functor terms (once we substitute the proper n-tuple for α). 5.6.10 f(α) g(α) h(α)

Note that an n-tuple of singular terms can include functor terms, as in the second and third examples at 5.6.9. ‘Functor term’ is defined recursively, which allows for composition of functions. For example, one can refer to the grandfather of x using just the functions for father, for example f(x), and mother, for example g(x). 5.6.11 represents ‘paternal grandfather’ and 5.6.12 represents maternal grandfather’. 5.6.11 f(f(x)) 5.6.12 f(g(x))

The use of punctuation (parentheses) in functor terms can multiply, but is sadly needed. For another example, if we take ‘h(x)’ to represent the square of x, then 5.6.13 represents the eighth power of x, in other words, ((x 2)2)2 .

A functor term is a functor followed by an n-tuple of singular terms in brackets.

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5.6.13 h(h(h(x)))

I have introduced only three functor letters. As with variables and constants (see 4.3), there are several different tricks for constructing an indefinite number of terms out of a finite vocabulary using indexing. But we won’t need more than the three letters here, so we will make do with only these. Even with just the three letters, we have an indefinite number of functors, since each of 5.6.14 is technically a different functor and can represent a different function. 5.6.14 f(a) f(a, b) f(a, b, c) f(a, b, c, d) and so on

The scope and binding rules are the same for FF as they were for M and F. The formation rules need only one small adjustment, at the first line. Formation Rules for wffs of FF FF1. An n-place predicate followed by n singular terms (constants, variables, or functor terms) is a wff. FF2. For any variable β, if α is a wff that does not contain either ‘(∃β)’ or ‘(∀β)’, then ‘(∃β)α’ and ‘(∀β)α’ are wffs. FF3. If α is a wff, so is ∼α. FF4. If α and β are wffs, then so are: (α ∙ β) (α ∨ β) (α ⊃ β) (α ≡ β) FF5. These are the only ways to make wffs. The semantics for FF are basically the same as for F, too. For an interpretation of FF, we insert an interpretation of function symbols at step 3. Semantics for FF Step 1. Specify a set to serve as a domain of interpretation. Step 2. Assign a member of the domain to each constant. Step 3. A  ssign a function with arguments and ranges in the domain to each function symbol. Step 4. A ssign some set of objects in the domain to each one-place predicate; assign sets of ordered n-tuples to each relational predicate. Step 5. Use the customary truth tables for the interpretation of the propositional operators.

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The function assigned in step 3 will be a function in the metalanguage used to interpret the function in the object language. I won’t pursue a discussion of metalinguistic functions, except to say that they work just like ordinary mathematical functions. Once you have the idea of how functions work in the object language, it will become clear how they work in the metalanguage. Let’s move on to the nuts and bolts of translation with functions.

Translations into FF and Simple Arithmetic Functions At 5.6.15, there is a translation key and some English sentences with their regimentations in FF. 5.6.15

Lxy: x loves y f(x): the father of x g(x): the mother of x o: Olaf



Olaf loves his mother. Olaf loves his grandmothers. Olaf’s father loves someone. No one is his/her own mother.

Log(o) Log(g(o)) ∙ Log(f(o)) (∃x)Lf(o)x (∀x)∼x=g(x)

While 5.6.15 shows some ordinary uses of functions, their most natural applications come in regimenting sentences of mathematics. Many simple concepts in arithmetic are functions: addition, multiplication, least common multiple. The most fundamental function in mathematics is the successor function. All other mathematical functions can be defined in terms of successor and other basic concepts. In fact, all of arithmetic can be developed from five basic axioms, called the Peano axioms. They are named for Giuseppe Peano, who published in 1889 a precise version of the axioms that Richard Dedekind had published a year earlier. Peano credited Dedekind, and sometimes these axioms are called the Dedekind-Peano, or even the Dedekind, axioms. 5.6.16

The Peano Axioms for Arithmetic

P1: Zero is a number. P2: The successor of every number is a number. P3: Zero is not the successor of any number. P4: No distinct numbers have the same successor. P5: I f some property may (or may not) hold for any number, and if zero has the property, and if, for any number, its having the property entails that its successor has the property, then all numbers have the property.

P5 is called the induction schema. It can be used to generate an indefinite number of axioms, one for each mathematical property. Mathematical induction is essential in metalogic, as well as in linear algebra and number theory.

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We can write the Peano axioms in FF using the given key, as I do at 5.6.17. 5.6.17

Peano’s Axioms in FF

a: zero Nx: x is a number f(x): the successor of x PA1. Na PA2. (∀x)(Nx ⊃ Nf(x)) PA3. ∼(∃x)(Nx ∙ f(x)=a) PA4. (∀x)(∀y)[(Nx ∙ Ny) ⊃ (f(x)=f(y) ⊃ x=y)] PA5. {Pa ∙ (∀x)[(Nx ∙ Px) ⊃ Pf(x)]} ⊃ (∀x)(Nx ⊃ Px)

Notice that the predicate ‘P’ as used in PA5 can stand for any property, like the property of being prime or the property of having a square. To write this axiom even more generally, one needs a stronger language, such as second-order logic. 5.6.18–5.6.21 present translations of some arithmetic sentences using functions. Note that in the following sentences, I take ‘number’ to mean ‘natural number’ (i.e., the counting numbers 1, 2, 3, . . .) and use the following translation key. o: one f(x): the successor of x f(x, y): the product of x and y Ex: x is even Nx: x is a number Ox: x is odd Px: x is prime 5.6.18 One is the successor of some number.   (∃x)[Nx ∙ f(x)=o] 5.6.19 The product of the successor of one and any other number is even.   (∀x)Ef(f(o), x) 5.6.20

If the product of a pair of numbers is odd, then the product of the successors of those numbers is even.   (∀x)(∀y){(Nx ∙ Ny) ⊃ [Of(x, y) ⊃ Ef(f(x), f(y))]}

5.6.21 There are no prime numbers such that their product is prime.   ∼(∃x)(∃y)[Nx ∙ Px ∙ Ny ∙ Py ∙ Pf(x, y)]

Summary Functors are not in the vocabulary of standard first-order logic. By adding functors to our language, we switch from F to FF. The addition facilitates some natural inferences. But some philosophers resist seeing functions as purely logical, and see them as mathematical. Mathematicians treat functions as kinds of relations, and relations as kinds of sets; we can define relations and functions in terms of sets. Set theory is ordinarily taken to be mathematics, not logic. But we are here supposed to be working with a purely logical, and not mathematical, language.

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Concerns that functions are mathematical, and not logical, should be allayed somewhat by noting that the work we are doing with functions here can be done in F with definite descriptions, though in more complicated fashion. Our last technical subject, in the next section, is derivations with functions. KEEP IN MIND

The work of functions can be done, less efficiently, with definite descriptions. FF is the result of adding functors to the language of F. We reserve ‘f ’, ‘g’, and ‘h’ as functor symbols; they don’t work as constants in FF.

TELL ME MORE  • How do we turn logical theories into mathematical ones? See 7.7: Logicism and 6S.13: Second-Order Logic and Set Theory.

EXERCISES 5.6 Use the given key to translate the following sentences into FF. For exercises 1–8, use: m: Mariel f(x): the mother of x g(x): the father of x Px: x is a person Sxy: x is a sister of y Txy: x takes care of y 1. Mariel takes care of her mother. 2. Mariel’s paternal grandmother takes care of Mariel. 3. Mariel takes care of her grandmothers. 4. Mariel’s sister takes care of Mariel’s grandfathers. 5. Mariel’s only sister takes care of Mariel’s grandfathers. 6. No one is his/her own mother. 7. Not everyone is the father of someone. 8. Some maternal grandmothers are sisters to someone.

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For exercises 9–16, use: t: two f(x): the square of x g(x): the successor of x f(x, y): the product of x and y Ex: x is even Nx: x is a number Ox: x is odd Px: x is prime 9. Two and its successor are prime numbers. 10. Not all odd numbers are prime. 11. The square of an odd number is odd. 12. The square of a number is not prime. 13. The product of even numbers is even. 14. The product of a number and its successor is not prime. 15. The product of an odd number and an even number is even. 16. The square of a number is the product of it with itself. For exercises 17–24, use: a: Ayo c: Conor j: Javier k: Katja m: Marquis o: Olivia s: Spencer f(x): the thing one place in front of x in line Gx: x is a graduate Lx: x is in line Px: x majored in philosophy Sx: x majored in sociology 17. Every graduate except Olivia is in line. 18. The graduate two places in front of Ayo majored in philosophy, as did Ayo. 19. In line, Ayo is one place in front of Conor, who is one place in front of Marquis. 20. In line, Javier is one place behind Katja, who is one place behind Marquis. 21. Every philosophy major is one place in front of some sociology major.

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22. Some philosophy majors are two spaces in front of some sociology majors. 23. Of all the graduates in line, none is one place in front of Olivia, and none is one place behind Spencer. 24. At most three sociology majors are two spaces in front of some philosophy major. For exercises 25–32, use: e: false p: proposition P q: proposition Q t: true f(x): the truth value of x f(x, y): the truth value of the conjunction of x and y g(x, y): the truth value of the disjunction of x and y Px: x is a proposition 25. The truth value of P is true, not false. 26. The conjunction of propositions P and Q is false. 27. No proposition is neither true nor false. 28. The truth value of the disjunction of some proposition with P is true. 29. If the conjunction of propositions P and Q is false, then either P is false or Q is. 30. If the truth value of the conjunction of P with every proposition is false, then P is false. 31. If the truth value of the disjunction of Q with every proposition is true, then Q is true. 32. If the truth value of the conjunction of any two propositions is equal to the truth value of the disjunction of those propositions, then either the original propositions are both true or they are both false. For exercises 33–40, use: b: Betsy h: Helena o: Oscar w: Will f(x): the mother of x f(x, y): the first child of x and y Px: x is a philosopher Mxy: x is married to y

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33. The first child of Betsy and Helena is a philosopher. 34. Betsy’s mother is the first child of Will and Helena. 35. No one is the first child of Oscar and Betsy. 36. The first child of Will and Oscar is married to the first child of Betsy and Helena. 37. Every philosopher’s mother is married to some first child. 38. If Betsy’s first child with someone is a philosopher, then Will is the first child of Oscar and Helena. 39. Some philosopher is married to the first child of two philosophers. 40. Exactly two philosophers are married to first children.

5.7: DERIVATIONS WITH FUNCTIONS Our final section concerns the derivations with functions that will simplify some of the inferences using definite descriptions. There are no new rules for producing derivations with functions, since they are just complex singular terms and act, in derivations, like other singular terms. We use instantiation and generalization rules as we have until now, with a couple of restrictions, focusing mainly on when you can and cannot introduce new functional structure into a formula.

Derivations and Functional Structure Functional structure

reflects the complexity of a functor term or of the n-tuple of singular terms in a functor term.

The functional structure of a singular term arises from the way in which functions may be embedded in other functions. The functional structure increases with the number of embedded functions. A precise definition is possible, but not necessary here. A simple constant or variable has no functional structure. The singular terms in 5.7.1 have increasing functional structure. 5.7.1

f(a, b) f(f(a), g(b)) f(f(g(a, b)), g(h(a, b, f(b))))

Using the instantiation and generalization rules with functions is straightforward, with no new restrictions if you don’t change the functional structure of the propositions with which you are working. You just have to be careful to manage constants and variables as usual. We consider a function as if it were either a constant or a variable, for the purposes of instantiating or generalizing, depending on the arguments of the function. A complex singular term acts like a variable if there are any variables

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in any of its argument places, or those of any of its embedded functions. Otherwise, it acts like a constant. If the arguments of a function are all variables, then you are free to use UG over the variables in that function. If the arguments of a function contain any constants, then you may not use UG. You may use UG on either ‘x’ or ‘y’ in 5.7.2, assuming that the proposition does not appear within an indented sequence in which the variables are free in the first line. 5.7.2

Af(x, y) ⊃ Bx(fx)

You may not use UG on ‘x’ and ‘y’ in 5.7.3, depending on whether the variables were free when ‘a’ was introduced. 5.7.3

Af(x, y, a)

For EI, we must continue always to instantiate to a new singular term. A functor is not a new singular term if any of its arguments or any of the arguments of any of its subfunctors have already appeared in the derivation or appear in the conclusion. The functor itself need not be new. At 5.7.4, you may not instantiate line 2 to ‘a’ or to ‘b’; use a new constant, as at line 3. 5.7.4

1. f(a)=b Premise 2. (∃x)Sf(x) Premise 3. Sf(c) 2, EI

Turning to complete proofs, the derivation at 5.7.5 uses a function merely as a singular term and does not alter the functional structure of any singular term. 5.7.5 QED

1. (∀x)[Px ⊃ Pf(x)] 2. (∃x)(Px ∙ Rxa) 3. Pb ∙ Rba 4. Pb 5. Pb ⊃ Pf(b) 6. Pf(b) 7. Rba 8. Pf(b) ∙ Rba 9. (∃x)[Pf(x) ∙ Rxa]

/ (∃x)[Pf(x) ∙ Rxa] 2, EI 3, Simp 1, UI 5, MP 3, Simp 6, 7, Conj 8, EG

Sometimes, though, a derivation requires us to add or reduce functional structure, as at 5.7.6. 5.7.6

1. (∀x)[Px ⊃ Pf(x)] 2. (∃x)[Pf(x) ∙ Qf(f(x))]

/ (∃x)[Pf(f(x)) ∙ Qf(f(x))]

In order to derive the conclusion of 5.7.6, we have to UI line 1 to ‘f (a)’. That will increase the functional structure of the terms in the premise. That’s acceptable, though, since the premise is universal. If a claim holds of anything, it holds of all functions of anything. So, the derivation at 5.7.7 is perfectly fine.

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5.7.7 QED

1. (∀x)[Px ⊃ Pf(x)] 2. (∃x)[Pf(x) ∙ Qf(f(x))] 3. Pf(a) ∙ Qf(f(a)) 4. Pf(a) 5. Pf(a) ⊃ Pf(f(a)) 6. Pf(f(a)) 7. Qf(f(a)) 8. Pf(f(a)) ∙ Qf(f(a)) 9. (∃x)[Pf(f(x)) ∙ Qf(f(x))]

/ (∃x)[Pf(f(x)) ∙ Qf(f(x))] 2, EI 3, Simp 1, UI 5, 4, MP 3, Simp 6, 7, Conj 8, EG

Given that you may increase functional structure when using UI, the inferences at 5.7.8 from the universal statement at the top to any of its instances below it are all acceptable. 5.7.8 (∀x)(Px ⊃ Qx) Pa ⊃ Qa Px ⊃ Qx Pf(x) ⊃ Qf(x) Pf(a) ⊃ Qf(a) Pf(x, y) ⊃ Qf(x, y) Pf(a, f(x), b) ⊃ Qf(a, f(x), b) Pf(f(g(f(a)))) ⊃ Qf(f(g(f(a)))) Pf(f(g(f(x)))) ⊃ Qf(f(g(f(x))))

Similarly, you can increase functional structure when using UG. All of the inferences at 5.7.9 are legitimate as long as the variables in the formula above the line still have their universal character (i.e., the formula is not within an indented sequence in which x is free in the first line). 5.7.9 Px ⊃ Qx (∀x)[Pf(x) ⊃ Qf(x)] (∀x)[Pf(x, g(y)) ⊃ Qf(x, g(y))] (∀x)[Ph(f(g(x, x))) ⊃ Qh(f(g(x, x)))]

You may not decrease functional structure with the universal rules. For an example of the problem with UG, consider the faulty derivation at 5.7.10 that decreases functional structure when using UG at line 3. 5.7.10 Uh-oh!

1. (∀x)Gf(x) Premise 2. Gf(x) 1, UI 3. (∀x)Gx 2, UG

But wrong!

The problem with 5.7.10 is clear if we interpret ‘Gx’ as ‘x is greater than 0’ and ‘f(x)’as the successor function for natural numbers. If we restrict our domain to the natural numbers including zero, then we have concluded that all natural numbers are greater

5 . 7 : Der i v at i ons w i t h F u nct i ons   3 7 7

than zero from the premise that all successors are greater than zero. But zero is not greater than zero! Decreasing functional structure is also unacceptable for UI. Imagine an interpretation of 5.7.11 that takes ‘Px’ as ‘x is even’ and ‘f(x)’ as ‘twice x’, and imagine again a domain of natural numbers. 5.7.11 (∀x)[Pf(f(x)) ⊃ Pf(x)]

On our interpretation, 5.7.11 says that if four times a number is even, then twice a number is even. That’s true. But if we decrease the functional structure when instantiating, as at 5.7.12, we get a false claim. 5.7.12 Pf(a) ⊃ Pa

5.7.12 says that if twice ‘a’ is even, then ‘a’ is even. If we interpret ‘a’ as any odd number, say 3, 5.7.12 is false even though 5.7.11 is true. So, when using universal instantiation and generalization rules, you can increase functional structure. But never decrease functional structure with the universal rules. Conversely, you may decrease functional structure with existential rules, both EI and EG, but you may never increase functional structure with them. Since existentially quantified sentences are so weak, merely claiming that some object in the domain has a property, we can EG at any point over any singular terms. ‘(∃x)(Px ∙ Qx)’ can be inferred from any of the statements listed above the horizontal line at 5.7.13, decreasing even very complex functional structure. 5.7.13 Pa ∙ Qa Pf(a) ∙ Qf(a) Pf(x) ∙ Qf(x) Pf(a, b, c) ∙ Qf(a, b, c) Pf(f(x), x, f(f(x))) ∙ Qf(f(x), x, f(f(x))) Pf(f(g(f(a)))) ∙ Qf(f(g(f(a)))) Pf(f(g(f(x)))) ∙ Qf(f(g(f(x)))) (∃x)(Px ∙ Qx)

Moreover, with nested functions, you can EG in different ways. All of the propositions below the line at 5.7.14 can also be acceptably inferred from the proposition at the top using EG. 5.7.14 Pf(f(g(f(a)))) ∙ Qf(f(g(f(a)))) (∃x)[Pf(f(g(f(a)))) ∙ Qf(f(g(f(a))))] (∃x)[Pf(f(g(x))) ∙ Qf(f(g(x)))] (∃x)[Pf(f(x)) ∙ Qf(f(x))] (∃x)[Pf(x) ∙ Qf(x)]

Decreasing functional structure using EI is also acceptable. Either inference from the quantified formula at 5.7.15 is acceptable.

3 7 8    C h apter 5  F u ll F i rst - O r d er L og i c

5.7.15 (∃x)[Pf(x) ∙ Qf(f(x))] Pf(a) ∙ Qf(f(a)) Pa ∙ Qf(a)

To see that you may not increase functional structure with existential rules, consider the illegitimate inference at 5.7.16. 5.7.16

1. (∃x)Qx Premise 2. Qf(a) 1, EI: but wrong!

Just interpret ‘Qx’ as ‘x is the last sale of the day’, ‘f(x)’ as the previous sale (in any given day) function, and ‘a’ as some sale. Then, 5.7.16 concludes that some sale that is previous to another is also the last of the day. So, you can’t decrease functional structure with EI. You can’t decrease functional structure with EG, either: a person may have a property without her/his mother or father or grandmother or firstborn child having it too! In brief, you may increase functional structure when using universal rules, but you may not decrease it. You may decrease functional structure when using existential rules, but you may not increase it. There is one last caveat about changing functional structure, which some proofs require. If you are changing the functional structure, make sure to change it uniformly. If you are replacing ‘f(x, g(a))’ with ‘x’ in one place in a wff, you must replace it with ‘x’ everywhere it appears in the wff. And you may not change functional structure for one singular term and not for another in the same instantiated or generalized formula. So, neither of the inferences 5.7.17 nor 5.7.18 is valid. 5.7.17 (∃x)[Ox ∙ Ef(x)] Oa ∙ Ea

No good!

5.7.18 Ox ⊃ ∼Ex (∀x)[Ox ⊃ ∼Ef(x)]

No good!

In 5.7.17, we might be concluding that some number is both odd and even from the premise that odd numbers have even doubles. And in 5.7.18, we might be concluding that no odd numbers have even successors from the claim that no odd numbers are even. Functional structure must be changed uniformly within a wff. So, in cases like (∃x)[Pf(x) ∙ Qh(f(x), g(x))], where you could, in theory, reduce the functional structure when instantiating because the quantifier is existential, the presence of the ‘g(x)’ means that you cannot, say, replace all of the ‘f(x)’s with ‘x’s.

Derivations with Functors Let’s return to argument 5.6.1. We saw at 5.6.6 that the conclusion follows if we regiment the argument using definite descriptions, as here.

5 . 7 : Der i v at i ons w i t h F u nct i ons   3 7 9

5.6.1

All applicants will get a job. Jean is an applicant. Jean is the first child of Dominique and Henri. So, some first child will get a job.

1. (∀x)(Ax ⊃ Gx) 2. Aj 3. (∃x)[Fxdh ∙ (∀y)(Fydh ⊃ y=x) ∙ x=j] / (∃x){(∃y)(∃z)[Fxyz ∙ (∀w)(Fwyz ⊃ w=x)] ∙ Gx}

But, as I said, invoking functions will make the derivation simpler. Let’s use a function ‘f(x, y)’ for ‘the first child of x and y’ to regiment the third premise and conclusion; the result is at 5.7.19. Notice how quickly and easily the derivation follows. 5.7.19 QED

1. (∀x)(Ax ⊃ Gx) 2. Aj 3. j=f(d, h) 4. Aj ⊃ Gj 5. Gj 6. Gf(d, h) 7. (∃y)Gf(d, y) 8. (∃x)(∃y)Gf(x, y)

/ (∃x)(∃y)Gf(x, y) 1, UI 4, 2, MP 5, 3, IDi 6, EG 7, EG

5.7.20 contains a derivation that uses some composition of functions. Note that ‘B’ is a two-place predicate, taking as arguments a variable and a functor term with a variable argument in the first premise, and taking as arguments two functor terms, each with variable arguments, in the conclusion. 5.7.20 QED

1. (∀x)[Ax ⊃ Bxf(x)] 2. (∃x)Af(x) 3. Af(a) 4. Af(a) ⊃ Bf(a)f(f(a)) 5. Bf(a)f(f(a)) 6. (∃x)Bf(x)f(f(x))

/ (∃x)Bf(x)f(f(x)) 2, EI to “a” 1, UI to “f(a)” 4, 3, MP 5, EG

In the short derivation 5.7.21, we instantiate to a two-place function, f(x, g(x)), one of whose arguments is itself a function. Since none of the arguments of any of the functions in 5.7.21 are constants, UG is permissible at line 3. 5.7.21 1. (∀x)∼Cx / (∀x)∼Cf(f(x), x) 2. ∼Cf(x, g(x)) 1, UI 3. (∀x)∼Cf(x, g(x)) 2, UG QED

3 8 0    C h apter 5  F u ll F i rst - O r d er L og i c

5.7.22 derives the conclusion of an argument that uses concepts from number theory in which functions play an important role. 5.7.22 1. If the product of a pair of numbers is odd, then the    product of the successors of those numbers is even. 2. Seven and three are odd numbers. 3. The product of seven and three is odd. So, the product of the successors of seven and three   is even. QED

1. (∀x)(∀y){(Nx ∙ Ny) ⊃ [Of(x, y) ⊃ Ef(f(x), f(y))]} 2. Os ∙ Ns ∙ Ot ∙ Nt 3. Of(s, t) / Ef(f(s), f(t)) 4. (∀y){(Ns ∙ Ny) ⊃ [Of(s, y) ⊃ Ef(f(s), f(y))]} 1, UI 5. (Ns ∙ Nt) ⊃ [Of(s, t) ⊃ Ef(f(s), f(t))] 4, UI 6. Ns ∙ Nt 2, Simp 7. Of(s, t) ⊃ Ef(f(s), f(t)) 5, 6, MP 8. Ef(f(s), f(t)) 7, 3, MP

Summary The derivation system we use with FF is basically the same as the one we use with F; you mainly have to be careful to obey the guidelines about altering functional structure. We have come to the end of our main technical work. Still, there are many further logical languages and systems, discussions of some of which are available as supplements to this book. KEEP IN MIND

The restrictions on instantiation and generalization rules for constants and variables are the same whether the singular terms are simple or complex. A complex singular term acts like a variable if there are any variables in any of its argument places, or those of any of its embedded functions. Otherwise, it acts like a constant. You may increase functional structure when using universal rules (UI or UG), but you may not decrease it. You may decrease functional structure when using existential rules (EI or EG), but you may not increase it. If you change the functional structure of a wff, you must change it uniformly throughout.

TELL ME MORE  • Are there more rules and different kinds of derivations? See 6.5: Modal Logics, 6S.11: Axiomatic Systems, and 6S.12: Rules of Passage.

5 . 7 : Der i v at i ons w i t h F u nct i ons   3 8 1

EXERCISES 5.7a Derive the conclusions of each of the following arguments. 1. 1. (∀x)[Ax ⊃ Af (x)] 2. Aa 3. f (a)=b / Ab 2. 1. (∀x)[Bx ≡ Bg(x)] 2. (∀x)g(x)=f (x, x) 3. Ba / Bf(a,a) 3. 1. (∀x)[Px ≡ Pf (x)] 2. f (a)=f (b) 3. Pa / Pb 4. 1. (∀x)[Px ⊃ Pf (x)] 2. (∀x)(Qx ⊃ Px) 3. Qa / Pf (a) 5. 1. (∀x)(∀y)(∀z)[f(x, z)=y ⊃ f(y, z)=x] 2. f (a, b)=c 3. Pc ∙ Pa / (∃x)[Pf (a, x) ∙ Pf (c, x)] 6. 1. (∀x)Hf(x) 2. a=f (b) ∙ b=f (c) 3. (∀x)(Hx ⊃ ∼Ix) / a=f (f (c)) ∙ ∼Ia 7. 1. (∀x)(∀y)f (x, y)=f (y, x) 2. a=f (b, c) 3. b=f (c, a) 4. a≠b 5. Pa ∙ Pb / (∃x)(∃y)(∃z)[Pf (x, z) ∙ Pf (y, z) ∙ x≠y] 8. 1. (∀x)(∀y)[f (x)=f (y) ⊃ x=y] 2. f (a)=g(c, d) 3. f (b)=g(c, e) 4. d=e / a=b 9. 1. (∀x)[Pf (x) ⊃ (Qx ≡ Rx)] 2. Pa ∙ Qf(a) 3. (∀x)f (f (x))=x / Rf(f(f(a))) 10. 1. Pa ∙ (∀x)[(Px ∙ x≠a) ⊃ Qax] 2. (∀x)(∀y)[x=f (y) ⊃ x≠y] 3. Pb ∙ b=f(a) / Qab

3 8 2    C h apter 5  F u ll F i rst - O r d er L og i c

11. 1. f(a, b)=c 2. (∀x)[(∃y)f(a, x)=y ⊃ Px] 3. (∀x){Px ⊃ [Qc ∙ g(x)=c]}

/ (∃x){Px ∙ (∃y)[Q y ∙ g(x)=y]}

12. 1. (∀x)[Bf(x) ⊃ (Cx ∙ Df(f(x)))] 2. (∃x)Bf(f(x)) 3. (∃x)Cf(x) ⊃ (∀x)Ex

/ (∃x)[Df(f(f(x))) ∙ Ef(f(f(x)))]

13. 1. (∀x)(∀y)[(Fx ∙ Fy) ⊃ Gf(x, y)] 2. (∀x)(∀y)[Gf(x, y) ≡ Gf(x, x)] 3. (∀x)[Gx ⊃ Gf(x)] 4. Fa ∙ Fb

/ Gf(f(a, a))

14. 1. (∀x)(∀y){Pf(x, y) ⊃ [(Px ∙ Py) ∨ (Qx ∙ Q y)]} 2. (∀x)Pf(x, f(f(x))) 3. (∃x)∼Qx / (∃x)Pf(f(a)) 15. 1. Pa ∙ (∀x)[f(x)=a ⊃ Px] 2. (∀x)(∀y)[(Qxb ∙ Q yb) ⊃ x=y] 3. f(b)=c 4. Qab ∙ Qcb

/ Pb

16. 1. (∃x){Px ∙ Qx ∙ (∃y)[Py ∙ Q y ∙ x≠y ∙ Pf(x)f(y)]} 2. (∀x)(∀y)[Pxy ⊃ (Rx ∙ Ry)] 3. (∀x)[Rf(x) ⊃ Rx] / (∃x){Rx ∙ Rf(x) ∙ (∃y)[Ry ∙ Rf(y) ∙ x≠y]} 17. 1. (∀x){[Px ∙ (∃y)(Py ∙ Fxy)] ⊃ (∀z)[(Pz ∙ Fxz) ⊃ z=x]} 2. (∀x)[Px ⊃ Pf(x)] 3. Pa ∙ Faf(a) / ∼(∃x)[Px ∙ Fax ∙ x≠f(a)] 18. 1. (∀x)(∀y)[(Pxy ∙ Qxy) ⊃ ∼f(x)=y] 2. (∀x)(∀y)[Qxy ≡ Qxf(y)] 3. f(a)=b ∙ f(b)=a 4. Pab

/ ∼Qaa

19. 1. (∀x)(∀y){Qf(x, y) ⊃ [(Px ∙ Q y) ∨ (Py ∙ Qx)]} 2. (∀x)[Px ⊃ Qf(x)] 3. (∀x)Qf(x, f(x)) 4. ∼Pa / Qa ∙ Pf(a) 20. 1. (∃x)(∃y){Px ∙ Qx ∙ Py ∙ Q y ∙ x≠y ∙ f(x)=y ∙ (∀z)(∀w){[Pz ∙ Qz ∙ Pw ∙   Qw ∙ z≠w ∙ f(z)=w] ⊃ (z=x ∙ w=y)}} 2. Pa ∙ Qa ∙ Pb ∙ Qb ∙ a≠b ∙ f(a)=b ∙ Sa ∙ Sb 3. Pc ∙ Pd ∙ f(c)=d ∙ ∼Sc ∙ ∼Sd ∙ c≠d / ∼(Qc ∙ Qd)

5 . 7 : Der i v at i ons w i t h F u nct i ons   3 8 3

EXERCISES 5.7b Translate each of the following arguments into FF. Then, derive the conclusion using our rules of inference. 1. If something is your father, then you are its child. Pavel has no children. So, Pavel is not the father of Andres. (a: Andres; p: Pavel; f(x): the father of x; Cxy: x is the child of y) 2. No number is equal to its successor. One and two are numbers, and two is the successor of one. So, one is not two. (a: one; b: two; f(x): the successor of x; Nx: x is a number) 3. The brother of Amanda and Amanda are children of  Nancy. Peter’s mother is a woman named Nancy. Something is your mother if, and only if, it is a woman and you are her child. So, Amanda and Peter share a mother. (a: Amanda; n: Nancy; p: Peter; f(x): the mother of x; Wx: x is a woman; Bxy: x is a brother of Amanda; Cxy: x is a child of y) 4. Anyone is happy on any day if, and only if, they are unhappy on the following day. Joyce is a person who will be happy in three days. Today is a day, and the day after any day is a day. So, Joyce won’t be happy in two days. (j: Joyce; t: today; f(x): the day after x; Dx: x is a day; Px: x is a person; Hxy: x is happy on day y) 5. Anyone who completes a task is proud on the following day. Friday is the day that the person Emma completed the task of her logic homework. Saturday is the day after Friday. So, Emma is proud on Saturday. (a: Friday; b: Saturday; e: Emma; l: Emma’s logic homework; f(x): the day after x; Dx: x is a day; Px: x is a person; Tx: x is a task; Cxyz: x completes y on day z) 6. The product of two and any odd is even. The sum of two and any odd is odd. Seven is odd. So, the product of two and the sum of two and seven is even. (a: two; b: seven; f(x, y): the product of x and y; g(x, y): the sum of x and y; Ex: x is even; Ox: x is odd) 7. One, two, and four are distinct numbers. The positive square root of four is two. Two is the sum of one and itself. So, the positive square root of some number is the sum of some other number and itself. (a: one; b: two; c: four; f(x): the positive square root of x; f(x, y): the sum of x and y; Nx: x is a number) 8. One, two, and four are distinct numbers. The sum of two and the sum of one and one is four. Two is the sum of one and itself. So, some number is the sum of the sum of some other number with itself and the sum of the latter number with itself again. (a: one; b: two; c: four; f(x, y): the sum of x and y; Nx: x is a number)

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9. Exactly one number is the sum of itself and itself. Zero is the sum of itself and itself. The number one is the successor of the number zero, and no number is its own successor. So, one is not the sum of itself and itself. (a: zero; b: one; f(x): the successor of x; f(x, y): the sum of x and y; Nx: x is a number) 10. Exactly two numbers are the products of themselves with themselves. The product of a number and itself is its square. The square of zero is zero. The square of one is one. Zero, one, and five are distinct numbers. So, the square of five is not five. (a: zero; b: one; c: five; f(x): the square of x; g(x, y): the product of x and y; Nx: x is a number)

KEY TERMS argument, 5.6 atomic formula, 5.2 definite description, 5.4 dyadic predicates, 5.1 functional structure, 5.7 functor, 5.6 functor term, 5.6 IDi, 5.5 IDr, 5.5

IDs, 5.5 n-tuple, 5.2 n-tuple of singular terms, 5.6 narrow scope of a quantifier, 5.1 polyadic predicates, 5.1 relational predicates, 5.1 triadic predicates, 5.1 wide scope of a quantifier, 5.1

Chapter 6 Beyond Basic Logic

6.1: NOTES ON TRANSLATION WITH PL In section 2.1, I presented some useful general guidelines for regimenting English sentences into PL. But since that discussion came before our exploration of the basic truth tables, some of the guidelines appeared dogmatic. Here, I discuss a few of the subtleties that underlie the guidelines of the early section on translation: the extensionality of PL, the importance of logical equivalence in translation, and the difference between inclusive and exclusive disjunction. I devote the separate next section to the oddities of the material conditional.

Logical Equivalence and Translation In general, our logic is more fine-grained than natural language. We can use it to make careful distinctions, ones that are trickier to make in English. As Frege observed, using formal logic is like looking at ordinary language through a microscope. It allows us to be precise. Still, every logical language has its limits. One limit of our logic is its extensionality. Extensionality is to be contrasted with intensionality (not intentionality). Roughly, intensions are meanings. Two phrases are intensionally equivalent when they have the same meaning. Two phrases are extensionally equivalent when they have the same truth value (for sentence-sized phrases) or are true of the same things (for smaller linguistic units). Consider, for example, the difference between the subsentential phrases 6.1.1 and 6.1.2. 6.1.1 6.1.2

Creature with a heart Creature with kidneys

As a matter of biology, creatures have hearts if, and only if, they have kidneys. So, 6.1.1 and 6.1.2 pick out the same creatures; they have the same referents. But the two phrases have different meanings. They are extensionally equivalent, but intensionally different. To see the difference between intension and extension at the level of sentences, remember that we provide the semantics for PL by giving truth conditions, using truth 385

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tables. As long as the truth conditions for two sentences are the same, we call the propositions logically equivalent. Our truth-functional logic does not distinguish between two logically equivalent propositions. Thus, our logic is extensional. Because our logic is extensional, sentences with different intensions, like 6.1.3 and 6.1.4, may be translated identically. 6.1.3 6.1.4

Quine is an extensionalist and Frege is not. It is not the case that either Quine is not an extensionalist or   Frege is.

To see that 6.1.3 and 6.1.4 are extensionally equivalent though intensionally distinct, let’s regiment them and look at their truth tables. 6.1.3r Q ∙ ∼F 6.1.4r ∼(∼Q ∨ F) Q

∙

∼

F

∼

(∼

Q

∨

F)

1

0

0

1

0

0

1

1

1

1

1

1

0

1

0

1

0

0

0

0

0

1

0

1

0

1

1

0

0

1

0

0

1

0

1

0

Since the two propositions have the same values for the main operator in their truth tables, despite whatever differences they might have in meaning, 6.1.3 and 6.1.4 are logically equivalent. As far as our truth-functional logic is concerned, we can use these two propositions interchangeably. They have the same entailments. They are consistent or inconsistent with the same propositions. The notion of an intension, or a meaning, like the concept of a proposition, is controversial. To help clarify or illustrate the concept of an intension, some philosophers and logicians have explored, fruitfully, possible worlds and their corresponding modal logics. In contrast, the concept of logical equivalence is the central concept in the characterization of logic as extensional. We can use it, for example, to help us understand the biconditional.

The Material Conditional and the Biconditional We use biconditionals to represent ‘if and only if ’ statements. By comparing the biconditional to the conjunction of two conditionals, as at 6.1.5, we can understand the relation between the biconditional and the material conditional.

6 . 1 : N otes on T ranslat i on w i t h P L    3 8 7

α ≡ β (α ⊃ β) ∙ (β ⊃ α)

6.1.5 α

≡

β

(α

⊃

β)

∙

(β

⊃

α)

1

1

1

1

1

1

1

1

1

1

1

0

0

1

0

0

0

0

1

1

0

0

1

0

1

1

0

1

0

0

0

1

0

0

1

0

1

0

1

0

Notice that claims of each form are logically equivalent. And the expression on the right is just the conjunction of ‘α if β’, on the right of the conjunction, and ‘α only if β’, on the left: α if, and only if, β.

Inclusive and Exclusive Disjunction In section 2.3, we adopted the inclusive disjunction as the semantics for ∨, despite some concerns that ‘or’ has both an inclusive and an exclusive sense. Decisions are often framed with an exclusive ‘or’: Will you take the Thursday lab or the Tuesday lab? Will you have the soup or the salad? Let’s use ⊕ as the symbol for exclusive disjunction (though we will use it only in this section). 6.1.6 thus shows the truth tables for inclusive and exclusive ‘or’. 6.1.6

Inclusive ‘or’

Exclusive ‘or’

α

∨

β

α

⊕

β

1

1

1

1

0

1

1

1

0

1

1

0

0

1

1

0

1

1

0

0

0

0

0

0

Using the concept of logical equivalence, we can show that ⊕ is definable in terms of  ∨, and thus that we do not need a special symbol for exclusive disjunction. We just need to provide a formula that yields the same truth table as ⊕, but which does not use that term. Such a truth table is at 6.1.7.

3 8 8    C h apter 6  Be y on d Bas i c L og i c

6.1.7 (α ∨ β) ∙ ∼(α ∙ β) (α

∨

β)

∙

∼

(α

∙

β)

1

1

1

0

0

1

1

1

1

1

0

1

1

1

0

0

0

1

1

1

1

0

0

1

0

0

0

0

1

0

0

0

Thus we can see that if we want to regiment a sentence of English as an exclusive ‘or’, we can just use the conjunction of ‘α ∨ β’ with ‘∼(α ∙ β)’, which, if you think about it, should strike you as sensible: you’ll take either the Thursday lab or the Tuesday lab, but not both. A good grasp of logical equivalence allows us also to clear up a related question about translation, about the use of disjunction for ‘unless’.

‘Unless’ and Exclusive Disjunction We ordinarily translate ‘unless’ using a ∨. Let’s see why. Consider the sentence 6.1.8 and think about what we want as the truth values of ‘unless’ in that sentence. 6.1.8

The car will not run unless there is gas in its tank.

We’ll start by translating the ‘unless’ as a ∨, and constructing a standard truth table for the proposition, at 6.1.9 6.1.9 ∼

R

∨

G

0

1

1

1

0

1

0

0

1

0

1

1

1

0

1

0

Now, let’s think about what we want as the truth values for the proposition expressed by 6.1.8.

6 . 1 : N otes on T ranslat i on w i t h P L    3 8 9

The car r uns.

The car will not run unless it has gas.

The car has gas.

1

1

1

0

0

1

0

0

In the first row, the car runs and has gas, so the complex proposition 6.1.8 should be true. In the second row, the car runs but does not have gas. In this case, perhaps the car runs on an alternative fuel source, or magic. The proposition 6.1.8 should thus be false in the second row. In the third row, the car does not run but has gas. Perhaps the car is missing its engine. This case does not falsify the complex proposition, which does not say what else the car needs to run. 6.1.8 gives a necessary condition for a car to run (having gas), but not sufficient conditions. Thus 6.1.8 should be considered true in the third row. In the fourth row, the car does not run and does not have gas. The proposition thus should be true in the fourth row. Considering our desired truth values for the sentence, we get a truth table for ‘unless’, at 6.1.10 6.1.10 The car r uns.

The car will not run unless it has gas.

The car has gas.

1

1

1

1

0

0

0

1

1

0

1

0

Notice that the truth table for ‘unless’, at 6.1.10, is precisely the same as the truth table for the ∨, at 6.1.9. Since the two truth tables are the same, we can use the ∨ to stand for ‘unless’; it gives us precisely what we want. Unfortunately, this felicitous result does not hold for all uses of ‘unless’. Let’s analyze 6.1.11 the same way we analyzed 6.1.8. 6.1.11

Liesse will attend school full time unless she gets a job.

Liesse at tends school.

Liesse will at tend school full time unless she gets a job.

Liesse gets a job.

1

1

1

0

0

1

0

0

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This time, we will work from the bottom up. In the fourth row, Liesse does not get a job but doesn’t go to school. The complex proposition is false, since it says that she will attend school unless she gets a job. In the third row, she gets a job and doesn’t go to school, and so the proposition should be true. In the second row, she attends school but doesn’t get a job, and so the proposition should be true. In the first row, Liesse gets a job but attends school anyway. What are your intuitions about the truth value of LS in this case? In my experience, most people who have not studied formal logic take 6.1.11 to be false in the first row. It’s clear that if the proposition is true and Liesse does not get a job, then she will attend school. Many people also believe that if the complex proposition is true and Liesse does get a job, then she will not attend school. Here, ‘unless’ is taken in what is sometimes called a stronger sense. In this case, the truth table for 6.1.11 should be 6.1.12. 6.1.12 Liesse at tends school.

Liesse will at tend school full time unless she gets a job.

Liesse gets a job.

1

0

1

1

1

0

0

1

1

0

0

0

The truth table for ‘unless’ as used in 6.1.11 seems to have the same truth conditions as exclusive disjunction, not for ∨. Unless thus appears to be ambiguous in the same way as ‘or’: there is an inclusive and exclusive ‘unless’. To regiment 6.1.11, then, it would be natural to use the form of 6.1.7, the exclusive disjunction, yielding 6.1.13. 6.1.13 (S

∨

J)

∙

∼

(S

∙

J)

1

1

1

0

0

1

1

1

1

1

0

1

1

1

0

0

0

1

1

1

1

0

0

1

0

0

0

0

1

0

0

0

There are even simpler ways of representing exclusive disjunctions. Notice that we understand 6.1.11 really as a biconditional: Liesse attends school if she does not

6 . 1 : N otes on T ranslat i on w i t h P L    3 9 1

get a job, and if she attends school she does not get a job. Thus we can use either ‘∼S ≡ J’ or ‘∼(S ≡ J)’, as we see at 6.1.14, since they are logically equivalent to 6.1.13 (and shorter too!). 6.1.14 ∼

S

≡

J

∼

(S

≡

J)

0

1

0

1

0

1

1

1

0

1

1

0

1

1

0

0

1

0

1

1

1

0

0

1

1

0

0

0

0

0

1

0

In other words, if you have a sentence that you wish to regiment as an exclusive disjunction, you can use a proposition of any of the forms: ∼α ≡ β, ∼(α ≡ β); or (α ∨ β) ∙ ∼(α ∙ β); or any alternative form that is logically equivalent to it. When faced with an unless, then, we ordinarily just take it to be a ∨. But if we are concerned about getting the truth conditions precisely correct, then we have to decide whether the sentence functions more like 6.1.8, and so deserves the inclusive disjunction, or more like 6.1.11, in which case we should write it with one of the acceptable forms for exclusive disjunction. Nothing in our logic can tell you which truth conditions you want in a translation. That is a matter of interpretation.

Summary The extensionality of our logic means that our main concern in translation is getting the truth conditions of our propositions right. There are always different, but logically equivalent, ways of regimenting a sentence of English into PL. The concept of logical equivalence is thus central to our work in translation. Generally, we seek the simplest translations. But the concept of simplicity is not clear and categorical. Using ⊕ for exclusive disjunction, for example, makes our language more complicated. But ‘P ⊕ Q’ is a shorter, and thus simpler, way of expressing ‘∼(P ≡ Q  )’ or ‘(P ∨ Q  ) ∙ ∼(P ∙ Q  )’. This tension in the notion of simplicity becomes more apparent as we think more about how many logical operators we really need to express the concepts and entailments of propositional logic. Perhaps the most pressing other questions about translation using PL surround the ⊃, the subject of the next section on conditionals.

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TELL ME MORE  • How is the extensionality of our logic contrasted with the intensionality of propositions? See 6S.7: The Propositions of Propositional Logic. • How can we use logical equivalence to show that some logical operators are unnecessary? See 6S.8: Adequacy. • How can logic be treated axiomatically? See 6S.11: Axiomatic Systems.

For Further Research and Writing 1. What is the difference between an intension and an extension? How does this difference underlie our work in logic? 2. Is the natural language ‘or’ inclusive or exclusive? Provide examples. 3. Are there alternatives to ∨ for translating ‘unless’ into PL? Provide examples, distinguishing between those that are logically equivalent to a straight ∨ and those that are not.

Suggested Readings Fitting, Melvin. “Intensional Logic.” In The Stanford Encyclopedia of Philosophy. http://plato .stanford.edu/archives/sum2015/entries/logic-intensional/.  Accessed January 25, 2016. Traces the history of intensional logics and presents some details of various approaches. Hurford, James. “Exclusive or Inclusive Disjunction.” Foundations of Language 11 (1974): 409–411. Hurford argues that some uses of ‘or’ are exclusive. Orlandini, Anna. “Logical, Semantic and Cultural Paradoxes.” Argumentation 17 (2003): 65–86. Orlandini connects the exclusive disjunction to some paradoxes. Sainsbury, Mark. Logical Forms: An Introduction to Philosophical Logic, 2nd ed. Oxford, UK: Blackwell, 2001. Chapter 2 has a lovely and engaging discussion of many aspects of translation with propositional logic.

6.2: CONDITIONALS The Material Interpretation of the Natural-Language Conditional There are lots of different kinds of conditionals in natural language. Among them are 6.2.1–6.2.6. 6.2.1 Indicative conditionals If the Mets lost, then the Cubs won. 6.2.2 Conditional questions If I like logic, what class should I   take next? 6.2.3 Conditional commands If you want to pass this class, do the   homework. 6.2.4 Conditional prescriptions If you want a good life, you ought to   act virtuously.

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6.2.5 Cookie conditionals If you want cookies, there are some    in the jar. 6.2.6 Subjunctive conditionals If Rod were offered the bribe, he    would take it.

As we saw in chapter 2, we use the material conditional and its standard truth table for indicative conditionals. The material conditional is true unless the antecedent is true and the consequent is false. α

⊃

β

1

1

1

1

0

0

0

1

1

0

1

0

Since the truth table for the material conditional is false only in the second line, we can think of it as saying that ‘If α then β’ is equivalent to ‘Not (α and not-β)’. An understanding of some other kinds of conditionals seems to depend on our analysis of indicative conditionals. 6.2.2 and 6.2.3 are not propositions as they stand: they lack truth values. 6.2.4 might have a truth value, though this is a controversial point. We can easily parse them all truth-functionally by turning them into indicatives. 6.2.2′ 6.2.3′ 6.2.4′

If you like logic, then you take linear algebra next. If you want to pass the class, you do the homework. If you want a good life, you act virtuously.

We can thus regiment 6.2.2′–6.2.4′ as material conditionals, just as we did for 6.2.1. Analyses of cookie conditionals like 6.2.5 as indicatives are moot. (Such sentences are sometimes called biscuit conditionals, because biscuits are cookies in England and elsewhere.) Such statements are often seen as frauds masquerading as conditionals. If 6.2.5 is true as intended, then there are cookies in the jar whether or not you want any. The speaker of 6.2.5 is likely to mean something like “You may take one of the cookies in the jar” or “If you would like a cookie, you may have a cookie.” The former is not a conditional; the latter is easily parsed as an indicative. 6.2.6 is problematic, and it is the subject of much of the remainder of this section. While we can interpret it as an indicative, material conditional, doing so leads to some problems, as we will see. The material conditional is probably the best truthfunctional option for representing the conditional as it appears in English and other natural languages. But, the natural-language conditional is more complex than the material interpretation. Thinking about a proper treatment of conditionals quickly

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leads to important questions regarding the nature of scientific laws, and the ways in which they are confirmed or disconfirmed. Indeed, discussion of the proper treatment of conditionals is a central topic in the philosophy of science. Recent work in the logic of conditionals has also led to sophisticated modal extensions of classical logic, called conditional logics. Conditional logics are beyond the scope of this text. Here, we will discuss a few of the subtleties of conditionals, especially subjunctive conditionals, and the challenges facing those who pursue their proper logical treatment. We’ll start with the so-called paradoxes of material implication.

Logical Truths and the Paradoxes of Material Implication The material interpretation of the conditional creates what are called the paradoxes of material implication, for the classical bivalent logic of this book. To understand the paradoxes of material implication, one has first to understand the nature and importance of logical truth. A logical theory, like any theory, is identified by its theorems. The theorems of a logical theory are called logical truths. 6.2.7 and 6.2.8 are two intuitive logical truths. 6.2.7 P ⊃ P 6.2.8 [(P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)]

6.2.7 and 6.2.8 have a natural obviousness that properly characterizes a theorem of logic, which is supposed to be the most obvious of disciplines. Many other tautologies are also obvious. Among the paradoxes of material implication are statements of forms 6.2.9, 6.2.10, and 6.2.11. We sometimes call them, too strongly, paradoxes. A paradox leads to a contradiction; the paradoxes of material implication are merely non-obvious or contentious logical truths. 6.2.9 α ⊃ (β ⊃ α) 6.2.10 ∼α ⊃ (α ⊃ β) 6.2.11 (α ⊃ β) ∨ (β ⊃ α)

The problems that arise from the so-called paradoxes of material implication appear once we start to interpret them. 6.2.9 says, approximately, that if a statement is true, then any other statement implies it. Remember, the truth table for the material conditional is true on every line in which the consequent is true. So, 6.2.12 is true on the material interpretation. 6.2.12

If Martians have infrared vision, then Barack Obama was president    of the United States in 2015.

6.2.10 says that if a statement is false, its opposite entails any other statement. So 6.2.13 is true on the material interpretation: since the antecedent is false, even an absurd consequent follows. 6.2.13

If George W. Bush was president of the United States in 2015, then    Venusians have a colony on the dark side of Mercury.

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Note that this principle underlies explosion, the characteristic of propositional logic that any statement follows from a contradiction and which we discussed in section 3.5. Since a contradiction is always false, anything follows from an argument that contains a contradiction. Lastly, 6.2.11 says that for any statement, β, any other statement either entails it or is entailed by it. In classical bivalent logic, every statement is either true or false. If a given statement is true, then, as in 6.2.9, any statement entails it. If a given statement is false, then, as in 6.2.10, it entails any statement. So 6.2.14 is true, according to the material interpretation of the conditional. 6.2.14

Either ‘Neptunians love to wassail’ entails ‘Saturnians love to    foxtrot’ or ‘Saturnians love to foxtrot’ entails ‘Neptunians    love to wassail’.

Indeed, 6.2.14 is not only true, but a law of logic. Further, either 6.2.15 or 6.2.16 is true. 6.2.15 6.2.16

‘It is raining’ entails ‘Chickens are robot spies from Pluto’. ‘Chickens are robot spies from Pluto’ entails ‘It is raining’.

If it is raining, then 6.2.16 is true, whatever the case is about chickens. If it is not raining, then 6.2.15 is true. According to a further law of logic, called excluded middle, either it is raining or it is not raining. So, one or the other of 6.2.15 and 6.2.16 must be true even though they are both absurd sentences. In sum, the paradoxes of the material conditional are two kinds of awkward results. First, statements of the form of 6.2.9–6.2.11, like 6.2.12–6.2.14, are laws of logic that are not obviously true. Second, some statements like either 6.2.15 or 6.2.16 are true, given the truth values of their component propositions, even though we do not intuitively see them as true.

Dependent and Independent Conditionals The paradoxes of material implication show that there is something funny going on with the ⊃. To begin to diagnose the problems with the material conditional, let’s distinguish between dependent and independent indicative conditional statements. A dependent conditional has a connection between its antecedent and consequent. I’ll leave exactly what I mean by a connection unstated, here, but 6.2.17–6.2.20 are examples of dependent conditionals. 6.2.17 6.2.18 6.2.19 6.2.20

If it is raining, then I will get wet. If I run a red light, then I break the law. If the car is running, then it has fuel in the tank. If I were to jump out of the window right now, I would fall to the   ground.

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Notice that the material interpretation seems acceptable for dependent conditionals like 6.2.17–6.2.20. Even when the antecedents are false, the connections between the antecedents and consequents expressed by the conditionals hold. Recall the example from section 2.3. 2.3.10

If you paint my house, then I will give you five thousand dollars.

If the antecedent of 2.3.10 is not true, if you do not paint my house, we can take the conditional to be true as a standing offer. A similar analysis holds for all of 6.2.17–6.2.20. The last two lines of the truth table are a little awkward, but, as I will show below, the material analysis is the only plausible truth-functional analysis of conditionals. In contrast, let’s consider some independent conditionals, ones that lack the connections we find in dependent conditionals like 6.2.17–6.2.20. The awkward instances of the paradoxes of the material conditional, above, are independent conditionals. So are 6.2.21–6.2.24. 6.2.21 6.2.22 6.2.23 6.2.24

If 2+2=4, then cats are animals. If 2+2=4, then cats are robots. If pigs fly, then Chicago is near Lake Michigan. If pigs fly, then Chicago is the capital of Canada.

Since ‘2+2=4’ is true and ‘cats are animals’ is true, 6.2.21 is true. Since ‘2+2=4’ is true and ‘cats are robots’ is false, 6.2.22 is false. Since ‘pigs fly’ is false, 6.2.23 and 6.2.24 are both true. All of these results seem counterintuitive. The paradoxes of material implication are awkward because they hold for any values of the propositional variables, whether the relation is dependent or independent. The oddities of the so-called paradoxes of the material conditional are at least partly related to the oddities of any independent conditionals. What are the truth values of such claims supposed to be anyway? The material conditional, as the rule of inference called material implication says, is just the disjunction of the denial of the antecedent with the consequent. The truth values of 6.2.21′–6.2.24′ seem less awkward because we are more familiar with independent disjunctions. 6.2.21′ 6.2.22′ 6.2.23′ 6.2.24′

Either 2+2≠4 or cats are animals. Either 2+2≠4 or cats are robots. Either pigs do not fly or Chicago is near Lake Michigan. Either pigs do not fly or Chicago is the capital of Canada.

True False True True

Thus, the oddity of the conditionals 6.2.21–6.2.24 is mitigated by remembering that ‘α ⊃ β’ is just another way of writing ‘∼α ∨ β’. We might accept the material analysis of the conditional merely for the benefits it yields to our analysis of dependent conditionals. The material interpretation returns a truth value for any conditional combination of propositions. It allows us to maintain the truth functionality of our logic: the truth value of any complex sentence is completely dependent on the truth value of its component parts. The dependent conditional is much more common than

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the independent one anyway, and most people don’t have strong feelings about the truth values of sentences like 6.2.21–6.2.24. The paradoxes of material implication, accepting counterintuitive claims as logical truths, may thus just be seen as the price we have to pay to maintain truth-functionality. One response to the paradoxes of material implication that you might be considering is to find a different truth table for the conditional. It’s worth a moment to see that this response is not promising. I will do so in two stages, first looking at the first two rows of the truth table for the material conditional and then at the second two rows.

Nicod’s Criterion and the First Two Rows of the Truth Table The first two lines of the truth table for the material conditional are mostly uncontroversial, especially in dependent conditionals like 6.2.17–6.2.20. Indeed, they represent what is known as Nicod’s criterion for confirmation of a scientific claim. Many scientific laws are conditional in form. Nicod’s criterion says that evidence will confirm a law if it satisfies both the antecedent and consequent of such a law. It also says that evidence will disconfirm a law if it satisfies the antecedent but fails to satisfy the consequent. Let’s take a sample law, Coulomb’s law, which says that the force on two particles is proportional to the absolute value of the product of the charges on each particle (q1 and q2) divided by the square of the distance between them (r). CL F=k |q1q2|/ r2.

We may analyze CL as a claim that if two particles have a certain amount of charge and a certain distance between them, then they have a certain, calculable force between them. We take evidence to confirm the law if it satisfies the antecedent and the consequent of that conditional. We take evidence to disconfirm the law if it were to satisfy the antecedent and falsify the consequent. If we were to find two particles that did not have the force between them that the formula on the right side of Coulomb’s law says should hold, and we could not find overriding laws to explain this discrepancy, we would seek a revision of Coulomb’s law. To take a simpler example, consider the claim that all swans are white. We may analyze that claim as ‘if something is a swan, then it is white’. When we find a white swan, which satisfies the antecedent and the consequent, it confirms the claim. If we were to find a black swan, which satisfies the antecedent but not the consequent, then it would falsify the claim. According to Nicod’s criterion, instances that do not satisfy the antecedent are irrelevant to confirmation or disconfirmation. A white dog and a black dog and a blue pen have no effect on our confidence in the claim that all swans are white. Call a conditional in which the antecedent is false a counterfactual conditional. Nicod’s criterion thus says nothing about counterfactual conditionals. We are considering alternatives to the material interpretation of the conditional. The point of mentioning Nicod’s criterion is to say that we should leave the first two lines of the truth table alone.

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The Immutability of the Last Two Rows of the Truth Table for the Material Conditional Given the first two rows, there are three possibilities for the third and fourth lines of the truth table for the conditional that are different from the material interpretation.

Option A

Option B

Option C

α

⊃

β

α

⊃

β

α

⊃

β

1

1

1

1

1

1

1

1

1

1

0

0

1

0

0

1

0

0

0

1

1

0

0

1

0

0

1

0

0

0

0

1

0

0

0

0

Option A gives the conditional the same truth values as the consequent. It thus makes the antecedent irrelevant. Option B gives the conditional the same truth values as a biconditional. Option C gives the conditional the same truth values as the conjunction. Thus, the truth table for the material conditional is the only one possible with those first two lines that doesn’t merely replicate a truth table we already have by other means. The conditional seems to have a different role in natural language from the biconditional, the conjunction, or a restatement of the consequent (except, in the latter case, for cookie conditionals). Thus, the material interpretation seems to be the best truth-functional option. To see the problem more intuitively, consider again a good counterfactual dependent conditional like 6.2.25, which we’ll call S. 6.2.25

If I were to jump out of the window right now, I would fall to the   ground.

Option A says that S is falsified when I don’t jump out the window and I don’t fall to the ground. Options B and C say that S is falsified when I don’t jump out of the window and I do fall to the ground. But neither case seems to falsify S as it is intended. The only time that S is falsified, as on Nicod’s criterion, is in the second line of the truth table, when I jump out of the window and hang in the air, like the woman pictured here. It looks like we have to stick with the original truth table if we want the conditional to be truth-functional.

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Subjunctive and Counterfactual Conditionals We have been considering whether and to what extent the material conditional stands for the natural-language indicative conditional. Our worries about independent conditionals were allayed by three considerations. First, we don’t have strong feelings about the truth values of many of those sentences, like 6.2.21–6.2.24, which seem rare and deviant. Second, the truth values of equivalent statements written with disjunctions seem plausible. Third, our desire to maintain truth-functionality entails that we have to give all complex propositions truth values on the basis of the truth values of their component sentences and the material conditional is the only option that respects our strong feelings about the first two rows of the table. The first two rows of the truth table for the material conditional capture our intuitions about dependent conditionals as well. Despite worries about the paradoxes of the material implication and the oddities of material interpretations of independent conditionals, the material interpretation seems to be the best way to maintain truth-functionality. Unfortunately, more problems beset the material conditional. Subjunctive conditionals, especially in their counterfactual interpretations, raise further problems. Consider again 6.2.6. If its antecedent is true, we know how to evaluate the sentence: if Rod takes the bribe, then 6.2.6 is true; if he refuses the bribe, then it is false. According to the material interpretation of the conditional, if Rod is never offered the bribe, then 6.2.6 is true. So far so good. But there are cases in which a conditional with a false antecedent should be taken as false. Compare 6.2.26, which is a subjunctive conditional similar to 6.2.6, with 6.2.27. 6.2.26 6.2.27

If I were to jump out of the window right now, I would fall to the   ground. If I were to jump out of the window right now, I would flutter to   the moon.

According to the material interpretation, 6.2.26 and 6.2.27 are true, since I am not jumping out of the window. But, we ordinarily take 6.2.26 to be true and 6.2.27 to be false. The difference between 6.2.26 and 6.2.27, and its inconsistency with the material interpretation of the conditional, has come to be known as the problem of counterfactual conditionals. Nelson Goodman, in his famous paper on the problem, contrasts 6.2.28 with 6.2.29. 6.2.28 6.2.29

If that piece of butter had been heated to 150°F, it would have   melted. If that piece of butter had been heated to 150°F, it would not have   melted.

Let’s imagine that we never heat that piece of butter, so that 6.2.28 and 6.2.29 both contain false antecedents. According to the material interpretation, since the

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antecedents of 6.2.28 and 6.2.29 are both false, or counterfactual, both sentences come out as true. But it seems that they should be taken as contraries. They can’t both be true. Indeed, just as we want to call 6.2.26 true and 6.2.27 false, we want to call 6.2.28 true and 6.2.29 false. We have already seen that there are no good options for alternative truth tables. If we want to distinguish among counterfactual conditionals, we may have to quit thinking of the natural-language conditional as a truth-functional operator.

Non-Truth-Functional Operators We might resolve the tension between 6.2.26 and 6.2.27 (and between 6.2.28 and 6.2.29) by claiming that ‘if . . . then . . .’ has two meanings. The, let’s say, logical aspects of the natural-language conditional may be expressed by the truth-functional ‘⊃’, encapsulated by the truth table for the material conditional. Other aspects of the natural-language conditional are not truth-functional. We could introduce a new operator, strict implication, ⇒, to regiment conditionals whose meaning is not captured by the material interpretation. Statements of the form ‘α ⊃ β’ could continue to be truth-functional, even though statements of the form ‘α ⇒ β’ would be non-­t ruth-functional. So, consider again sentence 6.2.21, “If 2+2=4, then cats are animals.” We could regiment it as ‘T ⊃ C’ in the standard way. We could, alternatively, regiment it as ‘T ⇒ C’. ‘T ⇒ C’ would lack a standard truth value in the third and fourth rows. We could leave the third and fourth rows of the truth table blank, neither true nor false. Or we could add a third truth value, often called undetermined or indeterminate. Such a solution would leave many conditionals, especially counterfactual conditionals, without truth values. Introducing ⇒ would entail giving up our neat bivalent semantics for propositional logic. Another option, deriving from work by the early-twentieth-century logician C. I. Lewis, and gaining popularity in recent years, is to interpret strict implication modally. Lewis defined ‘α ⇒ β’ as ‘◽(α ⊃ β)’. The ‘◽’ is a modal operator. There are many interpretations of modal operators. They can be used to construct formal theories of knowledge, moral properties, tenses, or knowledge. For Lewis’s suggestion, called strict implication, we use an alethic interpretation of the modal operator, taking the ‘◽’ as ‘necessarily’. So, on the modal interpretation of conditionals, a statement of the form ‘α ⇒ β’ will be true if it is necessarily the case that the consequent is true whenever the antecedent is. Modal logics are controversial. Some philosophers believe that matters of necessity and contingency are not properly logical topics. Other philosophers worry that our ability to know which events or properties are necessary and which are contingent is severely limited. One advantage of introducing a modal operator to express implication is that it connects conditional statements and scientific laws. A scientific law is naturally taken as describing a necessary, causal relation. When we say that event A causes event B, we

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imply that A necessitates B, that B could not fail to occur, given A. To say that lighting the stove causes the water to boil is to say that, given the stability of background conditions, the water has no choice but to boil. Thus, we might distinguish the two senses of the conditional by saying that material implication represents logical connections, whereas strict implication attempts to regiment causal connections.

Counterfactual Conditionals and Causal Laws As we have seen, the natural-language conditional often indicates a connection between its antecedent and its consequent. Consider again the dependent conditionals 6.2.17–6.2.20 and 2.3.10. 6.2.17 6.2.18 6.2.19 6.2.20 2.3.10

If it is raining, then I will get wet. If I run a red light, then I break the law. If the car is running, then it has fuel in the tank. If I were to jump out of the window right now, I would fall to the   ground. If you paint my house, then I will give you five thousand dollars.

The connection in 6.2.18 is mainly conventional. 2.3.10 refers to an offer or promise. The other three are fundamentally causal, depending on scientific laws or regularities. The truth of these complex sentences depends not only on the truth values of the simple component sentences and our logic, but also on the way that the world works. The truth values of 6.2.26 and 6.2.27 (and of 6.2.28 and 6.2.29) differ because of the nature of the physical world, not because of any logical properties of the sentences or their combinations. Our investigation of the logic of the conditional has taken us into questions about the relation between logic and science. Let’s consider a rudimentary law-like conditional like 6.2.30. 6.2.30

If this salt had been placed in water, it would have dissolved.

6.2.30 indicates a dispositional property of salt, one that we use to characterize the substance. Other dispositional properties, like irritability, flammability, and flexibility, refer to properties interesting to scientists. Psychological properties, like believing that it is cold outside, are often explained as dispositions to behave, like the disposition to put on a coat or say, “It’s cold.” Contrast 6.2.30 with 6.2.31. 6.2.31

This marble counter is soluble in water.

If we never place the counter in water, then 6.2.31 comes out true on the material interpretation. To be soluble is just, by definition, to have certain counterfactual properties. The counter is soluble just in case it would dissolve if placed in water. Similarly, my pajamas are flammable just in case they would burn if subjected to fiery conditions. The laws of science depend essentially on precisely the counterfactual conditionals that the logic of the material conditional gets wrong.

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Counterfactual conditionals thus do not seem to be merely formal-logical relations, at least not in terms of the logic of this book. Logicians have attempted, over the last century, to construct a formal logic for them. For an important example, see Robert Stalnaker’s “A Theory of Conditionals” and the other papers in Frank Jackson’s (1991) Conditionals collection. Views about the success of these projects vary. The problem of giving an analysis of the logic of conditionals is intimately related to the problem of distinguishing laws from accidental generalizations, as the comparison of 6.2.32 with 6.2.33 shows. 6.2.32 6.2.33

There are no balls of uranium one mile in diameter. There are no balls of gold one mile in diameter.

The explanation of 6.2.32 refers to scientific laws about critical mass. If you gather too much uranium in one spot, it explodes. The explanation of 6.2.33, in contrast, is merely accidental. It is physically possible to gather that much gold together, but it is impossible to collect the same amount of uranium. In order to know that difference, though, you must know the laws of our universe. The problem of distinguishing 6.2.32 from 6.2.33, the problem of knowing the laws of nature, is inextricably linked to the problem of understanding the logic of the natural-language conditional. We may use conditionals as truth-functional operators, sometimes. More commonly, especially in counterfactual cases, we use them to state connections between antecedents and consequents. So, a conditional will be true if the relevant connections hold among the antecedent and the consequent. It is false if such connections do not. Consider Nelson Goodman’s sentence 6.2.34. 6.2.34

If that match had been scratched, it would have lighted.

For 6.2.34 to be true, as for 6.2.17, 6.2.19, 6.2.20, or 6.2.30, there have to be a host of other conditions satisfied. We mean that conditions are such—i.e. the match is well made, is dry enough, oxygen enough is present, etc.—that “That match lights” can be inferred from “That match is scratched.” Thus the connection we affirm may be regarded as joining the consequent with the conjunction of the antecedent and other statements that truly describe relevant conditions. (Goodman, “The Problem of Counterfactual Conditionals,” 8, emphasis added)

When we assert a conditional, we commit ourselves to the relevant related claims. But to understand the claims to which we are committed, we must understand the relevant connections between the antecedent and the consequent. We must understand the general laws connecting them. In order to infer the consequent of 6.2.30, we need to presume causal laws governing the dissolution of salt in water. In order to infer the consequent of 6.2.34, we need to presume causal laws about the lighting of matches. Goodman thus argues that a proper analysis of counterfactual conditionals would include both logical and scientific aspects about inferences and causal laws.

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It seems that we have gone far from just understanding the logic of our language. We are now engaged in a pursuit of the most fundamental features of scientific discourse. Distinguishing between 6.2.26 and 6.2.27, or between 6.2.28 and 6.2.29, in contrast to the material interpretation, could import extra-logical features into our logic. While we believe that 6.2.27 and 6.2.29 are false, our reasons for those beliefs are laws of physics. If we were living on a planet with very little gravitational force, one on which buildings had limited force fields that kept us tethered to the floor inside, it might indeed be the case that if I jumped out of the window, I would fly to the moon rather than fall to the ground. If the melting point of butter were raised as a consequence of a change in the physical laws, we might be able to heat it to 150°F without melting it. Our beliefs about such matters seem empirical rather than logical.

Summary The material interpretation of the conditional has several advantages. It represents a logical conditional well and provides us with compositionality for conditional claims. ‘α ⊃ β’ serves neatly as a natural shorthand for ‘∼α ∨ β’. While the material interpretation leads to the so-called paradoxes of material implication, those results may be taken more as characteristics of the logical relation and less as problems. It is clear that not all conditional claims are best represented as logical relations or best interpreted materially. In particular, scientific claims that evoke regularities in nature, or laws, are not best understood as material conditionals. We ordinarily want our logic to be independent of the extra-logical facts and do not want to import the physical facts into our logic; we want our logic to be compatible with any way the world might be. Thus, we rest with the material interpretation of the naturallanguage conditional, giving up hope for a truth-functional analysis of the causal conditional and remembering that the natural-language conditional represents a strictly logical relation. TELL ME MORE  • What is the importance of logical truths such as the so-called paradoxes of material implication? See sections 3.8, 4.6, 5.3, and 6S.11 for formal work on logical truths. See 6S.7: The Propositions of Propositional Logic and 6S.9: Logical Truth, Analyticity, and Modality for more on the importance of logical truths. • What are some consequences of giving up bivalence? See 6.3: Three-Valued Logics. • What is the modal logic that we might use for a strict conditional? See 6.5: Modal Logics.

For Further Research and Writing 1. Contrast the following pair of counterfactual conditionals. If bin Laden didn’t plan the 9-11 attacks, then someone else did. If bin Laden hadn’t planned the 9-11 attacks, then someone else would have.

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The antecedents and consequents of these statements are nearly identical, but our estimations of the truth values and semantics of the two sentences are different. Discuss the similarities and differences between these sentences. Can we use the material conditional for these examples? Are there other options? See Bennett, 2003, and Jackson, 1987, for discussions of a relevantly similar pair of sentences. 2. Consider the following inference. If this is gold, then it is not water-soluble. So, it is not the case that if this is gold then it is water-soluble.

3.

4. 5.

6.

7.

8.

Intuitively, this argument seems valid. But if we regiment the argument in a standard way, we get an invalid argument. Discuss this problem in the light of the discussion of the material conditional. For possible solutions, you might look at the work of Lewis and Langford, 1932; Priest, 2008; or Goodman, 1983. In relevance logic, we insist that for a conditional to be true, its antecedent and consequent must be appropriately related. People working on relevance logics are mostly following C. I. Lewis’s suggestion concerning strict implication. Discuss the benefits of and problems with relevance logics. See Priest, 2008. C. I. Lewis and C. H. Langford, in their classic 1932 logic text, treat strict implication as different from the material conditional. What is the difference? What are modal operators, and how do they employ them? The philosopher Paul Grice, responding in part to the problems of the conditional, distinguished between the logical content of language, and other, pragmatic, features of language. Can we solve the problems of the material interpretation by appealing to pragmatics? See Grice, “Logic and Conversation.” Jennifer Fisher, Graham Priest, and Jonathan Bennett all have useful discussions of Grice’s suggestion. One option for treating the natural-language conditional is to introduce a third truth value, undetermined or indeterminate, for the last two rows of the basic truth table. Another is to leave those two rows blank. Both suggestions give up truth-functionality for the conditional. Explore these suggestions. What does it mean to give up truth-functionality? See section 6.3 on the introduction of three-valued logics. Nelson Goodman’s “The Problem of Counterfactual Conditionals” explores the relation between conditionals and scientific laws. What are scientific laws and how does our understanding of them inform our treatments of conditionals? See also his “New Riddle of Induction.” You might look at work by Hempel on scientific explanation, especially. Frank Jackson, David Lewis, and Robert Stalnaker provide extended technical treatments of conditionals. You might find some neat ideas in their difficult work; this topic is only for the most ambitious beginning logic student.

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Suggested Readings Bennett, Jonathan. A Philosophical Guide to Conditionals. Oxford: Oxford University Press, 2003. Perhaps the best detailed overview of the contemporary debate, Bennett’s work is challenging, but well written, without being overly technical. Fisher, Jennifer. On the Philosophy of Logic. Belmont, CA: Wadsworth, 2008. Chapter 8 connects conditionals with relevance logics. Goodman, Nelson. “The New Riddle of Induction.” In Fact Fiction and Forecast, 59–83. Harvard University Press, 1983. Goodman extends Hume’s problem of induction and connects it with difficulties in formulating statements of laws. Goodman, Nelson. “The Problem of Counterfactual Conditionals.” In Fact Fiction and Forecast, 3–27. Harvard University Press, 1983. Goodman’s outline of the problem connects the logic of the conditional with the philosophy of science. Grice, H. P. “Logic and Conversation.” In Conditionals, edited by Frank Jackson, 155–75. Oxford: Oxford University Press, 1998. Grice’s distinction between semantics and pragmatics grounds the possibility of treating some aspects of conditionals nonlogically. Harper, William, Robert Stalnaker, and Glenn Pearce. Ifs. Dordrecht: Reidel, 1981. Lots of advanced papers, some quite technical, capturing the main threads of the intense debate over modal analyses of conditionals in the 1970s, especially by Robert Stalnaker and David Lewis. Hempel, Carl G. Philosophy of Natural Science. Englewood Cliffs: Prentice Hall, 1966. An excellent introduction to the philosophy of science from one of the pioneers of applying logic to problems in science, especially to questions of explanation and confirmation. Hughes, R. I. G. A Philosophical Companion to First-Order Logic. Indianapolis: Hackett, 1992. A selection of very good, advanced papers. Dorothy Edgington’s “Do Conditionals Have Truth Conditions?” is especially relevant and accessible. Jackson, Frank. Conditionals. Oxford: Basil Blackwell, 1987. An engaging and accessible monograph. Jackson, Frank, ed. Conditionals. Oxford: Oxford University Press, 1991. A collection of papers on conditionals, focusing largely on work by Lewis and Stalnaker. Lewis, C. I., and C. H. Langford. Symbolic Logic. New York: Century, 1932. Reprinted by Dover Publications (New York), 1959. A classic text on logic. The notation might take some getting used to, but the use of modal operators in treating conditionals is interestingly precedental. Lewis, David. Counterfactuals. Malden, MA: Blackwell, 1973. The classic treatment of conditionals using possible worlds. Priest, Graham. An Introduction to Non-Classical Logic, 2nd ed. Cambridge: Cambridge University Press, 2008. Priest develops conditional logics neatly in chapters 1, 4, and 5, and makes brief but illuminating remarks along the way. Read, Stephen. Thinking About Logic. Oxford: Oxford University Press, 1995. The first chapter (“Truth, Pure and Simple: Language and the World”) of Read’s accessible introduction to the philosophy of logic discusses various views about propositions.

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Stalnaker, Robert. “A Theory of Conditionals.” In Studies in Logical Theory, edited by Nicholas Rescher, 98–112. Oxford: Blackwell, 1968. Stalnaker introduces the tool of possible worlds, or possible situations, to interpret the non-truth-functional conditional. Weiner, Joan. “Counterfactual Conundrum.” Nous 13, no. 4 (1979): 499–509. An accessible introduction to the contemporary debate, from a special issue of Nous devoted to laws and counterfactual conditionals.

6.3: THREE-VALUED LOGICS The first five chapters of this book use a bivalent semantics for the propositional operators. A bivalent semantics is one with just two truth values: truth (1) and falsity (0). For various reasons, some logicians have constructed semantics for propositional logic with more than two truth values. These semantics divide into two classes. The first type of nonbivalent semantics uses three truth values. The second type uses more than three truth values. Many interpretations that use more than three truth values use infinitely many truth values. We can take ‘0’ for absolute falsity, ‘1’ for absolute truth, and all real numbers between 0 and 1 as different truth values landing somewhere between absolute truth and absolute falsity. Such semantics are called fuzzy logics. Three-valued interpretations are generally called three-valued logics. That name is infelicitous, since the difference between bivalent and three-valued interpretations comes not in the object-language logic but in the metalanguage semantics. Still, I will use the term ‘three-valued logic’ for consistency with common usage. In this section, we will look at seven motivations, M1–M7, for three-valued logics. Some of these topics are examined in depth elsewhere in this text and so the discussion here will be brief. M1. Unproven mathematical statements M2. Future contingents M3. Failure of presupposition M4. Nonsense M5. Semantic paradoxes M6. The paradoxes of the material conditional M7. Vagueness

M1. Unproven Mathematical Statements Some philosophers introduce a third truth value to classify sentences whose truth values are unknown to us. In particular, mathematical sentences like 6.3.1 seem puzzling. 6.3.1

Every even number greater than four can be written as the sum of    two odd primes.

6.3.1 is called Goldbach’s conjecture, though Euler actually formulated it in response to a weaker hypothesis raised by Goldbach in 1742. Goldbach’s conjecture has

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neither been proved true nor disproved. It has been verified up to very large values. There are websites at which you can test any number. Using computers, people have verified that 6.3.1 holds up to at least 4 × 1018. There are also inductive arguments that make mathematicians confident that Goldbach’s conjecture is true. As the numbers grow, the number of different pairs of primes that sum to a given number (called Goldbach partitions) tends to grow. Even for the relatively small numbers between 90,000 and 100,000, there are no even numbers with fewer than 500 Goldbach partitions. It would be extremely surprising if the number suddenly dropped to 0. Still, many smart mathematicians have tried and failed to devise a proof of Goldbach’s conjecture. We might take Goldbach’s conjecture to be neither true nor false. We might do so, especially, if we think that mathematics is constructed, rather than discovered. If and when someone proves it, or its negation, then we could apply a truth value to the proposition. Until we have a proof, we could take Goldbach’s conjecture, and other unproven mathematical claims, to lack a truth value.

M2. Future Contingents Mathematical sentences like 6.3.1 are alluring for those who favor three-valued logics. But many mathematicians believe that Goldbach’s conjecture is either true or false. The reason we cannot decide whether it is true or false is that we have limited intelligence. But the sentence has a definite truth value. Such sentiments are not as strong in cases like 6.3.2. 6.3.2

There will be a party tomorrow night at Bundy Dining Hall.

Maybe there will be a party in Bundy tomorrow; maybe there will not be. Right now, we cannot assign a truth value to 6.3.2. The classic discussion of the problem in 6.3.2, called the problem of future contingents, may be found in Aristotle’s De Interpretatione regarding a sea battle. Since 6.3.2 is a contingent proposition, we can, at this moment, neither assert its truth nor its falsity. In things that are not always actual there is the possibility of being and of not being; here both possibilities are open, both being and not being, and consequently, both coming to be and not coming to be (De Interpretatione 9.19a9–13).

We know that one of the two truth values will apply, eventually. Either there will be a party tomorrow night or there will not be. Right now, though, 6.3.2 seems to lack a truth value. It is necessary for there to be or not to be a sea-battle tomorrow; but it is not necessary for a sea-battle to take place tomorrow, nor for one not to take place—though it is necessary for one to take place or not to take place (De Interpretatione 9.19a30–33).

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If the claim that there will be a sea-battle tomorrow has a truth value now, then the event is not contingent; it is already determined. Since the future is not determined, the truth values of statements about the future should also be undetermined. We can understand Aristotle’s claim better by considering 6.3.3–6.3.5. 6.3.3 6.3.4 6.3.5

Either there will be a sea-battle tomorrow or there will not be a   sea-battle tomorrow. There will be a sea-battle tomorrow. There will not be a sea-battle tomorrow.

Aristotle calls 6.3.3 true, indeed necessarily true, while withholding truth values from 6.3.4 and 6.3.5. If 6.3.4 and 6.3.5 are not true, and we only have two truth values, then they must be false. If they are false, we should be willing to assert their negations, 6.3.4′ and 6.3.5′. 6.3.4′ 6.3.5′

It is not the case that there will be a sea-battle tomorrow. It is not the case that there will not be a sea-battle tomorrow.

6.3.4′ and 6.3.5′ represent our acknowledgment of the contingency of the event. But, taken together, they form the contradiction at 6.3.6. 6.3.6

∼P ∙ ∼ ∼P

If we have a third truth value, we can assert both 6.3.4′ and 6.3.5′ without contradiction. In a three-valued logic, denying that a statement is true does not entail that it is false. It can be neither true nor false.

M3. Failure of Presupposition Neither 6.3.7 nor 6.3.8 are true propositions. 6.3.7 6.3.8

The king of America is bald. The king of America is not bald.

But 6.3.7 looks like the negation of 6.3.8. If we regiment 6.3.7 as ‘P’, we should regiment 6.3.8 as ‘∼P’. In a bivalent logic, since ‘P’ is not true, we must call it false, since we have only two truth values. Assigning the value ‘false’ to ‘P’ means that ‘∼P’ should be assigned ‘true’. But 6.3.8 is not a true statement. The problem is that in this case we want both a proposition and its negation to be false. But in a bivalent logic, the negation of a false proposition is a true proposition. Thus, in a bivalent logic, we can never deny both a statement and its negation, as we wish to do with 6.3.7 and 6.3.8. We think that 6.3.7 and 6.3.8 are both false because they both contain a false presupposition. 6.3.9–6.3.11 all contain failures of presupposition, the last even though it is not a declarative sentence. 6.3.9 6.3.10 6.3.11

The woman on the moon is six feet tall. The rational square root of three is less than two. When did you stop cheating on your taxes?

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Russell treated such problems with his analysis of definite descriptions, which we saw in section 5.4. Another response to the problem of presupposition failure is to call such propositions neither true nor false, in other words, assign a third truth value.

M4. Nonsense The distinction between syntax and semantics can be a motivation to adopt a third truth value. The syntax of a formal language tells us whether a string of symbols of the language is a wff. The correlate of syntax, in natural language, is grammaticality. But not all grammatical sentences are sensible. We might consider some grammatical but nonsensical sentences like 6.3.12 and 6.3.13 to lack truth values. 6.3.12 6.3.13

Quadruplicity drinks procrastination. (From Bertrand Russell) Colorless green ideas sleep furiously. (From Noam Chomsky)

In the syntax of English, 6.3.12 and 6.3.13 are well formed. But their well-­ formedness does not entail that we can assign truth values to them. If we adopt a three-valued logic, we can assign them the missing truth value and save falsity for sentences that are sensible as well as grammatical.

M5. Semantic Paradoxes There are a variety of semantic paradoxes. The most famous, 6.3.14, is called the liar. 6.3.14

6.3.14 is false.

6.3.14 is an example of a paradoxical sentence. If 6.3.14 is true, then it is false, which makes it true, which makes it false . . . 6.3.14 seems to lack a definite truth value, even though it seems to be a perfectly well-formed sentence. The liar is often called Epimenides’s paradox. Epimenides was a Cretan to whom the statement that all Cretans are liars is attributed. Since assigning truth or falsity to 6.3.14 leads to a contradiction, we might assign it a third truth value.

M6. The Paradoxes of the Material Conditional Statements of the form 6.3.15–6.3.17 are sometimes called paradoxes of the material conditional. We call them paradoxes, a name that is probably too strong. They are logical truths of PL even though they are not obvious. 6.3.15 α ⊃ (β ⊃ α) 6.3.16 ∼α ⊃ (α ⊃ β) 6.3.17 (α ⊃ β) ∨ (β ⊃ α)

The paradoxes of the material conditional have awkward consequences. 6.3.15 says, approximately, that if a statement is true, then anything implies it. This is because the truth table for the material conditional is true on every line in which the consequent is true. 6.3.16 says that if a statement is false, its opposite entails any other statement.

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6.3.17 says that for any statement β, either any other statement entails it or it entails any statement. Every statement must be either true or false. If a given statement is true, then any statement entails it. If a given statement is false, then it entails any statement. On the one hand, 6.3.15–6.3.17 are logical truths. We certainly cannot call their instances false. On the other hand, their instances are not the kinds of sentences that some people feel comfortable calling true. So, we might use a third truth value for them.

M7. Vagueness As a last motivation for three-valued logics, consider the phenomenon of vagueness. Many predicates admit of borderline, or vague, cases. For example, consider baldness. There are paradigm instances of baldness that are incontrovertible. There are also paradigm instances of non-baldness. But there are also cases in which we don’t know what to say.

6.3.18 6.3.19 6.3.20

The person on the left is bald. The person in the middle is bald. The person on the right is bald.

6.3.18 is true; 6.3.20 is false. But between these clear cases, there is a penumbra. We could, if we wish, give 6.3.19 a third truth value, expressing that it is neither determinately true nor determinately false. Whether the person in the middle is bald is unknown or indeterminate. Another option, perhaps more compelling, is to apply a logic in which there are infinitely many truth values, called a fuzzy, or many-valued, logic. In one relevant kind of fuzzy logic, any real number between 0 and 1 is a truth value. So, we could assign a value of 0 to 6.3.20, a value of 1 to 6.3.18, and any real number between 0 and 1, say .3 or .612 to 6.3.19.

Three Three-Valued Logics The rules for determining truth values of formulas in a logic are called the semantics. We provided semantics for propositional logic by constructing truth tables. Since we

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used only two values, true and false, our semantics is called two-valued. Our twovalued semantics is also called classical semantics. If we want to adopt a third truth value, which we might call unknown, or indeterminate, we must revise all the truth tables. I will call the third truth value indeterminate and use the Greek letter iota, ι, to indicate it. Remember, the idea is that we can ascribe ι to sentences that lack a clear truth value. There are two options for how to deal with unknown or indeterminate truth values in the new semantics. First, one could claim that any indeterminacy among component propositions creates indeterminacy in the whole. This is the principle underlying Bochvar semantics, which is sometimes called weak Kleene semantics. Second, one could try to ascribe truth values to as many formulas as possible, despite the indeterminate truth values. For example, a conjunction with one false conjunct could be ascribed falsity whether the other conjunct is true, false, or unknown. This is the principle underlying strong Kleene semantics and Łukasiewicz semantics. We proceed to look at these three different three-valued semantics. We will look at: (1) the rules for each, (2) how the new rules affect the logical truths (tautologies), and (3) how the new rules affect the allowable inferences (valid arguments). To show the semantics, I will present truth tables for the standard propositional operators. For simplicity, I will ignore the biconditional. BOCHVAR (OR WEAK KLEENE) SEMANTICS (WK) α

∼α

α

∙

β

α

∨

β

α

⊃

β

1

0

1

1

1

1

1

1

1

1

1

ι

ι

1

ι

ι

1

ι

ι

1

ι

ι

0

1

1

0

0

1

1

0

1

0

0

ι

ι

1

ι

ι

1

ι

ι

1

ι

ι

ι

ι

ι

ι

ι

ι

ι

ι

ι

0

ι

ι

0

ι

ι

0

0

0

1

0

1

1

0

1

1

0

ι

ι

0

ι

ι

0

ι

ι

0

0

0

0

0

0

0

1

0

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In Bochvar semantics, no classical tautologies come out as tautologies. Consider, under WK, the bivalent logical truths ‘P ⊃ P’ and ‘P ⊃ (Q ⊃ P)’. P

⊃

P

P

⊃

(Q

⊃

P)

1

1

1

1

1

1

1

1

ι

ι

ι

1

ι

ι

ι

1

0

1

0

1

1

0

1

1

ι

ι

1

ι

ι

ι

ι

ι

ι

ι

ι

ι

0

ι

ι

0

1

1

0

0

0

ι

ι

ι

0

0

1

0

1

0

These two classical tautologies, and all others, do not come out false on any line on WK. But they do not come out as true on every line. This result is generally undesirable, since the classical tautologies seem pretty solid. Tautologies are also known as logical truths. They are the theorems of the logic. For those motivated by the paradoxes of the material conditional, Bochvar semantics could be tempting. None of them come out as logical truths on WK. Other systems of logic, called relevance logics, attempt to keep most classical logical truths but eliminate the paradoxes of material implication. Unfortunately, WK seems too strong a reaction to the oddities of those so-called paradoxes; it eliminates all classical tautologies. One solution to the problem of losing logical truths in WK would be to redefine ‘tautology’ as a statement that never comes out as false. Redefined in this way, though, weakens the concept, making it ambiguous between truthful and indeterminate claims. Given the underlying tenet of WK, that indeterminacy is utterly infectious, such a definition of tautology seems inapt. Next, consider what WK does to validity. We defined a valid argument as one for which there is no row in which the premises are true and the conclusion is false. We

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could have defined a valid argument as one for which there is no row in which the premises are true and the conclusion is not true. Classically, these two definitions are equivalent. But in three-valued semantics, they cleave. If we take a row in which the premises are true and the conclusion is indeterminate as a counterexample to an argument, as Bochvar did, then some classically valid inferences come out invalid. Under classical semantics, ‘P / Q ∨ P’ is a valid inference. Q

∨

P

1

1

1

1

1

0

1

1

0

1

1

0

0

0

0

0

P

//

Under Bochvar semantics, the argument comes out invalid. The second row is a counterexample. Q

∨

P

1

1

1

1

1

ι

1

0

P

//

1 1

1

1

ι

ι

ι

0

ι

0

1

0

ι

0

0

1

0 0

0

0

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WK proceeds on the presupposition that any indeterminacy infects the whole. It thus leaves the truth values of many formulas undetermined. Such an interpretation might capture well the motivation for three-valued semantics from M4: nonsense. It is difficult to know how to assign a truth value to 6.3.21, for example. 6.3.21

Either two and two are four or numby dumby wumby.

Given what follows the ‘or’, we can understand why someone might want to call 6.3.21 nonsensical. Still, the disjunction of a true statement with one of indeterminate truth value is true no matter what that latter truth value is. Similarly, a conditional with a true consequent or a false antecedent will be true whatever the value on the other side. Such observations motivate strong Kleene three-valued semantics, which leaves fewer rows unknown. In the basic truth tables here, I’ve highlighted the cells in which K3 differs from WK. STRONG KLEENE SEMANTICS (K3) P

∼P

P

∙

Q

P

∨

Q

P

⊃

Q

1

0

1

1

1

1

1

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6 . 3 : T h ree - Val u e d L og i cs   4 1 5

K3 has a certain intuitiveness. But in order to compare WK to K3 properly, we should look at the differences on logical truths and inference patterns. Consider the same two tautologies, ‘P ⊃ P’ and ‘P ⊃ (Q ⊃ P)’, under Kleene semantics:

P

⊃

P

P

⊃

(Q

⊃

P)

1

1

1

1

1

1

1

1

ι

1

1

ι

1

1

0

1

1

0

1

1

ι

1

ι

ι

ι

ι

ι

ι

ι 0

1

ι

1

0

1

ι

0

1

1

0

0

0

1

ι

ι

0

0

1

0

1

0

While many more of the rows are completed, the statements still do not come out as tautologous under the classical definition of ‘tautology’. Łukasiewicz, who first investigated three-valued logics, tried to preserve the tautologies. ŁUKASIEWICZ SEMANTICS (L3)

There is only one difference between K3 and L3, which we can see in the fifth row of the truth table for the conditional.

4 1 6    C h apter 6  Be y on d Bas i c L og i c

P

∼P

P

∙

Q

P

∨

Q

P

⊃

Q

1

0

1

1

1

1

1

1

1

1

1

ι

ι

1

ι

ι

1

1

ι

1

ι

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0

1

1

0

0

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1

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1

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1

1

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1

ι

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0

0

ι

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0

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ι

0

0

0

1

0

1

1

0

1

1

0

0

ι

0

ι

ι

0

1

ι

0

0

0

0

0

0

0

1

0

One might wonder how we might justify calling a conditional with indeterminate truth values in both the antecedent and consequent true. After all, what if the antecedent turns out true and the consequent turns out false? Putting that worry aside, look at what this one small change does.

6 . 3 : T h ree - Val u e d L og i cs   4 1 7

P

⊃

P

P

⊃

(Q

⊃

P)

1

1

1

1

1

1

1

1

ι

1

ι

1

1

ι

1

1

0

1

0

1

1

0

1

1

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1

1

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1

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1

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ι

1

0

1

ι

0

1

1

0

0

0

1

ι

ι

0

0

1

0

1

0

Voila! L3 retains many of the classical tautologies that WK and K3 lost. In fact, L3 does not get all classical tautologies, including the law of excluded middle, ‘P ∨ ∼P’, but that is a law that some folks would like to abandon anyway. P

∨

∼

P

1

1

0

1

ι

ι

1

0

ι 0

1

The fact that L3 recaptures many classical tautologies is advantageous. It remains to be shown that the change in semantics is warranted. Is it acceptable to call true a conditional whose antecedent and consequent are both of the third truth value, undetermined or unknown or indeterminate? The justification for L3 over K3 seems impossible from the bottom up. The lesson of Łukasiewicz semantics is that we need not give up classical tautologies, logical truths, to have a three-valued logic. The fewer changes we make to the set of logical truths, the less deviant the logic is. But the semantics that allows us to retain these logical truths may not be as pretty as we would like.

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Lastly, consider the effect on validity of moving from WK to K3 or L3. Consider again the argument ‘P / Q ∨ P’. Bochvar

Kleene

Q

∨

P

P

1

1

1

1

1

ι

1

0

P

//

1

Łukasiewicz

Q

∨

P

P

1

1

1

1

1

1

ι

1

1

1

0

//

Q

∨

P

1

1

1

1

1

1

ι

1

1

1

1

1

0

1

1

1

ι

1

1

ι

//

1

ι

1

ι

ι

ι

ι

ι

ι

0

ι

0

ι

0

ι

0

1

0

ι

0

0

1

0

0

0

1

0

0

ι

0

0

0

Counterexample in row 2

1

0

0

0

1

0

0

ι

0

0

0

Valid—no counterexample

1

0 0

0

0

Valid—no counterexample

Both K3 and L3 semantics thus maintain some of the classical inference patterns that are lost in WK.

Problems with Three-Valued Logics All three-valued logics abandon some classical tautologies and classically valid inference patterns. This result may be acceptable, depending on one’s motivation for adopting a three-valued logic. But it is not clear that all the problems that motivated three-valued logics can be solved by adopting three-valued logics. For example, Bochvar hoped that his semantics would solve the problems of the semantic paradoxes. The liar sentence can be given a truth value in WK without paradox. That’s good. But consider 6.3.22, the strengthened liar. 6.3.22

SL is not true.

Suppose 6.3.22 is true. Then what it says, that 6.3.22 is not true, must hold. So, 6.3.22 is not true. It might be false, or it might be undetermined. In either case, 6.3.22,

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because it says that it is not true, turns out to be true. The paradox recurs. Adopting the three-valued logic does not eliminate the strengthened liar paradox. Another worry about three-valued logics concerns the interpretation of the third value. Thinking of it as unknown, for example, seems to involve a conceptual confusion. ‘Unknown’ seems to be not a third truth value, but the lack of a truth value. Instead of filling in such cells in the truth table, we could just leave them blank. Leaving certain cells of a truth table blank is part of what is called a truth-value gap approach. A truth table some of whose cells are left blank is called a partial valuation: only some truth values of complex propositions are computed on the basis of the truth values of component propositions. Faced with partial valuations, the logician may consider something called a supervaluation. A supervaluation considers the different ways to complete partial valuations and classifies formulas and arguments according to the possibilities for completion. Supervaluations can, in certain cases, recapture some of the missing values in a partial valuation. One serious worry about three-valued logics, present especially in work by Quine, is called the change of logic, change of subject argument. The basic idea of the argument is that in order to disagree with someone, you have to at least agree on what you are disagreeing about. There has to be some common ground on which you can stand, or else you are not really disagreeing at all. Consider two terms, and their definitions, which I will stipulate. Chair1 Chair2

desk chairs, dining room chairs, and such, but not recliners or   beanbag chairs all chair1 objects, and also recliners and beanbag chairs

Now, consider person 1, who uses ‘chair’ as chair1, and person 2, who uses ‘chair’ as chair2 . Imagine these two people talking about a beanbag chair. Person 1 affirms, ‘that’s a chair’, while person 2 denies that sentence. Since they are using the same term, it looks like they are disagreeing. But they are not really disagreeing about whether the beanbag chair is a chair. They are disagreeing about what ‘chair’ means. They are both correct in their claims about the beanbag. It is a chair1 and is not a chair2 . What looks like a disagreement is not really a disagreement; the subject has been changed. Quine presented the change of logic, change of subject argument in response to a proposal, related to the introduction of three-valued logics, to allow some contradictions in one’s language. The problem with accepting contradictions, as we saw in section 3.5, is that they lead to explosion, the inferential process by which any proposition follows from a contradiction. Those who attempt to embrace contradictions have to find a way to block explosion. Perhaps, it is suggested, we can so rig our new logic that it will isolate its contradictions and contain them. My view of this dialogue is that neither party knows what he is talking about. They think they are talking about negation, ‘∼’, ‘not’, but surely the notation ceased to be recognizable as negation when they took to regarding some conjunctions of the form ‘p ∙ ∼p’ as true,

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and stopped regarding such sentences as implying all others. Here, evidently, is the deviant logician’s predicament: when he tries to deny the doctrine he only changes the subject. (Quine, Philosophy of Logic, 81)

If we are considering debates over the correct logic, even claims of what it means to affirm or deny a sentence are under discussion. Debates over the correct logic seem to be more like the disagreement between chair1 and chair2 . The disputants do not agree on the terms they are using, and so are talking past each other. Imagine we are linguists, and we are headed to a newly discovered alien planet. We have to translate a completely new language into English. We start by assuming that the aliens obey the rules of logic. If we were to propose a translation of the alien language on which the aliens often made statements that translated into the form of ‘α ∙ ∼α’, we would not assume that these beings are often contradicting themselves. We would likely revise our translation to make it more charitable instead. We take the laws of logic as fundamental. We use them as common ground on which to base our translations. If we hypothesize that the alien is asserting a contradiction, we take that to be evidence against our translation rather than evidence against the alien’s intellectual capacity, for example. We need logic to serve as a starting point for the translation. We need common ground even to formulate disagreement. If we disagree about the right logic, then we have merely changed the subject.

Avoiding Three-Valued Logics I introduced three-valued logics in order to respond to some problems that arose with classical logic. M1. Unproven mathematical statements M2. Future contingents M3. Failure of presupposition M4. Nonsense M5. Semantic paradoxes M6. The paradoxes of the material conditional M7. Vagueness I mentioned that three-valued logics does not solve the problems of the semantic paradoxes. There are ways for the classical logician to deal with all of these problems, anyway. I will not discuss each of them here. But following are a few hints to how to solve them. M1, concerning sentences with unknown truth values, and M2, concerning propositions referring to future events, are related. In both cases, we can blame ourselves, rather than the world, for our not knowing the truth value. Thus, we can say that Goldbach’s conjecture is either true or false but we just do not know which. Similarly, we can say that either there will be a party at Bundy Dining Hall tomorrow or there will not. We need not ascribe a deep problem to truth values. Such sentences have truth values. We just do not know them (yet).

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Proponents of classical logic may deal with problems about time by appealing to a four-dimensional framework. We can take a God’s-eye point of view and think of the world as a completed whole, from the beginning until the end of time. Going four-­ dimensional, we add an implicit timestamp to all our claims. Instead of saying that it is snowing, for example, we say that it is snowing at 4:37 p.m., Eastern Standard Time, on December 31, 2021. Then, a statement about the future is true if it ends up true at that time. We need not see the logic as committing us to a determined future. We just know that statements about future events will eventually have truth values. There are also tense logics, which introduce temporal operators but maintain classical semantics, to help with time. For failures of presupposition, M3, we can use Bertrand Russell’s analysis of definite descriptions. In section 5.4, there is a more precise analysis of Russell’s solution. For now, consider again an example of failure of presupposition, 6.3.9, the claim that the woman on the moon is six feet tall. We can analyze the claim to make the assumption explicit, recasting it as 6.3.23. 6.3.23

There is a woman on the moon and she is six feet tall.

6.3.23 has the form ‘P ∙ Q’. ‘P’ is false, so ‘P ∙ Q’ is false. We can similarly recast 6.3.24 as 6.3.25. 6.3.24 6.3.25

The woman on the moon is not six feet tall. There is a woman on the moon and she is not six feet tall.

We regiment 6.3.25 as ‘P ∙ ∼Q’. P is false, so ‘P ∙ ∼Q’ is false. We thus do not have a situation in which the same proposition seems true and false. In both cases, P is false, so the account of the falsity of both sentences 6.3.24 and 6.3.25 can be the same. We thus lose the motivation for introducing a third truth value. For M4, nonsense, M5 and M6, paradoxes, and M7, vagueness, we can deny that such sentences express propositions. We may claim that just as some strings of letters do not form words and some strings of words do not form sentences, some grammatical sentences do not express propositions. This would be the same as to call them meaningless. This solution is a bit awkward, since it does seem that ‘This sentence is false’ is perfectly meaningful. But if it prevents us from having to adopt three-valued logics, it might be a useful move.

Summary There are many reasons to call our bivalent semantics for PL into question. Perhaps the most plausible alternative is some version of three-valued logic, though manyvalued logics also have interesting properties. In considering adopting a three-­valued logic, you should keep in mind the problem with bivalence you are trying to solve and whether an adoption of a three-valued logic solves or merely defers the problem. Moreover, it is difficult, in light of Quine’s change-of-logic, change-­of-subject argument, to know exactly how to argue for or against such a fundamental difference in logics.

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TELL ME MORE  • What are the paradoxes of material implication and logical truths? See sections 3.8, 4.6, 5.3, 6S.7, 6S.9, and 6S.11 on logical truth. See 6.2: Conditionals on the paradoxes of material implication. • What is the liar paradox? What problems does it raise for formal systems? See 7.5: Truth and Liars. • How does Russell’s theory of definite descriptions treat failure of presupposition? See 7.6: Names, Definite Descriptions, and Logical Form.

EXERCISES 6.3a 1. Construct truth tables for each of the following propositions, under classical semantics and each of the three three-valued semantics (Bochvar, Kleene, Łukasiewicz). Compare the results. 1. P ∨ ∼P 2. P ⊃ P 3. (P ⊃ Q ) ≡ (∼P ∨ Q ) Note: you can construct the truth table for the biconditional by remembering that ‘P ≡ Q’ is logically equivalent to ‘(P ⊃ Q ) ∙ (Q ⊃ P)’. 2. Use the indirect method of truth tables to test each of the following arguments for validity, under classical semantics and each of the three three-valued semantics (WK, K 3, and L3). Compare the results. 1. P ⊃ Q P / Q 2. P / ∼(Q ∙ ∼Q  ) 3. P / P ∨ Q

For Further Research and Writing 1. Write a paper comparing Bochvar semantics, Kleene semantics, and Łuka­ siewicz semantics. What differences do the different semantics have for classical tautologies? What differences do they have for classical inferences (validity and invalidity)? Be sure to consider the semantics of the conditional. Which system seems most elegant? This paper will be mainly technical, explaining the different semantics and their results.

6 . 3 : T h ree - Val u e d L og i cs   4 2 3

2. Do three-valued logics solve their motivating problems? Philosophers explore three-valued logics as a way of dealing with various problems, which I discuss in these notes. Consider one or more problems, and show how one of the systems tries to resolve the problem. For this paper, I recommend that you focus on Kleene’s semantics. If you try to deal with Epimenides, and the semantic paradoxes, you might want to focus just on that problem. 3. Bochvar used a so-called assertion operator, ⊢. Use of this operator allows us to recapture analogs of classical tautologies within Bochvar semantics. Describe the truth table for this operator. Show how it allows us to construct tautologies. How does the new operator affect the set of valid formulas? (It can be shown that on Bochvar semantics, any argument using only the standard operators that has consistent premises, and which contains a sentence letter in the conclusion that does not appear in any of the premises, is invalid. You might consider this result, and the effect of the new operator on it.) 4. Quine, in chapter 6 of Philosophy of Logic, calls three-valued logic deviant and insists that to adopt three-valued logic is to change the subject. Why does Quine prefer classical logic? Consider his maxim of minimum mutilation. Who can deal better with the problems, sketched at the beginning of these notes, that motivate three-valued logic? (You need not consider all of the problems, but you should provide a general sense of how each approach works.) 5. Do assertions about the future have a truth value? Consider both the bivalent and the three-valued alternatives. You might compare Aristotle’s view with that of Leibniz, who says that contingent truths are not necessary, even though they are certain. Alternatively, you could look at Haack’s discussion of the way Aristotle’s suggestion was pursued by Łukasiewicz. If you want to pursue an interesting technical discussion, Prior’s “Three-Valued Logic and Future Contingents” is written in Polish notation. 6. How should we understand the sentence ‘the king of America is not bald’? Consider Russell’s theory of descriptions, and contrast it with Strawson’s response. You might also consider the questions about whether there is a difference between logical and grammatical form and whether ordinary language has a logic. 7. Are there any people? Consider the problem of vagueness, and the many-­valued approach to its solution. See Unger’s paper.

Suggested Readings Aristotle. De Interpretatione. In The Complete Works of Aristotle, vol.1, edited by Jonathan Barnes. Princeton, NJ: Princeton University Press, 1984. On the sea-battle, and future contingents. Bochvar, D. A. “On a Three-Valued Logical Calculus and Its Application to the Analysis of the Paradoxes of the Classical Extended Functional Calculus.” History and Philosophy of Logic 2 (1981): 87–112.

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Chomsky, Noam. Syntactic Structures, 2nd ed. Berlin: De Gruyter, 2002. This book contains the discussion about colorless green ideas but not a defense of three-valued logics. Chomsky was arguing for a distinction between grammaticality and meaningfulness. Dummett, Michael. “The Philosophical Basis of Intuitionistic Logic.” In Philosophy of Mathematics: Selected Readings, 2nd ed., edited by Paul Benacerraf and Hilary Putnam, 97–129. Cambridge: Cambridge University Press, 1983. See also the selections by Heyting and Brouwer in this volume. The intuitionists believed that an unproven mathematical statement lacked a truth value. These articles are all pretty technical. Fisher, Jennifer. On the Philosophy of Logic. Belmont, CA: Wadsworth, 2008. Chapters 7 and 9 discuss three-valued logics, the liar, and paraconsistent logics, which block the explosion that results from contradictions. Haack, Susan. Deviant Logic, Fuzzy Logic: Beyond the Formalism. Chicago: University of Chicago, 1996. Chapter 4 contains a discussion of Aristotle’s view on future contingents as well as more recent applications. Haack, Susan. Philosophy of Logics. Cambridge: Cambridge University Press, 1978. Chapters 9 and 11 are accessible and relevant here. Kleene, Stephen. “On Notation for Ordinal Numbers.” Journal of Symbolic Logic 3, no. 4 (1938): 150–155. Leibniz, G. W. Discourse on Metaphysics. In G.W. Leibniz: Philosophical Essays, edited by Roger Ariew and Daniel Garber, 35–68. Indianapolis: Hackett, 1989. The early sections, especially 6–13, contain his distinction between certainty and necessity. Priest, Graham. An Introduction to Non-Classical Logic: From Ifs to Is, 2nd ed. Cambridge: Cambridge University Press, 2008. Chapter 7, on many-valued logics, and chapter 11, on fuzzy logics, are clear on the technical material and contain excellent further references. Prior, A. N. “Three-Valued Logic and Future Contingents.” Philosophical Quarterly 3, no. 13 (1953): 317–326. Putnam, Hilary. “Three-Valued Logic” and “The Logic of Quantum Mechanics.” In Mathematics, Matter and Method: Philosophical Papers, vol. 1. Cambridge: Cambridge University Press, 1975. Do we need three-valued logic in order to account for oddities in quantum mechanics? Quine, Willard van Orman. Philosophy of Logic, 2nd ed. Cambridge, MA: Harvard University Press, 1986. The discussion of deviant logics and changing the subject is in chapter 6, but chapter 4, on logical truth, is exceptionally clear and fecund. Quine, Willard van Orman. The Ways of Paradox. Cambridge, MA: Harvard University Press, 1976. The title essay is the source of the ‘yields a falsehood’ paradox and contains an excellent discussion of paradoxes. Read, Stephen. Thinking About Logic. Oxford: Oxford University Press, 1995. Chapter 6 discusses semantic paradoxes. Chapter 7 focuses on the sorites paradox and fuzzy logic. Russell, Bertrand. Introduction to Mathematical Philosophy. London: Routledge, 1993. See chapter 16, “Descriptions.” Contains a clearer discussion of Russell’s solution to the problem of some forms of failure of presupposition than the ubiquitous “On Denoting.”

6 . 4 : Metalog i c   4 2 5

Russell, Bertrand. “On Denoting.” In The Philosophy of Language, 5th ed., edited by A. P. Martinich, 230–38. New York: Oxford University Press, 2008. “On Denoting” is widely available. Strawson, P. F. “On Referring.” In The Philosophy of Language, 5th ed., edited by A. P. Martinich, 246–60. New York: Oxford University Press, 2008. An alternative to Russell’s theory of descriptions. Unger, Peter. “Why There Are No People.” Midwest Studies in Philosophy 4 (1979): 177–222. A consequence of vagueness. Williamson, Timothy. Vagueness. London: Routledge, 1994. Chapter 1 has a nice discussion of the history of vagueness, and chapter 4 discusses the three-valued logical approach to the problem.

6.4: METALOGIC We use language to make assertions and raise questions, among other activities. Some of our assertions are ordinary claims, whether true or false, about the world: the cat is on the mat, the sun is setting, some robots are intelligent, all animals are human. Sometimes we use language to make claims or ask questions about language itself: the English alphabet contains twenty-six letters; natural languages are compositional; are there thoughts that cannot be expressed in English? A similar phenomenon appears in formal logic. Sometimes we make ordinary assertions that could be about anything; sometimes we want to talk about the logic itself. Most of this book concerns the former task, how to use logic to make assertions (by translating sentences of natural language into logical formulas) and to make inferences (using our systems of natural deduction). Logicians also ask questions and make assertions about the logical systems we study: Are there limits to the abilities of our logical languages to express various concepts? What can we prove in our logic? Is our system of proof clean, or can we prove things that we shouldn’t be able to prove? How many rules do we need? The study of logical languages and systems of inference is called metalogic. There are essentially two branches of metalogic, corresponding to our two ways of doing logic. In proof theory, we study systems of inference. In model theory, we study the semantics of logical languages. It is difficult to understand very much of metalogic without having a firm grasp on how logic works in translation and derivation, the core topics of this book. Metalogic is thus a natural subject for an advanced formal logic course. Still, strictly speaking, some of the work of the first five chapters is metalogical. We specified the rules of inference in a metalanguage, and the truth tables are metalogical too. But often when we talk about metalogic, we are referring to claims about the power and limits of our logical languages and theories. We can peruse the important

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metalogical results in order to better understand the work we do here, including questions about adequacy and alternative notations, which I discuss in other sections. In this section, we will look, briefly and informally, at a few central metalogical concepts: consistency, soundness, completeness, and decidability.

Consistency Consistency for a theory is generally characterized in terms of derivability, in proof theory, or models, semantically. The proof-theoretic version concerns not being able to derive contradictions. A theory is consistent, proof-theoretically, if, and only if, there is no sentence α such that α and ∼α are theorems (i.e., can be proven). A theory is also said to be consistent if there is at least one wff that is unprovable, since if a theory is inconsistent, then, by explosion, every formula is provable. We saw the semantic definition of consistency, at least implicitly, in the work on semantics in chapters 4 and 5. A theory is consistent, semantically, if, and only if, there is an interpretation on which all sentences of the theory come out true; such an interpretation is called a model. We interpret PL by assigning truth values to simple propositions. PL is obviously consistent, since we take as theorems only those formulas that are tautologous, or true on every interpretation. Still, to show that all the theorems of PL are tautologous, we would have to show that every rule of inference and equivalence is truth-preserving. Such a proof, like many of the proofs of metalogic, proceeds by mathematical induction, which shows that the rules of inference are truth-preserving, no matter where they are used, no matter how long or complicated the formulas are to which they are applied. Proofs of the consistency of the various theories of predicate logic are complicated mainly by the details of interpretations for those theories.

Soundness Soundness for theories is different from soundness for arguments, which we saw in section 1.5. An argument is sound if it is valid and its premises are true. For theories, soundness is not being able to prove a proposition that we shouldn’t be able to prove. A theory is sound if, and only if, every provable argument is semantically valid and every provable proposition is logically true. Proofs of the soundness of PL again rely on mathematical induction. Be sure to understand what we mean by ‘every provable formula is valid’, or logically true. We often derive propositions that are not logical truths, on the assumption of premises. Such derivations are not proofs of their conclusions, but proofs that the conclusions follow from the premises. We can think of such derivations as proofs that

6 . 4 : Metalog i c   4 2 7

the conjunctions of premises entail the conclusion, as I showed in section 3.8, for example, when we converted arguments with premises into logical truths.

Completeness The converse of soundness is called completeness. A formal theory, like our system of propositional logic, is called complete when all the logically true wffs are provable. A theory is complete if, and only if, every valid argument and every logical truth is provable. PL, M, and F, are all both sound and complete, which is part of what makes them such important logical languages. Kneale and Kneale report that the Stoics, on whose work PL is based, presumed the completeness of their logic, though without the ability to prove it (Kneale and Kneale [1962] 2008, 172). The completeness of truth-functional propositional logic, like PL, was proved in 1920, by Emil Post. Kurt Gödel proved the completeness of first-order quantificational logic, which we see in the theory F, in his 1929 doctoral dissertation; the proof for M had been presented by Leopold Löwenheim in 1915. There are far more rules of inference and equivalence in this book than we need for a complete system of propositional or predicate logic. That’s largely because my goal in writing this book was not to construct a theory with the smallest vocabulary or rule set that I could. In austere systems, ones with few logical terms and few inference rules, translations from natural languages and derivations can be long and arduous. In contrast, the more rules one includes in a logical system, the easier the proofs become, and the more powerful the resulting system. Some inferential systems have two categories of rules: basic rules and derived rules. The basic ones are only those needed for completeness; the derived rules allow for friendlier derivations. In this book, for example, all of the rules in section 3.6 are derived; we don’t need them for the completeness of the system. Since we are not proving the completeness of the systems in this book, I did not make the distinction between basic and derived rules. We look at logic in two different ways in this book: semantically (or model-­ theoretically) and proof-theoretically (or syntactically). We have semantic ways of identifying logical truths and testing for validity: truth tables and interpretations. We have proof-theoretic ways of identifying logical truths and testing for validity: derivations in our natural deduction system. The proofs of completeness ensure us that our work is, in a way, redundant. Both kinds of methods yield the same results. In more sophisticated theories, proof separates from truth and validity. Kurt Gödel’s first incompleteness theorem shows that in theories with just some weak mathematical axioms, there will be true sentences that are not provable. Gödel uses arithmetic to allow a formal theory to state properties like provability within the theory. He constructs a predicate that ‘is provable’, that holds of sentences only with specific, statable arithmetic properties. Then, he constructs a theorem that says, truly, of itself

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that it is not provable. Since it is true, it is not provable. Thus, in theories that allow the Gödel construction, model theory (semantics) and proof theory provide different results. Logicians and mathematicians have explored the differences between complete and incomplete theories and have narrowed down the properties of theories that lead to incompleteness. One of the strongest complete theories is called Presburger arithmetic, a theory that includes addition, but in which divisibility and exponentiation are unrepresentable; those latter tools enable the Gödel construction that leads to incompleteness. There are other, related concepts of completeness, including syntactic completeness, in contrast to the version presented here, which may be called semantic completeness; see Hunter 1971, section 33. A theory is syntactically complete if no unprovable theorem could be added to the theory without generating inconsistency. PL is syntactically complete, but F is not. Notice that a theory may be complete without being sound and may be sound without being complete. If a theory were to be inconsistent and explosive, then it could be complete, since all formulas, valid and invalid, would be provable; but it would be unsound. Conversely, if our proof system for PL, say, were the same as it is in this book except that it did not allow conditional or indirect proof, it would be sound, allowing us to prove a subset of what we can prove now, but not complete, since no valid formulas would be provable; all derivations would have to start with some premises.

Decidability One last interesting metalogical property is called decidability. A theory is decidable if, and only if, there is an effective procedure for determining whether a formula is logically true and whether an argument is valid. Our definition of decidability relies on the concept of an effective procedure or method. Intuitively, it’s easy to see what an effective method is: it’s just an algorithm that can be followed mechanically. Sometimes such methods are called decision procedures. An important conjecture, called Church’s thesis, says that our intuitive notions of an effective procedure correspond to certain formal properties, that there are formal ways of specifying the intuitive notion of an algorithm or effective procedure. The formal methods use the concept of recursion, which I will not pursue here. PL and M are decidable theories. Interestingly, F is not decidable, as Alonzo Church proved in 1936 (see Hunter, 1996, section 52). There are procedures that will yield the validity of any valid formula, but, for any procedure, some invalid formulas will never be shown invalid. The undecidability of first-order logic is closely related to the famous halting problem in computer science; see Boolos, Burgess, and Jeffrey, 2007.

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Summary While proofs of the particular results discussed in this section are left for more advanced logic texts like the ones listed below, the results themselves have important consequences for the work in this book. What theories, precisely, do we identify as logical, rather than mathematical or set-theoretic? Many mathematical concepts can be expressed using the tools in this book, especially the language F and its special identity predicate. Is there a difference between mathematics and logic? Mathematics seems to talk about a domain of numbers, sets, spaces, knots, and graphs (and other mathematical objects). Logic has no special subject matter; it’s about everything. For those who believe that there is a definite division between logic and mathematics, the metalogical results are especially important in helping to characterize that dividing line. Exploring metalogic can be salutary for another reason. While many people enjoy working within logical systems, logicians work on metalogic, developing different logical theories and proving things about them, far more often than they work within an object language. If you fall in love with logic, as I did, it’s good to know how logicians really work. TELL ME MORE  • How does metalogic help us try to distinguish between logic and mathematics? See 7.7: Logicism.

For Further Research and Writing 1. Explore various metalogical results by consulting the sources below. Especially interesting for those who like the technical aspects of logic are the proofs of consistency and soundness for our logical theories and the more complicated (generally) proofs of completeness. 2. Describe the boundary between complete theories and incomplete theories. Books on Gödel’s incompleteness theorems are legion; Nagel and Newman’s is generally solid and accessible. Goldstein integrates more context. Hunter’s Metalogic approaches the important results formally. 3. The history of the important developments in logic and metalogic in the early twentieth century can be fun to explore. Two excellent and engaging books are Doxiadis and Papadimitriou, Logicomix: An Epic Search for Truth, which explores the early days of logic in a graphic novel (seriously), and Feferman and Feferman, Alfred Tarski: Life and Logic, which tracks the amazing life of one of the most important logicians of that (or, really, any) era.

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Suggested Readings Bergmann, Merrie, James Moor, and Jack Nelson. The Logic Book, 4th ed. Boston: McGrawHill, 2004. This is an introductory logic text, much like this book, but with an alternative inference system and more metalogic. It’s accessible if you can work with the different systems. Boolos, George, John Burgess, and Richard Jeffrey. Computability and Logic, 5th ed. Cambridge: Cambridge University Press, 2007. A logic text on computability, useful for a next course on logic or for someone with good technical skills. Doxiadis, Apostolos, and Christos H. Papadimitriou. Logicomix: An Epic Search for Truth. New York: Bloomsbury, 2009. A comic book about the exciting developments in logic in the late nineteenth and early twentieth centuries. Feferman, Anita Burdman, and Solomon Feferman. Alfred Tarski: Life and Logic. Cambridge: Cambridge University Press, 2004. An engrossing biography of a one of the most important and enigmatic logicians. Goldfarb, Warren. Deductive Logic. Indianapolis: Hackett, 2003. Another introductory logic text, but of a very different sort, and well written. The ‘Reflection’ sections are especially useful. Goldstein, Rebecca. Incompleteness: The Proof and Paradox of Kurt Gödel. New York: Norton, 2005. Goldstein provides compelling narrative context for Gödel’s work: personal, mathematical, and philosophical. Hofstadter, Douglas. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books, 1999. Hofstadter won a Pulitzer Prize for this examination of the self-referential connections among logic, mathematics, computer science, art, and music. Hunter, Geoffrey. Metalogic. Berkeley: University of California Press, 1996. Hunter’s book is an excellent place to find proofs of consistency, soundness, completeness, and an array of other metalogical concepts. The language is different from ours, and Hunter uses an axiomatic logical theory. But his work is clear and should be accessible to students who have mastered the concepts in this book. Hunter’s discussions of core concepts are especially lucid and helpful. Kneale, William, and Martha Kneale. The Development of Logic. Oxford, UK: Clarendon Press, (1962) 2008. The final chapter of Kneale and Kneale’s survey focuses on metalogic. While the material is somewhat dated, the authors provide detailed proofs, with interesting and useful historical information, and the writing is generally accessible. Mendelson, Elliott. Introduction to Mathematical Logic, 4th ed. Boca Raton, FL: Chapman & Hall/CRC, 1997. Perhaps the broadest, most careful, and fecund book on mathematical logic. Students working through even the smallest portions of this text will need some guidance. Nagel, Ernest, and James R. Newman. Gödel’s Proof, rev. ed. New York: New York University Press, 2001. The new edition of a classic presentation clears up some infelicities in the original and remains accessible and clear. Quine, W. V. Methods of Logic, 4th ed. Cambridge, MA: Harvard University Press, 1982. Quine’s classic and important logic text integrates more metalogic and philosophy of

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logic than most current logic books, including Quine’s erudite and idiosyncratic views and a good range of historical notes. It’s worth the effort to persevere through his annoying dot notation. Quine, W. V. Philosophy of Logic, 2nd ed. Cambridge, MA: Harvard University Press, 1986. Quine’s classic work defends the classical logic of this book as a canonical language; see especially chapter 5. Shapiro, Stewart. “Classical Logic.” In The Stanford Encyclopedia of Philosophy, edited by E. N. Zalta. Stanford University, Winter 2013. http://plato.stanford.edu/archives/win2013/ entries/logic-classical/. Shapiro provides a concise and robust introduction to logic with a substantial section on metalogic that presents proofs of many important results. Sider, Ted. Logic for Philosophy. Oxford: Oxford University Press, 2010. Sider’s text is broader than Hunter’s, and covers lots of advanced topics in greater detail than this book does. Van Heijenoort, Jean. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge: Harvard University Press, 1967. This collection contains many of the most important papers from the early days of modern mathematical logic, including Frege’s Begriffsschrift and Gödel’s completeness and incompleteness theorems. The papers tend to be difficult to read, but some are more accessible than you might expect.

6.5: MODAL LOGICS Modal logics are formal languages with a wide variety of applications. Modal operators, the central concepts of modal logics, are modifiers of propositions. There are lots of ways of modifying or characterizing a proposition. So, there are lots of modal logics. In this section, we will look a bit at propositional modal logic. The formal study of modern modal logics began with C. I. Lewis in the early twentieth century. Interest in modal logic became more serious with work from Ruth Barcan Marcus, starting in the 1940s, and Saul Kripke and David Lewis, starting in the 1960s. Today, there are many varieties of modal logics, many different formal systems and many different interpretations of the standard modal operators.

Modal Operators The sentences 6.5.1–6.5.3 all contain modifications of the proposition that the sun is shining. 6.5.1 6.5.2 6.5.3

It is not the case that the sun is shining. It is possible that the sun is shining. It is necessary that the sun is shining.

6.5.1 contains a sentential operator, negation, which is a function that takes truth values to their opposites. 6.5.2 and 6.5.3 contain sentential operators, too. These operators are called modal operators. The logic of these kinds of operators is called modal logic. Since modal operators are sentential operators, most of their interesting properties are available in propositional logic, though modal logics can naturally be extended

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into predicate logics, too. The tools that have been developed for modal logic are useful in a wide variety of logics, including conditional logics. The vocabulary for typical formal systems of propositional modal logic is the same as that for PL, plus two, interrelated modal operators. The formation rules for modal logic are also the same as those for PL, with one addition of a rule for the operators, which may appear in front of any wff. Formation Rules for Propositional Modal Logic (PML) PML1. A single capital English letter is a wff. PML2. If α is a wff, so is ∼α. PML3. If α and β are wffs, then so are (α ∙ β), (α ∨ β), (α ⊃ β), and (α ≡ β). PML4. If α is a wff, then so are ♢α and ◽α. PML5. These are the only ways to make wffs. The primary interpretation of the modal logic symbols are the operators we see at 6.5.2 and 6.5.3, taking ‘♢’ for ‘possibly’ or ‘it is possible that’ and ‘◽’ as ‘necessarily’ or ‘it is necessary that’. The result is called alethic modal logic. There are several other interpretations of the same modal operators including deontic interpretations, which are useful in ethics; temporal interpretations for logics of time; and epistemic interpretations for logics of belief or knowledge. Different interpretations lead to different logics, with different semantics and rules. We will focus on the alethic interpretation, but I will mention others along the way.

Alethic Operators and Their Underlying Concepts In alethic modal logic, the ‘♢’ is interpreted as ‘it is possible that’. If we take ‘S’ to stand for ‘the sun is shining’, then 6.5.2 is regimented as ‘♢S’. So, ‘♢∼S’ means that it is possible that the sun is not shining. ‘∼♢S’ means that it is not possible that the sun is shining. And ‘∼♢∼S’ means that it is not possible that the sun is not shining. That last sentence is, you may notice, equivalent in meaning to 6.5.3. In austere versions of modal logic, only one modal operator is taken as basic, and the other is introduced by definition, according to the equivalences 6.5.4 and 6.5.5. 6.5.4 6.5.5

◽ α ♢α

→ ∼♢∼α ← → ∼◽∼α ←

Studies of alethic operators trace back to Leibniz’s interest in possible worlds. Leibniz thought that we lived in the best of all possible worlds, and that this fact followed from the power and goodness of God. Our interest in this section is in the formalization of the tools with which Leibniz framed his theodicy and his distinction between contingency and necessity, which he grounded in a distinction between finite and infinite analysis. The distinction between necessary claims and contingent ones is really a central theme throughout much of the history of philosophy. Some philosophers, like Descartes, draw a firm line between the two. Others deny a real distinction. Spinoza, for example, takes all claims to be necessary. Mill is skeptical of any necessity.

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Today, many philosophers continue to have some interest in possibility and necessity. Philosophers of mind ask whether it is possible for beings physically precisely like us to lack consciousness. Philosophers of science wonder whether laws of nature are necessary or contingent. Metaphysicians seek the essential, or necessary, properties of people. Philosophers of language work on the connection between referring terms, especially names and natural kind terms, and necessary properties of things. We also have lots of ordinary beliefs about possibilities and necessities. Obama won the U.S. presidential election in 2012. But many of us believe that he could have lost that election and that Romney could have won. In the actual world, Obama won. In some other possible worlds, Romney won. Talk of possible worlds can just be a way of expressing such ordinary beliefs; in another possible world, Romney won the 2012 election. Modal logic gives us a way to regiment such claims. Still, there are lots of different ways to think about possible worlds and their meanings. The language of ‘possible worlds’ can make them seem like something out of science fiction. But talk about possible worlds can be just a way of thinking about other ways in which this world might be. When I say that there is a possible world in which I have a full head of hair and a possible world in which I am completely bald, I can just mean that I could be hairier or balder, not that there is some other person in some other universe who has those properties. But possible worlds are contentious and some people do think of possible worlds as different from this one. The world in which I am less bald may require other changes to the laws or regularities that led to my current lack of hair. It is not clear, say some folks, that you can just stipulate one change to our world without changing other factors. Consequently, we should not claim that the denizens of other possible worlds are ourselves. Instead, we can think of people and objects in other possible worlds as our counterparts, related to us, but not identical to us. The question of whether other possible worlds contain me or my counterpart is related to the question of how to think about the reality of other possible worlds. Some philosophers see them thinly, as just ways of talking about this world. Others, modal realists, think of possible worlds as real and distinct from the actual world. On modal realism, all possible worlds exist, just as the actual world exists. The questions of how to understand possible worlds and the objects in them are deep and interesting. Here, we focus on the formal tools of modal logic in order to frame such questions. We can use the language of possible worlds to interpret sentences of formal modal logic. We will see two kinds of modal logic semantics, which we will call Leibnizian and Kripkean.

Actual World Semantics To interpret sentences about the real world, the language of truth tables suffices. In section 2.3, we saw the semantics for the operators of PL using truth tables. Another way to convey the same information, one that will be useful in adapting to PML, is at 6.5.6 taking ‘V ’ for ‘the truth value of ’.

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V(∼α) = 1 if V(α) = 0; otherwise V(∼α) = 0 V(α ∙ β) = 1 if V(α) = 1 and V(β) = 1; otherwise V(α ∙ β) = 0 V(α ∨ β) = 1 if V(α) = 1 or V(β) = 1; otherwise V(α ∨ β) = 0 V(α ⊃ β) = 1 if V(α) = 0 or V(β) = 1; otherwise V(α ⊃ β) = 0 V(α ≡ β) = 1 if V(α) = V(β); otherwise V(α ≡ β) = 0 6.5.6

Consider the propositions P–S at 6.5.7. 6.5.7

P: The penguin is on the TV. Q: The cat is on the mat. R: The rat is in the hat. S: The seal is in the sea.

Suppose that we know the actual truth values of P–S, the values in our world, which we can call w1. 6.5.8

In w1, P is true, Q is true, R is true, and S is true.

Given the truth assignments at 6.5.8, we can easily translate the claims in exercises 6.5a and determine their truth values.

EXERCISES 6.5a Translate each of the following claims into English, and determine their truth values given the translation key and truth assignments above. 1. P ⊃ Q

3. ∼(R ∨ S)

2. P ⊃ R

4. (P ∙ Q ) ⊃ R

5. (Q ∨ ∼R) ⊃ ∼P

Semantics for Other Worlds Now, let’s consider another world, called w2 , in which the propositions at 6.5.7 have the truth values at 6.5.9. 6.5.9

In w2, P and Q are true, and R and S are false.

In other words, in w2 , the penguin is on the TV and the cat is on the mat, but the rat is not in the hat and the seal is not in the sea. To represent the differences between w1 and w2 , we need to extend actual world semantics to possible world semantics. In actual world semantics, valuations are one-place functions. In possible world semantics, valuations are two-place functions, of the proposition and the world at which we are considering the proposition. At each world, the valuations are just as in the

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actual world. But by indexing our formulas, we can consider statements whose truth values depend on the facts in other possible worlds. First, we consider a universe, U, which is just a set of worlds: {w1, w2 , w3, . . . ,w n} Then, we introduce the indexed rules at 6.5.10. 6.5.10 V(∼α, wn) = 1 if V(α, wn) = 0; otherwise V(∼α, wn) = 0 V(α ∙ β, wn) = 1 if V(α, wn) = 1 and V(β, wn) = 1; otherwise   V(α ∙ β, wn) = 0 V(α ∨ β, wn) = 1 if V(α, wn) = 1 or V(β, wn) = 1; otherwise   V(α ∨ β) = 0 V(α ⊃ β, wn) = 1 if V(α, wn) = 0 or V(β, wn) = 1; otherwise   V(α ⊃ β, wn) = 0 V(α ≡ β, wn) = 1 if V(α, wn) = V(β, wn); otherwise V(α ≡ β, wn)   = 0

At each world, the semantic rules are just as they are in the actual world. We introduce alternate worlds only in order to provide semantics for the modal operators, which we will do in the next subsection. First, let’s apply this general structure by considering a small universe of three worlds, at 6.5.11. 6.5.11 U = {w1, w2, w3}

At w1, P, Q, R, and S are all true. At w2, P and Q are true, but R and S are false. At w3, P is true, and Q, R, and S are false.

Let’s return to the sentences from exercises 6.5a, evaluating them for w2 and w3, as in exercises 6.5b.

EXERCISES 6.5b Determine the truth values of each of the following claims at w2 and w3, given the values at 6.5.11 (You should come up with ten answers, five for w2 and five for w3.) 1. P ⊃ Q

3. ∼(R ∨ S)

2. P ⊃ R

4. (P ∙ Q ) ⊃ R

5. (Q ∨ ∼R) ⊃ ∼P

Possible World Semantics (Leibnizian) We have looked at non-modal propositions in other possible worlds. We can now give some interpretation sentences using modal operators, like ‘♢P’. It is easy enough to

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translate such sentences. ‘♢P’ expresses the proposition that it is possible that the penguin is on the TV. But we need a semantics to determine their truth. 6.5.12 is a Leibnizian semantics for the modal operators. 6.5.12 V(◽α) = 1 if V(α, wn) = 1 for all wn in U V(◽α) = 0 if V(α, wn) = 0 for any wn in U V(♢α) = 1 if V(α, wn) = 1 for any wn in U V(♢α) = 0 if V(α, wn) = 0 for all wn in U So, ‘♢S’ is true, since there is at least one world in which the seal is in the sea, w1. Similarly, ‘◽Q’ will be false, since there is at least one world in which the cat is not on the mat, w3.

EXERCISES 6.5c Determine the truth values of each of the following propositions given the values at 6.5.11 and the semantics at 6.5.12. 1. ◽(P ⊃ Q )

4. ♢P ⊃ ♢Q

2. ♢(P ⊃ Q )

5. ♢[(Q ∨ ∼R) ⊃ ∼P]

3. ◽P ⊃ ◽Q

6. ♢P ⊃ [Q ⊃ ◽(R ∙ S)]

Different Worlds, Different Possibilities Leibnizian semantics is generally taken to suffice to express logical possibility. But there are more kinds of possibility than logical possibility. For example, it is logically possible for a bachelor to be married, for objects to travel faster than the speed of light, and for a square to have five sides. These claims are logically possible, even though they might be semantically impossible, or physically impossible, or mathematically impossible. Concomitantly, possibility might vary among worlds. In our world, it is impossible to travel faster than the speed of light or for the force between two objects to vary with the cube of the distance between them. Perhaps there are other possible worlds with different physical laws. A more subtle version of modal logic might be useful to express relations among different sets of possibilities. Consider two possible worlds. The first world is ours. The second world is like ours, except that there is some force that moves the planets in perfectly circular orbits. The law that says that all planets move in elliptical orbits holds in both worlds, since a

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circle is just a type of ellipse. So, w2 obeys all the laws of w1, but w1 does not obey all the laws of w2 .

Possible World Semantics (Kripkean) The modal logician handles variations among possibilities at different worlds in terms of accessibility relations. In our scenario above, w2 is accessible from w1, but w1 is not accessible from w2 . The accessibility relation thus provides a description of the relationships among different possible worlds. Accessibility relations entail that propositions that are possible at some worlds are not possible at other worlds. 6.5.13 shows an accessibility relation that we can stipulate among our worlds w1, w2 , and w3. 6.5.13 R = {, , , , , }

R says that all worlds are accessible from themselves, that world 1 is accessible from worlds 2 and 3, and that world 2 is accessible from world 3. Accessibility relations can be given diagrammatically, too. By providing accessibility relations among possible worlds, we extend our Leibnizian semantics into what is known as a Kripkean semantics.

W3 W2 W1 For alethic operators in a Kripkean semantics, a statement is possible not if there is a world in which it is true, but if there is an accessible world in which it is true. A statement is necessary at a world if it is true at all accessible worlds. Such semantics are shown at 6.5.14. 6.5.14 V(◽α, wn) = 1 if V(α, wm) = 1 for all wm in U such that    is in R V(◽α, wn) = 0 if V(α, wm) = 0 for any wm in U such that    is in R V(♢α, wn) = 1 if V(α, wm) = 1 for any wm in U such that    is in R V(♢α, wn) = 0 if V(α, wm) = 0 for all wm in U such that    is in R

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The introduction of accessibility relations turns modal logic into an enormous field. For example, an accessibility relation might be an equivalence relation: reflexive, symmetrical, and transitive. A relation is reflexive if every object bears the relation to itself. A relation is symmetric if a bearing R to b entails that b bears R to a. A relation is transitive if given that a bears R to b and b bears R to c, it follows that a bears R to c. If the accessibility relation on a universe is an equivalence relation, the universe is Leibnizian. But there are much more restrictive kinds of accessibility relations. For example, in the case above, every world is accessible from itself, so the relation is reflexive. The relation is not symmetric, though, since w2 is accessible from w1 , but w1 is not accessible from w2 . It is transitive. Let’s summarize our theory of our located animals in three different worlds at 6.5.15 and see how they fare in a Kripkean universe with the accessibility relation R. 6.5.15 U = {w1, w2, w3}

At w1, P, Q, R and S are all true. At w2, P and Q are true, but R and S are false. At w3, P is true, and Q, R, and S are false. R = {, , , , ,   }

Since the values of our propositions vary across worlds, it is convenient to indicate the world at which we intend to express the propositions. I’ll add a subscript to each formula. So, ‘P1’ means that the penguin is on the TV in world 1; ‘◽(P ⊃ Q )2’ means that in world 2, ‘P ⊃ Q’ is necessary; and ‘♢P3 ⊃ ◽P2’ means that if it is possible that the penguin is on the TV at world 3, then it is necessary that the penguin is on the TV at world 2.

EXERCISES 6.5d Determine the truth values of each of the following formulas, given the universe described at 6.5.15. 1. ◽(P ⊃ Q )1

4. ♢∼(Q ∨ R)2

6. ◽P1 ⊃ ◽Q  1

2. ◽(P ⊃ Q )3

5. ♢∼(Q ∨ R)3

7. ◽P3 ⊃ ◽Q  3

3. ♢∼(Q ∨ R)1

System S5 We can develop systems of inference for all modal logics. Leibnizian modal logic, ordinarily called S5, has a variety of rules. Most formal systems of modal logic are either

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axiomatic or use truth trees. 6.5.16 is an axiomatic presentation of S5, with five axiom schema and two rules of inference. 6.5.16

S5 Axioms and Rules

PL Duality K T 5

Any tautology of PL ♢α ≡ ∼◽∼α ◽(α ⊃ β) ⊃ (◽α ⊃ ◽β) ◽α ⊃ α ♢α ⊃ ◽♢α

Necessitation α / ◽α MP α ⊃ β, α / β

The rule of inference necessitation may look a little odd, until you recognize that the only propositions we prove in logic, without premises, are logical truths.

Other Modal Logics Different accessibility relations determine different modal systems. Different modal systems support different modal intuitions. For example, the alethic interpretation demands a reflexive relation. If an accessibility relation is not reflexive, then it would be possible for ‘◽P’ to be true in the same world in which ‘P’ were false. But if P is necessarily true, then it surely is true. A non-reflexive relation might support a deontic interpretation of the modal operators, though. On a deontic interpretation, ‘◽P’ can be used to express that P is morally obligatory; ‘♢P’ says that P is morally permissible. Since people sometimes fail to meet their ethical obligations, the accessibility relations for deontic logics need not be reflexive. Modal systems can be characterized both by their accessibility relations and by the modal claims they generate. Modal system K has the proposition K, above, as a characteristic axiom. The derivation system based on K has fewer rules, only K and PL (as axioms), modus ponens, and necessitation, though we can derive another rule, called regularity, 6.5.17. 6.5.17 Regularity α ⊃ β ⊢ ◽α ⊃ ◽β

We can show that regularity is valid in K by deriving, at 6.5.18, ‘◽P ⊃ ◽Q’ from an arbitrary ‘P ⊃ Q’. 6.5.18 1. P ⊃ Q 2. ◽(P ⊃ Q) 3. ◽(P ⊃ Q) ⊃ (◽P ⊃ ◽Q) 4. ◽P ⊃ ◽Q QED

/ ◽P ⊃ ◽Q 1, Nec Axiom K 3, 2, MP

Note that Reg is valid in both K and S5, since the rules used to derive it are present in both systems.

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Now, let’s do a slightly more complicated proof in K, of the modal claim ‘◽(P ∙ Q ) ⊃ (◽P ∙ ◽Q )’, at 6.5.19. 6.5.19 1. (P ∙ Q) ⊃ P PL 2. ◽(P ∙ Q) ⊃ ◽P 1, Reg 3. (P ∙ Q) ⊃ Q PL 4. ◽(P ∙ Q) ⊃ ◽Q 3, Reg 5. [◽(P ∙ Q) ⊃ ◽P] ⊃ {[◽(P ∙ Q) ⊃ ◽Q] ⊃ [◽(P ∙ Q) ⊃ (◽P ∙ ◽Q)]} PL 6. [◽(P ∙ Q) ⊃ ◽Q] ⊃[◽(P ∙ Q) ⊃ (◽P ∙ ◽Q)] 5, 2, MP 7. ◽(P ∙ Q) ⊃ (◽P ∙ ◽Q) 6, 4, MP QED

Note that the theorem used at line 5 has a simple instance: ‘(P ⊃ Q ) ⊃ {(P ⊃ R) ⊃ [P ⊃ (Q ∙ R)]}’. K is a weak modal logic. A slightly stronger logic, D, has the characteristic axiom at 6.5.20. 6.5.20 D ◽α ⊃ ♢α

Every theorem provable in K is provable in D, but D allows for more to be proven. Thus, it is a stronger logic, and more contentious. Let’s consider the meaning of the characteristic axiom D. In alethic interpretation, it means that if a statement is necessary, then it is possible. This seems reasonable, since necessary statements are all true. And, true statements are clearly possible. But, an interesting fact about D is that the theorem T, at 6.5.21, is not provable: 6.5.21 T ◽α ⊃ α

Thus, D seems like a poor logic for the alethic interpretation. But there are other interpretations of the modal operators, like the deontic one. The characteristic axiom of D seems true under the deontic interpretation since it is natural to believe that I am permitted to perform any action that I am obliged to do. And since the fact that an action is obligatory does not entail that people actually do it, that T is not provable in D supports using D for a deontic logic. Other interpretations of the modal logic symbols give us epistemic logics. For one epistemic logic, we take ‘◽P’ to mean that P is known, and ‘♢P’ to mean that P is compatible with things that are known. Hintikka’s epistemic logic takes the three axioms at 6.5.22. 6.5.22 K ◽(α ⊃ β) ⊃ (◽α ⊃ ◽β) T ◽α ⊃ α 4 ◽α ⊃ ◽ ◽α

Any logic with the T axiom will have a reflexive accessibility relation. Any logic with the 4 axiom will also have a transitive accessibility relation. Note that 4 is contentious in epistemic logic. It’s the KK thesis, that if you know some proposition, you also know that you know it, a claim that many epistemologists reject. A system characterized by K, T, and 4 is called S4.

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One can get a slightly stronger logic by adding a condition of symmetry to the accessibility relation. Then, we have S5, which takes K, T, 4, and B, at 6.5.23, as characteristic. 6.5.23 B α ⊃ ◽♢α

There are other interpretations of the modal operators. In some temporal logics, ‘◽P’ means that P is always the case, and ‘♢P’ means that P is the case at some point in time. Temporal logics can be extended to include tense operators, for the future and the past. The modal operators can also be used to express metalogical concepts. Hartry Field uses the modal operator to represent consistency, which is ordinarily not represented in an object language. On Field’s view, ‘♢(P ∙ Q ∙ R)’ means that P, Q , and R are consistent. He argues that the consistency of a set of sentences is actually a logical notion, and so we should have symbols in the language to represent it.

Modal Logic: Questions and Criticism To complete our discussion of modal logics, let’s return to the previous, alethic interpretation of the modal operators. We can ask at least two types of questions about alethic modal logics. The first type of question is about the status of possible worlds. What is a possible world? Do possible worlds exist? Do they exist in the way that the real world exists? Are they abstract or concrete objects? These are all metaphysical questions. The second type of philosophical question about possible worlds is epistemic. How do we know about possible worlds? Do we stipulate them? Do we discover them, or facts about them? Do we learn about them by looking at our world? Do we learn about them by pure thought? The metaphysical questions are linked to the epistemological questions. If possible worlds are real, and independent of the actual world, then knowledge of possible worlds seems difficult to explain. Indeed, many philosophers abjure the concepts of necessity and possibility precisely because it seems difficult to reconcile knowledge of them with our view of ourselves as natural things whose capacities for learning about the world are mainly sensory. Despite modal operators being used in the early days of modern logic, research on modal logics stalled for a long time, perhaps due mainly to such epistemic concerns about alethic modalities. As formal systems of modal logic, especially Kripkean systems, were developed and found to be robust and fruitful formal tools, philosophers have become more confident appealing to modality. Still, persistent worries about modalities raise serious questions. We end with a glance at one of the most influential criticisms of modalities, from Quine. Consider the claims 6.5.24 and 6.5.25. 6.5.24 6.5.25

Nine is greater than seven. The number of planets is greater than seven.

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6.5.24 and 6.5.25 have the same truth value. And we expect that they will have the same truth value, since one can be inferred from the other by a simple substitution, given 6.5.26. 6.5.26

The number of planets is eight.

Now, transform 6.5.24 and 6.5.25 into the modal claims 6.5.27 and 6.5.28. 6.5.27 6.5.28

Necessarily, nine is greater than seven. Necessarily, the number of planets is greater than seven.

6.5.27 and 6.5.28 have different truth values: the former is true and the latter is false. But 6.5.28 seems to follow from 6.5.26 and 6.5.27 by substitution of identicals, just as 6.5.25 followed from 6.5.24 and 6.5.26. The modal context invalidates the inference. Our logics PL, M, F, and FF are all extensional, allowing such substitutions. But modal logic is intensional, and so does not allow us always to substitute coreferential terms like the ones on the two sides of ‘is’ at 6.5.26. This intensionality makes modal logics highly controversial.

Summary The two modal operators can do a lot of work and lead to a thrilling variety of modal logics. They can be variously interpreted to construct alethic, deontic, epistemic, temporal, and other logics. They can be used in different systems of various strengths and subtle differences, with different characteristic axioms and rules, and they can illuminate lasting philosophical debates. The variety of modal logics can be dizzying as well as fecund. The ambitious student of modal logic should be prepared to approach the formal work carefully and patiently. TELL ME MORE  • What is intensionality? See 6.1: Notes on Translation with PL and 6S.7: The Propositions of Propositional Logic. • How does Leibniz invoke possible worlds? See 7.4: Logic and the Philosophy of Religion. • What do axiomatic systems of logic look like? See 6S.11: Axiomatic Systems.

For Further Research and Writing 1. One excellent, and potentially challenging, approach to enriching your logical studies is to work through one of the many recent modal logic textbooks. For those seeking illumination of philosophical questions, the books by Girle and Garson are especially useful and engaging. 2. Leibniz’s distinction between necessity and contingency rests on a distinction between finite and infinite analysis, which he discusses especially in sections 29–37 of Monadology. What is the difference between finite and infinite analysis? What is the result of analysis, in Leibniz’s view? Is such a distinction useful today?

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3. David Lewis and Saul Kripke each explore modal logics for their relevance to metaphysical questions, especially questions about names and objects and persons. Both Lewis’s modal realism and Kripke’s intuitive methods have been extensively criticized. Kripke claims that possible worlds are just ways of talking about alternative histories of this world, whereas Lewis believes that other possible worlds exist, concretely, just like the real world. Kripke believes that we exist in other possible worlds; Lewis argues that other worlds contain (or don’t) our counterparts. There are many different topics here worth further exploration. Sainsbury’s chapter on necessity may be useful. 4. Quine’s influential criticisms of modal logic were sometimes countered by a leading proponent of modal logic, Ruth Barcan Marcus. Discuss both Quine’s concerns about modalities and Marcus’s responses. Chapters from Haack, Read, and Grayling can each be helpful.

Suggested Readings Chellas, Brian. Modal Logic. Cambridge: Cambridge University Press, 1980. A classic. The exercises are often challenging. Fitting, Melvin, and Richard Mendelsohn. First-Order Modal Logic. Dordrecht, The Netherlands: Kluwer, 1998. An excellent modal logic textbook, with efficient and clear technical work. Garson, James. Modal Logic for Philosophers, 2nd ed. Cambridge: Cambridge University Press, 2013. An excellent modal logic textbook that uses natural deduction systems and trees, and integrates useful discussions of the philosophical relevance of the technical work. Girle, Rod. Modal Logics and Philosophy, 2nd ed. Montreal and Kingston: McGill-Queen’s University Press, 2009. Less comprehensively technical than Fitting and Mendelsohn or Garson, Girle’s book has an excellent survey of the applications of modal logic to philosophy and uses mainly truth trees. Grayling, A. C. An Introduction to Philosophical Logic. Malden, MA: Blackwell, 1997. See especially chapter 3, “Necessity, Analyticity, and the A Priori.” Haack, Susan. Philosophy of Logics. Cambridge: Cambridge University Press, 1978. Chapter 10 focuses on the philosophical debates around modal logic. Kripke, Saul. Naming and Necessity. Cambridge, MA: Harvard University Press, 1980. A broad and accessible philosophical exploration of the necessity and its consequences in logic, language, metaphysics, and philosophy of mind. Unlike Lewis, Kripke defends the view that the same individual can exist in various possible worlds. Leibniz, G. W. Monadology. 1714. In G.W. Leibniz: Philosophical Essays, edited by Roger Ariew and Daniel Garber, 213–25. Indianapolis: Hackett, 1989. Lewis, David. On the Plurality of Worlds. Oxford, UK: Blackwell, 1986. A defense of modal realism—that all possible worlds exist, in the same way that the actual world exists— from its most prominent proponent. In contrast to Kripke, Lewis defends the counterpart relations among individuals in different worlds.

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Marcus, Ruth. Modalities. New York: Oxford University Press, 1993. A wide-ranging and insightful collection of essays from one of the most important modal logicians of the twentieth century. Priest, Graham. An Introduction to Non-Classical Logic: From Ifs to Is, 2nd ed. Cambridge: Cambridge University Press, 2008. All of the nonclassical logics in Priest’s book are approached using techniques from modal logic. Quine, W. V. “Reference and Modality.” In From a Logical Point of View, 2nd ed., 139–159. Cambridge, MA: Harvard University Press, 1980. And W. V. Quine, “Three Grades of Modal Involvement,” in The Ways of Paradox and Other Essays, rev. and enlarged ed. (Cambridge, MA: Harvard University Press, 1976), 158–176. Two important anti-­ modality essays from a leading modal skeptic. See also Ruth Marcus, “A Backwards Look at Quine’s Animadversions on Modalities,” in Perspectives on Quine, edited by Robert Barrett and Roger Gibson, 230–244 (Cambridge: Blackwell, 1990); with Quine’s reply to Marcus. Read, Stephen. Thinking About Logic. Oxford: Oxford University Press, 1995. Chapter 4, “The Incredulous Stare: Possible Worlds,” has good discussions of the metaphysics of possible worlds and more good references. Sainsbury, Mark. Logical Forms: An Introduction to Philosophical Logic, 2nd ed. Oxford, UK: Blackwell, 2001. Chapter 5, “Necessity,” is a relatively gentle further introduction to modal logic, both propositional and predicate logic, with detailed discussions of deep philosophical questions.

Solutions to Exercises 6.5a 1. 2. 3. 4.

If the penguin is on the TV, then the cat is on the mat; true If the penguin is on the TV, then the rat is in the hat; true Neither the rat is in the hat nor the seal is in the sea; false If the penguin is on the TV and the cat is on the mat, then the rat is in the hat; true 5. If either the cat is on the mat or the rat is not in the hat, then the penguin is not on the TV; false

Solutions to Exercises 6.5b 1. 2. 3. 4. 5.

True at w2 , false at w3 False at w2 and w3 True at w2 and w3 False at w2 , true at w3 False at w2 and w3

Solutions to Exercises 6.5c 1. False 2. True

3. False 4. True

5. False 6. False

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Solutions to Exercises 6.5d 1. True 2. False 3. False

4. False 5. True

6. True 7. False

6.6: NOTES ON TRANSLATION WITH M In sections 4.1 and 4.2, we looked at translation between monadic predicate logic and English, focusing on the core aspects (how to use predicates, singular terms, and quantifiers) and some general rules for working with sentences with multiple predicates and multiple quantifiers. In this section, I discuss a few of the subtleties of translation in M using the tools of our inferential system for M presented in sections 4.4–4.6 and our semantics for M in sections 4.7–4.8. First, let’s look at a controversy about our use of conditionals as the main connectives in the subformulas of universally quantified formulas.

Universally Quantified Formulas and Existential Import In predicate logic, we ordinarily use conditionals with universally quantified expressions, as at 6.6.1. 6.6.1 All hippogriffs are aggressive. (∀x)(Hx ⊃ Ax)

In other words, we take 6.6.1 to say that if there are hippogriffs, then they are aggressive. We sometimes call this the Boolean interpretation of universally quantified formulas, after the nineteenth-century logician George Boole. But using conditionals with universally quantified propositions in this Boolean way reinforces a particular answer to a controversial question regarding the existential import of such claims. To see the controversial question, consider 6.6.2. 6.6.2

All orangutans are mammals.

The controversy is whether to understand 6.6.2 as 6.6.3 or as 6.6.4. 6.6.3 6.6.4

If something is an orangutan, then it is a mammal. There are orangutans, and they are all mammals.

6.6.3 is the Boolean interpretation. 6.6.4 may be called Aristotelian, since Aristotle believed that universal claims about existing things have existential import. To see the difference between the Boolean and the Aristotelian interpretations, notice that 6.6.3 may be taken as vacuously true but 6.6.4 could not. Orangutans are critically endangered. Were they, sadly, to become extinct, 6.6.3 would remain true but 6.6.4 would be false. So, is it better to understand 6.6.2 as 6.6.3 or as 6.6.4? The answer to this controversial question is settled in favor of the Boolean interpretation in standard classical logic, like the logic of this book. But that is not to say that the Boolean interpretation is the right one.

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‘And’s and ‘Or’s and Universally Quantified Formulas The English ‘and’ may be represented by a disjunction when it appears in the antecedent of a universally quantified formula, as at 6.6.5. That sentence may be regimented either as 6.6.6, in which the ‘and’ is a disjunction, or 6.6.7, in which it is not. 6.6.5 All planets and asteroids are rocky. 6.6.6 (∀x)[(Px ∨ Ax) ⊃ Rx] 6.6.7 (∀x)(Px ⊃ Rx) ∙ (∀x)(Ax ⊃ Rx)

To show that 6.6.6 and 6.6.7 are equivalent, we can do two derivations: 6.6.6 from 6.6.7 and 6.6.7 from 6.6.6. (The derivations are in an appendix to this section.) Since 6.6.6 and 6.6.7 are derivable from each other, either they are logically equivalent or our logic is inconsistent (which it’s not). The equivalence of these two ways of expressing common properties of different groups is useful but does not extend to related pairs of claims that invoke existential quantifiers instead of universal quantifiers. So 6.6.8 and 6.6.9 are not equivalent. 6.6.8 (∃x)[(Px ∨ Ax) ∙ Rx] 6.6.9 (∃x)(Px ∙ Rx) ∙ (∃x)(Ax ∙ Rx)

Let’s take ‘Px’ to stand for ‘x is a planet’, ‘Ax’ to stand for ‘x is an asteroid’, and ‘Rx’ to stand for ‘x is rocky’. Then 6.6.9 says that some planets are rocky and some asteroids are rocky. This claim requires at least two objects, a rocky planet and a rocky asteroid, while 6.6.8 requires only one object, either a rocky planet or a rocky asteroid. 6.6.8 follows easily from 6.6.9; again, I’ll put the derivation in the appendix. But 6.6.9 does not entail 6.6.8. There is a counterexample in a domain of one object, where Pa and Ra are true but Aa is false. When negations enter the picture, whether we can use the two forms equivalently depends on whether the sentence is existential or universal when the negation has narrow scope. 6.6.10 may look like a universal claim, in which case one might think that either 6.6.11 or 6.6.12 is an acceptable translation. 6.6.10 6.6.11 6.6.12

Not all planets and asteroids are rocky. ∼(∀x)[(Px ∨ Ax) ⊃ Rx] ∼(∀x)(Px ⊃ Rx) ∙ ∼(∀x)(Ax ⊃ Rx)

But 6.6.10 is really an existential claim: some planets or asteroids are not rocky. Using QE, we can show that 6.6.11 is equivalent to 6.6.13 and that 6.6.12 is equivalent to 6.6.14. 6.6.13 (∃x)[(Px ∨ Ax) ∙ ∼Rx] 6.6.14 (∃x)(Px ∙ ∼Rx) ∙ (∃x)(Ax ∙ ∼Rx)

As in the relevantly similar cases of 6.6.8 and 6.6.9, 6.6.13 and 6.6.14 are not equivalent. We can derive 6.6.13 from 6.6.14, just as we derived 6.6.8 from 6.6.9. But 6.6.14 does not follow from 6.6.13 since it requires two things where 6.6.13 does not. I don’t believe that there is an unequivocal answer to the question of which pair is the correct translation of 6.6.10. If the speaker of 6.6.10 intends a proposition that

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would be refuted if it turned out that all planets were rocky but some asteroids weren’t (or vice versa), then 6.6.12 and 6.6.14 are the correct pair. But if the speaker intended a proposition that would be refuted only if all planets and all asteroids were rocky, then 6.6.11 and 6.6.13 are the right ones. And if one wants a version of those with two quantifiers, one can use 6.6.15. 6.6.15 (∃x)(Px ∙ ∼Rx) ∨ (∃x)(Ax ∙ ∼Rx)

Again, I leave the proofs of the equivalence of 6.6.15 and 6.6.13 to the appendix.

Quantifiers, Domains, and Charity Some sentences of English are implicitly quantificational but without a clear statement of whether they are to be existential or universal, like 6.6.16. 6.6.16

Goats and cows produce milk.

If we take 6.6.16 to be existential, we want 6.6.17 (or an equivalent) as its translation in M. But if we take it to be universal, we want 6.6.18 (or an equivalent). 6.6.17 (∃x)(Gx ∙ Mx) ∙ (∃x)(Cx ∙ Mx) 6.6.18 (∀x)[(Gx ∨ Cx) ⊃ Mx]

There is no grammatical criterion for statements of this form, as we can see from 6.6.19–6.6.21, which share grammatical form. 6.6.19 6.6.20 6.6.21

Goats and cows have spots. Goats and cows are in the barn. Goats and cows are mammals.

6.6.19 and 6.6.20 are existential claims: some goats and cows have spots; some goats and cows are over there in the barn. But the claim about being mammals is likely to be intended as identifying a natural property of the two species that is more properly understood as universal. And similarly for 6.6.16, which is, in some sense, very close to 6.6.21. But what about the male goats and cows? They don’t produce milk. If we translate 6.6.16 as 6.6.18, we’re taking what seems like a true sentence and regimenting it as a false one. That violates one of the first rules of interpretation: the principle of charity. The principle of charity has a variety of formulations. At root, it is advice to try to understand the words of others as true unless you have good reason to think that they are not. When you’re given a sentence in a logic book to translate, there is no obvious speaker, no one to ask about the speaker’s intent in uttering a sentence. Still, we should practice charity. We don’t want to turn what looks like a true sentence into a false one. Maybe there’s a true sentence of M that we can use instead of 6.6.18. How about 6.6.22, which restricts the claim to female goats and cows? 6.6.22 (∀x){[(Gx ∨ Cx) ∙ Fx] ⊃ Mx}

6.6.22 is better at charity than 6.6.18, but it’s not better as a translation. The original sentence said nothing about female goats and cows. We’ve imposed this extra

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predicate on the translation. It looks as if we’re kind of stuck in a dilemma between a strict translation of the words and a charitable interpretation of them. So be it. That’s a problem with interpretation generally. Sometimes our best translations of other people’s words aren’t literal (what we sometimes call homophonic). Consider what it takes to translate a poem, like the Iliad. Translators do not ordinarily match words in English with words in ancient Greek, one by one. Such translations would be clunky and would impose idiomatic expressions of the original onto a language with a different set of quirks. Instead, we aim to match other factors, like rhyme or meter or feel, while preserving what we can of content. Also, sometimes we speak in euphemism. Your friend might say, “I’m going to the barn dance tonight,” when it’s common knowledge (among your friend group) that you use ‘going to the barn dance’ to mean ‘going to the library to study logic’ because you don’t want other people to know that you spend a lot of time at the library studying logic. A friend of mine often feeds his children “chicken nuggets”; but they’re made of soy, not chicken. So 6.6.22 might be preferable to 6.6.18 in real life. But it’s too much interpretation for a logic class. If you gave your instructor 6.6.22 rather than 6.6.18 on an assignment, s/he would likely be puzzled. Where did the ‘Fx’ come from? And things get worse: 6.6.22 doesn’t even get things right. Baby female goats don’t produce milk. Lactating cows stop producing milk if they aren’t milked, and there are ailments that will prevent milk production. Like all mammals, goats and cows must meet conditions other than being female in order to produce milk. If we’re interpreting charitably, we seem to have to expand past 6.6.22 to include all of these other conditions. The situation is actually not quite so bad. Maybe we can appeal to some sorts of dispositions. Let’s take ‘Dx’ to stand for the property of being disposed to produce milk, a simple predicate standing for the complex of properties that are required of a goat or a cow to produce milk: having given birth, having continued milking, and not suffering from mastitis or lameness or other problems that Wikipedia tells us can reduce or end milk production. Then 6.6.23 might seem better than 6.6.22. 6.6.23 (∀x){[(Gx ∨ Cx) ∙ Dx] ⊃ Mx}

I expect that you are not very surprised when I tell you that there are problems with 6.6.23 too. The speaker of the original sentence might use the given English words to mean any of a variety of claims. But it would be pretty deviant to use it to mean something like: If P then P. Or even: If you’re a goat or a cow, then if P then P. But that’s pretty close to what 6.6.23 does: For all x, if x is a goat or a cow disposed to make milk, then it makes milk. That’s nearly tautological. But the original sentence is nothing like a tautology. So 6.6.23 is no good either. We can help quite a bit by taking ‘Mx’ to stand for the property of being the kind of thing of which the female produces milk after giving birth; again, that requires a

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bit of complex charity, but that’s not the kind of thing we can ever really avoid. Interpretive questions are just below the surface, and sometimes they poke out without warning. At this point, you might just want to give up completely on charity. What’s the use of charity if it gives you such a headache? But charity is always a factor in our translations. Even taking the less controversial sentences for translation in this book as grammatical sentences of English requires charity. You might take them as Swahili nonsense, for example, and refuse to translate nonsense! Many uses of universal quantifiers for translation require some kind of charity. We rarely say anything about everything. More often with the universal quantifier, we make claims about all things within a particular domain, as at 6.6.24, where we are talking about only humans, not all things. 6.6.24 All humans are mortal. (∀x)(Hx ⊃ Mx)

Technically, in interpreting such a sentence, we start with the universal quantifier indicating everything and then narrow our domain with the antecedent of the conditional: we’re talking only about humans. Sometimes these restrictions of a domain are more subtle, and not explicitly stated, as at 6.6.25, literally regimented at 6.6.26. 6.6.25 Only executives have administrative assistants. 6.6.26 (∀x)(Ax ⊃ Ex)

6.6.25 seems false, unless we were to stipulate it as a definition of ‘executive’, and such a definition is implausible. To read 6.6.25 charitably, we are likely to want to restrict the domain to a particular institution in which only executives have administrative assistants. For example, the speaker of 6.6.25 is likely to mean something like 6.6.27, translated into M at 6.6.28. 6.6.27

At Metalogic Incorporated, only executives have administrative   assistants. 6.6.28 (∀x)[(Mx ∙ Ax) ⊃ Ex]

Still, there’s no indication of which institution we might be talking about in 6.6.25. Translating 6.6.25 as 6.6.28 is ill advised. Similarly, a charitable interpretation of 6.6.29 seems difficult to manage. 6.6.29

All presidents are American politicians.

Do we take this as a true sentence, and thus interpret ‘president’ as ‘president of the United States’? Or do we take this as a false sentence in which the speaker forgot that there are presidents of many different sorts.

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In practice, it often doesn’t matter. We work mainly on the surface grammar. And for the purposes of this book, 6.6.18 is probably the best choice for the original 6.6.16, and 6.6.25 is probably best as 6.6.26. Still, there are important cases in which the surface grammar of a sentence is really not what we usually use it to mean, as in 6.6.30. 6.6.30

All that glitters is not gold.

The surface grammar yields 6.6.31, with ‘Gx’ standing for ‘x glitters’ and ‘Ax’ standing for ‘x is gold’: nothing gold glitters. 6.6.31 (∀x)(Gx ⊃ ∼Ax)

But ordinary uses of 6.6.30 are usually better rendered as either 6.6.32 or 6.6.33: there are things that glitter that aren’t gold; you’d better not conclude from its glittering that what you’ve got is valuable. 6.6.32 ∼(∀x)(Gx ⊃ Ax) 6.6.33 (∃x)(Gx ∙ ∼Ax)

So, while we ordinarily translate according to the surface grammar, in cases like 6.6.30, where the usage is so obviously not according to the surface grammar, we have to invoke charitable interpretation. It would be nice if language were cleaner and easier to translate. But if English were precise in all cases, we wouldn’t need formal logic.

Summary Pretty much all work in philosophy requires interpretation and critical assessment. We first have to know what folks are saying before we can determine whether it is valid or sound, true or false. One of the advantages of formal language is that it can be more precise than natural language. Thus, regimenting sentences of English or other languages into formal languages requires us to disambiguate and clarify. Written sentences are often ambiguous or unclear. When we translate into logical languages, we have to make decisions about their likely intended meanings. Such meanings will not always be determinate. Our translations must be guided by general principles of charity and by our practical goals. Do we need relational predicates, or will monadic ones do for our purposes? Should we use functions or definite descriptions? Why are we formalizing our claims? Answers to questions about why we are translating into logical languages can help us frame our decisions about levels of precision and charity.

For Further Research and Writing 1. The examples in the translation exercises in chapters 3 and 4 are designed largely to have clear answers, and so are often not as unclear or ambiguous as language we ordinarily use. An excellent exercise is to try to translate

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philosophical texts you are reading in other classes, especially once you have the full expressive power of F at your disposal. 2. Discuss the purposes of translation into formal languages. What does such regimentation achieve? Are there disadvantages as well as advantages? For the more ambitious, you could look more closely into translation and charity, especially in the work of Quine and Davidson. 3. What is quantification? How does it differ from predication? Compare quantifiers and mathematical functions. See the suggested readings by both Dummett and Frege, and the chapter in Kneale and Kneale.

Suggested Readings Davidson, Donald. “Coherence Theory of Truth and Knowledge.” In Truth and Interpretation: Perspectives on the Philosophy of Donald Davidson, edited by Ernest LePore, 308–319. Oxford, UK: Blackwell, 1986. Davidson uses the principle of charity in responding to problems of skepticism. Dummett, Michael. “Quantifiers.” In A Philosophical Companion to First-Order Logic, edited by R. I. Hughes, 136–161. Indianapolis, IN: Hackett, 1993. A detailed examination of the nature of quantification in Frege’s work. Fisher, Jennifer. On the Philosophy of Logic. Belmont, CA: Wadsworth, 2008. Chapters 1 and 5 contain some useful observations on the utility of quantificational logics. Frege, Gottlob. Begriffsschrift. In From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, edited by Jean van Heijenoort, 1–82. Cambridge, MA: Harvard University Press, 1967. The preface is an important and engaging statement of the purposes of formal logic, and includes Frege’s eye and microscope analogies. Kneale, W., and M. Kneale. The Development of Logic. Oxford, UK: Clarendon Press, 1962. In this classic history of logic, chapter VIII on Frege’s logic contains a detailed and lucid discussion of Frege’s work on quantification. Quine, W. V. Methods of Logic, 4th ed. Cambridge, MA: Harvard University Press, 1982. Chapters 14–18 describe various alternatives to quantification, historical antecedents to Fregean quantification, and their limits. Quine, W. V. Word and Object. Cambridge, MA: MIT Press, 1960. Chapter 2, “Translation and Meaning,” is an influential work on the challenges of translation, especially from an unknown language, and contains some of Quine’s thoughts on charity. Sainsbury, Mark. Logical Forms: An Introduction to Philosophical Logic, 2nd ed. Oxford, UK: Blackwell, 2001. Chapter 4, “Quantification,” is a broad discussion of the uses of quantificational logic, with close attention to questions about translation. Strawson, P. F. “Logical Appraisal.” In A Philosophical Companion to First-Order Logic, edited by R. I. Hughes, 6–27. Indianapolis, IN: Hackett, 1993. This chapter contains insightful observations about logic and our goals in using it.

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Appendix to 6.6 DERIVING 6.6.7 FROM 6.6.6 1. (∀x)[(Px ∨ Ax) ⊃ Rx] Premise 2. ∼[(∀x)(Px ⊃ Rx) ∙ (∀x)(Ax ⊃ Rx)] AIP 3. ∼(∀x)(Px ⊃ Rx) ∨ ∼(∀x)(Ax ⊃ Rx) 2, DM 4. (∃x)∼(Px ⊃ Rx) ∨ (∃x)∼(Ax ⊃ Rx) 3, QE 5. (∃x)∼(∼Px ∨ Rx) ∨ (∃x)∼(∼Ax ∨ Rx) 4, Impl 6. (∃x)(Px ∙ ∼Rx) ∨ (∃x)(Ax ∙ ∼Rx) 5, DM, DN 7. (∃x)(Px ∙ ∼Rx) AIP 8. Pa ∙ ∼Ra 7, EI 9. (Pa ∨ Aa) ⊃ Ra 1, UI 10. Pa 8, Simp 11. Pa ∨ Aa 10, Add 12. Ra 9, 11, MP 13. ∼Ra 8, Com, Simp 14. Ra ∙ ∼Ra 12, 13, Conj 15. ∼(∃x)(Px ∙ ∼Rx) 7–14, IP 16. (∃x)(Ax ∙ ∼Rx) 6, 15, DS 17. Ab ∙ ∼Rb 16, EI 18. Ab 17, Simp 19. (Pb ∨ Ab) ⊃ Rb 1, UI 20. Pb ∨ Ab 18, Add, Com 21. Rb 19, 20, MP 22. ∼Rb 17, Com, Simp 23. Rb ∙ ∼Rb 21, 22, Conj 24. (∀x)(Px ⊃ Rx) ∙ (∀x)(Ax ⊃ Rx) 2–23, IP, DN QED DERIVING 6.6.6 FROM 6.6.7 1. (∀x)(Px ⊃ Rx) ∙ (∀x)(Ax ⊃ Rx) 2. (∀x)(Px ⊃ Rx) 3. Px ⊃ Rx 4. (∀x)(Ax ⊃ Rx) 5. Ax ⊃ Rx 6. (Px ⊃ Rx) ∙ (Ax ⊃ Rx) 7. (∼Px ∨ Rx) ∙ (∼Ax ∨ Rx) 8. (Rx ∨ ∼Px) ∙ (Rx ∨ ∼Ax) 9. Rx ∨ (∼Px ∙ ∼Ax) 10. Rx ∨ ∼(Px ∨ Ax) 11. ∼(Px ∨ Ax) ∨ Rx 12. (Px ∨ Ax) ⊃ Rx 13. (∀x)[(Px ∨ Ax) ⊃ Rx] QED

Premise 1, Simp 2, UI 1, Com, Simp 4, UI 3, 5, Conj 6, Impl 7, Com 8, Dist 9, DM 10, Com 11, Impl 12, UG

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DERIVING 6.6.8 FROM 6.6.9 QED

1. (∃x)(Px ∙ Rx) ∙ (∃x)(Ax ∙ Rx) 2. (∃x)(Px ∙ Rx) 3. Pa ∙ Ra 4. Pa 5. Pa ∨ Aa 6. Ra 7. (Pa ∨ Aa) ∙ Ra 8. (∃x)[(Px ∨ Ax) ∙ Rx]

Premise 1, Simp 2, EI 3, Simp 4, Add 3, Com, Simp 5, Conj 7, EG

DERIVING 6.6.15 FROM 6.6.13 1. (∃x)[(Px ∨ Ax) ∙ ∼Rx] Premise 2. ∼[(∃x)(Px ∙ ∼Rx) ∨ (∃x)(Ax ∙ ∼Rx)] AIP 3. ∼(∃x)(Px ∙ ∼Rx) ∙ ∼(∃x)(Ax ∙ ∼Rx) 2, DM 4. (∀x)∼(Px ∙ ∼Rx) ∙ (∀x)∼(Ax ∙ ∼Rx) 3, QE 5. (∀x)(∼Px ∨ Rx) ∙ (∀x)(∼Ax ∨ Rx) 4, DM, DN 6. (∀x)(Px ⊃ Rx) ∙ (∀x)(Ax ⊃ Rx) 5, Impl 7. (Pa ∨ Aa) ∙ ∼Ra 1, EI 8. (∀x)(Px ⊃ Rx) 6, Simp 9. Pa ⊃ Ra 8, UI 10. (∀x)(Ax ⊃ Rx) 6, Com, Simp 11. Aa ⊃ Ra 10, UI 12. Pa ∨ Aa 7, Simp 13. Ra ∨ Ra 9, 11, 12, CD 14. Ra 13, Taut 15. ∼Ra 7, Com, Simp 16. Ra ∙ ∼Ra 14, 15, Conj 17. (∃x)(Px ∙ ∼Rx) ∨ (∃x)(Ax ∙ ∼Rx) 2–16, IP, DN QED

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DERIVING 6.6.13 FROM 6.6.15 1. (∃x)(Px ∙ ∼Rx) ∨ (∃x)(Ax ∙ ∼Rx) Premise 2. ∼(∃x)[(Px ∨ Ax) ∙ ∼Rx] AIP 3. (∀x)∼[(Px ∨ Ax) ∙ ∼Rx] 2, QE 4. (∀x)[∼(Px ∨ Ax) ∨ Rx] 3, DM, DN 5. (∀x)[(Px ∨ Ax) ⊃ Rx] 4, Impl 6. (∃x)(Px ∙ ∼Rx) AIP 7. Pa ∙ ∼Ra 6, EI 8. (Pa ∨ Aa) ⊃ Ra 5, UI 9. Pa 7, Simp 10. Pa ∨ Aa 9, Add 11. Ra 8, 10, MP 12. ∼Ra 7, Com, Simp 13. Ra ∙ ∼Ra 11, 12, Conj 14. ∼(∃x)(Px ∙ ∼Rx) 6–13, IP 15. (∃x)(Ax ∙ ∼Rx) 1, 14, DS 16. Ab ∙ ∼Rb 15, EI 17. (Pb ∨ Ab) ⊃ Rb 5, EI 18. Ab 16, Simp 19. Pb ∨ Ab 18, Add, Com 20. Rb 17, 19, MP 21. ∼Rb 16, Com, Simp 22. Rb ∙ ∼Rb 20, 21, Conj 23. (∃x)[(Px ∨ Ax) ∙ ∼Rx] 2–22, IP, DN QED

Chapter 7 Logic and Philosophy

7.1: DEDUCTION AND INDUCTION The first five chapters of this book concern deductive inferences. Deduction may be contrasted with other forms of reasoning that we can call, broadly, inductive. Much of human reasoning is inductive, although deductive inferences are ubiquitous and so obvious that they often pass unnoticed. The best way to understand the difference between deduction and induction is to work through the first five chapters of this book and get a sense of deduction. Induction, in our broad sense, is all other reasoning. To characterize the difference a bit further, deductive arguments are necessary entailments from premises to conclusions. Inductions are often probabilistic attempts to gather a variety of evidence into simple, general claims. Abductions, or inferences to the best explanation, are sometimes contrasted with induction, but are also nonnecessary entailments that we can call inductive. Perhaps the difference between deduction and induction is best seen by paradigms. 7.1.1 is a deductive inference. 7.1.1

Polar bears are carnivorous. Polar bears are mammals. So, some mammals are carnivorous.

Notice that the conclusion of 7.1.1 is almost disappointing in its obviousness. One you have seen the premises of a deductive argument, the conclusion is rarely surprising. A natural response to the third sentence in 7.1.1 is, “Yes, you said that already.” In contrast, 7.1.2 is an inductive inference. 7.1.2

Forty-seven percent of Americans in a recent poll approve of the    way the Supreme Court does its job. There were 1003 adults polled. The margin of error for this poll is plus or minus 3 percent. So, between 44 and 50 percent of Americans approve of the way    the Court does its work. 455

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Inductive conclusions can be both surprising and contentious. 7.1.2 is not particularly surprising, but inductive inferences often are, as when an astronomer discovers water on a distant planet or a statistician notices a significant trend. Like ordinary reasoning, much of scientific reasoning is inductive and inductive inferences are often called logical even if they aren’t necessary entailments. Indeed, when people say that a conclusion is logical, they often mean that it is well supported by evidence, not that the conclusion follows from the premises. Abductive reasoning forms a central part of scientific reasoning. 7.1.3 is a classic abductive inference to the existence of electrons, negatively charged particles. 7.1.3

Cathode rays carry a charge of negative electricity. They are deflected by an electrostatic force as if they were   negatively electrified. They are acted on by a magnetic force in the same way that    magnetic forces would act on an electrified body in the path    of the cathode rays. So, the cathode rays are charges of negative electricity carried by    particles of matter.1

A classic, if ultimately misleading, way to think about the difference between deduction and induction is to contrast reasoning from the general to the particular (deductive) with reasoning from the particular to the general (inductive).2 Paradigmatic deductive reasoning often does start with general principles, and paradigmatic inductive reasoning often does begin with particular claims. 7.1.4 is a deductive inference from general to particular claims, while 7.1.5 is inductive, from particular claims to general ones. 7.1.4

All college students enroll in some classes. Megha is a college student. So, Megha is enrolled in some classes.

7.1.5

Juliet is a college student enrolled in some classes. Alexandra is a college student enrolled in some classes. Luka is a college student enrolled in some classes. . . .  All college students enroll in some classes.

Still, it is possible to reason inductively from the general to the particular and to reason inductively to the general. For example, from the general claim that every time I have been to CitiField, the Mets have lost, I might inductively infer that the next time I go to CitiField, the Mets will lose again. This inference would be inductive, but from the general to the particular. 1 2

See Peter Achinstein, The Book of Evidence (Oxford: Oxford University Press, 2001), 17.

See, for example, Bertrand Russell, The Problems of Philosophy (London: Oxford University Press, [1912] 1959), 79.

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We might think about the rules of deduction as preconditions for all inferences. Our work in this book provides necessary conditions for reasoning: no good reasoning violates the conditions described in the first five chapters. But ordinary inferences also, and more obviously, presume inductive reasoning, and there is little in the first five chapters about that. Inductive reasoning, including abduction, is largely in the domain of scientists: physical scientists who seek the fundamental laws and structures of the world, social scientists who gather information about how people and societies work, psychologists who study actual human reasoning patterns, and so on. Sometimes general principles of inductive reasoning are collected under the heading ‘inductive logic’, but that term is used mainly as shorthand for a panoply of practices that are also known as (sometimes ‘the’) scientific method. While scientific inferences generally proceed inductively, we often represent them deductively. We develop general laws from which particular claims may be deduced, for example. But the development of the general laws is rarely deductive. Scientists gather evidence and look for the best way to explain or represent it. The laws summa­ rize the data. Once we devise these general laws, we can infer new cases from them. So, if we know that the momentum of an object is equal to the product of the object’s mass with its velocity, and we know the mass and velocity of the object, we can deduce its momentum. Once we know that supply varies inversely with demand, we can deduce, given a glut of a product, that demand will decrease. Careful attention to the practice of scientists shows that there is no such thing as a unique scientific method. Scientists have many different kinds of practices: gathering data in various ways, hypothesizing general claims, invoking inferences to the best explanation, and so on. Much scientific reasoning is probabilistic. Calling it all ‘induction’ may be seen as a misleading overgeneralization. Still, there are some general principles underlying inductive practices and the particular methods of science. And there is a general philosophical problem underlying the pursuit of a single scientific method or the systematization of scientific reasoning. This problem, often attributed to the eighteenth-century philosopher David Hume, is called the problem of induction, and it is the most significant barrier to developing formal theories of induction parallel to the deductive logical theories in the first five chapters of this book.

Hume’s Problem of Induction It is more or less clear how we acquire and justify our beliefs about the world around us at the moment or in the recent past, using sense perception and memory, though there is much to be learned about those processes. It is more of a puzzle to think about how we can acquire justifiable beliefs about the unobserved. How do we know that the sun will rise tomorrow? How do we know that there is a heart beating in your chest? How do we know about distant planets or black holes from which light (and therefore information) cannot travel to us?

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In order to acquire beliefs about the future or unobserved, we make inductive inferences from our experiences, including predictions about the future. Hume presents a skeptical argument about such inductions. He starts by distinguishing matters of fact from relations of ideas. All the objects of human reason or enquiry may naturally be divided into two kinds, namely, relations of ideas, and matters of fact. Of the first kind are the sciences of geometry, algebra, and arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. That the square of the hypothenuse is equal to the square of the two sides is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. (Hume, Enquiry, IV.1)

Relations of ideas are, generally, necessary claims known by deductive methods, and so acquired a priori. Hume, writing before the development of modern logic, says that they are known by the operation of mere thought, guided by the principle of contradiction. One of Frege’s intentions in developing modern logic was to show that deductive inferences are independent of our psychology and may proceed syntactically, as our derivations in this book do. The importance of the principle of contradiction can be traced back at least to Leibniz’s work. (See Leibniz, Monadology, 31 et seq.) Following Leibniz, Hume believes that a statement can be known to be necessarily true only if its negation entails a contradiction. “We are possessed of a precise standard by which we can judge of the equality and proportion of numbers and, according as they correspond or not to that standard, we determine their relations without any possibility of error” (Hume, Treatise, I.3.1). While Leibniz and Hume had an informal understanding of the notion of a contradiction, we can understand the principle of contradiction now as the claim that any statement of the form α ∙ ∼α is false. We invoke the principle of contradiction in indirect proof and see the ill effects of contradiction in the explosive inferences of section 3.5. We know the mathematical claims that Hume cites because their negations are self-contradictory. Matters of fact, in contrast to relations of ideas, are contingent and acquired a posteriori, or empirically. Unlike relations of ideas, according to Hume, the contrary of every matter of fact is possible. We must have some sense experiences of an object or event to know