Systems of Formal Logic [Softcover reprint of the original 1st ed. 1966] 9401035490, 9789401035491

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SYSTEMS OF FORMAL LOGIC

L. H. HACKSTAFF

SYSTEMS OF FORMAL LOGIC

D. REIDEL PUBLISHING COMPANY / DORDRECHT-HOLLAND

ISBN-13: 978-94-010-3549-1 e- ISBN-13: 978-94-010-3547-7 001: 10.1007/978-94-010-3547-7

1966 Softcover reprint of the hardcover 1st edition 1966 All rights reserved No part of this book may be reproduced in any form, by print, photoprint, microfilm or any other means, without permission from the publisher

PREFACE

The present work constitutes an effort to approach the subject of symbolic logic at the elementary to intermediate level in a novel way. The book is a study of a number of systems, their methods, their relations, their differences. In pursuit of this goal, a chapter explaining basic concepts of modern logic together with the truth-table techniques of definition and proof is first set out. In Chapter 2 a kind of ur-Iogic is built up and deductions are made on the basis of its axioms and rules. This axiom system, resembling a propositional system of Hilbert and Bernays, is called P +, since it is a positive logic, i.e., a logic devoid of negation. This system serves as a basis upon which a variety of further systems are constructed, including, among others, a full classical propositional calculus, an intuitionistic system, a minimum propositional calculus, a system equivalent to that of F. B. Fitch (Chapters 3 and 6). These are developed as axiomatic systems. By means of adding independent axioms to the basic system P +, the notions of independence both for primitive functors and for axiom sets are discussed, the axiom sets for a number of such systems, e.g., Frege's propositional calculus, being shown to be non-independent. Equivalence and non-equivalence of systems are discussed in the same context. The deduction theorem is proved in Chapter 3 for all the axiomatic propositional calculi in the book. Eight propositional logics are worked out in varying detail and a number of others are remarked upon. Besides axiomatic systems, the book presents exposition and deductions in a number of systems of natural deduction (Chapters 4 and 6). It is shown how systems of natural deduction, equivalent with axiomatic systems previously developed, may be constructed, e.g. the classical propositional calculus, the calculi of intuitionism, of Fitch, etc. If it is deemed desirable to introduce systems of natural deduction prior to axiom systems, Chapter 4 has been so written that it may be introduced v

SYSTEMS OF FORMAL LOGIC

immediately following Chapter I without much disaccommodation. In Chapter 7 a functional calculus of the first order is constructed as an axiomatic system and a number of theorems are proved. The system resembles that of Russell's functional calculus of 1908, but it is restricted to one-place predicates and is built up on the basis of a propositional calculus distinct from Russell's system. In Chapter 8 this system is extended and some results of the second order functional calculus with identity are proved. In this chapter two methods of proving results in the calculus of classes are set out. Chapter 9 gives an exposition of the Russell paradox and the semantic paradoxes, indicating the manner in which these are avoided in the systems of the book. The decision problem for systems beyond the calculus of propositions is also briefly explored in this chapter. Chapter 10 is devoted to the presentation of three non-standard functional calculi (the intuitionist, the Johannson, and the Fitch systems). No use has been made of axiom schemata, though rule schemata are frequently used. All axiomatic systems are articulated on the basis of a finite number of axioms with explicitely formulated rules of substitution. Some sections of the book are more difficult than others, Chapter 5 gives proofs of consistency and completeness (or incompleteness) for a number of systems developed in earlier chapters. Its level of difficulty is such that it might profitably be passed over by students at the elementary level or those not interested in the refinements of symbolic logic. Other sections of an advanced nature are: 3.7 A Proof of the Deduction Theorem; 3.9 On Independence of Functors and Axioms in Four Systems; 4.9 A Decision Procedure for the System of Natural Deduction, PND; Appendix to Chapter 6; 9.4-6 Consistency and Completeness of Functional Systems. With the exception of some parts of the first of these, 3.7, which outline methods of derivation subsequently employed, these sections may be passed over by elementary students without effecting the intelligibility of the book. The book has been constructed with the intention of striking a mean between such books as that of Alonzo Church which, while systematically admirable, are too difficult for any but fairly advanced students, and those less admirable, but equally problematic, books which are so simplified as to insult the intelligence of all but the most obtuse undergraduate. It is hard to know whether this mean has been attained, though an approach resembling that of this book has been employed successfully by the author VI

PREFACE

for a number of years in teaching high ability students at the University of Missouri and summer students at The Colorado College. No one, however, should assume that all sections of the book are 'easy'. Some require concentrated study. The book is intended primarily for serious and intelligent readers. No previous study of logic or of higher mathematics is presupposed. It is hoped that such readers will find something of value in the work to justify whatever exertion it may cost. It is hoped that the book will also be of interest and profit to mature formal logicians. The book terminates, systematically speaking, with the proof of some results in the second order functional calculus with identity. A subsequent volume will cover not only systems beyond the first order functional calculus, but also systems of modal, deontic, and many-valued logic and the theory of recursive functions. An incomplete and error ladened version of the present book was made available in mimeographed form in 1959 for teaching purposes at the University of Missouri. It is hoped that the more egregious errors of the former Systems of Formal Logic have been eliminated here. That some of the less obvious errors were detected at all is due to the care and enthusiasm of a graduate class for which the prototype was written in 1959. Besides the prototype, the book also makes use of a number of sections of my Doctoral Dissertation, The Status of the Laws of Thought and Their Function in Systems, submitted to the Department of Philosophy of Yale University, 1958. In particular, the original versions of systems having either the law of excluded middle or the law of contradiction as axioms (cf. Chapters 6 and 10) were developed there. Likewise, some of the observations were first set out in the dissertation. Logicians and their books, the ideas of which have been freely used, deserve special notice here. The author is greatly indebted to Professor F. B. Fitch of Yale to whom he owes his first insight into the richness and significance of logic and, among other things, the elegant linear method of natural deduction adopted in Chapter 4, as well as a criticism of the prototype mimeograph. No logician can read the book without recognizing its debt to the works of Rudolf Carnap, W. V. Quine, and, above, all, Alonzo Church. The book of Church with its great logical richness and precision and its scholarly brilliance has been indispensible in the genesis of this work. Though credit is given where credit is due in most cases, there are probably a number of writers, including those mentioned VII

SYSTEMS OF FORMAL LOGIC

above, whose work has become so much a part of the author's logical equipment that he has failed properly to credit their work. To them, his greatest creditors, the author tenders his apologies. To J. M. Bochenski in particular and to the University of Fribourg in general, who made the work possible, special thanks are offered. Of my personal debts my greatest is by far to my wife who had the patience to tolerate IQng periods of concentration required by the book, the good sense not to interfere with work in progress, the kindness and industry required to type the manuscript, and the insight to criticize the result as a logician. Indeed over the years of reworking and revision she has contributed so much and in so many parts that she virtually deserves recognition as co-author. It goes without saying that the book is dedicated to her. The kind conscientiousness of Miss Joella Brereton who typed the manuscript for the prototype, and of Mrs. David D. Corney who helped in proofreading the present work should not go without mention. Funds for producing the prototype were granted by the Chairman of the Department of Philosophy of the University of Missouri, W. D. Oliver, and by the University itself. To these much thanks.

Wabash College Crawfordsville, Ind. November 1965

VIII

L. H. HACKST AFF

TABLE OF CONTENTS

1

Chapter 1 Introduction: Some Concepts and Definitions

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7, 1.8 1.9 1.10

Arguments and Argument Forms Symbolic Logic and its Precursors Symbolization Logical Functors and Their Definitions Tests of Validity Using Truth-tables Proof and Derivation The Axiomatic Method Interpreted and Uninterpreted Systems The Hierarchy of Logical Systems The Systems of the Present Book Abbreviations

Chapter 2 The System P +

2.1 2.2 2.3 2.4 2.5 2.6

Summary Rilles of Formation of P + Rules of Transformation of P + Axioms ofP+ Definitions of P + Deductions in P +

Chapter 3 Standard Systems with Negation (PLT, PLT', PLTF, PPM)

3.1 3.2 3.3 3.4 3.5 3.6

Summary Rules of Formation of PLT Rules of Transformation of PLT Axioms of PLT Definitions of PLT Deductions in PLT

1 8 11 14 22 29 36 43 45 46 47 48 48

48 49 50 50 51 94 94 94 95 95 96 96 IX

SYSTEMS OF FORMAL LOGIC

3.7 The Deduction Theorem 3.8 The System PLT' 3.9 Independence of Functors and Axioms Chapter 4 The System

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

PND.

Systems of Natural Deduction

106 116 122 130

Summary The Bases of the System PND Proof and Derivation Techniques in PND Rules of Formation of PND The Structure of Proofs in PND Rules of Transformation of PND Proofs and Theorems of the System PND Theorems of the Full System PND A Decision Procedure for the System PND A Reduction of PND

130 130 130 132 132 137 145 169 182 190

Chapter 5 The Consistency and Completeness of Formal Systems

193

5.1 5.2 5.3 5.4

Summary The Consistency of PLT' The Completeness of PLT' Metatheorems on P +

Chapter 6 Some Non-Standard Systems of Propositional Logic

193 198 200 203 207

6.1 Summary 207 6.2 What is a Non-Standard System? 207 6.3 The Intuitionistic System and the Fitch Calculus (PI 208 and PF) 6.4 Rules of Formation of PI 209 6.5 Rules of Transformation of PI 210 6.6 Axioms of PI 210 6.7 Definitions of PI 211 6.8 Deductions in PI 211 6.9 The Propositional Logic of F.B. Fitch 223 6.10 The Johansson Minimum Calculus 229 Chapter 7 The Lower Functional Calculus

7.1

Summary and Remarks

234

234

x

T ABLE OF CONTENTS

7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

Rules of Formation of LFLT' Transformation of LFLT' Axioms of LFLT' Definitions of LFLT' Some Applications and Illustrations Rules of Transformation of LFLT' Axioms of LFLT' The Propositional Calculus and LFLT' Deductions in LFLT'

Chapter 8 An Extension ofLF LT' and Some Theorems of the Higher Functional System. The Calculus of Classes

8.1 8.2 8.3 8.4 8.5

Summary and Modification of the Formation Rules of LFLT' The Lower Functional Calculus with Identity Quantification over Predicate Variables. The System 2FLT ' Abstraction and the Boolean Algebra The Boolean Algebra and Propositional Logic

Chapter 9 The Logical Paradoxes

9.1 Self Membership 9.2 The Russell Paradox 9.3 Order Distinctions, Levels of Language, and the Semantic Paradoxes 9.4 The Consistency of LFLT' 9.5 The Decision Problem 9.6 Consistency and Decision in Higher Functional Systems Chapter 10 Non-Standard Functional Systems

236 237 237 238 239 248 255 258 264 284

284 285 288 292 301 304

304 305 306 308 309 311 313

10.1 Summary 313 10.2 Intuitionistic and Johansson Functional Logics 313 10.3 The Fitch Functional Calculus of the First Order with 316 Identity (LFi;) Bibliography

344

Index

346 XI

CHAPTER 1

INTRODUCTION: SOME CONCEPTS AND DEFINITIONS

1.0 Arguments and Argument Forms Argumentation takes place whenever reasons are offered to support some statement. Thus when Galileo asserted the statement that the earth moves and offered reasons or evidence supporting this statement, he engaged in argumentation. Likewise, when a Delegate to the United Nations presents a proposal to the Assembly and proceeds to present considerations which, he hopes, will make the proposal plausible to his fellow delegates, he is engaged in argumentation. When a college student questions the grade given on work he has submitted and attempts to convince his teacher that it should be raised by giving reasons, he is engaged in argumentation. When the scientist, or the Delegate, or the student engage in argumentation they do so by offering a statement or series of statements which, they believe, support or ground the statement for which they are arguing. 1.01 Let us agree to call any statement supported by argumentation the conclusion of the argumentation. Let us agree to call any statement or set of statements used in argumentation to support a conclusion the premises of the argumentation. The sequence of statements including both premises and conclusion is called an argument. Unlike the ordinary meaning of the word "argument", the term as used here need not have any associations with controversy or debate. For instance, the premises and conclusions found in a book on the integral calculus are, according to this usage, arguments, though they have nothing to do with contention or disagreement. Thus a general definition of "argument" would be: An argument is any set of statements the conclusion of which is claimed to be supported by the premises.

Evidently, there are "good" (i.e. correct), and "bad" (Le. incorrect) 1

SYSTEMS OF FORMAL LOGIC

arguments. An argument is said to be "good" if the premises do in fact support the conclusion. An argument is said to be "bad" if the premises do not support the conclusion. The following is an example of a "good" argument. (1) If the earth rotates around the sun then the earth moves. (2) The earth rotates around the sun. (3) The earth moves.

A

In argument A statements (1) and (2) are the premises, and statement (3) is the conclusion. The following is a "bad" argument. B

(1) If the earth rotates around the sun then the earth moves. (2) The earth moves. (3) The earth rotates around the sun.

(1) and (2) of B are the premises and (3) is its conclusion. A is a "good" argument because (3) of A is actually supported by (1) and (2) of A. B is a "bad" argument because, despite the truth of (3) and the truth of (1) and (2), (3) is not supported by (1) and (2) of B. Thus to say that an argument is "bad" is not merely to say that its premises are false or that its conclusion is false, since both premises and conclusion of Bare true - but B is none the less a "bad" argument. The conclusion of A follows from its premises; but the conclusion of B does not follow from its premises. 1.02 It is useful now to replace the ambiguous terms "good" and "bad" as used above with more precise terminology. What has previously been called a "good" or correct argument will hereafter be called a valid argument; what has heretofore been called a "bad" or incorrect argument will be called an invalid argument. It is now possible to specify under what circumstances arguments are valid and invalid. 1.021 If the specific statements of arguments A and B are replaced by letters standing for statements it is relatively easy to see why A is valid and B is invalid. Replace the first premise, statement (1), of A and the first premise, statement (1), of B with the capital letter R. Replace the second premise of A with P. Replace the second premise of B with Q. Replace the conclusion of A with Q, and replace the conclusion of B with P. This replacement gives a preliminary skeleton form of A which looks like this: 2

INTRODUCTION

A'

(1) R (2) P (3) Q

And a preliminary skeleton form of B which looks like this: B'

(1) R (2) Q (3) P

But inspection of the first premise possessed in common by both A and B and represented in A' and B' as R reveals that this premise is made up of two statements connected by the form "If... then ... ". Now the statement following the word "If" is "the earth rotates around the sun", the second premise of A, represented by P in A'. The statement following the word "then" is "the earth moves", the conclusion of A, represented by Q in A'. Thus, to fill out the preliminary skeleton of A' and B', R is replaced by the form IfP then Q. By this process we obtain as a modification of A', the following skeleton form. A"

(1) If P then Q (2) P (3) Q

And as a modification of B' the skeleton form: B"

(1) If P then Q (2) Q (3) P

Here and hereafter in this section (terminating at 1.05), expreSSIOns numbered (1) and (2) are "the premises" and those numbered (3) represent "the conclusion". This convention is stipulated only for this section. 1.03 Now the reader is requested, for the time being, to forget all about the specific arguments A and B and to focus attention exclusively upon A" and B", thinking of P and Q as marks standing, not merely for the specific statements of arguments A and B, but for any statements. He may replace or substitute any statement for P, and any statement for Q as long as he always replaces the same statement for P and the same statement for Q in the skeleton. Thus he could replace or substitute the 3

SYSTEMS OF FORMAL LOGIC

statement "Socrates is human" for P in A", and "Socrates is mortal" for Q in A". The result of this substitution would be the specific argument, C

(1) If Socrates is human then Socrates is mortal. (2) Socrates is human.

(3) Socrates is mortal. The same substitution for the P's and Q's of B" would be: (1) If Socrates is human then Socrates is mortal. (2) Socrates is mortal. (3) Socrates is human.

D

If the reader compares C and D with A and B, he will discover that C and A, though about different subject matter, have the same form or structure as arguments. They are both represented by the skeleton form A". Likewise, comparison of D and B reveals that these two arguments are of the same form, the form represented by B". Let us agree to call any argument which is represented by the form A", substitution instances of the form A". Both A and C may be obtained by substituting the specific statements of A and C for the P's and Q's of A". Thus both A and Care substitution instances of A", differing with respect to subject matter but identical with respect to their form. Likewise, Band D are substitution instances of W. Evidently, there is an innumerable collection of arguments which are substitution instances of A" and an innumerable collection of arguments which are substitution instances of B". 1.031 All substitution instances of A", among them A and C, are valid arguments, while all substitution instances of B" are invalid. This follows because A" is a valid argument-form, while B" is an invalid argumentform. We define the notion of a valid argument-form or rule as follows: An argument-form is valid if its application will not allow the assertion of a false conclusion from the assertion of true premises.

An invalid argument-form or rule is defined as follows: An argument-form is invalid if its application will allow the assertion of a false conclusion from the assertion of true premises.

Put more briefly, a valid form of argumentation is one which never allows the deduction of a false conclusion from true premises, while an 4

INTRODUCTION

invalid form of argumentation is one which sometimes allows the deduction of a false conclusion from true premises. 1.032 If we apply these definitions, it is easy to see why B" is an invalid argument-form and hence, why B is an invalid argument. Consider the following substitution instance of B". (1) If John is a Christian then John believes in the existence of one God. (2) John believes in the existence of one God. (3) John is a Christian. The premise given in statement (1) is clearly true, and let us assume that (2) is true also. But in this case, the conclusion, (3) is false if, for instance, John is a Mohammedan, while the premises are true. Thus the premises

give no grounds for asserting the conclusion. Another case, where C is the name of some individual thing: (1) If C is a trout then C is a fish. (2) C is a fish. (3) C is a trout. 1.033 Premise (1) is clearly true, since no matter what C names, if C is a trout then C is a fish. Suppose premise (2) is true. Statement (3) does not follow, since C could be a shark or a halibut, or any other fish. Note that even if C were, say, a shark, premise I would still be true, since if C were a trout, C would be a fish. Thus, there are cases in which the application of B" would lead from true premises to a false conclusion. Hence B" is an invalid argument-form and B is an invalid argument, since it is a substitution on B". B" is unreliable. Substitution on itwould sometimes lead from premises all of which are true to a conclusion which is false. 1.034 It might be objected that, in both Band D, the conclusion is true; therefore the arguments must be "valid". But it is easy to see that the truth of the premises of Band D have no bearing on the truth of the conclusion in either of these arguments. In the case of argument D, both premises could be true and the conclusion false, e.g., in case Socrates is the name of a dog. The same applies to argument B. 1.035 No argument which has the form B" is valid, but every argument which is a substitution on A" is valid. By the definition of validity given on pp. 4f, if the application of an argument-form will never lead from

5

SYSTEMS OF FORMAL LOGIC

truth (i.e., true statements as premises) to falsehood (a false conclusion) then the form is valid. Now consider some substitutions on A". We have A and C as substitutions on A". Their premises are true and their conclusions are true likewise. But consider another case. (1) If John is a man then John is a fish.

E

(2) John is a man. (3) John is a fish. This argument is valid. The premise stated in (1) is false. Thus E cannot lead from true premises to a false conclusion, merely by virtue of the fact that (1) is false. Suppose the conclusion is true; then E is still valid, since it does not lead from truth to falsity. Suppose the conclusion is false. Then, again, the form A" has not led from truth to falsehood. Suppose,per impossibile, that both premises are true. Then the conclusion would have to be true also. Consider the argument (1) If Socrates is a shark then Socrates walks on four legs.

F

(2) Socrates is a shark. (3) Socrates walks on four legs. Supposing that Socrates is a man, all the statements of F are false. Thus statement (3), the conclusion, is false. But then the premises of F, statements (1) and (2) are false and hence have not led from truth to falsehood. Hence, F is valid. Again, consider the argument (l) If Moses is a monotheist then Moses is a Christian. (2) Moses is a monotheist. (3) Moses is a Christian.

G

Statement (3) is false, but so is (1). Thus G does not lead from true premises to a false conclusion. Thus if the premises ofa substitution instance of A" are true then so is the conclusion. If the conclusion is false, then so is at least one of the premises. Thus A" is valid in precisely the sense that if the premises of any substitution on it are both true, then so is the conclusion. 1.036 Thus A" is a valid form. In consequence, both of the substitution instances A and C are valid arguments. But B" is an invalid form. In consequence, Band D are both invalid arguments. 1.037 An argument is said to be invalid by virtue of its form, or, more 6

INTRODUCTION

briefly, logically invalid if the argument-form of which the argument is a substitution instance is invalid. An argument is said to be logically valid if the argument-form of which it is an instance is valid. The form A" is valid. Hence all substitution instances of A" are valid. Note that to say that an argument is valid is to say something about its formal structure; it is not to say that the premises of the argument are true, or that the conclusion is true. As inspection of the arguments above shows, valid arguments may have false premises and false conclusions. To say that an argument is valid is to say that if its premises are true, then its conclusion must be true. We are, of course, usually interested in more than the validity of arguments: we are interested also in whether their premises are in fact true. A valid argument with none but true statements as premises is called a sound argument. The aim of the various empirical sciences is to determine the factual truth or probability of premises. This is not the aim of the formal science of logic. Logic studies the conditions of validity - and no argument can be sound unless it is valid. 1.038 In summary, all valid forms of argument are truth-preserving, i.e., if their premises are true then their conclusions are true. All invalid forms of argument fail to be truth-preserving, i.e., from the fact that their premises are true it cannot be known that their conclusions are true. 1.04 Thus, given the arguments of Galileo, the U.N. Delegate, and the student, it is known that any of their arguments which has the form A" is valid, while every argument which has the form B" is invalid. If the argument is valid its premises support the conclusion; if it is invalid they have no logical bearing on the conclusion. 1.041 But an argument is not necessarily invalid if it fails to have the form of A". And there are other forms of invalid argumentation besides B". There are other forms of valid - and of invalid - argumentation. However, if any argument, like A, is valid by virtue of its form, it is said to be formally valid. Likewise, if any argument is, like B, invalid by virtue of its form, it is said to beformally invalid. An argument which is formally valid is said to be a case offormal validity. An argument which is formally invalid is said to be a case of formal invalidity. 1.05 Formal logic is the science whose subject-matter is formal validity as defined above. It studies the form of arguments in order to classify argument-forms into two mutually exclusive and exhaustive divisions: 7

SYSTEMS OF FORMAL LOGIC

one division being reserved for formally valid argument-forms, the other division for formally invalid forms. The object of formal logic is thus obtained when logical techniques and methods are available by means of which it is possible in principle to identify all formally valid arguments as formally valid, and formally invalid arguments as formally invalid. The rules of a formally valid logical system, i.e., a system all of whose laws are valid, are valid argument-forms. The laws or theorems of such a system are expressions in symbolic form, all of whose substitutioninstances are true. Such laws are called formally valid laws. If such a system is complete, all formally valid arguments will be substitution instances of its laws. The subject of the present book is formal logic in its modern symbolic form.

1.1 Symbolic Logic and its Precursors The systems to be considered in this book are all forms of what is variously referred to as "symbolic logic", "mathematical logic", and "logistic". These designations all refer to the same discipline: for convenience we will hereafter refer to this discipline as "symbolic logic". Symbolic logic is not, however, unique in its use of signs and symbols. The logic of Aristotle likewise uses symbols of various kinds to designate logical operators and relations. Aristotle's mature logic (the theory expressed in Prior Analytics and Posterior Analytics) employs variables in much the same way as does modern symbolic logic. In fact, it is difficult to see how a logic could be formulated at all without the use of some symbols, since words and sentences, spoken or written, are themselves symbols. The symbols used and the way they are employed in modern systems have certain important features which distinguish them from earlier logics, e.g., the modern notation is considerably more elegant, economical, and perspicuous, but this innovation is, perhaps, of minor importance compared with certain other features characteristic of "the new logic". 1.11 The logic pioneered by Leibniz in the seventeenth century and developed with increasing comprehensiveness since the time of Boole and De Morgan in mid-nineteenth century is superior to the great systems of Aristotle, the Stoics, and the logicians of the Middle Ages with respect to (1) the rigor and strictness of its proofs, (2) the com8

INTRODUCTION

pleteness and comprehensiveness of its systems, (3) the clarity and sophistication of its distinctions within the domain of logic, and (4) the perspicuity of its operations. We consider each of these four considerations in turn. 1.12 Rigor. An argument or proof is said to be rigorous, or strict, if it is the case that all items necessary for the demonstration of its conclusion are exhaustively listed, i.e., if no item essential to the demonstration is omitted. With respect to rigor of proof the techniques and operations of symbolic logic surpass earlier systems. The isolation of the requisite assumptions for rigorous proof is often not an easy task. Failure to so isolate items of proof led Aristotle to an incomplete analysis of the relations which hold in the so-called "square of opposition"; a similar failure by Euclid to stipulate all the assumptions of his geometry was not corrected until David Hilbert supplied the necessary postulates in this century. Many of Euclid's theorems in his original formulation are not valid, though they may be rendered so by recourse to Hilbert's supplementary postulates. Similarly, though Aristotle supposes the syllogism to be sufficient for all demonstration, it is evident that certain laws from the division of logic called "the propositional calculus", a division not included in the theory of the syllogism but rather presupposed by that theory, must be employed in the proof of certain syllogisms. These lapses from rigor have been rendered evident through the techniques of modern logical analysis. 1.13 Comprehensiveness. In antiquity, the study of formal logic was divided into two camps: the Aristotelian and the Stoic-Megarian. The Aristotelians, following the master, worked primarily with the logic of terms and the logic of classes, the logic of subjects and predicates, i.e., on part of that domain now called functional logic conceived of as independent of propositional logic. The Stoic-Megarian logicians focused attention on the relations between propositions, that part of logic now called the propositional calculus. (For a discussion of the relation between functional and propositional logic see Section 1.8 below.) For reasons which will not concern us here, these two approaches to logic were regarded as antithetical. We now know that the two domains are not opposed, but rather that propositional logic is a part of functional logic. While the Medievals were more fully cognizant of the relation between these domains, a full realization of the intimacy of their connection 9

SYSTEMS OF FORMAL LOGIC

awaited the development of symbolic logic. Similarly, the full development of the so-called functional logics beyond the lower functional calculus was the achievement of our time. None of the earlier logics were developed in such a way as to take adequate account of relations. The difficulty of expressing and proving many relational propositions motivated Leibniz to do some of his pioneer work in logical theory. 1.14 Clarity. Little will be said about this here, since the distinctions and clarifications brought about by modern developments will become evident in the sequel. The distinctions between the object-language and the meta-language, between proof and derivation, between formation and transformation rules, to be discussed below, may be mentioned in passing. 1.15 Finally, with respect to perspicuity, we may likewise be brief. For instance, the isolation of various meanings of the relation "if ... then ... " may be clearly indexed by the use of distinct signs. Hence, ... ::J ---

may be used for "material implication", while ... --3 ---

and ... --+ ---

and

...-)---

symbolize "strict implication", "intuitionistic implication" and "strong implication" respectively. While it is possible to indicate these distinct relations in words, the symbols are clearer and indicate by their difference of shape that different meanings of "if... then ... " are involved. Similarly it is easier to show briefly and perspicuously the relation between various connectives such as '::J' and '--3'. Suppose we want to describe the relation between the formula

[p--3q] and the formula

[p::Jq] where p and q are statements. In the language of words we can say: "The statement 'The statement p strictly implies the statement q' mate10

INTRODUCTION

rially implies the statement 'It is necessary that the statement p materially implies the statement q'; and the statement, 'It is necessary that the statement p materially implies the statement q', materially implies the statement, 'The statement p strictly implies the statement q'." In symbolic form we can express the same thing in the following terms: p--3q=. D .p::Jq

1.2 Symbolization

Symbolic logic is a kind of formal logic. A full system of symbolic logic would ideally encompass the entire field of formal logic, since its object is the same as that of formal logic generally taken (cf. 1.05). Symbolic logic is usually developed as a system. The term 'system' is not defined here, since this book may be regarded as a kind of extended definition of systems of logic. Suffice it to say for the present that unless logic were developed systematically, it would be impossible to attain its object, since there would be no way to determine whether or not it were complete in the sense that it contains as laws statements corresponding to all valid inference forms, and whether it were consistent, i.e., whether or not it contains as laws both a statement and the denial of that statement. 1.21 The reader has already been introduced in earlier sections to some of the symbols of symbolic logic. In Section 1.021, letters were used to represent statements, and in Section 1.14 some symbols representing the relation "if ... then ... " were introduced, along with letters representing statements. The logical motivation for replacing specific statements by letters standing for any statements may be clear to the reader already. But it is useful to specify this motive in a more precise way. Consider argument A in 1.01 and argument C in 1.03. The subject matter of logic is not the movement or rotation of the earth: this is the subject matter of the science of physics. Likewise, the subject matter of logic is not the humanity or mortality of Socrates - or of any other being. Logic is not directly concerned with mortality statistics or historical questions of who or what Socrates - or anything else - was. Logic's subject matter is formal validity, the formal relations which obtain between any statements. Hence, the irrelevancies of the earth, the sun, rotation, Socrates, mortality and the like are dispensed with by representing any statements by letters, thus fixing attention on the logical form all such arguments possess in 11

SYSTEMS OF FORMAL LOGIC

common; for this is what is relevant to the science of logic and to the logician qua logician. 1.211 The letters used in the argument-forms A" and B" in 1.021 and in the formulae of 1.15 are called variables, more specifically propositional variables. Let us agree that the small letters from the middle of the alphabet, p, q, r, s, t, are to be propositional variables. If there is need of more propositional variables than these five, further variables with subscripts may be introduced, PI, ql, rl, Sl, II, P2, ... , to fill the need.

Propositional variables are symbols for which statements expressing propositions may be substituted. Thus A and B of 1.01 are substitution instances of A" and B" of 1.02, according to the previous usage. By "proposition" is understood the meaning of a declarative sentence or statement. Propositions are not marks on paper or spoken sentences or symbols, but what marks on paper, spoken sentences and symbols mean. The discovery of the variable appears to be due to Aristotle. From the point of view of logic, mathematics and other deductive sciences, this was one of the most important of all discoveries, since it freed these disciplines of the burden of irrelevant subject matter and allowed the development of a completely formal logic. It is important to note that a propositional variable is not merely an abbreviation for some specific statement or other, but is a symbol for which any statement may be substituted. 1.212 The first premise of A" is the statement-form or formula, If p then q when we replace the capital letters with small ones in accordance with the specification of 1.211 concerning which symbols are to be propositional variables. Two things are to be noted in connection with this expression. (1) The expression is not, strictly speaking, a statement, since p and q are not statements, but variables. The expression becomes a statement when statements are substituted for its variables. None the less, expressions of such a form having propositional variables will be, for convenience, referred to as formulae or statements indiscriminately. (2) The expression "If p then q" is systematically ambiguous, since the vague "if. .. then ... " of ordinary English occurs in it. If the object of logic is to be obtained, then this ambiguity must be avoided, since

12

INTRODUCTION

there are some meanings of "if... then ... " which would make statements formed by means of it true, while there are other meanings which would make what appeared to be the same statements false. Hence, if we had in our logic only the vague "if... then ... " used above, without further specification, it would be impossible to evaluate for validity some arguments using it. Hence systems of symbolic logic replace the ambiguous "if. .. then ... " of ordinary language by a symbol whose meaning is constant or invarient from statement to statement. 1.213 The variables in a system of symbolic logic are logical signs. However there are other logical signs which are parts of every system of symbolic logic. These signs are called the constants, or junctors, or connectives, (the names are used interchangeably) of the system. Every logical system has at least one such sign; many have more than one. Each functor of a system has an invarient meaning. It always means one and the same thing. In this it differs from the variables. The symbols ::), --3, ~, and -) of 1.15 are functors representing distinct meanings of "if. .. then ... ". Functors differ from variables in that substitution does not apply to them (except in a special way to be discussed in Chapter 3). Likewise, a variable standing alone may constitute a formula of a logical system; but a functor standing alone cannot; the functors of a system serve to generate formulae from formulae: thus by means of the functor for material implication, ::), the formula p::) q, i.e., "p materially implies q" can be constructed from the variables (formulae) p and q. But ::) standing alone, read " ... materially implies ... ", is not a formula and cannot be converted, by itself, into a statement expressing a proposition. For this reason a distinction is made between proper symbols of a system, i.e., symbols which standing alone are formulae of the system, and improper symbols, i.e., symbols which do not constitute formulae when standing alone. The variables are among the proper symbols, and the functors, among the improper symbols of the systems constructed in this book. Propositional variables (or statements) are said to be the arguments of the functors of the logic of propositions. (This is a quite distinct meaning of the word "argument", differing from the use of the term previously given.) If the symbol F is a functor then the argument of F is any expression which is determined by F in a sense soon to be specified. 1.22 The usual functors of propositional logic, the logic with which we begin, are the following: 13

SYSTEMS OF FORMAL LOGIC

(A) (B) (C)

& v

(D) (E)

(the (the (the (the (the

negation Junctor) conjunction Junctor) material disjunction Junctor) material implication Junctor) material equivalence Junctor)

the names being found at the right of the symbols. These, then, are the standard propositional functors (or constants, or connectives). By means of these functors complex formulae are constructed from simple formulae. A formula is simple if it is made up of a single variable. Any formula in the logic of propositions which is not a simple formula is a complex formula. Thus p is a simple formula, while '" p and p::::> q are complex formulae. The functor '" stands to the left of its argument; thus: '" p. This expression is read "not p". The remaining functors are dyadic, i.e., they require two arguments, while '" is monadic, i.e., requires only one argument. The functors (B), (C), (D), (E), stand between their arguments, thus:

p&q read, "p and q";

pvq read, "p and/or q";

p::::>q read, "p materially implies q";

p=-q read, "p is materially equivalent with q".

1.3

Logical Functors and Their Definitions

1.31 From the point of view of logic, the most important property of propositions is their truth-value, i.e., truth and falsehood. A true proposition (or a statement expressing a true proposition) is said to have the truth-value truth. A false proposition (or a statement expressing a false proposition) is said to have the truth-valueJalsehood. We will adopt here the view of Gottlob Frege that declarative statements have propositions as their meaning, but that they are not names of propositions; they name 14

INTRODUCTION

or denote truth-values. Now, given any meaningful statement, i.e., any statement expressing a proposition, the statement has one or the other (but not both) of the values truth or falsehood. Hence for any possible combination of two independent statements substitutable for the propositional variables p and q there are only four possible combinations of truth-values: both are true; or, the first is true and the second is false; or, the first is false and the second is true; or, both are false. Let us agree that the capital letter "T" designates the truth-value truth, and the letter "F" designates the truth-value falsehood. Given this, a table can be set up which indicates the four possibilities above outlined. p q T T

T F F T F F Clearly the four possibilities are exclusive, since if p and q have anyone of the combinations of truth-values, they cannot have any other without contradiction. 1.32 The five functors of standard or classical propositional logic can now be defined in terms of the above table. The functors are said to be extensional or truth-functional. A logic in which the only functors are extensional functors is said to be an extensional logic. What this means is that, for any complex statement or formula formed out of such functors with the help of propositional variables or statements, the truth-value of the complex statement can be determined by determining the truth-values of the simple statements or formulae which are its constituents. Put in another way, one can always discover the truth-value of a complex statement or formula, in which the only functors are extensional functors, if one knows the truth-value of its simple component statements. The extensionality of the functors makes the tabular, truth-functional definitions given below possible. 1.321 The following truth-table defines negation: TTl (1)

(2)

(A) (B) p i "'p T I F

FIT

15

SYSTEMS OF FORMAL LOGIC

Given any statement p there are but two possibilities: p has the truthvalue T, or p has the truth-value F. The first truth-table (TTl) defines negation. The horizontal divisions of the table below the horizontal line are called rows. The vertical divisions are called columns. When p has the value T, (row 1, column A), '" p has the value F (row 1, column B), i.e., when p is true, "'pis false. When p has the value F (row 2, column A), '" p has the value T (row 2, column B), i.e., when p is false, "'pis true. Thus, substituting onp, if the statement, "F. D. Roosevelt was President of the United States", is true, then ,its negation, "It is not the case that F. D. Roosevelt was a President of the United States", or "F. D. Roosevelt was not a President of the United States", is false. Likewise, if "Douglas MacArthur was a President of the United States", is false, then its negation or denial, "It is not the case that Douglas MacArthur was a President of the United States" is true. 1.322 The definition of the conjunction functor is given in TT2. TT2

P q !p&q T T T

T F F T

F F

F F

F

This means that a conjunction of statements, p&q, is true only if both p and q are true. It is false otherwise. The column to the left ofthe vertical line is a repetition of the preliminary table given in 1.31. In future tables this column will be dropped, the sequence of alternative values being given under the propositional variables in the formula itself. TT2' illustrates this practice. TT2'

p & q T T T T F F

F F T

F F F The functor & is the strongest in the system, since it is true only when both conjoined statements are true. One functor, A, is said to be stronger than another, B, if there are fewer circumstances under which A gives a true statement, according to the truth-tables, than B. The functor '&'

16

INTRODUCTION

corresponds with the functor "and" in ordinary language. The middle column of the table TT2' is called the index column. The index column gives the value of the formula for the various values of its variables. 1.323 The table defining the inclusive or material disjunction functor is, TT3

p v q T T T

T T F F T T F F F

The table specifies that a disjunctive statement is true in every case except when both of the statements connected by the or-symbol, 'v', are false, i.e., a disjunctive statement p or q is true when both (or all) of the disjoined statements are true, and when one or the other is true: it is false when both or all of the disjoined statements are false. Note that the disjunction symbol is more inclusive than the customary "either ... or ... " of ordinary language. In most - but not all - cases when we say "either p or q" we mean not that one or the other or both of the statements so disjoined are true, but only that one or the other, but usually not both, is true. The "either/or" of ordinary language is called exclusive disjunction. However, there are perfectly "ordinary" uses of inclusive disjunction found in the language of every day. An example is the statement, "You can buy a hat either at Macy's or at Gimbels'" i.e., "You can buy a hat at Macy's or you can buy a hat at Gimbels'." When a statement of this kind is made the informant does not mean to exclude either possibility (or both). To assert such a statement is to use inclusive disjunction. The statement might be more clearly phrased, "You can by a hat at Macy's and/or you can buy a hat at Gimbels"'. Exclusive disjunctions are very common in everyday use. If someone says, "I will either spend the afternoon at the movies or I will spend the afternoon reading", he is using exclusive disjunction. The table for exclusive disjunction, where W is the symbol for this relation, is TT3'

p W q

T F T T T F F T T F F F 17

SYSTEMS OF FORMAL LOGIC

Thus an exclusive disjunction is true only if one or the other, but not both, disjuncts is true, and false if both are true or both are false. Most logical systems do not introduce a separate functor, 'o/!, for exclusive disjunction. The reason for this - merely apparent - poverty on the part of such logical systems, is that it is easy to define a formula using the constants v and & which adequately represents the exclusive disjunction of two statements. This formula is

[pvq]&[ "'pv "'q] An even simpler formula representing exclusive disjunction between any two statements, p and q, is

using the equivalence functor of section 1.325. The reader may wonder why weak, or inclusive, or material disjunction, as it is variously called, is employed in most symbolic systems in preference to the stronger functor' 'o/! '. Besides the fact that' 'o/! 'can be defined in terms of the standard functors, there is another important reason for using the weaker' v '. Part of the following discussion is due to I.M. Copi, in his Symbolic Logic. The constant, 'v', used in a statement, gives the partial common meaning for virtually all recognizable uses of the word "or". We are willing to grant the truth of a disjunctive statement if one or the other of the constituent statements is true, or, in some cases, if both are true. But we would not be likely to suppose any properly used statement involving "or" to be true if both the disjoined statements (hereafter called disjuncts) are false. Thus for all disjunctions which are asserted, at least one disjunct is true, and for all disjunctions which are denied both (or all) disjuncts are false. The material disjunction gives this common property of disjunctive statements. Likewise, all valid arguments which employ stronger forms of disjunction remain valid when the weaker material form is substituted for them in the arguments. Finally, it is advantageous to develop a system possessing the weakest possible functors ("possible" here is to be defined in terms of validity and completeness: the functors must at least be strong enough to preserve validity; they must not be such as to allow the derivation of false conclusions from true premises, since this is the minimum condition for

18

INTRODUCTION

validity). When weak or inclusive or material functors are used, it is possible to do two things which cannot be done if relatively stronger, more exclusive constants are used: (1) In many cases, the stronger constants can be defined by recourse to combinations of the weaker ones, as was noted earlier, and (2) Many valid arguments can be constructed and results proved by the use of the weaker functors which are not available if we use relatively stronger functors. As the reader will see, there are cases in which it is logically interesting and advantageous to construct a logic involving stronger constants than the standard ones (cf. Chapter 6). 1.324 The tabular definition of the functor for material implication is, TT4

q

p

::>

T T F F

TIT F!F T T T F

When the truth-table given above defining ::> is adopted the system in which the proofs are developed using this sign is called a system of material implication. A very large number of logical systems use this concept of logical implication. The sign " ... ::> ---" is read, " ... materially implies ---", or, "if ... then ___ ". The reader will already have noticed that material implication like material disjunction is not identical with all the every day meanings of "implies". This definition of implication renders true the following statements, which would not, by the ordinary meaning of "implication" be thought to be implicative relations: A B C D

'''Roosevelt is alive', implies, 'Khrushchev is alive'." '''Athens is a city in Greece', implies, 'The days are long in Switzerland during the summer'." '''Bertrand Russell wrote the Phaedo', implies, 'Alfred North Whitehead wrote Candide'," '''D. D. Eisenhower is a logician', implies, 'Winters in Fribourg are damp'."

In most of the ordinary, legal, journalistic, and philosophical uses of "implies", none of the antecedents (the initial statements) would be 19

SYSTEMS OF FORMAL LOGIC

thought to imply the consequents (the subsequent statements) in the statements A-D. Ordinary language sanctions, E

"'Aristotle is mortal', implies, 'Aristotle will die',"

as an implication since there is an evident connection of meaning between the antecedent and the consequent. But it would be unlikely to accept, F

'''Aristotle is mortal', implies, 'Plato is famous',"

since no such evident (or, for that matter, non-evident) connection is to be found. Yet all these "strange implications" are true, given the material interpretation of implication, since they conform to the condition defining implication given in the truth-table. This situation, though initially strange, follows naturally when it is remembered that the set of functors given here are all truth-functional. The implication hook, '::>' or horseshoe, as it is sometimes called, is a truth-functional functor; thus the meanings of the constituent statements are not taken into consideration, only their falsity or truth. It can be seen that there are a number of common alternative meanings given to the ambiguous word, "implication".

Formal and Strict Implication A formal implication is an implication which is true for all values, or substitutions on its variables, e.g., G

x is human implies x is mortal, for all values of x.

Taking any individual entity in the world, x, whatever x may be, if x is human then x is mortal. Put otherwise, we have a formal implication (x) [(1)x::>Px]

where (1) and P are arbitrary properties predicated of x when and only when there is no entity x such that "x is a (1)" is true and such that "x is a P" is false. A kind of implication related to formal implication has been developed by the American logician C. I. Lewis. It is called strict implication. The 'strict implication p~q

holds between two propositions if and only if the conjunction p&"""q

20

INTRODUCTION

is not merely false, but logically false, i.e., inconsistent or self-contradictory. And example would be H

x is an Euclidean triangle strictly implies x has three sides.

One cannot simultaneously affirm the left or antecedent side of this expression and deny its right or consequent side without contradiction when one has supplied the name of an individual for the x. While all strict implications are formal implications, it does not follow that all formal implications are strict implications. Causal or Synthetic Implication

An example of causal implication is, I

If I put my hand in boiling water I will receive a burn.

Evidently, implications of this sort - frequently used in predictiontheory - are of a different type than, e.g., strict implications, since the fact that the consequent follows from the antecedentcannot be discovered merely by resort to formal logic and definitions, but must be established empirically. Yet this is a perfectly common use of the terms "if... then ---", and "implies". There are, of course, a number of other uses of the term "implication" which the reader may suggest or work out for himself. Now are there any properties common to all forms of implication? Evidently there are; and it happens - as was the case with material disjunction - that material implication represents the functor which gives, using Copi's terms, the partial common meaning of all usual forms of implication. This holds in two ways. (1) The minimum condition for a valid implicative relation, as was noted before, is that it does not allow the derivation of a false conclusion from true premises. This condition, the validity condition, is clearly satisfied by material implication. (2) All implicative relations are subject to the logical law or rule called modus ponens. The argument form A" gives the form ofthe rule of modus ponens. This is an extremely important rule, as the reader will see in the remaining chapters. It will first be stated as a rule, then as a law. RULE

If 'p

implies q' is asserted and if 'p' is asserted, then 'q' may be asserted.

21

SYSTEMS OF FORMAL LOGIC

Since material implication satisfies the validity condition it evidently satisfies the condition of the modus ponens principle. Modus ponens is sometimes called "the rule of inference". 1.325 The definition of material equivalence is, TT5

P _

T T F F

T F F T

q T F T F

A material equivalence, p == q, holds when it is the case that p::::> q and q::::> p, thus it can be defined by the formula

1.4 Tests of Validity Using Truth-tables Truth-tables provide a mechanical method of testing the validity of some (though not all) complex formulae formed out of extensional connectives and variables. A formula which is true for every value of its propositional variables, i.e., a law of propositional logic, is called a tautology. The class of logical truths, often abbreviated (following Carnap) L-truths, has as members all formulae which are true for all values of their variables. Such formulae are also called "valid formulae". The class 0.( tautologies, i.e., the class of formulae which are true for all substitutions on their propositional variables, is thus a sub-class of the class of logical truths. (If a formula A is a tautology then A is a logical truth; but it does not follow that if A is a logical truth, A is a tautology in the sense defined, since there are variables of kinds other than propositional variables.) The class of logical falsehoods, abbreviated L-falsehoods, has as members all formulae which are false for all values of their variables. Logical falsehoods are also called contradictions, since a contradictory formula is one which is always false. The class of logical contingencies or logical indeterminacies, abbreviated L-contingencies or L-indeterminacies, has as members all formulae which are true for some substitutions on their variables and false for other substitutions, i.e., they are sometimes true, and sometimes false, depending upon the substitutions. Systems of logic

22

INTRODUCTION

of the usual sort are designed to include as theorems only such formulae as are logically true.

Truth-tables provide a method for determining the validity (or invalidity) of all formulae formulable using propositional variables and constants alone. 1.41 Clearly, all simple formulae are contingencies or are contingent according to the definition given above. This holds because the truthtable for a simple formula will always be of the form P T F

thus p is true when a true statement is substituted and false if a false statement is substituted for it. This does not mean that there are no simple statements which are, of themselves, necessarily true, but only that no variable standing alone ever has the value T for all substitutions on it. 1.42 Likewise, all complex formulae made up of a negation symbol with a single variable as argument are contingent. This follows for the same reason as did the result of 1.41. 1.43 Consideration of the tables TT2 through TT5 show that formulae of the form

p&q pvq

p=>q p=q

are logically contingent since, as the index columns of the tables show, these formulae are true under some substitutions for their variables and false for others. 1.44 Now consider the formulae A B

C D E

[p=>q]=>[q=>p] [[P=>q]&[q=>r]]=>[p=>r] [q=> [ '" [p v '" p]]] p&"'p "'[p&"'p] 23

SYSTEMS OF FORMAL LOGIC

F G

pV"'p [p ::>[q ::>[r ::>[s ::>[p => p]]]]]

The method of truth-tables provides an effective technique for deciding whether the formulae A-G are tautologies, contradictions, or contingencies. A method or technique for problem solving or testing is said to be effective if by the application of the method it is always possible in a finite number of applications of the method to solve any arbitrarily selected problem within its domain. The domain of problem solving for the truth-table method is to determine, for formulae made up of propositional variables and the five standard functors, whether the formulae are tautologies, contingencies, or contradictions. In its domain, the truthtable method is effective. 1.441 The brackets, [, ], in the formulae A-G serve as punctuation marks showing which variables and functors belong together. In applying the truth-table method the first thing to do is to discover the main functor of the formula to which the method is to be applied. There is always one and only one main functor in any well formed formula. The method is as follows. One looks for that connective which either is not enclosed within brackets, (e.g., the second implication sign in A, and the third hook in B) or which is enclosed only by the brackets which enclose the entire formula, e.g., the first implication sign in C. Then one determines the truth-value of the formulae to the left and right of the main functor by application of the truth-tables set out above. Thus one discovers the value of p => q and q => p in A. Then on the basis of the results in the index columns for these formulae, the value of the formula itself may be computed, by application of the same process. It is necessary, of course, always to give the same value to every occurrence of the same variable in every row. (Thus, if q is given the value T in a row it must be given the value T at every place in the row.) In the following the full expression to the left of an implication sign is called the antecedent of the implication. The full expression to its right is called the consequent of the implication. Thus in A the antecedent is p=>q, and the consequent is q=>p. A formula with two occurrences of propositional variables, the same or distinct, is said to have the length 2; a formula with three occurrences of propositional variables, the length 3; a formula with n occurrences of propositional variables, the length n. Thus formula A has the length 4, and formula B

24

INTRODUCTION

has the length 6. The truth-values under the main connective are determined last. Thus a test of A by the truth-table method would appear as follows. 1 Test

[p::J q]::J [q::J p]

TTA

[p

(1) (2) (3) (4)

T T T T F F F T T F T F

::J

q]

::J

[q

::J

T T T T F T F T F T F T

p] T T

F F

The table shows that [p::J q]::J [q::J p] is a contingent formula since there exists one truth-condition under which it is false, namely the condition given in row 3 of TTA. Test

[[p::J q] &[q::J r]]::J [p::J r]

TTB (1) (2) (3) (4) (5) (6) (7)

[[p

(8)

T T T T

F F F F

::J

q] & [q

T T T T T F F F F F F F

::J

r]]::J [p

T T T T F F F T T F T F T T T T T T T T F T F F T F T F T T T F T F T F

T T T T T T T T

T T T T

::J

r]

T T

F F T T

F F F T T F T F F T T F T F

1.442 This formula is a logical truth since it has only T's under the main

connective. When a compound proposition involves more than two bracketed expressions one proceeds with the expressions (as in B, where these expressions are connected by '&'), in the same way as one would do in a smaller expression. He locates the main connective in the lesser expression [[p::Jq]&[q::Jr]], and determines its truth-value, and proceeds to apply the same technique to the full formula, calculating the truth-value of the formula's main functor by reference to the truth-values 1 The third letter in each of the indices of the truth-tables given below corresponds to the letter assigned to the formula, given in 1.44, which the truth-table tests. Thus TTA applies to Formula A, etc.

25

SYSTEMS OF FORMAL LOGIC

of the full antecedent and the full consequent connected by the main functor of the complete formula. 1.4431 There is a mechanical way of determining the number of rows which a truth-table must have to exhaust the possibilities of truth-value assignments, for any arbitrarily selected formula. First ascertain the number of distinct variables in the formula: thus p::J P has only one distinct variable, with two occurrences; the formula A (p. 23), has two distinct variables, each with two occurrences; B has three distinct variables, each with two occurrences. Second, take the number 2, i.e. the number of truth-values, and attach to 2 as its power the number of distinct variables in the formula. Thus ifthere is one distinct variable we have 21(=2); if the formula has 2 distinct variables we have 22( =4); if the formula has 3 distinct variables we have 23(=8); if the formula has n distinct variables we have 2n. It is always possible to determine the number of rows in a table for a formula by discovering the value of 2n, where n is the number of the distinct variables in the formula. Thus the table for p::J P has 2 rows; the table for A has 4 rows; the table for B has 8 rows, and the table for D has 16 rows. 1.4432 There is also a mechanical method for distributing truth-values. Suppose a formula has two distinct variables; thus, by 1.4431, it has four rows. In that case divide the number of the rows (4) by 2, giving the number 2. Suppose the first variable is p. Under every p column write 2 rows of T's, and the remaining rows, F's. Suppose the second variable to be q. Divide the number of T's in the first column by 2, thus obtaining 1. Thus write in the column under q one T, then an F, then a T, then an F. The sequence will be one T, followed by one F, and so on. If the number of distinct variables is 3, then the number of rows is 8; thus, applying the same procedure divide 8 by 2, giving 4. Thus there will be a sequence of 4 T's under the first variable, followed by 4 F's. Divide 4 by 2, gaining 2; thus write 2 T's followed by 2 F's, followed by 2 T's, followed by 2 F's in the column under the second distinct variable. One applies the same procedure to the third distinct variable as applied before. Thus, following this rule, we have the following table for p::J[q::Jr], a contingency.

26

INTRODUCTION

(1)

(2) (3) (4) (5) (6) (7) (8)

P T T T T F F F F

::J T F T T T T T T

[q ::J r] T T F F T T F F

T F T T T F T T

T F T F T F T F

This procedure may be mechanically applied to any propositional formula. 1.444 The formula C involves the negation functor. It has a truth-table having the following form. Test

[q ::J ['" [p v '" p]]]

TTC (1) (2)

[q ::J ['" [p v '" p]]]

(3)

(4)

T F T F

F T F T

F F F F

T T F F

T T T T

F F T T

Note that the formula as a whole is a contingent one, as shown by the mixture of values under'::J'. However, the consequent alone represents a logical falsehood, since all the truth-stipulations under '",' in the formula are F's. Since the manner in which negation is dealt with in the truth-table may not be completely clear from TTC we will present an extended truth-table indicating the procedure. It will help the reader if he remembers that the truth-table tests do nothing but incorporate and work with the previously presented truth-table definitions. From the definition of ' ",' we can always infer that if '" p has the value T then p has the value F, and if p has the value T then'" p has the value F. For our example we use the denial of the famous law of contradiction. The law is symbolized '" [p & '" p]. By the definition of negation, we can see that the denial of this principle is '" [ '" [p & '" p]]. Test

'" ['" [p & '" p ]]

TTD (1) (2)

'" ['" [p & '" p]] FIT I T \ F \ F(T) F T F F T(F) 27

SYSTEMS OF FORMAL LOGIC

The procedure here is merely to exchange for the truth-value the formula would take if unnegated, the opposite truth-value when it is negated. If the truth-value for p is T then the value of ""pis F, and vice versa. The following are truth-table tests of the principles of contradiction and excluded middle. TTE

"" [p & ""p] TIT F F T F F T

TTF

p

v ""p

!I~I~ The main functor in TTD and in TTE is negation. The contradictory or denial of any logical falsehood is a logical truth, and the denial of any logical truth is a contradiction.

1.445 The following is a truth-table test of G. TTG (1)

(2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

[p ;::, [q ;::, [r ;::, [s ;::, [p ;::, p]]]]] T T T T T T T T F F F F F F F F

T T T T T T T T T T T T T T T T

T T T T F F F F T T T T F F F F

T T T T T T T T T T T T T T T T

T T F F T T F F T T F F T T F F

T T T T T T T T T T T T T T T T

T F T F T F T F T F T F T F T F

T T T T T T T T T T T T T T T T

T T T T T T T T T TTT TTT T T T TTT TTT F T F F T F F T F F T F F T F F T F F T F F T F

The above somewhat complicated, but none the less obvious, tautology 28

INTRODUCTION

illustrates the degree of complexity possessed by a full truth-table for four distinct variables. A table testing a formula with five distinct variables, e.g., the tautology, [[p:::> q]:::> [[q:::> r]:::> [[r:::>s]:::> [[s:::> t]:::> [p:::> t]]]]]

has 32 rows, while one with six distinct variables has, by the rule set out earlier, 64 rows, and one with seven distinct variables has 128 rows, and there are plenty of tautologies with this many distinct variables, e.g., the formula, [q2:::> [p:::> [q:::> [[r:::>s]:::> [t:::>Pl]]]]]:::> [[q2:::>p]:::> [q2:::> [q:::> [[r:::>s]:::> [t:::>Pl]]]]]

This is a tautology having seven distinct variables. 1.46 In conclusion, truth-tables, though useful in testing formulae of propositional logic (for which they provide an effective test) do not test all truths of logic of a level higher than propositional logic. Aformula which, when submitted to a truth-table test, has none but T's in its index column is called a "truth-table tautology" or "tautology" for short.

1.47 The logics considered in this book are extensional logics. To say that a logic is extensional is the same as to say it is truth-functional. One of the aims of the present book is to consider the properties of certain important systems of extensional logic. EXERCISE

Construct some well formed formulas (wffs) of propositional logic and test them for validity by means of truth-tables. 1.5 Proof and Derivation 1.51 It is awkward and inelegant to test every tautology of propositional logic by the method of truth-tables. There is, for instance, no reason to submit a formula of the form

A

[p:::>q]:::>[p:::>q]

to a truth-table once one has established the formula,

B

p:::>p

29

SYSTEMS OF FORMAL LOGIC

as a tautology. A is a clear case of B. But if A needs no truth-table test when B is established, there must be some reason for regarding A as established once B is established. The reason is not far to seek. A is available from B by substitution, substituting p =:l q for each of the two occurrences of p in B. Substitution is a method of proof, as will be seen below. A proof in symbolic logic, like a truth-table, is a structure designed to verify formulae. 1.52 In 1.01 an argument was defined as any sequence of statements the conclusion of which is claimed to follow from the premises. As noted there, arguments are formally valid or invalid. A proof is a kind of formally valid argument. A proof may be defined as follows: A proof is a sequence of statements or symbols representing statements such that each element (i.e., integral statement or formula) in the sequence is either an axiom or is an immediate consequence of earlier elements in the sequence. A proof is said to be a proof of the last element in the sequence.

1.521 To explain: given the definition, the sequence of the statements in a proof can be arranged in serial order, a number being correlated with each integral statement of the proof; these are called the step numbers of the proof. We call the expression to the right of a step number an element of the proof. Thus if A, B, C, D, are taken as representing statements or formulae, the structure would have the following appearance. (1) (2)

A B

(3)

C

(4)

D

A, B, C, D, represent the elements of the proof. Any argument, including

an argument that is a proof, is said to have a certain length. The length of the proof is the sum of the elements of the proof. It is given by the step number to the left of the final element of the proof. Thus the skeleton proof above has the length 4. It is said to be a four step proof By the definition, the proof skeleton would be said to be a proof of step 4, i.e. of the statement D. 1.522 The definition specifies under what circumstances a statement 30

INTRODUCTION

may become a step in a proof. There are only two alternatives specified in the definition. Alternative 1: the statement is an axiom. An axiom is a statement which is taken as primitive in a system: it itself is not proved in the system, but is used as a starting point for the proofs of other statements in the system. In principle one is free to choose any statements he wishes as axioms; logical truths, contingencies, or even contradictions. But from the point of view of logic as an interpreted system, the object of which is to possess only logical truths as its proper formulae (i.e., not as a mere play of symbols, but as a system of valid inferences and valid formulae), the choice of axioms is restricted to logical truths. None the less, as the reader will see in the ensuing chapters, the field of choice is still very wide (though not unlimited). The reader may wonder how it is possible to determine the logical truth of a formula taken as axiomatic in a system, if it cannot be proved in the system. The answer is that the system is not the sole criterion of logical truth. There are others, e.g., the truth-tables or, failing these, metalogical insight. But in the system itself the axioms are unproved. By the definition, one of the reasons which would justify the appearance of any of the statements A-D in the proof skeleton of 1.521 is the fact that it is an axiom of the system in which the proof occurs. Alternative 2: the statement is a step in the proof because it is an immediate consequence of the earlier steps. There are two ways in which one statement may be an immediate consequence of others in the sequence. Suppose we have as axioms of a system S the two truthtable-tautologies At A2

r~p

[p=>q]=>[[q=>r]=>[p=>r]]

Axiom 1 (At) is the law of identity; A2 is the second syllogistic law. Since these are axioms they may be introduced as steps in any proof in the system S. Now consider the following proof in the system S, call it s. s (1)

~[p => q] => Ceq=> r] => [p => r]]

A2

(2) (3) (4)

~[p => p] => rep=> r] => [p=> r]]

1, substitutingp for q

~p=>p

Al

~[p=>r] => [p=>r]

2, 3, modus ponens

s is valid. This proof in the system S illustrates the principle ways in which one statement is an immediate consequence of earlier steps. The 31

SYSTEMS OF FORMAL LOGIC

symbol "I-" is placed before those formulae of the system which are axioms or are provable from the axioms. 1.5221 Substitution. The rule of substitution is used to obtain step 2 in s from step 1. Using earlier terminology, the second step is a substitution instance of the first. It is obtained by simply replacing q at every occurrence by p. It is easy to see that, if I is a tautology, 2 must be a tautology as well. This follows because a tautology has the truthvalue truth for all values of its variables, i.e., no matter what statements are substituted for them. Hence the substitution of p for q, or, for that matter, of any formula simple or compound for q cannot effect the tautologousness of step 1. The result of consistent substitution on a tautology is always a tautology, since no matter what is substituted the formal structure of the formula is preserved; and it is by virtue of its formal logical structure that a formula is a tautology. Substitution is consistent when every occurrence of the same variable is replaced by the same formula. Thus to preserve consistency of substitution, q was replaced by p at both its occurrences in step I of s. The rule of substitution is not validly applied unless substitution is consistent. Note: substitution can be carried out only on simple formulae, i.e., single variables, within complex formulae. It is, for instance, illegitimate to substitute the variable q for the formula '" p, (though in this case q could be substituted for p in the formula, giving '" q). It is always legitimate to substitute a complex formula for a simple formula. Thus,

is also a valid substitution on step I of s. Because every consistent substitution on a tautology is also a tautology (never a contingency or a contradiction) the rule of substitution is said to be tautology preserving. Thus one way in which a statement can be an immediate consequence of another is by being the result of the application of the rule of substitution upon the other. A formal statement of the rule of substitution is found in Chapter 2. 1.5222 Modus ponens. This is the rule formulated as A" in 1.021. It is illustrated in the proof s in the deduction of the fourth from the second and third steps. A step in a proof is an immediate consequence of earlier steps if it is obtained from those steps by the application of the rule of

32

INTRODUCTION

If A and B are statements, then if A::J B is asserted and if A is asserted, then B may be asserted.

modus ponens:

Is the rule of modus ponens tautology preserving? Is it such that it will not allow the proof of a contingency or a contradiction from a law oflogic? It was shown that the rule of substitution is tautology preserving: no formula is obtainable from substitution on a logical truth which is not likewise a logical truth. Consider, then, the rule form (1)

A::JB

(2) (3)

A B

1, 2, -::J

If A::J B represents a formula in a proof and if it is not itself obtained by modus ponens, then it must be either an axiom or obtainable from an axiom by substitution. The same remark holds for any formula represented by A in step 2. Thus steps 1 and 2 must be laws of logic. Now suppose that the formula represented by B in step 3 is not a law of logic. In that case, it must be either a logical contingency or a logical falsehood. If, however, it is not the case that B is a law of logic, then either A::J B or A or both are not laws of logic. (a) If A is logically true, or tautological, and B is not, then A::JB is not a law oflogic. (Why?) But if A::JB is not a logical law, then it is neither an axiom of a valid logical system S, nor a valid substitution on an axiom of S, since all axioms of S, and hence all valid substitutions on them, are laws of logic. (b) If A::JB is a law and B is not, then A is not a law; otherwise A::JB could not be a law. (Why?) If B is not a law, then it is neither an axiom nor a valid substitution on a axiom of S. (c) But if either A::JB, or A, or both are not logical laws, then they cannot be introduced as steps in a proof. Thus if both A::JB and A are laws then B must be a law also. Hence the answer to the question above is affirmative, and modus ponens is a tautology preserving rule. EXERCISE

Show that the logical law corresponding with the modus ponens rule (cf 1.324 at the bottom of page 21) is a tautology by using a truth-table.

33

SYSTEMS OF FORMAL LOGIC

Note that modus ponens is validly applied only f(the expression represented by A in the statement of the rule is the entire antecedent of the implication A=> B. Thus the application of modus ponens to steps 2 and 3 is valid since p => p is the antecedent of step 2. However, the following would be incorrect applications. l' 2'

[[p=>p] => [p=>r]] => [p=>r] p=>p [p=>r] => [p=>r]

3'

[[p => p] => p] => [r => [p => r]] p=>p [p => [r=> [p => r]]]

I" 2"

3"

for the reasons given above. Step l' is, like step 1 of s, a tautology, but the argument following it is vitiated by an incorrect application of modus ponens. Step I" is also a tautology, its correlated proof being vitiated in the same way. Though all three (steps 1, 1', and I") are tautologies, they ate distinct formulae, as shown by the bracketing. 1.53 It can be seen that if the axioms of a system are tautologies and if the rules of the system are modus ponens and substitution, then every formula proved from the axioms with the help of these rules is a tautology, since both rules preserve tautologies. The kind of proof here outlined is called axiomatic proof, since proofs developed by means of this technique begin from axioms. A system which is articulated by means of axiomatic proofs is called an axiomatic system. Many of the systems developed in this book are axiomatic systems. A formula proved from axioms of a system by application of its rules is called a theorem or thesis of the system. All theorems of the kind of system described at the beginning of this paragraph are tautologies. All arguments which proceed by valid application of the rule of substitution upon theorems of such a system are valid arguments. Because it is through the application of the rules of a system that its axioms give rise to theorems, rules of this kind, e.g., modus ponens and substitution, are described as transformation rules. 1.531 Let us agree to reserve the word "proof" for valid arguments of the kind described above. Hence invalid forms such as those given in 1.5222 are called, not "invalid proofs", but "invalid arguments in the form of a proof." 1.532 In most axiomatic systems, theorems proved from the axioms of

34

INTRODUCTION

the system are used, in exactly the same way as the axioms, as steps in proofs. Indeed, in many systems, little direct use is made of the axioms at later stages of the development of the systems, proofs being developed almost exclusively from theorems proved from the axioms. Such proofs are said to be abbreviations of full proofs, the full proofs being demonstrations of the same theorems directly from the axioms. But it is tedious to repeat again and again the proofs of theorems already established in order to advance to new theorems. Thus, previously proved theorems are used on a par with the axioms of the system in developing proofs. This practice is justified by the proof given in 1.5222 that anything implied by a tautology (or logical truth) using substitution and modus ponens is itself a tautology (or logical truth). Clearly, no invalidity can enter into the system by virtue of the application of this abbreviating device. Thus, as a concession to the brevity of human life, this device is adopted. Of course, no theorem can be proved by the use of this abbreviating technique which could not be proved without it, articulating the system by use only of full proofs. As a consequence of this, we will add to the definition of "proof" given in 1.52 as follows: A proof is a sequence of statements or symbols representing statements, such that each element of the sequence is either an axiom or is a previously proved theorem or is an immediate consequence of earlier elements in the sequence. 1.54 A derivation is an argument resembling a proof but differing from

a proof in that it begins not from a fixed set of axioms initially set out, but from hypotheses or assumptions; these may be logically true, or contingent, or contradictory without effecting the validity of the laws of the system in which the derivation takes place. In a derivation no limitation whatever with respect to truth or falsehood or contingency is placed on its hypotheses. In the case of a formal axiomatic proof, it is possible to assert the last element (indeed any element) of a proof sequence by itself as a theorem, without making it conditional upon the axiomatic statements with which the sequence began. However, in a derivation, the last member is not in general a theorem, but becomes a theorem by the adjunction of the hypothesis from which it is derived as its condition. Thus if the formula 'B' is derived from the hypothesis 'A', the resulting theorem is not 'B', but 'A:::>B'. 35

SYSTEMS OF FORMAL LOGIC

A

1 2

hypothesis

3 B (In the scheme above it is assumed that there is some way of obtaining B from A by the rules of the system.) Most of the deductions in the systems of Chapter 4 are derivations, while most of those of Chapters 2 and 3 are proofs. 1.55 In the proof s, the reader will find to the right of every element a citation giving the justification for the element's presence in the proof, in accordance with the definition of "proof". The sequence of such justifications at the right of the deduction is called the justification column. Every element in a proof must have an entry in the justification column giving the reason for its legitimate presence. This may be done by giving the number of the axiom represented by the element, if it is an axiom; or by giving the theorem number of the element, if it is a previously proved theorem; or by citing one of the rules of the system by which the element was deduced and the step or steps to which the rule was applied obtaining the element. A similar practice is observed in the development of derivations. The justification column gives an effective method by means of which every argument may be checked for validity, since it is always possible to check the steps to ascertain whether or not they have been validly deduced. (There is a simpler way to prove the theorem s, [p => r] =(p => rJ. Its discovery is left to the reader.)

4

1.6 The Axiomatic Method

In 1.5 an informal treatment was given of axiomatic proof. The axiomatic method involves the notion of axiomatic proof as a part, but it is broader than a proof technique. The axiomatic method was devised to obtain the high standards of rigor, clarity and completeness required of a system oflogic. 1 There follows an outline of how a system would be constructed in accordance with the axiomatic method. 1 The reader will discover a number of deductions in the following chapters which are incomplete; hence not rigorous; hence not proofs. These have been left "open" for the reader to "fill in" as exercises.

36

INTRODUCTION

1.61 The object-system and the meta-system. Observation of what has gone before discloses two sorts of statements or formulae occurring in this chapter. The first kind includes the formulae of symbolic logic themselves, e.g., the formulae made up of propositional variables and functors found in the proof s of 1.522. These are said to be formulae of the object-system. The second kind of statement includes statements about the formulae, e.g., the abbreviated statements to be found in the justification of the proof s, stating that step I is axiom Al of the system S, and that step 2 is derived from step one by substitution. Other such statements are the assertion that the argument s of the system S is valid and the assertion that the arguments of 1.5222 are invalid. These statements are statements about the formulae of the object system S, i.e., they have these formulae as their object. Thus they are said to be statements of the meta-system which "talks about" the object-system S. It is a matter of some importance to distinguish between the statements of a system and statements about statements in the system or about the system itself. For unless some such distinction is made, the resulting logic can be shown to be contradictory, i.e., it has as laws both the assertion and the denial of the same statement (cf. Chapter 9). And by the laws of most logics it is possible to infer any arbitrary statement as a theorem, if a contradiction is a theorem. The difficulties which give rise to this distinction are examined in a later chapter. Put briefly, the distinction comes to this: no statements or formulae in the object-system are about themselves or about the object-system. All statements about the objectsystem are made in the meta-system correlated with the object-system. In the meta-system of a symbolic object-system are formulated: (1) statements about which expressions of the object-system are well formed; such statements formulate, so to speak, the grammer of the system: these statements are called the rules of formation or the formation rules of the system; (2) statements of the rules of transformation of the system, i.e., statements specifying under what circumstances one formula is a direct consequence of another; (3) statements about which well formed formulae of the object-system are axioms; (4) statements of what expressions or symbols of the system are to be regarded as abbreviations for others; these are the definitions of the system. The signs T and F in the truthtables are metalinguistic signs, since in their usual interpretation they specify truth-values of possible substitutions on the formula correlated

37

SYSTEMS OF FORMAL LOGIC

with the tables. The meta-system also contains theorems of its own, theorems about the object-system: these are called meta-theorems. The result proved in 1.5222, that the application of modus ponens to tautologies of the object-system always gives rise to tautologies, is a metatheorem. 1.62 It is useful to have variables in the meta-system through which it is possible to name and talk about the formulae of the object-system. Let us agree that the bold capitals from the beginning of the alphabet A, B, C, D, AI, ... stand for well formed formulae of the object-system (or the objectlanguage). Let us agree that bold small letters from the beginning of the alphabet stand for the propositional variables of the object-Ianguage l , unless they are otherwise specified, a, b, c, d, al, ... Let us agree that small Greek letters from the beginning of the Greek alphabet,

a,

p, y, ~, al, ...

are to stand either for a formula substituted for a propositional variable, or for truth values, as specified in the text. Let us agree that the following bold Greek capitals are to stand for any well formed expression of the system concerned, irrespective of its character:

r, A, 0, E, rl, ... A certain degree of flexibility will be retained for these meta-variables or syntactical variables; their use will be specified in each occurrence. The functors, :::), &, v, ==, "", are always used in the meta-system (or meta-Ianguage 2) to name themselves. Thus the meta-expression

A:::)B designates the relation of material implication obtaining between any two well formed expressions of the object language. The reader will see the utility of these symbols when he begins to work with the formation 1

2

"Object-language" and "object-system" are synonyms. "Meta-system" and "meta-language" are synonyms.

38

INTRODUCTION

and transformation rules of subsequent systems. Indeed, they have already been used in the statement of the rule of modus ponens in 1.5222. 1.63 In 1.61 four special classes of meta-systematic statements were noted. These four are: the rules offormation; the rules of transformation; the axioms; the definitions of a system. These four are called the primitive basis of any axiomatic system. It is from these that the object-system is constructed. The reader has already seen how the axioms and the rules of transformation function in the development of proofs of a system. This leaves the rules of formation and the definitions to be discussed. 1.6311 The formation rules of a system specify which symbols in the system are primitive symbols, i.e., which symbols are the basic units, not defined in the object-system itself, out of which formulae are constructed. In symbolic systems these usually fall into two classes, the primitive functors and the primitive variables of the systems. Thus the system of propositional logic, P +, introduced in Chapter 2, has the propositional variables, p, q, r, s, t, PI, ... as primitive variables, or proper symbols, and the functors, [, ], :::::>,

&, v,

=,

as primitive functors, or improper symbols. It must always be possible to determine, given any symbol of a system, whether that symbol is a primitive symbol of the system. 1.6312 As in ordinary language, so with symbolic systems, not every collection of symbols gives a statement or formula. The expression "And to but" is not a sentence of English. Likewise "[ v :::::> [& ]pq]" is not a formula of any known symbolic system: it is not a well formed formula. In natural languages, some combinations of words make sentences, while others do not. Likewise in symbolic systems, some combinations of symbols are well formed and some are not. Let us agree to abbreviate the expression "well formed formula" by the expression "wff", and the expression "well formed" by the expression "wf". The rules of formation give an effective specification of the conditions under which any formula formed out of the primitive symbols of a system is wf or is a wjJ. Thus by the rules of formation of the system, P +, the expression

39

SYSTEMS OF FORMAL LOGIC

is a wff. But the formula

[pq]rv & is not. It should be noted that from the fact that a formula is a wfJ of a system it does not follow that it is a theorem of that system - any more than it follows that a sentence in English is true simply because it is grammatically correct. 1.632 The definitions of a system state which wfJs are to be regarded as abbreviations for others. Thus in the system of Russell and Whitehead, Principia Mathematica, the wfJ

is an abbreviation for the wfJ

[ "'p v q] In the system PLT' of Chapter 3 of this book the formula

[pvq] is an abbreviation for the wfJ

Likewise, certain other symbols are introduced by definition. For instance, the symbol "I-" is defined to mean that the wfJfollowing it is a theorem of the system concerned. 1.64 Given the primitive basis of a system, the theoremhood or nontheoremhood of a wfJ purportedly deduced in it may always be effectively determined. The wfJ is a theorem if it has been deduced solely by virtue of the elements of the primitive basis. If the wfJhas not been so deduced, it has not been shown to be a theorem of the system. If the deduction has been carried out in such a way that the deduction is a proof of its last element by virtue of application of the primitive basis of the system, then its last element is a theorem of the system. Thus the primitive basis of a system gives an exhaustive specification of conditions under which a wfJ is a theorem of the system. 1.65 Besides the primitive transformation rules of a system, other rules are often introduced on the basis of theorems proved through the primitive transformation rules. These rules add nothing to the system

40

INTRODUCTION

which could not be obtained by the primitive rules, but serve to abbreviate proofs. They are not, of course, part of the primitive basis of a system. 1.66 Besides the forms of abbreviation already mentioned, one more abbreviative technique is here introduced and applied to the systems developed in the sequel. By the rules of formation, expressions are wffs only if suitably punctuated by brackets. Thus the second axiom of the system P + of the following chapter must be bracketed,

Axiom 1 of P + would be bracketed,

It is usual in symbolic systems to abbreviate this proliferation of bracketing by the use of certain conventions. The bracketing of the first of the above formulae may be abbreviated,

Here it is taken as understood that the main functor is the functor to the left of the largest constellation of dots or points. Thus the third implication sign is the main connective. The subordinate brackets are filled in according to the same principle. If there are two sets of points of the same number, e.g., in the formula

c

p~ .p~q~:p~ .p~q~:p~p,

it is understood that the first occurrence of a set of points, ":", is superior to the second, and that the main functor of the formula is to its left, while the main functor of the consequent of the formula is to the left of the second occurrence of the same set, ":". Thus the full bracketing of C would give,

where the third implication sign is the main functor of the formula and the sixth implication sign is the main functor of the consequent of the formula; the first implication sign is the main functor of the antecedent of the formula. Besides the convention instituted concerning the number of points and their replacement by brackets, another convention, the convention of association to the left is instituted. In the absence of points, 41

SYSTEMS OF FORMAL LOGIC

the brackets are to be restored first to the leftmost wff. Thus the formula D

p~q~q

by the convention of association to the left, is to be restored as D'

[[p~q]~q]

Likewise, the formula E

p~q~r~s

is an abbreviation of the wff E'

[[[p~q]~r]~s]

The expression, F

p&q~r~ .s~t~:pvq~ .r~s~:. p~ "'q~ .q~ "'p~:r~r

is an abbreviation for the wff

F'

[[[[[P&q] ~r] ~ [s~t]] ~ [[P v q] ~ [r~s]]] ~ [[[p~ "'q]~[q~ "'p]]~[r~r]]]

While the formula F is not strictly speaking a wff, since it does not possess the full bracket punctuation of F', F and formulae like it using the point punctuation will be considered proper abbreviations for their fully bracketed counterparts. 1.67 It will, perhaps, be convenient to list these elements of axiomatic systems and their correlated meta-systems in compendious form. 1. Rules of Formation for an axiomatized system S

(a) Rules stipulating the functors and variables of the system S. (b) Rules stating the kinds of things which may be substituted for the variables, e.g., "variables p, q, r, ... are propositional variables". (c) Rules stipulating the condition under which combinations of the symbols of S are meaningful. A meaningful combination of the symbols of S is said to be a well formed formula, abbreviated wff, of S or to be well formed, abbreviated wI, in S. II. Rules of transformation or of inference

(a) These are the rules of operation on logical expressions which function 42

INTRODUCTION

in such a way as to allow the derivation of unproved statements from proved ones (or from hypotheses). (b) Most standard axiomatic systems possess the rule of substitution and a form of the modus ponens rule.

III. Axioms The starting points from which the theorems of S are proved with the help of rules of inference and/or definitions. IV. Definitions (a) The definitions of a system state which compound expressions are posited as equivalent to others, or (b) which expressions are to be regarded as abbreviations for others. An instance of this is the Principia M athematica definition of material implication,

[p:::>q]=Df. ["'pvq] which means both that either of these expressions can be substituted for one another whenever they occur in a proof and that p:::> q is an abbreviation for '" p v q. 1.7 Interpreted and Uninterpreted Systems 1.71 Axiom systems can be developed either as interpreted or as uninterpreted systems. An interpreted system is one in which the constants have been given a definite meaning and proper classes of objects substitutable for the variables have been specified; in other words, when Formation rule Ib is observed. But for some purposes it is convenient to drop this rule and to consider systems as pure uninterpreted calculi; many very interesting results have been obtained by such an approach results which would probably not have been proved had logicians not been willing or able to take this further step in the direction of abstraction from subject-matter. In pure calculi we do not assume any interpretation for the constants. Nor do we concern ourselves with the sorts of thing which may be substituted for the variables: though, for convenience we may call them "propositional variables", "predicate variables", and similar terms associated with the principal interpretation of the system,

43

SYSTEMS OF FORMAL LOGIC

these are merely labels which are taken to imply no interpretation. For all consistent uninterpreted systems there exist one or more interpretations which would render the theorems of the systems true. Clearly, the rules of formation and transformation along with the axioms and definitions of an uninterpreted system S are merely stipulations instructing us as to how we may derive further signs from signs (namely from the axioms). (It is an error to call a theorem of an uninterpreted system "true" or "false" in any ordinary sense, since these properties belong to propositions, not to marks on paper.) 1.72 Axiomatic systems have a number of interesting properties which will be discussed later; they are roughly defined here, following Church. An axiom system S is consistent if it is impossible in S to derive anything logically absurd or selfcontradictory as a theorem.

Consistency is the minimum condition for a satisfactory system of any kind, since from an inconsistency in the system any and every proposition is derivable I , and we do not desire to prove every proposition in our logic. An axiom system S is complete if every statement not inconsistent with the axioms of S is a theorem of S, i.e., for every wff of S, {3, either {3 is an axiom or theorem ofS or the addition of {3 to the axioms ofS would make S inconsistent.

One further definition of completeness: An axiom system S is complete if every proposition not in conflict with the interpretation of S is a theorem of S.

Completeness is a property which belongs to some systems and not to others. That the question of the completeness of logical systems is of no small importance will be made clear in Chapter 5. An axiom pI of a system S is independent if and only if, in the system of statements obtained by omitting pI from the axioms of S, pI is not a theorem. The axiom pI is non-independent if it may be deduced as a theorem in the system derived from the set of axioms of S which does not include pl. 1

The Johansson system of Chapter 6 is an exception.

44

INTRODUCTION

It is easy to define an extension of this notion applied to sets of axioms as a whole. An axiom set A of a system S is independent if no axiom pl of A may be omitted from A and be derived from the remaining axioms, less Pl, as a theorem. An axiom set A of S is called non-independent with respect to axiom pl of A, ifPl is derivable from A-pl (read "A minus Pl") as a theorem.

A number of axiom sets of well known logics have been shown to be non-independent, e.g., the axiom set of Principia Mathematica. Other axiom sets are provably independent. Two systems, S and Sf, are said to equivalent under the following conditions: S and Sf are equivalent if every theorem of S is a theorem of Sf, and every theorem of Sf is a theorem of S. Otherwise S and Sf are non-equivalent. A system S is said to include a system Sf as a sub-system (but not be included in Sf), if all the theorems of Sf are theorems of S, but not all theorems of S are theorems of Sf. 1.8 The Hierarchy of Logical Systems In this book we will be concerned with two hierarchical orders of logic. The first, the one which we will be concerned with here, is one of increasing completeness and specificity of logical concepts. The second is a hierarchy introduced by the authors of Principia M athematica, introduced to secure logic and mathematics against contradiction; it falls within the hierarchy here to be discussed, forming a kind of complex hierarchy within a hierarchy. This hierarchical order, the theory of types, is discussed in Chapter 9. We set out the orders or divisions of logic in the following outline: 1. The propositional calculus (often called the sentential calculus); 2. The calculus of classes (the monadic predicate calculus); 3. The lower functional calculus (the restricted predicate calculus); 4. The extended functional calculus (the extended predicate calculus). 1.81 It will be noted that logics of probability and inductive logic have

45

SYSTEMS OF FORMAL LOGIC

been ignored in the above listing. This situation is likewise reflected in the content of this book in which nothing will be said about these disciplines. The deductive systems we will be considering are more fundamental than inductive or probabilistic logics in the precise sense that the laws of deductive systems are required for the development of the logics of probability and induction. Hence the reader must have a thorough grounding in the techniques of deductive inference before he can profitably study these further systems. The attainment of this goal is precisely the aim of the present book. 1.82 The hierarchy 1-4 is based strictly on the order, (a) of precedence, and (b) of inclusiveness. The propositional calculus is the simplest and most fundamental of the numbers of the order. Its laws are required for the development of the higher and more complex members. But in itself it is entirely inadequate to express all forms of valid reasoning. The calculus of classes is wider than the propositional calculus, using laws and rules of that calculus in the demonstration of its results. But likewise the calculus of classes is of limited scope and requires supplementation by a further developed logic. This progression continues up through the extended functional calculus. Higher logics on the hierarchy are usually conceived of as including the lower logics they presuppose. Thus the extended functional calculus is thought to include the propositional calculus, the calculus of classes and the lower functional calculus. 1.9 The Systems of the Present Book The present book is designed to introduce the reader to a number of logical systems, and to allow him to exercise his powers of deduction in a number of distinct contexts. The main systems considered here are: (1) P +, a propositional calculus without negation. (2) PLT, a full propositional calculus, built up on the basis of P +. (3) PLT', a full propositional calculus, obtained by dropping the redundent axioms of PLT and adding three definitions. (4) PLTF, a full propositional logic due to Frege. (5) PPM, a full propositional logic due to Russell and Whitehead. (6) PI, an intuitionistic propositional calculus built up from P +. (7) PF, a propositional logic due to F. B. Fitch. (8) PMIN, the minimum propositional calculus ofIngebrigt Johansson.

46

INTRODUCTION

(9) PILNC, PILEM, PILT, PIDN, PIDM; full propositional logics based on the axiom set of PI. (10) PND, a system of natural deduction for the propositional calculus. (11) PNDF, a system of natural deduction for the Fitch propositional calculus. (12) PNDI, an intuitionistic system of natural deduction for the propositional calculus. (13) PNDMIN, a system of natural deduction for the minimal propositional calculus. (14) LFLT, LFLT', LFI, LF}% LFMIN, various systems of the lower functional calculus. (15) A system of the calculus of classes, or Boolean Algebra.

1.10 Abbreviations Abbreviations frequently used are divided into three classes. Class 1. Abbreviations of this class abbreviate names for the divisions on the hierarchy of logical systems mentioned earlier. (1.1) APC, for the assertoric propositional calculus, or, more simply, the propositional calculus. (1.2) LFC, for the lower functional calculus. (1.3) AFC, for the assertoric functional calculus, or, more simply, the functional calculus (lower or higher.) Other abbreviations for divisions of logic are defined in the text. Class 2. Abbreviations for logical laws. (2.1) LEM, for the law of excluded middle, [p v '" p] (2.2) LNC, for the law of contradiction, sometimes called the law of non-contradiction, '" [p & '" p ] Other such abbreviations are defined when they occur. Class 3. Abbreviations for particular developments or titles of books in which such developments occur. (3.1) PM, for Principia Mathematica of Whitehead and Russell. (3.2) F, for Fitch's system in Symbolic Logic. (3.3) I, for the systems of intuitionist logic, (when the letter 'I' is used 9Y itself it denotes the system of A. Heyting). Other abbreviations for systems oflogic will be explained as they occur.

47

CHAPTER 2

THE SYSTEM P+

2.1 Summary In this chapter the axiomatic system P + is set out.! It has eleven axioms and two primitive rules of transformation. The primitive basis of P + is first presented; then a number of theorems of the system are proved. The systems of logic in the present book are all based on the axiom set ofP +. P +, unlike most of the remaining systems developed in the following chapters, is a logic devoid of negation. Hence it is described as a positive logic, a system including as theorems only those results of propositional logic which may be derived without resort to negation-functors of any kind. The present chapter should be studied in connection with Chapter 1 in which definitions and concepts, used here without explanation, are presented.

2.2 Rules of Formation of P + 2.21 The system possesses six improper symbols as primitive functors. (1) (2) (3)

[ ]

(4) (5)

v &

(6) These symbols are named in accordance with the terminology introduced 1 A system similar to P +, the Positive Logik, was first set out by Hilbert and Bemays in their monumental Grundlagen. (The system is not equivalent with the one set out by Church called P +; Church's P + is a positive implicational propositional calculus, Church also includes a system similar to the present P +.) The idea of using their version of P + as a foundation logic is also due to Hilbert and Bemays.

48

SYSTEM WITHOUT NEGATION

in Chapter I. (This qualification will be understood in later systems.) 2.22 The system possesses an infinite list of propositional variables,

taken from the middle of the alphabet. These are proper symbols. 2.23 An expression is a well-formed formula (wff) ofP+ only under the following circumstances: 2.231 A proper symbol standing alone is wf 2.232 If the expressions A and B are wI, then: (a) (b)

[A:JB]

(c) (d)

[A&B] [A=B]

[A vB]

are wffs. 2.233 The method of abbreviating the occurrence of brackets by means of points in a wff (described in Chapter 1) is adopted here. An object-linguistic expression is a wff of P + if and only if it conforms with the conditions specified in 2.231 and 2.232 above or can be made to do so by recourse to writing out an abbreviated expression in full in accordance with 2.233. The method of finding the principal functor of a wff is described in Chapter 1. The expressions "antecedent", "consequent", used here are those defined in the previous chapter. 2.3 Rules of Transformation of P + 2.31 From the expressions [A:J B] and A, B may be inferred as conclusion: (modus ponens).

(The application of this procedure is indicated in the justification column correlated with a proof in P + by the meta-symbol' - :J', to the right of the conclusion, preceeded by the index numbers of the two expressions from which the inference is made; thus: (1) (2) (3)

1,2, -:J 49

SYSTEMS OF FORMAL LOGIC

The occurrence of this symbol in the justification column signalizes that the rule of modus ponens has been used.)

2.32 From the expression A with occurrences of the variable, a, the expression B may be inferred by substitution of the expression Pfor a at every occurrence of a in A: (Substitution). (When the rule of substitution is applied, this is indicated in the justification column by writing A, + S, Pia, read, " ... results by substitution of Pfor a in A," where the index number of A is regarded as an abbreviation for A.) No inference in P + is valid unless it proceeds in accordance with modus ponens or substitution. 2.4 Axioms of P +

2AI 2A2 2A3 2A4 2AS

2A6 2A7 2A8 2A9 2AlO 2All

p=:J • q=:Jp p=:J .q=:Jr=:J:p=:Jq=:J .p=:Jr p&q=:Jp p&q=:Jq p=:J .q=:J .p&q p=:J .pvq q=:J .pvq p=:Jr=:J . q=:J r=:J .p v q=:Jr p=.q=:J .p=:Jq p=.q=:J .q=:Jp p=:Jq=:J .q=:Jp=:J .p=.q

(The numbers 2At, etc. correlated with the axioms of P +, are called "the index numbers of the axioms". The occurrence of such an index number in the justification column is regarded as an abbreviation for the axiom for which it is the index number.) No expression of the system is an axiom of P + unless it has the form of one of the expressions 2AI through 2AIl.

2.5 Definitions of P + 2.51 "r A" = df "A is a theorem of P +." "r" is a meta-symbol of P +, not a primitive functor of the object language. "rA" may be understood 50

SYSTEM WITHOUT NEGATION

to mean that A is provable in P + or that A holds for all values of its variables. 2.52 "~" = df" A is a theorem to be proved". The underline, like 1-, is a symbol in the metalanguage of P +. 2.53 "TH" = df "theorem". When the symbol "TH" occurs at the terminus of a proof in P + it is understood to abbreviate the theorem proved in the proof, and to refer back to the underlined expression ~ preceding the proof. Its occurrence indicates that the theorem to be proved is now proved; thus it corresponds with the traditional expression QED. 2.54 A proof in P + corresponds with the definition offered in Chapter 1. The articulation ofP+ in this chapter includes no derivations. A technique of introducing derivations into the system is reserved for Chapter 3 (see Section 3.7f). 2.55 Theorems of this chapter are numbered in accordance with the following conventions: "2Tn" = dj, "the nth theorem of the second chapter." It is understood that in specifying index numbers of theorems, a decimal may be added to the second integral to indicate related theorems. Thus 2T1.3 would be the third sub-theorem of theorem 2T1. 2.6 Deductions in P + Theorems in implication. (The proofs of the theorems 2TOI-2T06 are of little systematic importance, but are introduced to familiarize the reader with the applications of the transformation rules of the system.)

2.61

2TOI p~ .p~p 1. I-p~ .q~p 2. I-p~ .p~p 2T02 p~ .p~q~p 1. I-p~ .q~p 2. I-p~ .p~q~p 2T03 p~ .p~p~:p~p~ 1. 2. 3.

I-p~ .q~r~:p~q~ .p~r I-p~ .p~r~:p~p~ .p~r

I-p~ .p~p~:p~p~ .p~p

2AI 1, +S,p/q 2Al 1, +S,p~q/q .p~p

2A2 1, +S,p/q 2, +S,p/r

In subsequent deductions simultaneous substitution in a single step will

51

SYSTEMS OF FORMAL LOGIC

be permitted, i.e., substitutions on two or more variables at the same step. Using this modification, the proof of 2T03 could be shortened by one step.

1.

\-p=> .q=>r=>:p=>q=> .p=>r \-p=> .p=>p=>:p=>p=> .p=>p

2.

2A2 1, +S,p/q,p/r

This modification is derivable from the original rule of substitution, in such a way that the derived rule of substitution will not allow the demonstration of any results not available by the primitive rule. In consequence of this, the derived rule of substitution will be instituted into the system. It will be stated as follows: If \-A then \-B, +S, ~l/al, ~2/a2, ... ~n/an This may be interpreted, "If A is a theorem, then the resultant 'B' of substituting ~l for aI, ~2 for a2, ... , ~n for an is a theorem." That is, any number of substitutions may be made in a single step, provided the consistency-requirement for substitution stipulated in Chapter 1 is observed. The reader should be prepared to show why, if no invalidity is introduced by the primitive rule of substitution, 2.32, no invalidity can be introduced by the derived rule presented above. The use of both the primitive and derived rules will be symbolized by the original indexing, n, +S, ... 2T04

p=> .p=>q=>p=> :p=> .p=>q=> :p=>p

1. \-p=> .q=>r=>:p=>q=> .p=>r 2. \-p=> .p=>q=>p=>:p=> .p=>q=>:p=>p 2T05

p=>p=> .p=>p

1. \-p=> .p=>p=>:p=>p=> .p=>p 2. \-p=> .p=> P 3. \-p=>p=> .p=>p

2T06

2A2 1, +S,p=>q/q,p/r

2T03 2TOI

1,2, - =>

p=> .p=>q=>:p=>p

1. \-p=> .p=>q=>p=>:p=> .p=>q=>:p=>p 2. \-p=> .p=>q=>p 3. \-p=> .p=>q=>:p=>p

2T04 2T02

1,2, - =>

2TOl through 2T06 illustrate the processes of modus ponens and of substitution, and in particular the stipulation that any well-formed formula 52

SYSTEM WITHOUT NEGATION

of P + may be substituted for a propositional variable in the axioms of P + as long as the substitution is consistently carried through, i.e., when the same expression P is always substituted at every occurrence of the variable substituted upon. Thus in 2TOI p is substituted for the variable q of step 1 at every place in which q occurs. Since q occurs at only one point in step I, p is substituted at that one place. Likewise, in 2T02 the wff p => q is substituted in step 2 for the only occurrence of q in step 1. The expression 'q=>r=> .p=>q=> .q=>r' is also, of course, a substitution on 2A I, substituting q => r for p and p => q for q. The expression p=>:p=>.q=>r=>:p can be conceived as a substitution on either 2Al or on 2T02. If it is regarded as a substitution on 2AI we substituted p=> .q=>r for q; if as a substitution on 2T02, q=>r was substituted for q in that formula. But while both 2TOI and 2T02 are deductions by substitution from 2AI, neither is a deduction by substitution from the other, since there is no consistent substitution on either which would transform the one into the other. Likewise, though 2TOI and 2T02 are deductions-by-substitution from 2AI, 2AI is not a deduction-by-substitution from either 2TOI or 2T02. Hence, in general, substitution is not a symmetrical relation, i.e., from the fact that B is a substitution on A it does not follow that A is a substitution on B. There is one case in which substitution is symmetrical. This occurs when the following circumstance obtains: if B is a substitution on A and for every occurrence of the same variable a in A there is an occurrence of the same variable b in B and for every occurrence of a distinct variable in A there is an occurrence of a distinct variable in B, and B has the same logical form as A, then A is a substitution on B. This circumstance may be illustrated by the following deductions. 2T07

r=> .s=>r

1. I-p=> .q=>p 2. I-r=> .s=>r

2Al 1, +S, rip, s/q

p=> .q=>p

1. I-r=> .s=>r 2. I-p=> .q=>p

2T07 1, +S, p/r, q/s

Taken as a proof, the second is clearly circular, since step 1 is derived in 2T07 by recourse to 2Al, which is step 2 of the second proof. What is shown here is that in cases such as described above, the relation of

53

SYSTEMS OF FORMAL LOGIC

deduction-by-substitution is symmetric, 2T07 being a substitution on 2AI and 2AI being a substitution on 2T07. Such expressions will be called variants of each other. We can now give a definition of the relation "variant of" as holding between expressions. B is a variant of A if and only if B is derivable from A by substitution and A is derivable from B by substitution. Clearly, if B is a variant of A, then A is a variant of B. Likewise, trivially, A is a variant of itself. Thus 2T07 is a variant of 2AI, as is q=:J .p=:Jq. But 2TOI and 2T02 are not variants of 2AI, though of course they are deductions-by-substitution from 2AI. 2T03 and 2T04 give further illustrations of substitution, this time on

2A2. 2T05 and 2T06 illustrate the application of the rule of modus ponens. The ru1e of modus ponens, like that of substitution, involves a technique of deriving statements from statements. A formula having n variables (whether these variables be the same or distinct) will be said to have the length n. Thus 2AI is said to have the length 3. It will be noted that application of the rule of substitution to a formula of length n will always result in a formula of at least length n, never of a length less than n. 2TOI gives a case in which the substitution on an expression of length n has the length n. 2T02, however, illustrates a circumstance in which substitution gives rise to an expression such that, while the statement substituted on has the length 3, (=n), the resulting expression has the length 4, (n+ I). In general, the function of substitution is to generate expressions of lengths equal to or greater than n from expressions having the length n. Indeed, if the system P+ did not possess the rule of substitution it would be impossible to derive in it any statements longer than the axioms. For instance, the logically true statement p=:Jq=:J .q=:Jr=:J :p=:Jq=:J .p=:Jr

would not be derivable in P+. Since one of the functions of the rule of substitution in deduction is to expand expressions of length n into expressions of lengths greater than n, i.e., to introduce expressions into expressions, it will be useful to call substitution a rule of introduction, or, more briefly, simply an introduction rule. As noted above, substitution is the only introduction rule of the system P +. Modus ponens is the elimination rule corresponding to substitution. It is by application of modus ponens that an expression of a length less than n is derivable 54

SYSTEM WITHOUT NEGATION

from an expression of the length n. From the expression A::::> B and A, B is derivable. Modus ponens allows, from the assertion of A and the assertion of A::::> B the assertion of B, independent of its condition. Thus, in effect, the implication is eliminated, A and B being independently assertable. If P + did not possess modus ponens, or an equivalent, it would be impossible to deduce in it any statements, e.g.,

shorter than the axioms. An expression B is said to be an immediate consequence of an expression or series of expressions r if B is derivable from A by the rule of substitution, or B is derivable from r by the rule of modus ponens. In citing justification for a step B in an argument it is essential to specify two preceding steps if the justification for B resides in an application of the rule of modus ponens. The citation of a single step is both necessary and sufficient, if the justification for B resides in an application of the rule of substitution. 2.611 On construction of axiomatic proofs. In general there is no simple set of heuristic rules which, applied mechanically, without the insight of the logician, will guarantee the ability to construct a proof. There are, however, certain procedural techniques which are of help. They are listed in summary here. 1. Know the evidence. In the case of the present system, what is the evidence? Basically, it is the set of axioms 2Al-2All of P +. It is to be strongly recommended that the reader who wishes to work with this system know the axioms by heart. Otherwise, he will be forced to interrupt his train of thought by constant inspection of the axiom list, and will be unlikely to recognize elements of steps in proofs as potential substitutions on the axioms. 2. Know the rules. Here the rules are -::::> and + S. It is also advisable to master the derived rules such as simultaneous substitution, and other rules presented below, since they help to simplify the intellectual (and the mere pen and ink) labor of working out the steps. 3. Know "the problem". By "the problem" here I mean the specific conditions of solution imposed by the form or structure of the theorem to be proved. Ifwe set aside the mainly illustrative theorems 2TOl through

55

SYSTEMS OF FORMAL LOGIC

2T07 and consider the law of identity, p~p, as our first theorem, thus provable solely by recourse to the axioms and primitive rules, we can formulate the problem it sets in the following way: We analyse the theorem: The law is one in implication. It has two occurrences of the same variable, the antecedent and the consequent being the same. The two axioms in implication are both longer than the theorem; how is it possible to obtain this shorter theorem from the longer ones? Likewise, none of the axioms has the exact form of the theorem in any of its wf parts: how is it possible to introduce a formula having the form of the theorem into one or the other of the two implication axioms? With which axiom should one begin? By formulating "the problem" in a clear and comprehensive way, hypotheses will suggest themselves for its solution which otherwise might escape the eye. It is useful now to apply the procedures already discussed in the deduction of the law of identity for the APe, p ~ p; we know that there are two axioms having as functors none but implications. Hence we form the hypothesis that we may be able to obtain the theorem p~p, which is also a pure implication, by use of these axioms. They are, of course, 2AI and 2A2. Since the theorem has a length shorter than that of either of the axioms, the solution of the general problem of how to obtain it is, in part, solved by the recognition that modus ponens is the only rule which allows the deduction of result of shorter length from those oflonger length. Hence, solution will require at least one application of modus ponens. Likewise, since neither of the axioms has a wi part possessing the required form, we are forced to substitute; hence we already know that we will require application of substitution at least once, as well as at least one application of modus ponens. These discoveries allow us to discover the possible length-at-a-minimum of the proof. How? (1) If we state the axiom on which substitution is to be made in its primitive form and, (2) if we then make appropriate substitution on it, we have already two steps. (3) We then require a statement of a law (in this case, an axiom) which will give the antecedent of the second step. (4) If the consequent of the second step is the desired result (the theorem), it can be obtained as step 4 by application of modus ponens to steps 2 and 3. Thus the results so far show that the proof must be at least 4 steps long, though of course it may well be longer if more than one application of modus ponens, or more than one appli56

SYSTEM WITHOUT NEGATION

cation of substitution is required. But now the basic structure of the proof is beginning to suggest itself. Now we must discover the axiom with which to begin. This is a matter of considering reasonable hypotheses. One might be tempted to obtain a formula having the desired theorem as a wfpart by substitutingp for q in 2AI, which would give

(A)

p~ .p~p

This substitution does give what we wanted in the consequent, but reflection shows that this hypothesis leads to a blind alley. We can never obtain the antecedent, a bare p, which is contingent, as a theorem. Hence we cannot detach p ~ p from its condition. Similarly, if we try substituting p ~ p for pin 2AI or in (A) above, we obtain the unpromising results

(B)

p~p~ .q~ .p~p

and

all of which require the very theorem we are trying to prove as second premise in order to use modus ponens. Hence, since these hypotheses lead to nothing, the hypothesis suggests itself that we try the second axiom. Again, it is tempting to substitute p for all the variables of 2A2 giving (D)

p~ .p~p~:p~p~ .p~p

But it is easy to see that this substitution breaks down. Since, though we can obtain its antecedent by substitution on 2A1, (cf. (A) above), we are left with the intransigent (E)

p~p~ .p~p

which encounters the same difficulties as did (B) and (C). Another possibility is the following substitution on 2A2:

(F)

p~ .p~r~:p~p~ .p~r

This result, again, incorporates the desired result as a wf part of it, but leads nowhere. (1) The antecedent is unavailable by substitution on any

57

SYSTEMS OF FORMAL LOGIC

of the axioms; in fact, it is contingent; thus we cannot detach its consequent from it; (2) even if we could detach the consequent from the antecedent, the result desired is in the wrong position. We want it as the right terminal formula, to the right of a major implication - since modus ponens always detaches the right-hand expression from a left-hand expression. Though all the results (A) through (F) are perfectly legitimate laws of logic, none of them serves our present need. The failure of this hypothesis, however, suggests another, substituting p for r in 2A2.

This hypothesis is promising: it is a law of logic, a clear substitution on 2A2; the desired result is in the "right" position; if the axioms of the system have been mastered and memorized, the antecedent of (G) is immediately recognized as 2AI. Thus, applying modus ponens to (G) and 2A1, we can obtain

At first (H) looks unpromising, since p::J q, its antecedent, is contingent. How, then, can p::Jp be detached from it by modus ponens? Of course, it cannot, as it stands. Is there any way to convert p::J q, by substitution our only recourse here - into an axiom? We note that q has only one occurrence in (H). Thus we can substitute anything for it without effecting the structure of the remainder of the formula. A little reflection on the freedom and limitations of substitution discussed in 1.5221 shows that any wff may be substituted for q. Since we wish to eliminate the antecedent by application of modus ponens and since the only way in which such application can take place arises when the antecedent is a law, either an axiom or a theorem of the system, the problem at this point is to substitute on the antecedent of H in such a way that these conditions are fulfilled. This leaves open a variety of alternatives. We note that q is the right hand side of the antecedent. It takes only a little insight to see that we can do the following, substituting the axiom p::J .q::Jp for q of (H).

58

SYSTEM WITHOUT NEGATION

Then we can get the antecedent of (I) by substituting on 2AI in the following way:

(J)

p=> .q=>p=>:p=> .p=> .q=>p

Now since 2AI is itse((the antecedent of (J), we can obtain the desired antecedent of (I),

(K)

p=> .p=> .q=>p

by modus ponens applied to (J) and 2Al. Our desired theorem, p=>p, can then be obtained by application of modus ponens to (I) and (K). (ii) An exactly similar though more complicated procedure would result from substituting 2A2 for q in (H), giving:

applying the same method as before. (iii) But there is also an easier method, namely that of substituting the expression q => P for the variable q of (H). This gives:

Since the antecedent of (M) is recognized as 2AI, this axiom, (M), and the application of modus ponens give 2T1.

In formal sciences like logic, it is usual to prefer, among two or more valid ways to obtain the same result, the one which is least complicated. Thus, considering this "aesthetic" criterion, the third method would be preferred, though all are valid. In general, a finished proof shows no signs of the heuristic methods, the procedural techniques, the tries and hypotheses used to obtain it. Since formal logic as a science is concerned with proofs of the results themselves, not the various "tries" at proofs and the insights which allow the proofs, as things-being-developed, to proceed, the finished proof simply embodies the definition previously presented: "a sequence of statements or signs for them such that each element in the sequence is either an axiom or is an immediate consequence of earlier elements in the sequence." The following is a "finished proof" of 2Tl. 59

SYSTEMS OF FORMAL LOGIC

2T1 p::::J P (The law of identity for the APC)l 1. f-p::::J .q::::Jr::::J:p::::Jq::::J .p::::Jr 2A2 2. f-p::::J .q::::Jp::::J:p::::Jq::::J .p::::Jp I, +S,p/r 3. f-p::::J .q::::Jp 2AI 4. f-p::::Jq::::J .p::::Jp 2,3, -::::J 5. f-p::::J .q::::Jp::::J:p::::Jp 4, +S, q::::Jp/q 6. f- TH 5, 3, - ::::J The 'reasons' given in the justification column for step 6 are given the present order because the rule of modus ponens states that an expression B follows from the expressions A::::JB, and A. Thus the implication p::::J .q::::Jp::::J:p::::Jp (representing A::::JB) is cited first, while the expression forming the antecedent of this implication, p::::J •q::::J p, (representing A) is cited second, in keeping with the structure of the rule. Hereafter, the qualification "for the APe" will be omitted, since all laws of P + are laws of the Assertoric Propositional Calculus. 2Tl.l

q::::J .p::::Jp

1. f-p::::J .q::::Jp 2. f-p::::Jp::::J .q::::J .p::::Jp 3. f-p~p

2AI 1, +S,P::::Jp/p

2T1 2,3, -::::J

4. f-TH

(I.e., anything implies the law of identity.) It will be noticed that the theorem f-p::::Jp is used as a step in the proof of 2T1.1. This procedure is justified, since any statement derivable from the axioms, i.e., any theorem, is a truth of logic, and hence may be asserted as such along with the axioms. It should be kept in mind, however, that when a theorem ofthe system is cited as a step, the entire proof of that theorem enters as an element of the full proof. EXERCISES

Show that the following wffs are theorems of P +. This law should not be confused with another related one, namely (x) [x ~ xl, i.e., "every thing is identical with itself". The propositional law of identity, in its main interpretation, means "every proposition implies itself". 1

60

SYSTEM WITHOUT NEGATION

1.

p~ .s~ .r~t~:p~ .s~ .r~t

2.

s~ .p~r~ .p~r

3.

q~ .r~ .p~p

A deductively important result in the system is the first syllogistic law, proved below, indexed 2T2. 2T2

q~r~ .p~q~ .p~r

Again it is useful to work through the procedures which might be employed in obtaining this result. What is the problem? Knowledge of 2A2 shows that its consequent is identical with the consequent of 2T2 above. But unlike 2A2, 2T2 has in its antecedent two variables connected by one implication sign, not three variables connected by two implication signs. Further examination and comparison of the antecedents of the two formulae shows, also, that while the implication q~r of 2A2 is implied by p, thus havingp~ .q~r, the same antecedent implication, q~r, of 2T2 is not conditioned by p or by any other formula. The problem, then, is to discover a method, using the rules and the evidence now available, to eliminate the conditioning p from the left side of the antecedent of 2A2. We note that the theorem to be proved is greater in length than 2AI and 2T1, and shorter than 2A2. The solution. The evidence has now been supplemented by the proof of a theorem, the law of identity, which can now, by our definition of proof in Section 1.532, be used just like an axiom in the deVelopment of future proofs. Here we can make the same assumption about the sufficiency of axioms in pure implication for proof of this result, since it occurs early in the system and since, like the law of identity, it is a law in pure implication. Taking the hint implicit in the identity of consequents in 2T2 and 2A2, the most reasonable recourse is to opt for 2A2 as the first step ..Thus we have, (A)

p~ .q~r~:p~q~ .p~r

Now we have a choice: (i) We could take the bold step of applying (A) to itself, i.e., substituting p~ .q~r for p, p~q for q, andp~r for r in 2A2, giving: (B)

p~ .q~r~:p~q~ .p~r~:. p~ .q~r~:p~q~:.p~ .q~r~:p~r

61

SYSTEMS OF FORMAL LOGIC

But this is initially unpromising. Why? Because, though we can get rid of the antecedent by 2A2 and modus ponens, this leaves us with (C)

p~ .q~r~:p~q~:.p~ .q~r~:p~r

and if you test both the antecedent and the consequent separately with the truth-tables of Chapter 1, you discover that both are contingent, though (C), as a whole is logically true, Hence, this hypothesis seems unrewarding, though subsequent complex substitutions might lead somewhere. (ii) Equally unpromising is a substitution on the law of identity, giving, (D)

p~. q~r~ :p~q~.p~r~ :.p~. q~r~.p~q~.p~r

Application of modus ponens to (D) and (A) gives step (A) again. (iii) The third alternative is to substitute on 2Al in something like the following way, (E)

p~ .q~r~:p~q~ .p~r~:.

_ _ ~:

p~ .q~r~:p~q~ .p~r

where we substitute 2A2 for p and find some suitable substitution for q, here represented by " ". The problem is just what to substitute for q, since we know we can eliminate the antecedent of (E) by modus ponens. This hypothesis is, at least, suggestive, since it gives a logical truth as the right-hand side of the consequent of (E). Thus this hypothesis appears, prima jacie, the most rewarding, compared with the others. Again we are faced by a choice, this one somewhat more complicated, though not essentially so. The choice to try (E) is conditioned by two things: (i) what to substitute for q, and (ii) what the evidence (the axioms and theorems) is. If we ask again, "What is the problem", certain limitations on reasonable choice immediately make themselves felt. We wish to obtain

The substitutions for q, or " " in (E), of any of our "known" laws (either axioms or theorems already proved) lead to blind alleys. But examination of (E) plus knowledge of the axioms, plus imagination and ), in (E). Why? We know insight suggests substituting q~r for q, ( 62

SYSTEM WITHOUT NEGATION

that 2A2 holds; it is an axiom of P +. We know that q~r is the desired, and as yet unobtained, antecedent for the law we are trying to prove. And, most important, if we know axiom 2A2 we know also that it distributes the left-hand side of the antecedent of the axiom in such a way that it becomes the antecedent of both the left-hand and the right-hand sides of the consequent of 2A2; thus schematically we have ~---.:..~ - ~ ... ~:p~ - ~ .p~ ...

p~.

([p~ [q~r]] ~ [[p~q] ~ [p~r]])

Hence, if we substitute (E)

q~r

for the " ___ " of E we have:

p~ .q~r~:p~q~ .p~r~: .q~r~: p~ .q~r~:p~q~ .p~r

and, of course, since the antecedent of E is identical with 2A2 we have (F)

q~r~:p~ .q~r~:p~q~:p~r

Again we have a choice. But if we have grasped 2A2, and, thus, its consequence schematically diagrammed above, the choice is, for all reasonable purposes, limited to the choice of one alternative, namely that of substituting q~r for p, p~ .q~r for q and p~q~ .p~r for r in the axiom 2A2. This substitution will give (F) as antecedent and (G)

q~r~ .p~ .q~r

which is a substitution on 2Al, as the left side of the consequent and 2T2 as its right side. This choice is reasonable since the hypothesis embodied in the choice discloses that both the antecedent of the whole expression namely (F), already known, and the left-hand side of the consequent are tautologies, logical truths. Now it is merely a matter of applying modus ponens twice to obtain the desired law stated below as 2T2. Note that the antecedent p of the left side of the antecedent of 2A2 is dispensed with by incorporating it into a logical law which conditions the desired result, and which is eliminable by application of modus ponens. Thus the problem initially posed is solved. The proof of this theorem does not require recourse to 2T1, the law of identity. The resulting formal proof is as follows. 63

SYSTEMS OF FORMAL LOGIC

2T2

q:;)r:;) .p:;)q:;) .p:;)r (First syllogistic law)1

A 1. f-p:;) .q:;)r:;):p:;)q:;) .p:;)r 2. f-p:;). q:;) r:;):p:;) q:;) .p:;) r:;):. q:;)r:;) :p:;) . q:;) r:;) :p:;)q:;) .p:;)r

2A2 2Al, +S, p:;) .q:;)r:;):p:;)q:;) .p:;)r/p, q:;)r/q

3. f-q:;)r:;):p:;) .q:;)r:;):p:;)q:;) .p:;)r

2,1, -:;)

4. f-q:;)r:;):p:;) .q:;)r:;):p:;)q:;) .p:;)r:;):. q:;)r:;) .p:;) .q:;)r:;):q:;)r:;) .p:;)q:;).

2A2, +S, q:;)r/p, p:;) .q:;)r/q,p:;)q:;) .p:;)r/r

p:;)r

5. f-q:;)r:;) .p:;) .q:;)r:;):q:;)r:;). p:;)q:;) .p:;)r

6. f-q:;)r:;) .p:;) .q:;)r 7. f-TH

4,3, -:;) 2Al,

+ S, q:;) rip, p/q

5,6, -:;)

This proof may appear rather formidable to the reader, especially at steps 2 and 4. This appearance derives in large part from the length of the expressions involved, in which one tends to "get lost". In fact, the deduction is of elementary simplicity, step 2 being merely a substitution on the first axiom, and step 4 a substitution on the second. Inspection reveals that the proof from step 1 through 5 has the following structure:

A A:;)B B

B:;)C C This structure suggests the following abbreviation of the proof:

B 1. f-p:;) .q:;)r:;):p:;)q:;) .p:;)r (:;))

2A2

1 The term "Syllogistic" is derived from the Prior Analytics of Aristotle. The first and second syllogistic laws, though resembling two of Aristotle's syllogisms are not identical with them. For proof of some Aristotelian syllogisms in modem dress cf. Chapter 7.

64

SYSTEM WITHOUT NEGATION

2.

~q~r~:p~ .q~r~: p~q~ .p~r

1, 2Al, +S,

(~)

q~r/q,

p~ .q~r~:p~q~ .p~r/p, -~

3.

~q~r~ .p~ .q~r~:q~r~.

2,2A2, +S,

p~q~ .p~r

q~r/p,

p~ .q~r/q,p~q~ .p~r/r, -~

4.

~q~r~ .p~ .q~r

5.

~TH

2Al, +S, 3,4, -~

q~r/p,p/q

The derived rule of theorem introduction. This proof-structure will be frequently used in the sequel. It is based upon the fact, shown in Chapter 1, that any expression implied by an axiom (or a theorem) is a theorem; thus, since step 2 is implied by step 1, by appropriate substitution on 2Al and modus ponens (as exhibited in the first three steps of proof A of 2T2) we may pass directly to step 2 of proof B from step 1 of this proof. This is called the application of the derived rule of theorem introduction. The fact that such a direct passage has been made is signalized by the symbol "( ~ )". This symbol may be used if and only if (a) the element to its immediate left is an axiom or a theorem, and (b) the element immediately beneath it in the series follows from it by an axiom or theorem of the system and the application of one or more primitive rules of transformation of the system. (Thus in Proof B of 2T2, step 2 follows from step 1 by the application of axiom 2Al, substitution and modus ponens. Step 3 follows from step 2 by the application of 2A2, substitution and modus ponens.) The use of this technique is solely abbreviative; when it is used it is meant to indicate only that the full proof can be carried out. In some cases, the analogous results, incorporating the antecedent or the consequent of the theorem to be proved, must themselves be proved by the rules and evidence of the system. A case is given in the second syllogistic law, where the desired identity of parts is obtained only in the second step of the proof. Below the reader will find a proof of 2T3 using the abbreviating techniques just described, followed by a full proof of the same result.

2T3 1. ~q~r~

p~q~ .q~r~ .p~r .p~q~ .p~r

(~)

(Second syllogistic law) 2T2 65

SYSTEMS OF FORMAL LOGIC

2. I-q::>r::> .p::>q::>:q::>r::> .p::>r (::»

1,2A2, +S, q::>r/p, p::>q/q,p::>r/r, -::>

3. I-p::>q::> .q::>r::> .p::>q::>: p::>q::> .q::>r::> .p::>r

2,2T2, +S, q::>r::> .p::>q/q

4. I-p::>q::> . q::> r::> .p::>q

q::>r::> .p::>r/r,p::>q/p, -::> 2AI, +S,p::>q/p, q::>r/q

5.I-TH

3,4,

-::>

In reading the 'reason' for, e.g., step 2 above, one would proceed as follows: "Step 2 follows, by application of modus ponens, upon step 1 and the substitution q::> r for p, p::> q for q, and p::> r for r made on axiom 2A2." (The symbol (::» relates consecutive steps in the proof, of course, not the step and the reasons given for it.) A full proof includes explicitly the steps referred to in the justification column of the abbreviated proof. Here is a full proof of 2T3. The reader should carefully observe the correspondence between this proof and the citations in the justification column of the foregoing proof. 1. I-q::>r::> .p::>q::> .p::>r 2. I-q::>r::> .p::>q::> .p::>r::>: q::>r::> .p::>q::>:q::>r::> .p::>r

2T2 2A2, +S, q::>r/p, p::>q/q, p::>r/r

3. I-q::>r::> .p::>q::>:q::>r::> .p::>r 4. I-q::>r::> .p::>q::>:q::>r::> .p::>r::>:. p::>q::> .q::>r::> .p::>q::>:. p::>q::> .q::>r::> .p::>r

2, I,

-::>

2T2, +S, q::>r::> .p::>q/q, q::>r::> .p::>r/r,p::>q/p

5. I-p::>q::> .q::>r::> .p::>q::>: p::>q::> .q::>r::> .p::>r 6. I-p::> q::> . q::> r::> .p::> q 7.I-TH

4,3, -::> 2AI, +S,p::>q/p, q::>r/q 5,6, -::>

If anyone objects to the above method of abbreviation, he may always reconstruct any proofs using it in such a way that they do not use it simply by writing out in full the steps specified in the justification column of the abbreviated proofs. Another way of proving 2T3 is given below.

2T2

66

SYSTEM WITHOUT NEGATION

1, 2A2,

+ S, q::;; rip, p::;; q/q,

p::;;r/r, -::;; 3. f-p::;;q::;;:q::;;r::;; .p::;;q::;;: q::;;r::;; .p::;;r (::;;)

2, 2A1, +S, q::;;r::;; .p::;;q::;;: q::;;r::;; .p::;;r/p,p::;;q/q, -::;;

4. f-p::;;q::;; . q::;; r::;; .p::;;q::;;: p::;;q::;; . q::;; r::;; .p::;;r

3,2A2, +S,p::;;q/p, q::;;r::;; .p::;;q/q, q::;;r::;; .p::;;r/r, -::;;

5. f-p::;;q::;; .q::;;r::;; .p::;;q

2Al, +S,p::;;q/p, q::;;r/q

6. f-TH

4,5, -::;;

It may be noted that, since we have now proved the two syllogistic laws of the APe as theorems of the system P +, it is always possible to derive as a theorem an expression of the form A::;; C from two theorems of the form A::;; B, B::;; C, or from two theorems of the form B::;; C, A::;; B. The deductions would appear as follows, where it is assumed that the premises are theorems.

A 1. f-B::;;C 2. f-A::;;B 3. f-B::;;C::;; . A::;; B::;; .A::;;C 4. f- A::;; B::;; . A::;; C 5. I-A::;;C B 1. I-A::;;B 2. I-B::;; C 3. I-A::;;B::;; .B::;;C::;; .A::;;C 4. I-B::;;C::;; .A::;;C 5. I-A::;;C

TH TH 2T2,

+ S, B/q, C/r, A/p

3, 1, - ::;; 4,2, -::;; TH TH 2T3, +S, A/p, B/q, 3, 1, - ::;;

C/r

4,2, -::;;

A sequence similar to this is illustrated in the second proof of 2T3, steps 3 through 7. By virtue of this fact, it will be regarded as legitimate to extend the method of abbreviation previously introduced, as follows. When a circumstance of one or the other of the two forms described immediately above obtains, the form will be described as the series for material implication, or, more briefly, the implicative series. When an implicative series obtains between two or more elements of a proof,

67

SYSTEMS OF FORMAL LOGIC

an implication may be established between the extreme components. Thus the following inferences will be regarded as following from this derived rule. 1. I-A::::)B 2. I-B::::)C 3.I-A::::)C

TH TH 1,2, +IS

1. I-B::::)C 2. I-A::::)B 3.I-A::::)C

TH TH 1,2, +IS

1. 2. 3. 4.

TH TH TH 1,2,3, +IS

I-A::::)B I-B::::)C I-C::::)D I-A::::)D

The specification n, + IS when cited in the justification column to the right of a step, A, of a proof is read, "A follows from the steps .. . n, by the rule of the Implicative series." When this derived rule is cited, it is understood that it refers back to 2T2 or 2T3 and the requisite number of applications of the rule of modus ponens. Thus the full proof of 2T2 or of 2T3 or both enter into any proof in which the derived rule of the implicative series is employed. The application of the rule of the implicative series is illustrated in the proof of 2T4. For the last time in this chapter a procedural description is offered along with some comments on the "growth" of a system. The next theorem proved is indexed 2T4. 2T4

p::::) .q::::)r::::):q::::) ,p::::)r

This theorem is frequently used hereafter. By virtue of the rule of the implicative series (+ IS) and, of course, the two "syllogistic" laws on which it is based, the system, taken subjectively as a developing structure, is much enriched. If we take only the second syllogistic law and its related rule we see that the following schema, related to + IS always obtains: p::::) _ _ _ ::::) , _ _ _ ::::)r::::) ,p::::)r

Thus by the second syllogistic (or by the derived rule, + IS) we can,

68

SYSTEM WITHOUT NEGATION

under certain defined circumstances, connect two formulae which were not connected by implication in earlier steps. The basic characteristic of the second syllogistic law for the APC is called transitivity; the relation

disclosed by the law of identity, that of a formula to itself, is called reflexivity.

By virtue of the transitivity of implication we can prove another useful law, 2T4, stated above. What is the problem? We wish to obtain q:::> .p:::>r from p:::> .q:::>r. We observe that the formula 2T4 is of the same length as is 2T3. But it is of a different form: while, e.g., the antecedent of 2T3 has two variables, its consequent having four, the antecedent of 2T4 has three variables, its consequent having an equal number. Comparison of the theorem 2T4 with evidence now available shows that there are no axioms and no theorems identical with it in form. Likewise, the form exhibited in 2T4 is neither that of transitivity nor that of reflexivity: rather it allows us, simultaneously, to replace the left-side of the antecedent, p, with the left side of the consequent, q, and the left side of the consequent with the left side of the antecedent,p. The problem is to discover, (1) how to obtain a consequent of the form q:::> .p:::>r, a consequent with three variables, rather than with the usual two or four, and (ii) how to obtain the "reversal of roles" between p and q while leaving the position of r uneffected. The solution. The first premise is determined by observing that, while other hypotheses may be attractive, we have in 2A2 an antecedent which is identical with the antecedent of the theorem. This would not have been decisive prior to the proof of 2T2 and 2T3 and, as consequence, the introduction of the derived rule of the implicative series (+ IS). Now it is crucial, for we know that we can establish, from the fact that a law (theorem or axiom) A implies some other formula B, and that the formula B implies the formula C, the implicative law A:::> C. Thus it is reasonable to take as our first step axiom 2A2, (A)

p:::> oq:::>r:::> :p:::>q:::> op:::>r

in the hope that its consequent can be made to yield the desired result. Now we ask again, "What is the problem?" Obviously, to obtain an implication between q as antecedent and p:::> r as consequent, as conditioned by the main antecedent p:::> q:::> r. But now the problem narrows: o

69

SYSTEMS OF FORMAL LOGIC

we have p => r as the right-hand side, i.e., the consequent of the consequent of 2A2. Thus the problem is the specific one of (i) introducing q in such a way that (ii) it is the antecedent of p => r, and that (iii) we eliminate the unwanted intervening formula p => q. Consultation of the evidence reveals only one method of introducing a new variable (or formula) into a formula; that method is by application of substitution to 2At. Thus the reasonable choice is the following substitution on 2At. (B)

p=>q=> .p=>r=>:q=> .p=>q=> .p=>r

Allowing us to obtain by (C)

+IS

p=> .q=>r=>:q=> .p=>q=> .p=>r

This step has succeeded in solving the first part of the problem, having introduced q; but the other two parts remain unsolved. There is only one way in which we can distribute the q in the antecedent of the consequent of (C). Naturally, by application of 2A2 and substitution. (D)

q=> .p=>q=> .p=>r=>:q=> .p=>q=>:q=> .p=>r

Then it is easy to see the application of (E)

+ IS, giving

p~ .q~r~:q~ .p~q~:q~ .p~r

These moves give the solution to the second of the three parts of the problem, q now being correctly positioned. But we are still faced with the third part, since there is now an intervening tautology, q=> .p~q, between the desired antecedent and consequent of the law we are attempting to prove. It is easy to see that if we could distribute the antecedent of (E) over the left and right hand sides of the consequent, giving, (F)

p=> .q=>r=>:q=> .p=>q=>:.p=> .q=>r=>:q~ .p=>r

the third part of the problem would be close to solution. Is it possible to obtain the result (F) by evidence presently available? If one knows the axioms and has only a little imagination, he will see that the answer is affirmative. Substitutingp=> .q=>r for p; q=> .p=>q for q; and q=> .p=>r for r in 2A2 we see that (E) implies (F): thus we can obtain (F) from (E) by theorem introduction. From (F) the distance to the theorem is short. q=> .p=>q is a theorem. Is there a way to obtain an implication with this theorem as consequent and p ~ . q ~ r as antecedent? The answer is again 70

SYSTEM WITHOUT NEGATION

clearly affirmative, since by the very axiom 2Al of P +, any wff implies a theorem or axiom. We can give the deduction needed here as a special case of this general principle, substituting q=> .p=>q for p and p=> .q=>r for q in 2AI: (G)

q=> .p=>q=>:p=> .q=>r=> .q=> .p=>q

It is easy to see that we can obtain the right side of (G) by 2AI and application of modus ponens. Thus the theorem can be obtained from this result, (F) and modus ponens. p=> .q=>r=>:q=> .p=>r (law ofcommutation)l

2T4

1. I-p=> .q=>r=>:p=>q=> .p=>r 2. I-p=>q=> .p=>r=>:q=> .p=>q=> .p=>r 3. I-p=> .q=>r=>:q=> .p=>q=> .p=>r 4. I-q=> .p=>q=> .p=>r=>:q=> .p=>q=>: q=> .p=>r 5. I-p=> .q=>r=>:q=> .p=>q=>: q=>.p=>r (=» 6. I-p=> .q=>r=> :q=> .p=>q=>:. p=> .q=>r=>:q=> .p=>r 7. I-q=> .p=>q (=» 8. I-p=> .q=>r=>:q=> .p=>q

9.I-TH

2A2 2Al, +S,p=>q=> .p=>r/p 1,2, +IS

2A2, +S, q/p,p=>q/q,p=>r/r

3,4, +IS 5,2A2, +S,p=> .q=>r/p, q=> .p=>q/q, q=> .p=>r/r, - => 2Al, +S, q/p,p/q 7,2Al, +S, q=> .p=>q/p, p=> .q=>r/q, - =>

6,8, - =>

This proof exhibits the application of both derived rwes of the system. For the reader desiring to see how a proof of the same theorem, not using the rule of the implicative series would be constructed, the following reconstruction of steps 1 through 3 of that proof is given: the reconstruction of the remainder of the proof is left to the reader as an exercise. 1. 2.

I-p=> .q=>r=>:p=>q=> .p=>r I-p=>q=> .p=>r=>:q=> .p=>q=> .p=>r

2A2 2Al, +S,p=>q=> .p=>r/p

1 The systematic importance of the law of commutation will be fully evident to the reader as he proceeds through the deductions of the book. It is one of the most important principles used in determining the completeness and independence of axiom systems.

71

SYSTEMS OF FORMAL LOGIC

2a.

~p::> . q::> r::>:p::> q::> .p::> r::>:. p::>q::> .p::>r::>:q::> .p::>q::> .p::>r::>:. p::> . q::> r::>: q::> .p::> q::> .p::> r 2T3, + S, p::> . q::> rip, p::>q::> .p::>rlq, q::> .p::>q::> .p::>rlr 2b. ~p::>q::> .p::>r::> :q::> .p::>q::> .p::>r::>:. p::> .q::>r::>:q::> .p::>q::> .p::>r 2a, 1, -::> 3. ~p::> .q::>r::>:q::> .p::>q::> .p::>r 2b,2, -::>

It may be remarked that the proof of 2T4 could be still further shortened by eliminating step 3 and deducing step 5 by the application of the rule of the implicative series upon steps 1, 2, and 4; since these have the form:

the rule allows the deduction of A::> D as conclusion. In the following proofs, it will be made a practice to shorten proofs as much as possible by reference to the derived as well as the primitive rules of the system, where no confusion thereby results. The reader should, however, be prepared to reconstruct in full any proof in which the derived rules have been applied, since it is only the effective possibility of such reconstruction which renders application of the derived rules consistent with the specification, made in section 2.3, that no inference is valid in P + unless it proceeds in accordance with the two primitive rules of the system. p::> .p::> q::>:p::> q (Law of simplification for implication)

2T5 1. 2.

~p::>

3. 4.

~p::>p

.p::>q::> :p::>p::> .p::>q .p::>q::>:p::>q

~p::>p::>:p::>

(::»

2A2, +S,p/q, q/r 1,2T4, +S,p::> .p::>q/p, p::>p/q,p::>q/r, -::> 2T1 2,3, -::>

~TH

2T6

p::> .p::>q::>q (Law of assertion)!

1.

.p::>q (::»

~p::>q::>

2T1, +S,p::>q/p

1 The resemblance of the law of assertion (and 2T33) to the rule of modus ponens is obvious. But the law cannot replace the rule, since the former is a statement of a truth while the latter describes something to be done, a procedure of deduction.

72

SYSTEM WITHOUT NEGATION

3. 4.

~p:::>

.p:::>q:::>p

~TH

A shorter proof of 2T6 is 1. ~p:::>q:::> .p:::>q (:::» 2. ~p:::> .p:::>q:::>q

1,2A2, +S, p:::>q/p, pjq, qjr, -:::> 2T02 2,3, +IS

2T1, +S,p:::>q/p 1,2T4, +S,p:::>qjp,p/q, q/r, -:::>

2T7

p:::>r:::> .p:::> .q:::>r (Law of interpolation)

1. ~p:::>r:::> .q:::> .p:::>r 2. ~q:::> .p:::>r:::>:p:::> .q:::>r

3.

~TH

2Al, +S,p:::>rjp 2T4, +S, q/p,pjq 1,2, +IS

It may be noted that the proof of 2T6 applies the form of the rule of the implicative series based on 2T2, the first syllogistic law, while that of 2T7 uses the form of the same rule based upon 2T3, the second syllogistic law. The citation, +IS, is introduced regardless of which of these two forms is applied, since inspection yields an effective method for determining which form has been applied. 2T8 1. 2.

p:::>:p:::> .p:::>q:::>:q

.p:::>q:::>:p:::>q (:::» ~p:::>:p:::> .p:::>q:::>:q ~p:::>

2T9

2T5 1,2T4, +S,p:::> .p:::>q/p, p/q, q/r, -:::>

r:::> .p:::> .r:::>s:::>:r:::>p:::> .r:::>s

.p:::> .r:::>s:::>:r:::>p:::> .r:::> .r:::>s 2A2, + S, rip, p/q, r:::>sjr .r:::> . r:::>s:::>: .r:::> . r:::>s:::> :r:::>s: . r:::>p:::> .r:::>s 2T3, +S, r:::>p/p, r:::> .r:::>s/q, r:::>s/r 3. ~r:::> .p:::> .r:::>s:::>:.r:::> .r:::>s:::>:r:::>s:::>:. r:::> p:::> . r :::> s ( :::> ) 1, 2, + IS 4. ~r:::> .r:::>s:::>:r:::>s:::>:. 3,2T4, +S, r:::> .p:::> .r:::>s/p, r:::> .r:::>s/q, r:::>p:::> .r:::>s/r,

1. 2.

~r:::>

5.

~r:::>

6.

~TH

~r:::>p:::>

-:::>

.r:::>s:::>:r:::>s

2T5, 4,5,

+ S, rip, s/q -:::>

73

SYSTEMS OF FORMAL LOGIC

2TlO 1. ~p~ 2. ~q~ 3. ~q~

p~ .q~ .r~s~:q~ .r~p~ .r~s

.q~ .r~s~:q~ .p~ .r~s .p~ .r~s~:q~ .r~ .p~ .r~s

2T4, +S, 2T7, +S,

r~s/r q/p,p~ .r~s/r,

r/q

.r~ .p~ .r~s~:.r~ .p~ .r~s~:

r~p~ .r~s~:.q~ .r~p~ .r~s

2T3, +S, q/p,

r~ .p~ .r~s/q,

r~p~ .r~s/r

4.

~p~.q~.r~s~:. r~ .p~ .r~s~:r~p~ .r~s~:.

q~ :r~p~ .r~s

5.

1,2,3, +IS

(~)

~r~ .p~ .r~s~:r~p~ .r~s~:. p~ .q~ .r~s~:.q~ .r~p~ .r~s

4, 2T4,

+S,p~ .q~ .r~s/p,

r~ .p~ .r~s~ :r~p~ .r~s/q,

q~:r~p~ .r~s/r,

6.

rr~ .p~ .r~s~:r~p~ .r~s

7.

~TH

2T9 5,6, -

-

~

~

EXERCISES

Implication, P + 1. Show that any step derivable by simultaneous substitution may be derived in a reconstructed proof using only the primitive rule of substitution. 2. Verify, "Any step in a proof derivable by the rule of theorem introduction, ( ~ ), is derivable by use of only the primitive rules of the system." 3. Verify, "Any step in a proof derivable by the rule of the implicative series is derivable by use of only the primitive rules of the system. 4. Verify, "Introduction of a previously proved theorem as a step in a proof will not lead to invalidity." 5. Give a full proof of the second syllogistic law, making no use of derived rules. 6. Give a full proof ofthe law of commutation, making no use of derived rules. 7. Prove the following as theorems of P +. (Use derived rules where convenient.) 2T11

74

q~ .r~ .p~p

SYSTEM WITHOUT NEGATION

2T12

r:::J .S:::Jt:::J .p:::Jq:::Jr

2T13

p:::J .q:::Jr:::J :p:::Jq:::Jp:::J .p:::Jq:::Jr

2T14

p:::J .q:::Jr:::J .p:::Jq:::Jr

2T1S

p:::Jq:::Jr:::Js:::Jt:::J .p:::J .q:::J .r:::J .S:::Jt (This exercise is more

difficult than its predecessors.)

2.62 Theorems olP+ in implication and conjunction. In proof of theorems with conjunction use will be made, not only of the axioms 2A3, 2A4, 2AS, but of the axioms and theorems proved in Section 2.61 on implication. The axioms for conjunction are: 2A3 2A4 2AS

p&q:::Jp p&q:::Jq p:::J .q:::J .p&q

2T16

p&q:::J .q&p (Law of the symmetry ofconjunction,firstform)

1. f-p&q:::Jq 2. f-q:::J .p:::J .q&p 3. f-p&q:::J .p:::J .q&p (:::J) 4. f-p:::J .p&q:::J .q&p S. f-p&qwp 6. f-p&q:::J .p&q:::J .q&p 7. f-TH

2A4 2AS, +S, q/p,p/q 1,2, +IS 3,2T4, +S,p&q/p,p/q, q&p/;, -:::J 2A3 4, S, +IS 6,2TS, +S,p&q/p, q&p/q, -:::J

The converse of this law follows by an analogous procedure. 2T17

q&p:::J .p&q (Law of the symmetry of conjunction, second

form) 1. f-q&p:::Jp 2. f-p:::J .q:::J .p&q 3. f-q&p:::J.q:::J.p&q (:::J) 4. f-q:::J .q&p:::J .p&q

S. f-q&p:::Jq 6. f-q&p:::J .q&p:::J .p&q (:::J) 7. f-TH

2A4, + S, q/p, p/q 2AS 1,2, +IS 3,2T4, +S, q&p/p, p&q/r, -:::J 2A3, + S, q/p, p/q 4, S, +IS 6,2TS, +S, q&p/p, p&q/q, -:::J 75

SYSTEMS OF FORMAL LOGIC

The following is a simpler proof of 2T17. 1. 2.

~p&q=> ~q&p=>

.q&p .p&q

2T16 1, +S, q/p,p/q

p&q&r=> .p&.q&r (Associative law for conjunction, first form) ~p&q&r=>r 2A4, +S, p&q/p, r/q ~r=> .q=> .r&q 2A5, +S, rip ~p&q&r=>.q=>.r&q (=» 1,2, +IS ~q=> .p&q&r=> .r&q 3,2T4, +S,p&q&r/p, r&q/r, - => ~p&q&r=> .p&q 2A3, + S, p&q/p, r/q ~p&q=>q 2A4 ~p&q&r=>q 5,6, +IS ~p&q&r=> .p&q&r=> .r&q (=» 4,7, +IS ~p&q&r=> .r&q 8,2T5, +S,p&q&r/p, r&q/q, - => ~r&q=> .q&r 2T16, +S, rip ~p&q&r=> .q&r 9,10, +IS ~p&q=>p 2A3 ~p&q&r=>p 5, 12, +IS ~p=> .q&r=> .p&.q&r 2A5, +S, q&r/q ~p&q&r=> .q&r=> .p&.q&r (=» 13, 14, +IS ~q&r=> .p&q&r=> .p&.q&r 15,2T4, +S,p&q&r/p, q&r/q,p&.q&r/r, - => ~p&q&r=> .p&q&r=>. p&.q&r (=» 11, 16, +IS ~TH 17,2T5, +S,p&q&r/p, p&.q&r/q, - =>

2T18 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18.

The proof of the second form of the associative law for conjunction, 2T19

p&.q&r=>:p&q&r

is left to the reader as an exercise. Though the proofs of 2T18 (and 2T19) are somewhat lengthy, they involve no operations more difficult than those used for the proof of 2T16, namely the use of the three axioms for conjunction, the four rules of the system, and three theorems already proved in the system. Further, it may be shown that such deductions 76

SYSTEM WITHOUT NEGATION

may be abbreviated by recourse to subsequently proved theorems. The associative law and the law of the symmetry of conjunction show that the order of the conjuncts in a conjunctive expression is irrelevant to its truth value. One further illustration of this fact is given below.

p&q&r::J .q&p&r

2T20

1. 2. 3. 4. 5. 6. 7.

8. 9.

~p&q&r::Jr

.p&q ~p&q::J .q&p ~p&q&r::J .q&p ~q&p::J .r::J .q&p&r ~p&q&r::J

~p&q&r::J.r::J.q&p&r ~r::J

(::J)

.p&q&r::J .q&p&r

~p&q&r::J

~p&q&r::J

2A4, + S, p&q/p, r/q 2A3, + S, p&q/p, r/q 2T16 2,3, +IS 2A5, + S, q&p/p, r/q 4,5, +IS 6,2T4, +S,p&q&r/p, r/q,

q&p&r/r, - ::J .p&q&r::J .q&p&r (::J) 1, 7, +IS .q&p&r 8,2T5, +S, p&q&r/p, q&p&r/q, -::J

In an analogous way it is easy to prove: 2T21 2T22 2T23 2T24

p&q&r&s::J .r&p&s&q r&p&s&q::J .p&q&r&s p&q&r&s::J .s&r&q&p s&r&q&p::J .p&q&r&s

These theorems (and an infinite number of others resembling them) are all based upon exactly the same characteristics of conjunction illustrated in the proofs of 2T16-2T20. 2T25

1.

p&p::Jp

~p&p::Jp

2T26 1.

~p::J

2.

~TH

2T27

p::J .p&p .p::J .p&p (::J)

2A5, +S,p/q 1,2T5, + S, p&q/q, -

::J

p&q::J .p::Jq

1.

~p&q::Jq

2. 3.

~q::J

~TH

2A3, +S,p/q

.p::Jq

2A4 2A1, +S, q/p,p/q 1,2, +IS 77

SYSTEMS OF FORMAL LOGIC

The converse of 2T27 does not hold in general. 2T28

p:::J . q:::J r:::J :p&q:::Jr (Law of importation)

1. I-p:::J .q:::Jr:::J:p&q:::J .p:::J .q:::Jr 2. I-p&q:::J .p:::J .q:::Jr:::J:p&q:::JP:::J. p&q:::J .q:::Jr (:::J) 3. I-p&q:::Jp:::J:p&q:::J .p:::J .q:::Jr:::J: p&q:::J .q:::Jr

2Al, +S,P:::J .q:::Jr/p,p&q/q 2A2, +S,p&q/p,p/q, q:::Jr/r 2,2T4, +S,p&q:::J .p:::J. q:::Jr/p,p&q:::Jp/q, p&q:::J .q:::Jr/r, -:::J 2A3 3,4, -:::J

4. I-p&q:::Jp 5. I-p&q:::J .p:::J .q:::Jr:::J:p&q:::J .q:::Jr 6. I-p&q:::J .q:::Jr:::J: p&q:::Jq:::J .p&q:::Jr (:::J) 2A2, +S,p&q/p 7. I-p&q:::Jq:::J:p&q:::J .q:::Jr:::J:p&q:::Jr 6,2T4, +S,p&q:::J •q:::Jr/p, p&q:::Jq/q,p&q:::Jr/r, -:::J 8. I-p&q:::Jq 2A4 9. I-p&q:::J .q:::Jr:::J:p&q:::Jr 7,8, -:::J 10. I-TH 1,5,9, +IS

Having proved the law of importation, it is easy to prove the two correlates-with-conjunction of the first and second syllogistic laws. 2T29

q:::Jr&.p:::Jq:::J:p:::Jr (First syllogistic law with conjunction)

1. I-q:::Jr:::J .p:::Jq:::J .p:::Jr (:::J) 2. I-TH

2T30

2T2 1,2T28, +S, q:::Jr/p,p:::Jq/q, p:::Jr/r, -:::J

p:::J q &. q:::J r:::J :p:::J r (Second syllogistic law with conjunction)

1. I-p:::Jq:::J . q:::J r:::J .p:::Jr (:::J) 2.I-TH

2T3 1,2T28, +S,P:::Jq/p, q:::Jr/q, p:::Jr/r, -:::J

Further theorems of P+ in conjunction and implication: 2T31

p&q:::J .q:::Jr:::J:p&q:::Jr (Law of cancellation)

2T32

p:::Jq&.p:::Jr:::J :p:::J .q&r (Law of composition)

78

SYSTEM WITHOUT NEGATION

2T33

p&.p~q~ :q

2T34

p~q~ .p&r~

2T35

p ~ r &. q ~ s ~ :p & q ~ . r &s (The Praec/arum theorema of

(Law of assertion, with conjunction)

.q&r (Law of the factor)

Leibniz) 2T36

p&q~r~ .p~ .q~r

(The Law of exportation)

EXERCISES

Prove: 2T19 (use the law of composition), 2T21, 2T22, 2T23, 2T24. These expressions are given on p. 76f above; also 2T31-2T36 above. Prove: p~r~ .q~r~ .p&q~r andp~q~ .p~r~:p~ .q~r.

2.63 Theorems of P + with disjunction. The axioms of P + in which the functor for disjunction occurs are: .pvq .pvq

2A6 2A7 2A8

p~

2T37

pvp~p

1. 'rp~p~ 2. 'rp~p 3. 'rp~p~ 4. 'rTH

.p~p~

2T38

p~.pvp

q~

p~r~ .q~r~ .pvq~r

.pv p~p

.pvp~p

2A6, +S,p/q

1. 'rTH

2T39

2A8, +S,p/r,p/q 2T1 1,2, - ~ 3,2, -::::>

p&q~

.pvq

1. 'rp&q~p 2. 'rp~ .pvq 3. 'rTH

2A3 2A6 1,2, +IS

The converse of 2T39 does not hold in general. 2T40

p&q&r~.pvqvr

1.

'rp&q&r~r

2.

'rr~

3. 'rTH

.pvqv r

2A4, + S, p&q/p, r/q 2A7, +S, r/q,pvq/p 1,2, +IS 79

SYSTEMS OF FORMAL LOGIC

2T41

pvq~

.qv p

1. I-p~.qvp~:q~.qvp~:pvq~.qvp2AS, +S,qvp/r 2. I-p~.qvp 2A7, +S,p/q,q/p 3. I-q~.qvp~:pvq~.qvp 1,2,-~ 4. I-q~.qvp 2A6, +S,q/p,p/q 5. I-TH 3,4, - ~ The converse, 2T42, q v p~ .p v q, follows by substitution on 2T41. 2T43

p~r&.q~r~:pvq~r

1. I-p~r~ 2.I-TH

.q~r~

.pv q~r

(~)

2AS 1, 2T2S, +S,p~r/p, q~r/q,pvq~r/r, - ~

The converse is proved as follows: 2T44 1. 2. 3. 4. 5. 6. 7.

pvq~r~ .p~r&.q~r

I-p~ .pvq~:pvq~r~ .p~r

I-p~

2T3, +S,pvq/q

2A6

.pvq

I-pvq~r~ .p~r

1, 2,-~

I-q~ .pvq~:pvq~r~ .q~r

2T3,

I-q~.pvq

2A7

I-pvq~r~ .q~r

4,

+ S, q/p, p v q/q

5,-~

I-pvq~r~ .p~r~: pvq~r~ .q~r~:

pvq~r~ .p~r&:

2A5,

pvq~r~ .q~r

+S,pvq~r~ .p~r/p,

pvq~r~ .q~r/q

S.

I-pvq~r~ .q~r~: pvq~r~ .p~r&: pvq~r~ .q~r

9.

7,

3,-~

S,

6,-~

I-pvq~r~ .p~r&:

pvq~r~.q~r

(~)

to.I-TH

9,2T32,

+S,pvq~r/p,

p~r/q, q~r/r,

2T45 q~r~ .pvq~r~ .p~r 1. I-p~ .pvq 2. I-p~ .pvq~:pvq~r~ .p~r

so

-

2A6 2T3, +S,pvq/q

~

SYSTEM WITHOUT NEGATION

3. 't-pvq=:lr=:l .p=:lr (=:I)

2, 1, - =:I 3,2Al, +S,pvq=:lr=:l .p=:lr/p, q=:lr/q, -=:I

4. 't-TH 2T46

p v q=:lr=:l . q=:l r=:l .p=:lr

2T46 follows from 2T45 by 2T4, substitution and modus ponens. The theorems 2T43-2T46 are transformation-forms of the axiom 2A8. They show that the converse of this axiom, the commutation of the converce, the conjunctive form and its converse are theorems of the system. 2T47

qvr=:l.qv.pvr

2A6, + S, q/p, p v r/q 2A7, +S, r/q 2A7, +S,pvr/q, q/p 2,3, +IS

1. 't-q=:l.qv.pvr 2. 't-r=:l .pv r 3. 't-p v r=:l .qv .pvr 4. 't-r=:l.qv.pvr 5. 't-q=:l .qv .pvr=:l:r=:l .qv .pvr=:l: q=:l .qv.p v r&.r=:l .qv .p v r 6. 't-r=:l .qv .pvr=:l: q=:l .qv .pvr&:r=:l .qv .pvr 7. 't-q=:l .qv .pvr&:r=:l .qv .pvr

8. 't-TH

2A5, +S, q=:l .qv .pvq/p, r=:l .qv .pvr/q

( =:I)

5, 1, - =:I 6, 4, - =:I 7, 2T43, + S, q/p, r/q, qv .pvr/r, -

2T48

=:I

pv .qvr=:l .qv .pvr

1. 't-p=:l.p v r

2. 't-p v r =:I • q V •P v r

2A6, +S, r/q 2A7, +S,pvr/q, q/p 1,2, +IS 2T47

3. 't-p=:l .qv .pvr 4. 't-qvr=:l .qv .pvr 5. 't-p=:l .qv .pvr=:l:qvr=:l .qv .pvr=:l: pv .qvr=:l .qv .pvr 2A8, +S, qv .pvr/r, qvr/q 6. 't-q v r=:l .qv .pvr=:l: 5,3, - =:I pv .qv r=:l .qv .pv r 6,4, - =:I 7. 't-TH

The remaining disjunctive theorems corresponding to the associative laws for conjunction are left to the reader as excercises.

81

SYSTEMS OF FORMAL LOGIC

2T49

pvqvr=:!.pv.qvr

2T50

pv.qvr=:!:pvqvr

2T51

pvqvr=:!.rvqvp

2T52

rvqvp=:! .pvqvr

2T53

p V q V r V s =:! . r V p V S V q

2T54

rvpvsvq=:!.pvqvrvs

2T55

pvqvrvs=:!.svtvqvp

An infinite number of further laws of this form are provable. It may be noted that the theorem 2T48 is analogous with the axiom of Principia Mathematica numbered 1.5, Russell and Whitehead's Principle of Association. The Principle of Permutation, 1.4 in PM corresponds with 2T41; the Principle of Tautology, 1.2, with 2T37; the Principle of Addition, 1.3, with 2A7. We will now proceed to prove an analogue of the final Principia axiom, 1.6, the Principle of Summation. 2T57 1. 2. 3. 4.

5. 6. 7.

q=:!r=:! .p V q=:! .p V r (The Law of Summation)

(=:!)

2A6, +S, rjq ~q=:!r=:! .p=:!.p V r I, 2AI, +S,p=:! .pvrjp, q=:! rjq, - =:! ~q=:!r=:!:r=:! .pvr=:!:q=:! .pvr (=:!) 2T3, +S, qjp, rjq,p v rjr, ~r=:! .pvr=:!:q=:!r=:!:q=:! .pvr 3,2T4, +S,q=:!rjp,r=:! .p=:!rjq, q=:! .p V rjr, - =:! ~r=:!.p V r 2A7, rjq 4, 5, - =:! ~q=:!r=:! .q=:! .p V r ~p=:!.pvr

.p=:! .p V r=:!: q=:!r=:! .q=:! .p V r=:!: q=:!r=:! .p=:! .p v r&: q=:! r=:! . q=:! .p V r 2A5, + S, q=:! r=:! .p=:! .p v rjp, q=:!r=:! .q=:! .pv rjq 8. ~q=:!r=:! .q=:! .pv r=:!:q=:!r=:! .p=:!. pvr&:q=:!r=:! .q=:! .pvr 7,2, - =:! 9. ~q=:!r=:! .p=:! .pvr&: q=:!r=:!.q=:!.pvr (=:!) 8, 6, - =:! 10. ~q=:!r=:!:p=:!.pvr&:q=:!.pvr 9,2T32, +S, q=:!rjp, p=:!.p v rjq, q=:! .pv rjr, - =:! ~q=:!r=:!

82

SYSTEM WITHOUT NEGATION

11.

~p:::>

.pvr&:q:::> .pvr:::>:. pvq:::>.pvr

12.

~TH

2T43, +S, p v rjr 10,11, +IS

Thus we are able to prove all the axioms of Principia Mathematica, for the propositional calculus, as they are set out in Part 1, Section A, *1, "Primitive Ideas and Propositions." Likewise, all the axioms of P + are axioms or theorems of PM. The following correspondences are pointed out, where the number under the heading "PM" is the index number of the axiom, theorem, or definition corresponding to the axiom number given under the heading "P+". P+

2AI 2A2 2A3 2A4 2AS 2A6 2A7 2A8 2A9 2AW 2AIl

PM *2.02 *2.77 *3.26 *3.27 *3.2 *2.2 *1.3 *3.441 *4.01 *4.01 *4.01 2

By virtue of this correspondence of axioms it might be thought, following the doctrine of the first Chapter, that the two systems are equivalent. This view, however, is patently false. This becomes clear when one notes that the axioms, as set forth in the section of Principia cited above, are all abbreviations. The law of summation, e.g., as stated in Principia is an abbreviation for [ '" [ '" q v r] v [ '" [p v q] v [p v r]]] 1 2A8 is nowhere proved explicitely in PM, but it may be derived from *3.44 and the law of exportation, *3.3, hence it is a theorem of PM. 2 4.01 is a definition,p=q=.p=> q&.q=>p, Dj. Thus 2A9,2AI0 and 2All are derivable in PM by *4.01, *3.26, *3.27, and modus ponens.

83

SYSTEMS OF FORMAL LOGIC

obtained from this expression by the use of the definition of material implication, number *1.01 in PM,

[[P::>q]=[ -pvq]] Df Likewise, numerous conjunction theorems are derived by the definition of logical product, number *3.01 in PM.

[[p&q] = [ -[ -pv -q]]] Df The definition of material implication and the definition oflogical product are not, of course, elements of P +; indeed, they cannot even be formulated in P + by use of the present formation rules of the system. Thus PM and P + are not equivalent systems. The observations above suggest, however, that an extension of P + could be constructed which would be equivalent with PM. This possibility will be explored in the following chapter. For the purpose of shortening the remaining proofs of this section and of easing the labor of the exercises, two further derived rules will be here instituted. By axiom 2AS, the conjunction A&B, follows if A and if B. Hence, it will be regarded as legitimate to derive A&B directly from a proof of A and a proof of B, thus, I-A I-B I-A&B

1.

2. 3.

TH TH 1,2, *2AS

citing as reason the step numbers of the two conjuncts, and the axiom number preceded by an asterisk to signify that the derived rule is being used, thus *2AS. Similarly, it will be permissible to infer A vB::> C directly from the two expressions A::> C and B::> C, thus, I-A::>C I-B::>C I-AvB::>C

1.

2. 3.

TH TH 1,2, *2A8

citing as reason the step numbers of the two implications from which the conclusion follows by this rule, and the axiom number *2A8, the asterisk signifying that this is the rule derived from 2A8. As before, whenever these derived rules are used, the user must always be able to supply the abbreviated steps if his inference is challenged. The following are proofs using the two derived rules.

84

SYSTEM WITHOUT NEGATION

2T57 p&.qvr=> .p&qv .p&r 1. ~p&.qvr=>:p 2A3, +S, qv rjq 2. ~p=> .q=> .p&q 2A5 3. ~q=> .p&q=> :p&q=> .p&q v .p&r=>: q=> .p&qv .p&r 2T3, +S, qjp,p&qjq, p&qv.p&rjr 4. ~p&.qvr=>:p&q=> .p&qv .p&r=>: q=> .p&q v .p&r (=» 1-3, +IS 5. ~p&q=> .p&qv .p&r=>: p&.qv r=>:q=> .p&qv .p&r 4,2T4, +S,p&.qv rjp, p&q=> .p&qv .p&rjq, q=> .p&qv .p&rjr, - => 6. ~p&q=> .p&q v .p&r 2A6, +S,p&qjp,p&rjq 7. ~p&.qvr=>:q=> .p&qv .p&r 5,6, - => 8. ~p=> .r=> .p&r 2A5, +S, rjq 9. ~r=> .p&r=>:p&r=> .p&qv .p&r=>: r=> .p&qv .p&r 2T3, +S,p&rjq,p&qv .p&rjr 10. ~p&.qv r=>:p&r=> .p&qv .p&r=>: r=>.p&qv.p&r (=» 1,8,9, +IS 11. ~p&r=> .p&qv .p&r=>:p&.qvr=>: r=> .p&q v .p&r 10,2T4, +S,p&.qv rjp, r=> .p&qv .p&rjr, p&r=> .p&qv .p&rjq 12. ~p&r=> .p&qv .p&r 2A7, +S,p&r/q,p&q/p 13. ~p&.qv r=>:r=> .p&qv .p&r 11, 12, - => 14. ~p&.qv r=>:q=> .p&qv . p&r&:.p&.qvr=>: r=>.p&qv.p&r (=» 7,13, *2A5 15. ~p&.qv r=>: .q=> .p&qv . p&r&:r=> .p&q v .p&r 14,2T32, +S,p&.qvr/p, q=> .p&qv .p&rjq, r=> .p&qv .p&rjr, - => 16. ~q=> .p&qv .p&r&:r=> .p&qv. p&r=>: .qv r=> .p&qv .p&r 2A8, + S, qjp, rjq, p&qv.p&r/r 17. ~p&.qvr=>:qvr=> .p&qv. p&r (=» 15,16, +IS 85

SYSTEMS OF FORMAL LOGIC

18. I-TH 2T58 p&qv.p&r-=:;:p&.qvr 1. I-p&q-=:;p 2. I-p&q-=:;q 3.l-q-=:;.qvr 4. I-p&q-=:; .qv r 5. I-p&q-=:;p&.p&q-=:;.qvr (-=:;) 6. I-p&q-=:; .p&.qv r

7. 't-p&r-=:;p 8. I-p&r-=:;r 9.l-r-=:;.qvr 10. I-p&r-=:; .qvr 11. I-p&r-=:;p&.p&r-=:;.qvr (-=:;) 12. 't-p&r-=:; .p&.qv r

13. 't-TH 2T59 pv.q&r-=:;:pvq&.pvr 1. 't-p-=:;.pvq 2.l-p-=:;.pvr 3. I-p-=:;.pvq&:p-=:;.pvr (-=:;) 4.l-p-=:;.pvq&.pvr

17,2T3I, +S,qvr/q, p&q v .p&r/r, --=:; 2A3 2A4 2A6, + S, qlp, rlq 2,3, +lS 1,4, *2A5 5,2T32, +S,p&q/p, p/q, q v r/r, - -=:; 2A3, +S, r/q 2A4, +S, r/q 2A7, +S, r/q, q/p 8,9, +lS 7,10, *2A5 11,2T32, +S,p&r/p, p/q, qv rlr, --=:; 6, 12, *2A8

2A6 2A6, +S, r/q

13.I-TH

1,2, *2A5 3, 2T32, + S, p v q/q, pv rlr, --=:; 2A3, + S, qlp, rlq 2A7 5,6, +IS 2A4, + S, q/p, r/q 2A7, +S, r/q 8,9, +IS 7, 10, *2A5 11,2T32, +S, q&r/p, pvqlq,pvr/r, --=:; 4, 12, *2A8

2T60 p&qv p~ .p 1. 't-p&q-=:;p

2A3

5. I-q&r-=:;q 6. 't-q-=:;.pvq 7. I-q&r-=:;.p v q 8. I-q&r-=:;r 9. 't-r-=:;.pvr 10. 't-q&r-=:;.p v r 11. I-q&r-=:; .pvq&:q&r-=:; .pvr (-=:;) I2.l-q&r-=:;.pvq&.pvr

86

SYSTEM WITHOUT NEGATION

2.

~p:::::Jp

2T1

3.

~TH

1,2, *2A8

Further theorems with disjunction. 2T61

p:::::J .p&qv .p

2T62

p&sv .q&sv .r&s:::::J:pvqvr&s

2T63

p&.qv .p:::::Jqv .p:::::J .p:::::Jq:::::J:p&q

2T64

p&qv.r&s:::::J:pvs

2T65

pvq&.pvr:::::J:pv.q&r

2T66

pvr&.p:::::Jq&.r:::::Js:::::J:qvs

Further results in disjunction which will serve as exercises: 2T49-2T56.

2.64 Theorems of P+ with equivalence. The axioms of P + in which the functor for equivalence occurs are: 2A9 2AW

2All

p=q:::::J .p:::::Jq p=q:::::J .q:::::Jp p:::::Jq:::::J .q:::::Jp:::::J .p=q

2A9 and 2AtO are read, respectively, "If p is equivalent with q then p implies q", and "If p is equivalent with q then q implies p." Thus if one can prove an equivalence, p=q, then one can infer p:::::Jq and also q:::::Jp. 2All is read, "If p implies q then if q implies p then p is equivalent with q." Thus if one has both A:::::JB and B:::::JA as elements in a proof then A=B follows by substitution on 2All and two applications of modus ponens. It should be obvious that P+ possesses the following as theorems: 2T67 2T68

P:::::Jq&·q:::::JP:::::J:p=q

2T69

q:::::Jp:::::J .p=q:::::J .p:::::Jq

2T70

p=q:::::J .q:::::Jp:::::J .p:::::Jq

2T71

p=q:::::J .q=p

2T72

p=p

q:::::Jp:::::J .p:::::Jq:::::J .p=q

Proofs of the last two theorems will be offered, the remaining four being left to the reader. 87

SYSTEMS OF FORMAL LOGIC

2T71

p=.q-::l .q=.p

1.

~p=.q-::l

2. 3. 4. 5.

~p=.q-::l

~p-::lq-::l

.q-::lp .p-::lq .p-::lq-::l .q=.p .p-::lq-::l .q=.p (-::l) .p=.q-::l .q=.p

6.

~p=.q-::l

.p=.q-::l .q=.p (-::l)

7.

~TH

~q-::lp-::l

~p=.q-::l

2T72

p=.p

1.

~p-::lp-::l

.p-::lp-::l .p=.p

2. 3.

~p-::lp

4.

~TH

~p-::lp-::l

2AlO 2A9 2All, + S, q/p, p/q 1,3, +IS 4,2T4, +S,p=.q/p, P-::lq/q, q=.p/r, --::l 2,5, +IS 6, 2T5, + S, p =. q/p, q=.p/q, --::l 2All, +S,p/q

2T1 .p=.p

1,2, --::l 3,2, --::l

By virtue of 2A11 it is always possible to establish an equivalence wherever both an expression and its converse have been established as theorems. Thus we have the following proof of 2T73 in the system. 2T73

p=. .p&p

1.

~p-::l

2. 3. 4.

~p-::l

~p&p-::lp

5.

~TH

.p&p-::l:p&p-::lp-::l:p=' .p&p .p&p ~p&p-::lp-::l .p=. .p&p

2All, +S,p&p/2 2T26 1,2, --::l 2T25 3,4, --::l

In general, whenever an expression and its converse have been proved as theorems of P + there will always be a five step proof of equivalence exactly analogous with that of 2T73. Since a long series of such proofs discloses nothing informative about the process of deducing equivalences, the following convention will be instituted: If a theorem, A -::l B, and its converse, B -::l A, have been established such that it is possible to set up a five step proof of the form: 1. 2. 3.

~A-::lB-::l

.B-::lA-::l .A=.B

~B-::lA-::l

88

2All, + S, A/p, B/q

TH

~A-::lB

.A=.B

1, 2,

-:::>

SYSTEM WITHOUT NEGATION

TH 3,4, -=>

4.I-B=>A 5.I-A=B

the equivalence may be stated and the justification given by citing the theorem numbers of the implications from which it is derived by 2AII and modus ponens, followed by the expression *2All, referring to the form above. It is clear that no results are available by recourse to *2AII which are not available by going through the five step procedure specified above. (The citation of *2All constitutes an assertion that such a five step proof is available; hence it is justifiable only if such a proof can be constructed. There is always an effective test as to whether or not this is possible.) 2T74

p=.pvp

1. I-TH 2T75

2T38, 2T37, *2AII

p&q= .q&p

1. I-TH 2T76

2T16, 2T17, *2All

pvq=.qvp

1. I-TH 2T77

2T41,2T42, *2All

p&q&r= .p&.q&r

1. I-TH

2T18, 2T19, *2All

(It should be noted that if the reader were challenged concerning 2T77

he would have to be able to produce the proof of2T19, left as an exercise.) 2T78 1. I-TH

pvqvr= .pv .qvr 2T49, 2T50, *2All

(An analogous remark applies here, the reader this time having to supply proofs of both sides of the equivalence.) An infinite number of further equivalences concerning the associative laws for conjunction and for disjunction are provable in the system. It follows, of course, that for any pair of theorems in the system, the one of which is the converse of the other, an equivalence may be derived by a process analogous with that employed above. Instead of citing further equivalences derivable by application of *2All to theorems already proved in the text or in the exercises it will be useful to conclude this

89

SYSTEMS OF FORMAL LOGIC

section with two proofs not involving the citation of *2All, i.e., equivalences proved "from scratch". 2T79

1.

~q-::::;q

2.

~p-::::;q-::::;

3. 4. 5.

~p-::::;q-::::;

p-::::;q= .pvq=q

~p-::::;q-::::; ~p-::::;q-::::;

(-::::;) .q-::::;q .p-::::;q . q-::::;q&:p-::::;q-::::; .p-::::;q (-::::;) .q-::::;q&.p-::::;q

2T1, +S, q/p 1, 2Al, +S, q-::::;q/p, p-::::;q/q, --::::; 2T1, +S,p-::::;q/p

2,3, *2A5 4,2T32, +S,p-::::;q/p, q-::::;q/q,p-::::;q/r, --::::;

6. 7. 8. 9.

~q-::::;q&.p-::::;q-::::;:pvq-::::;q

.pvq-::::;q ~q-::::;.pvq (-::::;) ~p-::::;q-::::; .q-::::; .pvq

~p-::::;q-::::;

2T43, +S, q/p, q/r, p/q 5,6, +IS 2A7 8, 2Al, +S, q-::::; .p v q/p, p-::::;q/q, --::::;

10.

.pvq-::::;q&: p-::::;q-::::;.q-::::;.pvq (-::::;) 11. ~p-::::;q-::::; .pvq-::::;q&.q-::::; .pvq ~p-::::;q-::::;

12. 13. 14. 15. 16. 17. 18. 19.

~pvq-::::;q&.q-::::;

21.

~TH

~p-::::;q-::::;

.pvq-::::;:pvq=q .pvq=q

~pvq=q-::::;.pvq-::::;q

7,9, *2A5 1O,2T32, +S,p-::::;q/p, pvq-::::;q/q, q-::::;p V q/r, --::::; 2T67, +S,pvq/q

11,12, +IS 2A9, +S,pvq/p 2A3, +S,pvq/q, q/r 2A6 15, 16, --::::; 14,17, +IS

.pvq-::::; .pvq-::::;q-::::; .p-::::;q ~p-::::; .pvq ~pvq-::::;q-::::; .p-::::;q ~pvq=q-::::; .p-::::;q ~p-::::;q-::::; .pvq=q-::::;; pvq=q-::::; .p-::::;q-::::;:p-::::;q= .pvq=q 2A11, +S,p-::::;q/p,pvq=q/q 20. ~pvq=q-::::; .p-::::;q-::::; .p-::::;q= .pvq=q 19,13, --::::; ~p-::::;

2T80

20, 18, --::::; p=q&.q=r-::::;:p=r

1.

~p=q&.q=r-::::;:p=q

2. 3.

~p=q-::::;

.p-::::;q

~p=q&.q=r-::::;:p-::::;q

90

2A3, +S,p=q/p, q=r/q 2A9 1,2, +IS

SYSTEM WITHOUT NEGATION

6. I-p=q&.q=r=> :q=>r

2A4, +S,p=q/p, q=r/q 2A9, + S, q/p, r/q 4,5, +IS

7. I-p=q&.q=r=>:p=>q&:. p=q&.q=r=>:q=>r (=» S. I-p=q&.q=r=>:p=>q&.q=>r

3,6, *2A5 7,2T32, +S,p=q&q=r/p,

4. I-p=q&.q=r=> :q=r

5. I-q=r=> .q=>r

p=>q/q, q=>r/r, - =>

9. 10. 11. 12. 13. 14. 15. 16. 17.

I-p=>q&.q=>r=>:p=>r I-p=q&.q=r=>:p=>r I-p=q&.q=r=>:q=r I-q=r=> .r=>q I-p=q&.q=r=> :r=>q I-p=q&.q=r=>:p=q I-p=q=> .q=>p I-p=q&.q=r=>:q=>p I-p=q&.q=r=>:r=>q&:. p=q&q=r=> .q=>p (=»

2T30 S,9, +IS 2A4, +S,p=q/p, q=r/r 2AlO, + S, q/p, r/q 11, 12, +IS 2A3, +S,p=q/p, q=r/q 2AlO 14, 15, +IS

IS. I-p=q&.q=r=>:r=>q&.q=>p

13,16, *2A5 17,2T32, +S, r=>q/q, q=>p/r,

19. I-r=>q&.q=>p=>:r=>p

p=q&q=r/p, - => 2T30, + S, rip, p/r

20. I-p=q&.q=r=> :r=>p

IS, 19, +IS

21. I-p=q&.q=r=>:p=>r&:. p=q&.q=r=>:r=>p (=» 22. I-p=q&.q=r=>:p=>r&.r=>p

10,20, *2A5 21,2T32, +S,p=>r/q,

23. I-p=>r&.r=>p=>:p=r 24.I-TH

p=q&.q=r/p, r=>q/r, - => 2T67, +S, r/q

22,23, +IS

The following observations may be useful in connection with the proofs of 2T79 and 2TSO. (1) Both may be regarded as concatenations of two proofs: the first proof in 2T79, steps 1 through 13, is a proof of p=>q=> .pvq=q, the second, steps 14 through IS, of p v q =q => .p => q; the remainder consists of the establishment of the theorem from these two results plus 2All and modus ponens. 2TSO is structured in an analogous way, but its terminal steps (21-24) are accomplished in a different manner. (2) Inspection of the truth-table for equivalence 91

SYSTEMS OF FORMAL LOGIC

A T T F F

T F F T

B T F T F

discloses that expressions A and B are equivalent if and only if they have the same truth value under all substitutions on their variables. Hence equivalent formulae A, B, may be substituted for each other at any point in an expression, r, without changing the value of the expression. This may be shown as follows. Three cases exist with respect to r. Case 1. r is a contingency. Ifr is a contingency and A is a well-formed part of r and if B is equivalent with A, then (a) if A is a contingency, then B is a contingency; hence if B is substituted for A at any or every point in r then r remains contingent. (b) if A is L-true, then B is L-true; hence if r is a contingency having A as a well-formed part, then r remains a contingency if B is substituted for A at any or at every point. (c) if A is L-false, then B is L-false; hence if B is substituted for A at any or at every point, then if r is a contingency with A as a well formed part, then r is a contingency with B as a well formed part. Case 2. r is L-True. The reasoning in the three sub-cases is exactly analogous with that of Case 1.

Case 3. r is L-False. The reasoning in the three sub-cases is exactly analogous with that of Case 1. Hence we institute a special rule of substitution, to be used in the following chapters. If A and B are equivalent, they may be substituted for each other at any point in an expression r. The application of this rule will be signalized by, (1) the expression "SE" (substitution of equivalences), (2) the number of the step to which it is applied, and (3) the expressions substituted and substituted upon. EXERCISES

2T67 2T68

92

p-:=>q&.q-=:JP-=:J:p=q q-=:Jp-=:J .p-=:Jq-=:J .p=q

SYSTEM WITHOUT NEGATION

2T69 2T70 2T81 2T82 2T83 2T84 2T85 2T86 2T87 2T88 2T89 2T90

q::Jp::J .p=.q::J .p::Jq p =. q::J . q::J p::J .p::J q p&q::J .p=.q p&qv .p&r=.:p&.qv r pvq&.pvr=.:pv.q&r p=.q&.r=.s=.:p&r=. .q&s p::J .q::J .p::Jr=.:p&q::Jr p::Jq=. .p=. .p&q p::J q::J .p::J r =':p::J .p::J . q::J r p&qv p=.p p=.q&.r=.s=.:pvr=. .qvs p::J q&.p::J r::J:p::J . q=. r

93

CHAPTER 3

STANDARD SYSTEMS WITH NEGATION

3.1 Summary In Chapter 2 the reader was introduced to proofs in the propositional calculus through the positive logical system, P +. In this chapter, two new systems are developed. The first of these systems is called PLT. PLT is a full propositional calculus with negation. It is constructed by adding a twelfth axiom to the axiom-set of P +. A further system is also outlined, called PLT" PLT' obtains the full propositional calculus by a reduction of the redundant axioms of PLT. 3.11 P + lacks negation. Hence, if a full classical logic is to be built upon its basis, rules or axioms involving negation must be introduced. A system PL'!:' is here developed having the law "'p'::) "'q'::) .q'::)p

as an axiom. This law is called the law of transposition (sometimes called the law of contraposition).

3.2 Rules of Formation of PLT The rules of formation for PLT are those of P +, except that the system possesses a seventh improper symbol. 3.21 (7) "'. 3.22 The same as in 2.22. 3.23 An expression is a well-formed formula (wff) of PLT only under the following circumstances: 3.231 A proper symbol standing alone is wf 3.232 If the expressions A and Bare wf then (a) (b) (c) 94

[A '::) B] [A vB] [A&BJ

STANDARD SYSTEMS WITH NEGATION

(d) [A:=B] (e) ",A are wf The same bracketing abbreviations as introduced for P + are used in PLT. 3.3 The Rules of Transformation of PLT The rules of transformation for PLT are those of P+. 3.31 Modus ponens. 3.32 Substitution. The same abbreviative symbols, -::l, and + S, respectively, will be retained for PLT. 3.4 Axioms of PLT

3AI 3A2 3A3 3A4 3A5 3A6 3A7 3A8 3A9 3AlO 3All 3AI2

p::l .q::lp p::l .q::lr::l:p::lq::l .p::lr p&q::lp p&q::lq

p::l .q::l .p&q p::l .pvq q::l .pvq p::lr::l .q::lr::l .pvq::lr p=.q::l .p::lq p:=q::l .q::lp

p::lq::l .q::lp::l .p=.q '" p::l '" q::l . q ::l pI

1 A number of other axioms (or sets of axioms) besides the law of transposition could have been employed to convert P + into a full classical logic with negation. For instance the replacement of Al2 with the two laws '" '" p'::J P (Law of double negation) p'::J . q'::J '" p'::J"'q (Kant's Satz des Widerspruc!zs) gives a full propositional calculus as does A12's replacement by p v '" p (Law of excluded middle) '" p'::J .p'::J q (Law of the denial of the antecedent) p'::J"'p'::J"'p (Special law of reductio) The reader will find it instructive to work with these axiom sets as well. The law of transposition, '" p '::J ' " q '::J • q '::J P is not a theorem in the logics of Fitch, Heyting and Johansson (cf. Chapter 6).

95

SYSTEMS OF FORMAL LOGIC

No expression of the system is an axiom unless it has the form of one of the expression, 3AI through 3A12. 3.5 Definitions of PLT The definitions ofPLT are those of P +. It is also to be understood that all abbreviative techniques and derived rules developed in P + apply to PLT. 3.6 Deductions in PLT 3.601 All theorems of P + are theorems of PLT. If a wjJ, A, is a theorem of P +, then it is provable by recourse to the axioms and rules of P +. But PLT possesses all the axioms and rules of P +; therefore, if a wff, A, is a theorem of P + then it is a theorem of PLT. Since this is the case it is useful and time saving to refer back to proofs in P + instead of proving all the results of P + again in PLT. We will institute the following convention in this regard: a theorem 2Tn of P + becomes a theorem of PLT by the simple expediant of changing the first integral from 2 to 3. Thus 2T1 of P + is 3T1 of PLT. A proof in P + becomes a proof in PLT by recourse to changing the integral referring to axioms or previously proved theorems from 2 to 3. Thus the sequence of theorems 2T1 through 2T90 of P + are the theorems 3T1 through 3T90 of PLT. Thus 90 theorems are already proved in PLT. (Some of the following theorems ofPLT are unprovable in the restricted propositional calculi presented in Chapter 6. The following indices are used, where A is a theorem of PLT:

A(-F) = A, a theorem ofPLT, is unprovable in Fitch's calculus. A{-MIN)

= A, a theorem ofPLT, is unprovable in Johansson's minimum

calculus. At- H , -MIN) = A, a theorem ofPLT, is unprovable in both the Heyting intuitionistic logic and the minimum calculus. A(-F, -E, -MIN) = A, a theorem of PLT, is unprovable in the propositional calculi of Fitch, Heyting, and Johansson. The superscripts in parentheses are not part of the theorem, but are metalogical indices.)

96

STANDARD SYSTEMS WITH NEGATION

3T91

"'p=> .p=>q(~MIN) (Law of the denial of the antecedent).

3A1, +S, "'p/p, "'q/q 3A12, +S, q/p,p/q 1,2, +IS

1. I-",p=>. "'q=> "'p 2. I-"'q=> "'P=> .p=>q

3. I-TH

"If A is false then anything follows from it."

3T92

'" "'p=>p(~H.

~MIN)

(Law of double negation (first form))

1. I- '" '" p => . '"P => '" '" '"P 2. I-",p=> '" '" "'p=>. '" ",p=>p 3.1-",,,,p=>.,,,,,,p=>p

(=»

4. I-TH

3T91, + S, '" pip, '" '" '" p/q 3A12, +S, '" "'p/q 1,2, +IS 3,3T5, +S, '" "'p/p,p/q, - =>

"If it is false that A is false, then A."

3T93

p => '" '" p (Law of double negation (second form))

1. 1-", '" "'P=> "'p=> .p=> '" "'p 2. I- '" '" '" p => '" p 3.I-TH

3A12, +S, '" "'plp,p/q 3T92, + S, '" pip 1,2,-=>

"If A, then it is false that A is false."

3T94

p=> . "'p=>q (Law of the denial of the antecedent (second

form)) 1. I-p=>"''''p 2. I- '" '" p=> . '" p=> q

3.I-TH

3T93 3T91, +S, "'p/p 1,2, +IS

3T95 p=>q=>. "'q=> "'p(~F) (Converse law of transposition) 1. 1-", ",p=>p=> .p=>q=>. '" "'p=>q 3T3, +S, '" "'p/p,p/q, q/r 2. I- '" '" p => p 3T92 3.l-p=>q=>."''''p=>q 1,2,-=> 4. I-q=> '" "'q=>. '" "'p=>q=>. '" '" p => '" '" q 5. I-q =>

'" '" q 6. I- '" '" p => q => . '" '" p => '" '" q 7. 1-", "'p=> '" "'q=>. "'q=> "'p 8.I-TH

3T2, + S, '" '" q/r, '" "'pip 3T93, + S, q/p 4,5, - => 3A12, +S, "'p/p, "'q/q 3,6,7, +IS

97

SYSTEMS OF FORMAL LOGIC

In an analogous way 3T96

p::J "'q::J .q::J "'p.",pvq 3. 'rp&.p=>q=>:"'pvq (=» 4. 'rp=> .p=>q=>. "'pvq 5. 'r"'p=>."'pvq (=» 6. 'r",p=>.p=>q=>.",pvq

-H, -MIN)

3T33 3A7, +S, "'p/p 1,2, +IS 3,3T36, +S,p=>q/q, '" p v q/r, - => 3A6, +S, "'pip 5,3T7, +S, "'pip, "'pvq/r, p=>qlq, - =>

7. 'rp=> .p=>q=>. "'pvq=>: "'p=> .p=>q=>. "'pvq=>: p=>q=>."'pvq 8. 'r",p=> .p=>q=>. "'pvq=>: p=>q=>."'pvq 9. 'rp=>q=>."'pvq

100

3T104, +S,p=>q=>. "'p v qlq 4,7, - =>

6,8, - =>

STANDARD SYSTEMS WITH NEGATION

3T105b

'" p V q::::J .p::::Jq 3T91 3Al, + S, q/p, p/q 1,2, *3A8

1. I-"'P::::J .p::::Jq 2. I-q::::J .p::::Jq 3.I-"'pvq::::J·P::::Jq

3T105

p::::Jq=.. "'p V q(-F,

-H, -MIN)

1. I-p::::J q::::J . '" P V q 2. I-",p V q::::J .p::::Jq 3. I-P::::Jq::::J. "'pvq::::J:"'pvq::::J .p::::Jq::::J: p::::Jq=.."'pvq 4. l-"'pvq::::J .p::::Jq::::J:p::::Jq=.. "'pvq 5.I-TH 3T106

3All, +S,P::::Jq/p, "'pvq/q 3, 1, - ::::J 4,2, -::::J

'" .pvq::::J:"'p

1. I-P::::J.pvq 2.I-TH 3T107

3T1 05a 3T105b

(::::J)

3A6 1,3T95, +S,pvq/q,

-::::J

"'.pvq::::J:"'q

The proof is analogous with that of 3T1 06. 3T108

'" .P::::Jq::::J:",p::::Jq(-F,

1. l-"'pvq::::J.P::::Jq (::::J) 2. 1-", .P::::Jq::::J:"'. "'pvq

3. 1-",. "'p v q::::J:'" "'p 4. 1-", "'P::::JP 5.1-"'."'pvq::::J:P 6. I-P::::J. "'P::::Jq 7.I-TH

-H, -MIN)

3T105b 1,3T95, +S, "'pvq/p, P::::Jq/q, -::::J 3T106, + S, '" pip 3T92 3,4, +IS 3T94 2,5,6, +IS

There is a shorter proof of the same theorem 1. 1-",. "'pvq::::J:'" "'p 2. I- '" '" p::::J . '" '" p V q 3. 1-",. "'p v q::::J:'" "'pvq 4. I-TH

3T106, +S, "'pip 3A6, + S, '" pip 1,2, +IS 3, SE,p::::Jq/"'pvq, "'P::::Jq/'" "'pvq

This proof employs the rule of substitution for equivalences stated in 2.7.

It will be recalled that this rule allows the substitution of equivalent expressions at any point in a wfJ, as was done above. 101

SYSTEMS OF FORMAL LOGIC

There are a number of further theorems related to 3Tl08. Some of these are found in the exercises. The following result, analogous with 3T1OS, is left to the reader. 3Tl09

""P-:::Jq= .pvq (3Tl09a ""P-:::Jq-:::J .pvq)(-F. -H.

-MIN)

(3Tl09b P v q-:::J. ""p-:::Jq)(-MIN) 3Tl1O

P-:::Jq-:::J,.., ,p&""q(-F)

1. 2.

~p-:::J,""q-:::J"".p-:::Jq

3.

~TH

(-:::J) ~p&""q-:::J"" .p-:::Jq (-:::J)

3T98 1,3T28, +S, ""q/q, ,.., .p-:::Jq/r, --:::J 2,3T96, + S,p&""q/p,P-:::Jq/q,

There are sixteen implications, giving rise to a set of eight equivalences, relating conjunction to disjunction and vice versa in logics with full negation such as the present system. This set is called "The Law of De Morgan" (see the list on page 103). 3Tlll is one of the implications which gives rise to this important set. 3Tlll 1. 2. 3.

~,..,

4.

~TH

,.., .pvq-:::J:""p&""q

.pvq-:::J:""P ~,.., .pvq-:::J:""q ~,.., .pvq-:::J:""p&:. "".pvq-:::J:""q (-:::J)

3T106 3Tl07

1,2, *3AS 3, 3T32, + S, ,.., .p v q/p, ""p/q, ""q/r, --:::J

A further De Morgan principle: 3Tl12

"". ,..,p v q-:::J:p&""q(-H.

is easily proved thus, 1. ~"'" ""pvq-:::J:,..,""P

2. 3. 4.

S. 6.

-MIN)

3Tl06, +S, ""p/p 3T92 ~""""p-:::Jp 1,2, +IS ~"",""pvq-:::J:p ~"'" ,..,p v q-:::J:""q 3Tl07, +S, ""p/p ~"'" ,..,p v q-:::J:p&:. ""pvq-:::J""q (-:::J) 3,4, *3AS S,3T32, +S, "". ,..,p v q/p, ~TH p/q, ,.., p/r, - -:::J

102

STANDARD SYSTEMS WITH NEGATION

Having proved these results we return to the converse of 3TlIO, 3Tl13, which is proved below. 3Tl13

'" .p&",q-=:J:p-=:Jq(-H.

-MIN)

1. ~ '" p v q -=:J •P -=:J q ( -=:J ) 2. ~"'.p-=:Jq-=:J:",."'pvq

3Tl 05b 1,3T95, +S, "'p v q/p,P-=:Jq/q,

3. ~"" "'p v q-=:J:p&"'q 4. ~"'.p-=:Jq-=:J:p&",q (-=:J)

3Tl12 2,3, +IS 4,3T97, +S,P-=:Jq/p, p&"'q/q, --=:J

5.

~TH

This may be converted into'" .p&"'q-=:J: '" p v q by application of3T105a. By virtue of 3Tl13 and 3Tl09, all members of the left hand column below imply the right hand members: by virtue of 3Tlll all members of the right hand column imply the left hand members. '" .p&q ",.",p&q

"'pv"'q pv"'q "'pvq pvq

'" .p&"'q "'. "'p&"'q

By virtue of other laws of the system the following w.tJs imply each other. '" .pvq "'."'pvq '" .pv "'q

"'p&"'q p&"'q ",p&q p&q

This whole collection of equivalences is known as the Law of De Morgan. 1 3Tl14

p&q= '" .p-=:J "'q

3Tl14a

p&q-=:J '" .p-=:J "'q

1.

~p-=:J

2. ~P -=:J

.p-=:J "'q-=:J "'q '" q -=:J '" q -=:J • q -=:J

3. ~p-=:J.q-=:J"'.p-=:J"'q 4. ~TH

'" •P -=:J '" q

(-=:J)

3T6, + S, "'q/q 3T96, +S,P-=:J "'q/p 1,2, +IS 3,3T28, +S, ",.p-=:J"'q/r, --=:J

3Tl14b 1 For the theoremhood of the sublaws of the law of De Morgan in F,I, and other weakened systems see Chapter 6.

103

SYSTEMS OF FORMAL LOGIC

The proof of 3T114b and hence of the equivalence, 3T114, is left to the reader as an exercise. 3T115

pv "'p(-F.

-H. -MIN)

1. ~p-=:J P 2. ~P =:J P =:J • ' " P v P 3. ~ '" p v P 4. ~ '" p v P =:J • P V '" P 5. ~TH

(Law of Excluded Middle) 3T1 3T105a, +S,p/q 2, 1, - =:J 3T42, + S, '" p/q 4,3, - =:J

This is a proof of the celebrated law of excluded middle (LEM), hallmark of classical logic. In none of the non-classical systems of this book is LEM provable as a theorem. Since this is a theorem we have, of course, its double negation as a theorem also: 3T116 3T117

'" .p&", p(-F) (Law of Contradiction)

1. ~p&"'p=:Jp 2. ~p&"'p=:J "'p 3. 'rP&"'P=:JP=:J .p&"'P=:J "'P=:J '" .p&"'p 4. ~p&"'p=:J "'P=:J '" .p&"'p 5. ~TH

3A3, +S, "'p/q 3A4, +S, "'p/q 3T102, +S,p&"'p/p,p/q 3,1, - =:J

4, 2, -

=:J

Of the law of contradiction, Aristotle wrote, it is the "indemonstrable [principle] which must be known if one is to know anything at all", and it is "the principle upon which all men base their proofs."l The reader may snort with iconoclastic glee to discover the principle demonstrated in PLT, and at a rather late number at that. (In the present system, the law could not have been proved prior to the demonstration of the law of reductio ad absurdum or a similar negation law.) Some thinkers have even gone so far as to insist that LNC cannot serve as an axiom at all. This contention, however, is incorrect, as will be proved by the actual construction of two systems (P2 m1n and PILNC) where LNC is an axiom. It seems to the author that there is good reason for stifling 1

Aristotle, Metaphysics, BI, 995b78.

104

STANDARD SYSTEMS WITH NEGATION

the snort, since he believes that Aristotle's position is as sound as his adversaries', if it is rightly understood. Three final deductions are presented prior to the proof of the deduction theorem. 3T118

p-=:Jq-=:Jq-=:J .pvq(-F,

-H, -MIN)

1. I--P-=:J. -pvq-=:J:-pvq-=:Jq-=:J. -P-=:Jq 3T3, +S, -pip, -pvq/q, q/r 2. I--P-=:J. -pvq 3A6, +S, -pip

1,2,--=:J

3.I--pvq-=:Jq-=:J.-P-=:Jq 4. I--P-=:Jq-=:J .pvq 5. I--pvq-=:Jq-=:J .pvq

6. I-TH

3T109a 3,4, +IS 5, SE, +S,P-=:Jq/-pvq

3T119 pvq-=:J .p-=:Jq-=:Jq 1. I-p-=:J .p-=:Jq-=:Jq 2. I-q-=:J .p-=:Jq-=:Jq 3.I-TH

3T6 3Al, +S, q/p,P-=:Jq/q 1,2, *3A8

Thus by 3T1l9 and 3T118, 3All and two applications of modus ponens we have 3T120

pvq= .p-=:Jq-=:Jq

3T121

P-=:Jq-=:Jp-=:Jp(-F,

----

-H, -MIN)

(peirce's Law)

I. I-P-=:Jq-=:JP-=:J:-P-=:J - .p-=:Jq 2. I--P-=:J - .p-=:Jq-=:J:-p-=:J -. -pvq 3. I--P-=:J -. -pvq-=:J:-p-=:J .p&-q

3T95, +S,P-=:Jq/p,p/q

8.I-TH

1,2,3,6,7, +IS

2, SE, -pvq/P-=:Jq

3, SE,p&-q/-. -pvq, -pvq/P-=:Jq 4. I-p&-q-=:JP-=:J:-P-=:J .p&-q-=:J:-P-=:JP 3T2, +S,p&-q/q,p/r, -pip 5.l-p&-q-=:Jp 3A3, -q/q 4,5, --=:J 6. I--P-=:J .p&-q-=:J:-P-=:JP 7.I--p-=:Jp-=:Jp 3T105

This law has the somewhat curious characteristic that, although it contains no ingredient negation functor, it is unprovable in any system without resort to negation laws (unless that system has Peirce's law or a correlate as an axiom). Having got thus far a useful and important metatheorem will be proved for PLT. This result is called The Deduction Theorem. 105

SYSTEMS OF FORMAL LOGIC

3.7 The Deduct jon Theorem l In the comment following the proof of theorem 2T06, the notion of a variant was set out. This concept is of some importance in this section. It will be recalled that B is a variant of A if two (or more) occurrences of the same variables in A are distinct variables in B. Thus wffs which are variants of each other differ not with respect to their logical form, but only (if at all) with respect to their alphabetical form. B does not, of course, differ in any logical respect from A in the order and character of its functors. Thus variants of 3AI, p~ .q~p, are q~ .p~q; r~ .q~r; r~ .t~r: but p~ .q~r; p~ .p~r; p~ .p~p; r~ .p~p are not variants of 3Al since they do not conform to the above specified conditions. If B is a variant of A, then A is a variant of B. Likewise, if C is a variant of Band B is a variant of A, then C is a variant of A. Trivially, every wff is its own variant. In view of this last, the following terminology will be instituted: the expression "variant of axiom N (of theorem N)" will mean the same thing as "axiom N (theorem N) or a variant of axiom N (theorem N)". Every variant of an axiom is, of course, a theorem. In fact, if the present axioms of PLT were replaced by variants of them, the theorems would remain the same. Likewise, every variant of a theorem is a theorem, the same deductions being available from it as from the theorem of which it is a variant. Following Church, a variant proof is defined as a finite sequence of wffs such that each is either a variant of an axiom or is inferred from the foregoing wffs by one of the primitive rules of transformation of PLT. A derivation, following earlier instituted terminology, is defined as a finite sequence p of wffs BI, B2 ... B n , each of whose members either is, or follows from, the hypotheses AI, A2, A3, ... , An. Each of the members of the sequence p must be either (1) one of AI, A2, A3, ... , An, or (2) a variant of an axiom or (3) inferred by the rule of modus ponens from previous elements in the derivation Bf and Bg, or (4) inferred by the (primitive) rule of substitution from an earlier element in the derivation, Bg. The four circumstances under which a derivation obtains are referred to in the following as "the four conditions", or more simply, "the con1 The proof of the deduction theorem here given owes much to Church, though the method of presenting derivations is different than his.

106

STANDARD SYSTEMS WITH NEGATION

ditions". In the case of condition (4), a restriction must be carefully observed in order to avoid invalidity: the restriction is as follows: When substitution is carried out the variable for which substitution is made must not be one of the variables occurring in any of the hypotheses. For though '" p results by substitution of '" p for p in the wil p, it is not in general the case that '" p follows from the hypothesis p. The importance of the variant emerges at this point. Suppose q is a variable occurring in one or more of the hypotheses of a derivation. By virtue of its so appearing we cannot substitute for q in expressions occurring in the ensuing deduction. Inspection of the axioms reveals the devastation this restriction would reck if it were not for the device of the variant which can be inserted in the proof in place of the axiom when necessary to observe the restriction. Thus, when q occurs in the hypotheses and when an axiom of the form p ~ . q ~ p is needed for substitution, it is possible to write p ~ . r ~ p, or, if p also occurs in the hypotheses, s~ .r~s, since these are variants of 3A1. Hence it is possible to observe the restriction in the strictest sense, thus escaping invalidity consequent on its violation, and at the same time to preserve the requisite freedom of substitution on axioms (and theorems). If Bn is the final member of the sequence of wffs ~, then the derivation is said to be a derivation ofBn from the hypotheses Al, A2, A3 , .•. , An. As before, the expression

is defined to mean: there is a derivation of Bn on the hypotheses A!, A2, A3, ... , An. The following illustrates the concept of derivation: the metalogical form to be proved is p~q, q~r,p~r

1. 2. 3. 4.

5.

p~q q~r

p q r

hypothesis hypothesis hypothesis 1,3, - ~ 2,4, -~

This derivation shows that the wff r is derivable from the hypotheses p ~ q, q ~ r, and p. Step 1 is in the series by virtue of condition (1) above,

107

SYSTEMS OF FORMAL LOGIC

i.e., it is a hypothesis of the sequence. Steps 2 and 3 are in the sequence for the same reason. Step 4 is in the sequence by virtue of condition (3), i.e., by modus ponens, from steps 1 and 3; step 5 is in the sequence by virtue of the same condition, applying steps 2 and 4. Thus we have shown that the metalogical expression p'::)q, q'::)r, pl-r is justified. (This expression is classed as metalogical because it asserts a property of the object language, namely that from the expressionsp'::)q, q'::)r, andp, r is derivable in the object language. It will be noted that this expression has the same form as the rules of the system. Hence it could be alternatively conceived of as a derived rule. Rules, of course, are expressed in the metalanguage.) Now a proof of the deduction theorem may be given. This proof is on a more advanced level and the reader may omit it, moving directly to Section 3.71, "Proof by the method of the deduction theorem". The deduction theorem: If AI, A2, A3, ... , Anl-B, then AI, A2, As, ... , An-II-An'::)B is read, "If there is a proof of B from the hypotheses AI, A2, A3, ... , An, then there is a proof of An'::)B from the hypotheses AI, A2, As, ... , An-I." The antecedent of the deduction theorem gives us the assumption that a derivation ofB can be constructed on the hypotheses AI, A2, As, ... , An. The problem is to show that if this is the case, it is also the case that a derivation of An'::) B can be constructed from the same hypotheses, less An. On the basis of the preceding analysis of derivations, we know that if AI, A2, As, ... , Anl-B, then there is a sequence p of wffs BI, B2, B3, ... , Bn, (Bn being the same as B), such that each member of p is a member of Pby virtue of one of the four conditions set out above. Knowing this, we construct the sequence, S, of wffs: An'::)BI, An '::) B2, An '::) Bs, ... , An'::) Bn. (The n subscript to A is, of course, the step number of the last hypothesis in the sequence of hypotheses of the derivation. The n subscript to B is the number of the step at which B is derived.) Ifit is possible to supply additional wffs in the sequence, S, such that the resultant of this process is a derivation of An '::) Bn, on the hypotheses AI, A2, As, ... , A-n, then the problem set in the preceding paragraph is solved, since the sequence will be a derivation of precisely the form required. Thus it will be shown that - and how - such additional wffs may be supplied. (The reader should keep in mind that the desired result, An'::) Bn, is the same as An'::) B.)

108

STANDARD SYSTEMS WITH NEGATION

If, for some specified w./J, An:::lB n, n subscript to B is greater than 1, there are five and only five cases which can arise consistent with the conditions set out before. Case 1. Bn is one of the hypotheses, A!, A2, A3, ... , An-I. If Bn is, e.g., A3 then we have that A3:::l . An:::l Bn, which, since A3 is the same as Bn, is the same as A3:::l . An:::l A3. In order to derive An:::l Bn, we supply, in the sequence S, a sequence of four steps before it; An:::lBn being the fifth step. The steps are as follows: first step, supply 3A 1 or an appropriate variant; second step, substitute An for the second variable of step 1 by the primitive rule of substitution; third step, substitute A3 for the first variable and Bn for the third variable in step 2, (this gives a substitution on 3Al or a variant, since A3 and Bn are the same expression); fourth step, supply the expression A3, an hypothesis; fifth step, An:::lBn may be inferred from steps three and four by modus ponens. Case 2. Bn is the same wff as An. If Bn is the same wff as An, then the expression An:::l Bn is the same as An:::l An. Hence An:::l Bn is derivable in six steps by a variant of the proof of 3TI, the five steps leading up to this result being inserted in the sequence, S, before An:::l Bn. Case 3. Bn is a variant of an axiom. In this case we have that Bn:::l . An:::l Bn as in case 1. Thus there are four steps to be inserted before An:::l Bn in the sequence S as in case 1. Step I: a variant of 3Al; step 2: substitute An for the second variable in step 1; step 3: substitute Bn for the first and third variables in step 2; step 4: supply the variant of the axiom Bn. An:::l Bn then follows by modus ponens. Case 4. Bn is inferred by substitution of an earlier derived element on the proof, call it Bg (the necessary restriction being observed). Bg is a member of the sequence p, BI, B2, B3, ... , Bn. Hence An:::lBg is a member of the sub-sequence S. But if this is the case, no insertions need be made in the sub-sequence S in order to derive An:::l Bn, since exactly the same substitution by means of which Bn is inferred from Bg is sufficient to make the inference from An:::lBg to An:::lB n. Case 5. Bn is inferred by modus ponens from earlier derived elements in the proof, Bf and Bg. Bf, the major premise, will have the form Bg:::l Bn. If Bf did not have this form, Bn could not be inferred from Bf and the minor premise Bg, by modus ponens. Now unless Bf and Bg are themselves inferred by operations of modus ponens (in which case the present argument applies to them as well), we know that An:::l Bf and An:::l Bg, 109

SYSTEMS OF FORMAL LOGIC

by the previous arguments. To derive An::>Bn, in this case, the following five steps are supplied in the sequence S. Step 1: supply a variant of 3A2; step 2: for the first, fourth and sixth variables of step 1 substitute An; step 3: for the second and fifth variables of step 2 substitute Bg; step 4: for the third and seventh variables of step 3 substitute Bn: this gives the expression An::>. Bg ::> Bn::> : An::> Bg ::> . An::> Bn. Inspection of this result discloses that the consequent of the antecedent is the form which the major premise, Bf, of the pertinent modus ponens operation, has. Hence step 4 can be rewritten: An::> Bf::> . An::> Bg ::> . An::> Bn; step 5 : supply An::> Bg ::> . An::> Bn, which may be inferred from step 4 and the minor premise An::>Bf, (which is already a member of the sequence S), by modus ponens; step 6 is An::>Bn, the desired result, which follows by modus ponens from step 5 and the minor premise An::>Bg already in the sequence S. When n is equal to 1 we have as an immediate consequence of the above considerations the following correlative form of the deduction theorem: If An~ B then An::> B Thus the deduction theorem is proved for PLT.

3.71 Proof by the method of the Deduction Theorem. By virtue of the proof of the deduction theorem the reader may be introduced to a new style of proof, a style which may be described as proof by the method of the deduction theorem. It should be carefully noted that this method as presented is not independent of the method of axiomatic proof, since it makes explicit reference to axioms 3Al and 3A2 as well as to 3Tl. By the deduction theorem AI, ... , An-I~An::>B may be inferred from AI, ... , An-I, An~B. To illustrate: if we have p::>q, q::>r,p~r we may pass, by the deduction theorem, to p::>q, q::>r~p::>r. Keeping to the illustration, it can be seen that there is no need to terminate applications of the deduction theorem at this point. If we submit the result of the first application of the theorem, p::> q, q::> r~p::> r to yet another application of the deduction theorem, the result is p::>q~q::>r::> .p::>r. It takes little imagination to see that one further application of the same theorem to this result will give ~p::>q::> .q::>r::> .p::>r, the second syllogistic law. The first syllogistic law is available in an analogous way. 110

STANDARD SYSTEMS WITH NEGATION

Now it is quite easy to give a verification of the result One merely sets up the following hypothetical proof:

p~q, q~r,

p~r.

1.

p~q

2.

q~r

hypothesis hypothesis hypothesis 1,3, - ~ 2,4, -~

3. p 4. q 5. r

which shows that r is indeed derivable from the hypotheses p~q, q~r, and p; which is, of course, precisely what p~q, q~r, p~r means. Hence, if such a derivation may be constructed, the corresponding form ... ~--­ is verified, where everything to the left of the assertion sign is an hypothesis, and everything to the right of the sign is verifiably derivable from the elements to its left. If, then, we can establish an expression of the form ... ~---, we can establish a theorem of the object-language by successive applications of the deduction theorem. The following is a proof of the second syllogistic law. ("hyp" abbreviates "hypothesis" here, and hereafter, in the justification column.)

3T3 DI

1. 2. 3. 4.

p~q~ .q~r~ .p~r

1. p~q 2. q~r 3. p 4. q 5. r p~q, q~r,p~r p~q, q~r~p~r p~q~q~r~ .p~r

~TH

hyp hyp hyp 1,3, 2,4, -

~

~

Dl I,DT 2,DT 3,DT

When the expression "Dn" is found in the justification column of a proof by the deduction theorem, it is to be read, "from the result of derivation number n." "DT" means, "by the deduction theorem". As a matter of fact, any proof derivable by the full axiomatic method of PLT is derivable by the method of the deduction theorem, and usually the proof will be simpler. However, instead of regarding such proofs as belonging to a distinct system, it is useful to regard them as alternative 111

SYSTEMS OF FORMAL LOGIC

proofs of the same theorems as those proved by the full axiomatic method,

since the deduction theorem is a metatheorem of PLT. For the sake of illustration, five more proofs will be presented using the method of the deduction theorem. 3T4 D2

p:::J .q:::Jr:::J:q:::J .p:::Jr

1. p:::J .q:::Jr 2. 3. 4. 5.

q p q:::Jr r

1. p:::J . q:::J r, q,p~r 2. p:::J . q:::J r, q~p:::Jr 3. p:::J .q:::Jr~q:::J .p:::Jr

4.

~TH

3T30 D3

1.

3.

1. p:::J .q:::Jr 2. p&q 3. p&q:::Jp 4.p 5. p&q:::Jq 6. q 7. q:::Jr 8. r .q:::Jr~p&q:::Jr

~TH

3T45 D4

-:::J

D2 I,DT 2,DT 3,DT

hyp hyp 3A3 3,2, 3A4 5,2, 1,4, 7,6,

-:::J

-:::J -:::J -:::J

D3 I,DT 2,DT

q:::Jr:::J .pvq:::Jr:::J .p:::Jr

1. q:::Jr 2. 3. 4. 5. 6.

112

-:::J

p:::J .q:::Jr:::J:p&q:::Jr

p:::J.q:::Jr,p&q~r

2. p:::J

hyp hyp hyp 1,3, 4,2,

pvq:::Jr p p:::Jpvq pvq r

hyp hyp hyp 3A6 4,3, 2,5,

-:::J -:::J

STANDARD SYSTEMS WITH NEGATION

1. q-;::)r,pvq-;::)r,p'rr

D4

2. q-;::)r,pvq-;::)r'rp-;::)r

1, DT 2,DT 3,DT

3. q-;::)r'rp v q-;::)r-;::) .p-;::)r 4. 'rTH

3T91

"'p-;::) .p-;::)q

Though it is easy to prove this result from the two hypotheses '" p, p, and the deduction theorem, a shorter proof is available by beginning with the single hypothesis, '" p. D5

1. "'p

2. r-;::) .q-;::)r 3. 4. 5. 6. 7.

"'p-;::) .q. -;::) "'p "'p-;::). "'q-;::) "'p "'q-;::) "'P "'q-;::) "'r-;::) .r-;::)q "'q-;::) "'p-;::) .p-;::)q 8. p-;::)q

1. "'p'rp-;::)q 2. TH

hyp Variant of 3AI (cf condition 2, p. 106) 2, + S, '" plr 3, +S, "'qlq 4,1, --;::) Variant of 3Al2 6, +S,plr

7,5, --;::) D5 1, DT

The above is a variant proof from hypotheses using the primitive rule of substitution. The following two proofs are included to illustrate three points: (1) The use of variants in derivations (cf. step 6 of D6 and step 4 of D7). Here, as in D5, variants of axioms and theorems previously proved are introduced in order to avoid violation of the restriction on the rule of substitution forbidding substitution upon variables having occurrence among the hypotheses. The use of a variant of an axiom or a theorem is symbolized in the justification column by prefixing the axiom - or theorem - number of the formula by the letter "V". Thus, if a formula B, introduced into a derivation, is a variant of axiom 3An, we write in the justification column the expression V3An. An analogous usage is instituted in the case of variants of theorems, e.g., V3Tn. (2) The use of simultaneous substitution (cf. step 9 of D6). In the proof of the deduction theorem only simple substitutions were permitted, since 113

SYSTEMS OF FORMAL LOGIC

the proof was carried out from the primitive basis of the system given in the four conditions (p. 106). However, considerations analogous with those given in 2.61 (p. 51f) and prescribed in exercise 1 (p. 74), are sufficient to show that simultaneous substitution is validity-preserving when used as an abbreviative rule in derivations. The restriction prohibiting substitution on variables occurring in the hypotheses of a derivation must, of course, always be observed. (3) The use of logical operations in sequences employing the deduction theorem. It is permissible to treat elements to the right of the assertion sign, 1-, in steps obtained by the deduction theorem as premises for further deductions. Thus, if we have that AI-B

and that C is logically deducible from B, we may obtain AI-C.

See, for example, step 3 in the deduction theorem sequence following derivation D7. (Can the reader supply a metalogical justification for this procedure?) 3Tl22 D6

1. p~q 2. p~r 3. p 4. q 5. r 6. r~ '" "'r

7. "''''r 8. s~. "'t~ '" .s~t 9. q~. '" "'r~ '" .q~ "'r 10. "''''r~'''.q~'''r 11. '" .q~ "'r 1. p~q,p~r,pl-'" .q~ "'r 2. p~q,p~rl-p~ '" .q"'r 3. p~ql-p~r~ .p~ '" .q~ "'r 4.I-TH 3Tl23

114

hyp hyp hyp 1,3, -

2,

~

3,-~

V3T98 6,5, V3T98

~

8, + S, qls, '" rlt 9,4, - ~ 10, 7, ~ D6 1, DT 2, DT 3, DT

ST ANDARD SYSTEMS WITH NEGATION

The deduction theorem proof of 3Tl23 makes use of the technique of indirect proof We assume as hypotheses not only the antecedent and the antecedent of the consequent of the theorem, but also the denial, - r, of the very formula, r, which we wish to obtain. We then show that the hypotheses - including - r - allow the derivation of r itself. Thus, from the assumption that - r, in combination with the other hypotheses, its contradictory is deducible. Hence - r is inconsistent with the other hypotheses. D7

1. p-=:Jq-=:Jr

2. 3. 4. 5. 6. 7. 8.

p-=:Jr

-r S-=:Jt-=:J. -t-=:J-S p-=:Jr-=:J. -r-=:J-p

-r-=:J-p -p -P-=:J .p-=:Jq 9. p-=:Jq

10. r

hyp hyp hyp VT95 4. + S, pis, rlt

5,2,

--=:J

6,3, --=:J 3T91

8,7,

--=:J

9,1,

--=:J

1. p-=:Jq-=:Jr,p-=:Jr, -rf-r

D7

2. 3. 4.

I,DT

p-=:Jq-=:Jr,p-=:Jr~-r-=:Jr p-=:Jq-=:Jr,p-=:Jr~-s-=:Js-=:Js

VTlOI

p-=:Jq-=:Jr,p-=:Jr~-r-=:Jr-=:Jr

3, +S,rls 4,2, --=:J 5,DT 6,DT

5. p-=:Jq-=:Jr, p-=:Jr~r 6. p-=:Jq-=:Jr~p-=:Jr-=:Jr 7. TH

The reader's attention is directed to steps 3 through 5 of the deduction theorem sequence. How are these steps to be understood? N.B. The system PLT' is the next system to be developed. A further abbreviative device is here introduced in order to facilitate derivations in P LT ,. We know that any step obtained by substitution on an axiom or theorem is also obtainable from substitution on a variant of that axiom or theorem (cf. 3.7, pp. 106ft} Knowing this, it is tedious to write out the variant itself in order to obtain the requisite substitution in the ensuing steps. From now on it will be permissible to abbreviate as follows: in order to obtain, e.g., the law P-=:J. -P-=:J - .p-=:Jp by sub115

SYSTEMS OF FORMAL LOGIC

stitution in a derivation which has occurrence of p and q in its hypotheses, we will merely state the formula, justifying it by the citation "V3T98, +S,p/q". Here the occurrence of the prefix "V" indicates that we intend to substitute on a variant of 3T98 in order to avoid invalidity; none the less, the citation of actual substitutions refers to the original form of the law; this is a legitimate abbreviation since it is always possible to write down a full variant of the axiom or theorem as an intervening step and then to obtain the desired formula from it by substitutions. (In this case the variant would be p-;:) .t-;:) '" .p-;:)t). No formula may be substituted upon in this way unless it is a law. 3.8 The System PLT' 3.81 As the final task of the chapter, a system PLT' will be formulated. This system, due primarily to Lukasiewicz, is of considerable importance in succeeding sections of this work.l Introductory to the accomplishment of this task the reader is requested to inspect the following theorems of PLT: 3Tl20

pvq== .p-;:)q-;:)q

3Tl14

p&q==. '" .p-;:) "'q

and the following truth-tables: p

v

q

p

-;:)

q

-;:)

q

T T F F

T T T F T T FIF

T T F F

T F T T

T F T F

T T T F

T F T F

p

& T F F F

T T F F

q

T F T F

p

T F F F

-;:)

q

TFF(T) TTT(F) F T F (T) F T T (F)

See his presentation in Untersuchung ilber den Aussagenkalkul (with Alfred Tarski). The system was originally derived by simplification of the system of Frege given on pp. 125f.

1

116

STANDARD SYSTEMS WITH NEGATION

P - q T T T T F F F F T F T F

P ::::> q ::::> T T T T F F T F F T F F T T T T F T F F

q ::::> p

F T T F F T T T F F F T

T T

F F

The tables show that the right-hand formulae are equivalent to the corresponding left-hand formulae. The system PLT' has the following primitive basis.

3.82 Formation rules. The system possesses four improper symbols.

3.821 (1)

[

(2)

]

(3) (4)

::::>

3.822 The system possesses an infinite list of propositional variables which are proper symbols. 3.823 An expression is a wff of PLT' only under the following circumstances. 3.8231 A proper symbol standing alone is wf 3.8232 (a) If A and Bare wf then [A::::> B] is wf (b) If A is wI, then '" A is wf (The point system of abbreviating brackets is adopted.)

3.83 Transformation rules. 3.831 Modus ponens. 3.832 Substitution. 3.84 Axioms ofPLT" PLT,Al p::::> .q::::>p PLT,A2 p::::> .q::::>r::::>:p::::>q::::> .p::::>r P LT,A3 "'p::::> "'q::::> .q::::>p 3.85 Definitions OfPLT" 3.851 rA=dJ, "A is a theorem of PLT'''' 3.852 The expressions in the left column are abbreviations for those in the right hand column: 117

SYSTEMS OF FORMAL LOGIC

Dl

pvq

D2 p&q D3 p=q (See the truth-tables in 3.81.)

-.p-:::J-q - .p-:::Jq-:::J - .q-:::Jp

3.86 Deductions in PLT" The deductions in this system will be restricted to proofs of the axioms of PLT plus proofs of four theorems for use in Chapter 5. The first form given will be a proper theorem ofPLT', the second, (in parentheses), a reduction by the definitions of PLT' of these theorems to forms of the axioms of PLT. The numbering will correspond to that of the axioms of PLT. Hence PLT,T3 will correspond to 3A3; PLT,T4 to 3A4, etc. l (In the deduction of the following we will be at liberty to use any of the theorems 3T1 through 3T15, and 3T91 through 3T104, since these can be proved using none but the axioms of PLT'. Where convenient, the method of the deduction theorem will be used. The original index numbers for the system PLT will be used for convenience.)

PLT,T3

- .p-:::J -q-:::J:p

(T124)

(p&q-:::Jp), i.e., PLTA3 or 3A3 1. I--P-:::J .q-:::J -p 2. I-q-:::J -P-:::J .p-:::J -q 3.I--p-:::J.P-:::J-q(-:::J) 4. I-TH

PLT,Al, +S, -pip 3T96, +S, qlp,p/q 1,2, +IS 3,3T97, +S,P-:::J -q/q,

PLT,T4

(T125)

- .p-:::J -q-:::J:q

--:::J

(p&q-:::Jq), i.e., PLTA4 or 3A4 1. I--q-:::J .p-:::J -q(-:::J) PLT,Al, +S, -q/p,p/q 2. I-TH 1,3T97, +S, q/p,P-:::J -q/q, --:::J

p-:::J .q-:::J - .p-:::J -q

(T126)

(p-:::J .q-:::J .p&q)

l.p 2. q 3. P-:::J-q

hyp hyp hyp

1 It is for this reason that the set of theorems of PLT' begins with PLT,T3 rather than with PLT,Tl.

118

STANDARD SYSTEMS WITH NEGATION

4. '" q 5. r:::::J. "'S:::::J '" .r:::::Js 6. r:::::J. "'r:::::J '" .r:::::Jr 7. q:::::J. "'q:::::J '" .q:::::Jq 8. "'q:::::J '" .q:::::Jq 9. '" .q:::::Jq Thus: 1. p, q,P:::::J "'q~'" .q:::::Jq 2. p, q~p:::::J "'q:::::J '" .q:::::Jq (:::::J) 3. p, q~q:::::Jq:::::J '" .p:::::J "'q 4. p,q~q:::::Jq 5. p,q~"'.p:::::J"'q 6. p~q:::::J"'.p:::::J"'q 7. TH p:::::J .p:::::Jq:::::Jq

3,1, -:::::J V3T98 5, +S, r/s 6, +S,q/r 7,2, -:::::J 8,4, -:::::J DLT,5 1, DT 2,3T96, +S,P:::::J "'q/p, q:::::Jq/q, -:::::J V3T2, + S, q/p 3,4, -:::::J 5,DT 6,DT (Tl27)

(p:::::J .pvq)

1.

3T6

~TH

q:::::J.p:::::Jq:::::Jq

(Tl28)

(q:::::J.pvq)

1.

3A1, +S, q/p, P:::::Jq/q

~TH

PLT,T8 DLT,8

p:::::Jr:::::J . q:::::J r:::::J .p:::::Jq:::::Jq:::::Jr (p:::::Jr:::::J .q:::::Jr:::::J .pvq:::::Jr) 1. p:::::Jr 2. q:::::Jr 3. p:::::Jq:::::Jq 4. '" r 5. q:::::Jr:::::J. "'r:::::J "'q 6. "'r:::::J "'q 7. "'q 8. p:::::Jr:::::J. "'r:::::J "'p 9. "'r:::::J "'p 10. "'p 11. "'P:::::J .p:::::Jq

(Tl29)

hyp hyp hyp hyp V3T95, + S, q/p, r/q 5,2, -:::::J 6,4, -:::::J V3T95, + S, r/q 8,1, -:::::J 9,4, -:::::J 3T91 119

SYSTEMS OF FORMAL LOGIC

12. p'=:J q

11, 10, -

13. q 14. q=:J. "'q=:Jr

3, 12, -

15. "'q=:Jr 16. r

Thus: 1. p=:Jr, q=:Jr,p=:Jq=:Jq, "'r, ~r 2. p=:Jr, q=:Jr,p=:Jq=:Jq~"'r=:Jr 3. p =:Jr, q =:Jr, p =:Jq =:Jq~ '" r =:Jr =:Jr 4. p =:Jr, q=:Jr, p =:Jq=:Jq~r 5. p =:Jr, q =:Jr~p =:Jq =:Jq =:Jr 6. p=:Jr~q=:Jr=:J .p=:Jq=:Jq=:Jr

7.

~TH

'" .p=:Jq=:J '" .q=:Jp=:J:p=:Jq

=:J =:J

V3T94, +S, q/p, r/q 14, 13, - =:J 15, 7, - =:J

DLT,8 1, DT V3TlOl, +S, rip 3,2, - =:J 4,DT 5,DT 6,DT (Tl30)

(p=q=:J .p=:Jq)

1. ~'" .p=:Jq=:J:p=:Jq=:J '" .q=:Jp (=:J) 2. '" .p=:Jq=:J '" .q=:Jp=:J :p=:Jq

3T91, +S,P=:Jq/p, '" .q=:Jp/q 1,3T97, +S,P=:Jq/p, p=:Jq=:J '" .P=:Jq/q, - =:J

PLT,TlO

(Tl31)

'" .p=:Jq=:J '" .q=:Jp=:J:q=:Jp (p=q=:J .q=:Jp)

PLT,Al, +S, '" .q=:Jp/p, P=:Jq/q 1,3T97, +S, q=:Jp/p, p=:Jq=:J '" .q=:Jp/q, - =:J PLT,Tll

p=:Jq=:J .q=:Jp=:J '" .p=:Jq=:J '" .q=:Jp (p=:Jq=:J .q=:Jp=:J .p=q)

1.

~q=:Jp=:J

(Tl32)

'" '" .q=:Jp

2. ~P =:J q =:J : '" '" • q =:J P =:J : 3.

~

'" .p=:Jq=:J '" .q=:Jp (=:J) '" '" .q=:Jp=:J :p=:Jq=:J: '" .p=:Jq=:J '" .q=:Jp

120

3T4, +S,P=:Jq/p, '" '" .q=:Jp/q, '" .p=:Jq=:J "'q=:Jp/r, - =:J

STANDARD SYSTEMS WITH NEGATION

4. 't-q:::>p:::> .p:::>q:::> - .p:::>q:::> - .q:::>p (:::»

5. 't-TH

1,3, +IS 3T4, +S, q:::>p/p,p:::>q/q, -.p:::>q:::>-.q:::>p/r,

-:::>

Thus with the aid of the deduction theorem we are able to prove Axioms 3A3 through 3All (when these are reduced, by definition, to expressions having only implication and negation and brackets as functors) as theorems of PLT" The shrewd reader will ask, "What right have we to use the deduction theorem in PLT' when, as a matter of fact, it was proved only as a metatheorem of PLT?!" This is a good question, since it discloses an important fact about the deduction theorem. Inspection of the demonstration of this metatheorem shows that it requires only three laws; (p:::>p; p:::> .q:::>p; and p:::> .q:::>r:::>:p:::>q:::> .p:::>r), and two rules, (modus ponens and substitution). Hence it can be proved as a metatheorem of any axiomatic system in which modus ponens and substitution are primitive (or derived) rules and in which the three requisite laws are among the axioms or theorems. Thus, since PLT' is such a system, the deduction theorem is provable as a metatheorem for PLT" Proofs of four further theorems of PLT' are given below. These theorems are useful for further deductions. PLT,T12 p:::>p 1. 't-p:::> .q:::>p:::>:p:::>q:::> .p:::>p 2. 't-p:::> .q:::>p 3. 't-p:::>q:::> .p:::>p 4. 't-p:::> .q:::>p:::>:p:::>p

5. 't-TH PLT,T13

-p:::> .p:::>q

1. 't--p:::>.-q:::>-p 2. 't--q:::> -p:::> .p:::>q

3. 't-TH

PLT,A2, +S,p/r PLT,AI 1,2, -:::> 3, +S, q:::>p/q 4,2, -:::> PLT,Al, +S, -pip, -q/q PLT,A3, +S,p/q, q/p 1,2, +IS

(First prove lemma 1, p:::> .p:::>q:::>q, as in 2T6; then prove lemma 2, p:::>q:::>. -q:::> -p, exactly as in 3T95 ofPLT.)

121

SYSTEMS OF FORMAL LOGIC

1. 2.

~p:::::>q:::::>q:::::>.

3.

~TH

~p:::::>

(from lemma 1) (from lemma 2), 1,2, +IS

.p:::::>q:::::>q "'q:::::> '" .p:::::>q

(First prove lemma 3, r:::::> s:::::> "'q:::::>q:::::>q, as in 3TlOl.)

• s:::::>

t:::::> . r:::::> t, a variant of 3T3; then lemma 4,

l. p:::::>q 2. '" p:::::> q 3. p:::::>q:::::> "'q:::::> "'p

4. "'q:::::> "'p 5. "'q:::::> "'p:::::>. "'p:::::>q:::::>. "'q:::::>q

6. "'p:::::>q:::::> "'q:::::>q 7. "'q:::::>q 8. "'q:::::>q:::::>q 9. q

1. p:::::>q,

2. 3.

"'p:::::>q~q

p:::::>q~ '" p:::::> q:::::>q ~TH

+S, p:::::> q/p

hyp hyp (from lemma 2)

3,2,

-:::::>

(from lemma 3), "'p/s, q/t

5,4, 6,2,

+ S,

'" q/r,

-:::::> -:::::>

(from lemma 4) 8,7, -:::::>

DLT,15 1, DT 2,DT

3.9 Independence of Functors and Axioms l Returning to the immediate question, it will be shown that: (1) The primitive functor == of P + is not independent. (2) The primitive functors &, v, == of PLT are not independent. (3) The primitive functors of PLT' are independent. A functor is said to be independent if, when the functor is omitted, the truth-table defining it is no longer to be found among the truth-tables of the wffs constructable by use of the remaining functors. If the defining truth-table of a functor 1 The most systematic researches into independence were carried out by E. Post in his monograph The Two-Valued Iterative Systems of Mathematical Logic, 1941.

122

STANDARD SYSTEMS WITH NEGATION

can be constructed using only primitive functors other than the functor in question, that functor is said to be non-independent or not independent. (1) == is non-independent in P +. This follows since the following truthtable is constructable using only the primitive functors of P +: => and &.

p => q &. q => P TTTTTTT TFF F FTT FTTFTFF FTFTFTF Thus the truth-table for equivalence is among the truth-tables of the wffs of P + even if == is suppressed. A question for the reader: are any other functors of P + non-independent?

(2) The primitive functors, &, v, == are non-independent in PLT. Inspection of the truth-tables at the beginning of this section, p. 116, shows that the truth-tables for these functors may be duplicated by tables for wffs having as functors only '" and =>, plus punctuation. Hence these functors are non-independent. (3) The primitive functors =>, "', ofPLT' are independent. If => and '" are independent, then neither can be suppressed without the result that the truth-table for each is not among the truth-tables of the resulting wffs. Since => and", are, besides punctuation, the sole functors ofPLT', it follows that if either is non-independent, its truth-table must result from wffs constructed solely out of the other, plus punctuation. That '" is independent is clear from the fact that from implication alone, it is impossible to construct a wff having a truth-table with none but 'F's' under the main connective, while, of course, with negation we have,

"'.p => p T T T F A proof that no such truth-table can be had with implication alone is available by induction.

123

SYSTEMS OF FORMAL LOGIC

The table for implication is, of course, p

:::>

q

T T F F

T F T T

T F T F

The table shows that an implication holds whenever the consequent is true and whenever the antecedent is false. It is false only when the antecedent is true and the consequent is false. There are three cases.

Case 1. The index column of the above table shows that in the case of an implication having two variables as arguments, the associated table has a mixture of values in the index column, not 'F' exclusively. Hence, no wff in which implication is the main connective and in which the arguments are single propositional variables has an associated truthtable with 'F' exclusively in the index column. Case 2. If the wff is an implication having a variable and a wff other than a variable as arguments then there are two sub-cases: Sub-case a: The variable is the consequent of the implication. If this is so, then the associated truth-table will assign the value 'T' to the implication in each row at which the value 'T' is assigned to the variable, a circumstance which must occur at least once. Sub-case b: The variable is the antecedent of the implication. If this is so, then the associated truth-table will assign the value 'T' to the implication in every row at which the variable is assigned the value 'F', which must occur at least once. Hence no implication having a variable and a wff other than a variable as arguments has an associated truth-table with 'F' exclusively in the index column. Case 3. If the main implication of the wff has two wffs other than variables as arguments then analogous reasoning is applied to each side of the implication: e.g., if the antecedent has two variables as arguments for an implication sign we apply Case I and thus the entire expression is assigned the value 'T' whenever its antecedent has the value 'F', which 124

STANDARD SYSTEMS WITH NEGATION

must be at least once. The same argument applies when the implication has n variables, by successive application of Case 1 and Case 2. Hence there is no wff of PLT' constructed only out of implication and punctuation to which there is correlated a truth-table having none but 'F's' in its index column. Hence '" is independent. The independence of :::> is demonstrable from the fact that, since PLT' has no functors other than :::>, "', [, ], and since no functor of PLT' but :::> allows the construction of wffs of lengths greater than 1, if :::> were suppressed, no wff of a length greater than 1 could be constructed in PLT" A set of primitive connectives for the propositional calculus is said to be complete if all truth-tables of two or more columns are found among the truth-tables of the wffs which result from the formation rules of the system. The following will not be proved, since they are beyond the elementary level: they are well known results. (1) '" and :::> (plus punctuation) constitute a complete set of independent functors for the propositional calculus. (2) '" and v (plus punctuation) constitute a complete set of independent functors for the APC. (3) '" and & (plus punctuation) constitute a complete set of independent functors for the APC. A set of axioms is described as independent if there is no axiom in the set which may be deduced as a theorem when that axiom is suppressed. Given this definition, it does not follow that the deduction of implicativenegative forms of 3A3-3All as theorems ofPLT' shows the set 3AI-3A12 of PLT to be non-independent. The deductions in fact show that these axioms are redundant, given the definitions, p&q=df· '" .p:::> "'q p V q=dj.p:::> q:::> q p=q=dj. '" .p:::>q:::> '" .q:::>p

which is a very different matter, since these definitions are not part ot the primitive basis of PLT. However, consider the following system of Frege. This system will 125

SYSTEMS OF FORMAL LOGIC

be called PLTF.1lt has the same definitions, formations and transformation rules as PLT' and possesses the following six axioms. PLTF Al PLTFA2 PLTFA3 PLTFA4 PLTFA5 PLTFA6

p~ .q~p p~ .q~r~:p~q~ .p~r

p~ .q~r~:q~ .p~r

p~q~."'"'q~"'"'p

"'"''''"'p~p

P~"'"''''"'P

It can be seen that, while PLTFAI and PLTF A2 are axioms of PLT', PLTF A3 through PLTF A6 are theorems of PLT'. Two things suggest themselves. First, inspection of the axiom set of PLTF suggests that it may be equivalent with PLT'. (We already know, that PLT' contains PLTF since all the axioms of the latter are either axioms or theorems of the former. The present conjecture is to the effect that PLTF also contains PLT', and that the systems are, consequently, equivalent.) Second, one is tempted to conjecture that at least one of the axioms of PLTF is nonindependent. Both of these conjectures are in fact correct. This will be shown by a proof of the expression "'"'p~ "'"'q~ .q~p as a theorem of PLTF. 1. We know from inspection of the proofs of 3T1, 3T2, and 3T3 that the law of identity and the two syllogistic laws for the APe are provable by resort only to PLTF Al and PLTF A2 and the rules of PLTF. Hence it will be permissible to use these laws: they will be numbered PLTF n, (law of identity); PLTF T2, (first syllogistic law); and PLTF T3, (second syllogistic law).

2.

"'"'p~ "'"'q~ .q~p

is proved thus:

1. I-"'"'p~ "'"'q~. "'"' "'"'q~ "'"' "'"'p

PLTF A4, +S, "'"'pip, "'"'qlq

2. I- "'"' "'"'P ~ P ~ . "'"' "'"' q ~ "'"' "'"' p ~ . "'"''''"'q~p

PLTF T2, + S, "'"' "'"' plq, plr, "'"' "'"'qlp

1 This is Frege's propositional calculus, published in the Begriffschift (1879). In that it is the first formulation of the propositional calculus as a full axiomatic system it has considerable historical significance. Even more remarkable, this first system is complete and consistent. This is easy to prove, since PLTF is equivalent with PLT' and the latter system is proved complete and consistent in Chapter 5. Thus if PLT' is complete and consistent then so is PLTF. The symbolization of the axiom set is a modern one replacing Frege's own difficult symbolism.

126

ST ANDARD SYSTEMS WITH NEGATION

3.

~,. . "'''p~p

PLTF A5

'" "'q~ '" "'p~. '" "'q~p 5. ~"'p~ "'q~. '" "'q~ '" "'p~: '" ""q~ '" "'p~. '" ""q~p~: "'p~ "'q~. '" "'q~p

4.

~

2,

3,-~

PLTF T3, +S, "'p~ "'q/p,

'" "'q~p/r, '" "'q~ '" "'p/q '" "'p~. '" "'q~p~: "'q~. '" "'q~p

6.

~'" "'q~

7.

~ "'p~ "'q~:'" "'q~p

"'p~

8. ~q~ '" "'q~ . '" "'q~p~ 9. ~q~"''''q 10. ~'" "'q~p~ .q~p

1l.

~"'p~ "'q~.

'"

'"

.q~p

5,1, - ~ 6, 4,-~ PLTF T3, + S, q/p,p/r, '" "'q/q PLTF A6 8, 9,-~

"'q~p~:

"'q~p~ .q~p~:

"'p~ "'q~ .q~p

PLTF T3, +S,

'" "'q~p/q, 12.

"'q/q,

~'" "'q~p~ .q~p~: "'p~ ""q~ .q~p

13.

"'p~

q~p/r

~TH

11,7,-~

12,10, -

~

This demonstration shows that axiom PLT,A3 is provable as a theorem of PLTF and thus that the two systems are equivalent. We know also that the third axiom of PLTF may be suppressed and proved as a theorem of the system. The proof would be similar to that of 3T4. Thus, PLTF A3 is redundant, i.e., non-independent. It will be shown in Chapter 5 that both PLT and PLT' are complete and consistent. EXERCISES

1 Prove as theorems of PLT, without using the deduction theorem:

l. '" .pvq~:"'pv "'q 2. '" .p~ q~: '" p~ "'q 3. '" .p~q~:q~p 4. p= "'q= .q= "'p 5. pvq~pv .pvq~q 6. The full Law of De Morgan 7. p&"'p~q 127

SYSTEMS OF FORMAL LOGIC

2 Using the deduction theorem, prove as theorems:

A. ofPLT 1. "'p::::;,q::::;,. "'p::::;, "'q::::;,p 2. ",.p&"'p 3. pv "'p

B. ofPLT'

1. p::::;,q::::;, .p::::;, "'q::::;, "'p 2. "'p::::;,q::::;, .p::::;,q::::;,q 3. p::::;, "'q::::;,. '" .p::::;,q In carrying out the proofs of this number do not use theorems which have not been proved by the method of the deduction theorem until you have proved them by that method. 3 Consider the following set of axioms "'.pvpv:p "'qv.pvq "'.pvqv:qvp "':pv .qvrv:qv .pvr "':"'qvrv '" .pvqv .pvr

the set given in PM expressed in the primitive constants of the system, '" and v. (a) Show that", and v are both independent functors. (b) In a system with '" and v as primitive, what correlative rule would replace modus ponens, serving the same function? (c) Translate the set, using an additional functor and a definition in such a way that negation does not appear in the axioms. 4 Does the fact that '" p::::;, '" q::::;, . q::::;, p is provable in PLTF give reason for believing that PLTFA4-PLTFA6 are non-independent? 5 Using the axiom set developed in 3c, the rules of substitution and modus ponens, plus the following definition: (definition of material implication) p::::;,q=dj. "'pvq

prove: (a) the equivalence of this system, PPM, with PLT" (b) the equivalence of PPM with PLTF.

128

STANDARD SYSTEMS WITH NEGATION

6 In 1926 Paul Bernays published his discovery that the fourth axiom of PM was non-independent. Reproduce this work by showing that the fourth axiom is deducible as a theorem in a system PPM-4 having the other four axioms of PPM as its axioms.

129

CHAPTER 4

THE SYSTEM PND SYSTEMS OF NATURAL DEDUCTION

4.1

Summary

In this chapter a technique of deduction which differs from that of P + and PLT is presented. The system which results from the application of this method is called PND. The rule technique and the rules of deduction are first presented, then some deductions of theorems proved axiomatically in the preceding chapters are carried out using the new technique. A decision procedure is developed for PND. Finally, a reduction of PND, called PND (Red), is presented. 4.2 The Bases of the System PND

The system of propositional logic presented below is based on methods of deduction due to Gerhard Gentzenl, Stanislaw Jaskowski 2, and Fredric Fitch. 3 Systems based on such methods are called, collectively, "systems of natural deduction". The subscript ND in the rubric PND abbreviates "natural deduction". 4.3 Proof and Derivation Techniques in PND Proofs in PND differ in several respects from those in the corresponding axiom system PLT. (1) PND employs only derivations. 4 While it, like the foregoing axiom 1 Gerhard Gentzen, 'Untersuchungen tiber das logische SchlieBen', In the Mathematische Zeitschrift 39 (1934) 176-210,405-431. The most important passages: pp. 183-190. 2 Stanislaw Jaskowski, 'On the Rules of Suppositions in Formal Logic', Studia Logica, No.1, Warsaw, 1934. 3 Fredric Fitch, Symbolic Logic, Ronald, New York, 1952. 4 The term "derivation as used here corresponds, in part, with the use given in Section 3.7. A derivation is a deduction from hypotheses. A more specialized use of "derivation" is instituted in 4.54.

130

SYSTEMS OF NATURAL DEDUCTION

systems, is formulated in terms of rules of formation, rules of transformation and definitions, it possesses no axioms. (2) In P+ and PLT the axioms of the systems constitute the deductive starting points from which theorems are deduced in accordance with the rules of the system. In place of axioms, proofs in PND begin with hypotheses from which conclusions are derived through the application of the rules of PND upon the hypotheses. It is to be understood that an hypothesis of a derivation in PND may be any expression framed in accordance with the rules of formation of the system, i.e., it may be a logical truth, a contingency, or even a contradiction. This being the case, an hypothesis of a proof in PND is not asserted, but assumed. It is not necessarily a thesis of the system. (3) Likewise, while every element in a valid deduction in P + and PLT is itself a theorem, this does not hold of derivations in PND. In fact, in most derivations in this system only the final step of the deduction is a theorem of the system. In PND a hypothesis or hypotheses, H, gives rise, by application of the transformation rules, to a conclusion, C, said to be true on the hypothesis H. But in most cases the element C is not itself a theorem. For instance we have

as theorem T3 of P +. A corresponding derivation in PND gives the conclusion true on the hypothesis p-:::Jq

But clearly neither the hypothesis nor the conclusion are theorems in the sense earlier defined. As will be shown, one more step in a derivation of PND will give rise to the theorem corresponding in this system to T3 of P +. Hence it is important to keep the distinction between a conclusion of a deduction, which mayor may not be asserted in PND, and a theorem, which is, by definition, asserted in PND, clearly in mind. (4) While most axiom systems have a minimum of primitive transformation rules, P+ and PLT, for instance, having only two, the rules of systems of natural deduction are, comparatively, quite numerous. The system PND has eighteen of such rules, (though this number may be

131

SYSTEMS OF FORMAL LOGIC

reduced), while another well known system, PNDC, has not less than thirty. The necessity for this proliferation of transformation rules will soon become clear. 4.4 Rules of Formation of PND The rules of formation for PND are the same as those for PLT. 4.5 The Structure of Proofs in PND The construction of proofs and derivations in the system is carried out by means of the following rules and conventions: 4.51 The use of "index lines" and proof echelons. (By a "proof" in PND we understand a valid deduction in PND which terminates with an item to the immediate right of a theorem echelon; see below. Bya "derivation" in PND we understand a valid deduction in PND which terminates with an item to the immediate right ofthe main (hypothetical) echelon; see below.) To every proof in the system there corresponds a vertical line running the entire length of the proof, located to the immediate right of the numbers assigned to the steps in the proof. This line is called the theorem echelon. Thus if the proof is six steps long, the configuration of the theorem echelon would be as follows:

1 2 3 4 5 6 TH In this case the only step which belongs uniquely to the theorem echelon is the theorem of the proof, stated to the immediate right of the echelon at step 6'. No expression may be introduced into the theorem echelon except an expression proved as a theorem by application of the rules of the system to the foregoing steps of the deduction (or an expression already proved as a theorem in another proof of the system). 4.52 The main hypothesis of the proof is located on an echelon to the immediate right of the theorem echelon. This vertical line is called the main echelon of the proof. It has the following configuration.

132

SYSTEMS OF NATURAL DEDUCTION

I 2

-

3

""lhypotheSiSbarl

.... main echelon I

4

1

5 6 TH I

theorem echelon

1

As contrasted with the theorem echelon, the main echelon never extends over the entire length of the proof of the theorem, but always terminates at least one step short of the extension of the theorem echelon. Likewise, the main echelon possesses a connected horizontal line, called the hypothesis bar. The hypothesis bar is placed immediately under the step which constitutes the main hypothesis of the proof, thus separating the hypothesis from which the deductions take place from the deductions themselves. The following deduction illustrates these conventions.

p&q

2 q 3 qvr 4 p&q::J.qvr Here step I is the main hypothesis, step 4 the theorem, and steps 2 and 3 the deductions from the main hypothesis which justify the theorem. Following terminology previously introduced, step 3 is called the conclusion of the main echelon, as contrasted with the theorem, step 4. (Rules supplying justification for steps I through 4 will be given in section 4.62.) 4.53 Many deductions in PND involve more hypotheses than the main hypothesis. Indeed, in principle, a proof in PND may have any finite (or denumerably infinite) number of hypotheses. Following Fitch, we call the hypotheses other than the main hypothesis subordinate hypotheses. Each subordinate hypothesis is correlated with a vertical echelon similar to that of the main echelon which usually begins one step number after that of the main hypothesis, and to its right. An echelon correlated with a subordinate hypothesis is called a subordinate echelon. To each echelon of a proof in PND corresponds one and only one hypothesis. Hence the 133

SYSTEMS OF FORMAL LOGIC

first subordinate echelon of a proof in PND usually begins at step 2 of the proof and terminates one step short of the conclusion of the ,main hypothesis, thus: (hypothesis of the main echelon) 2

I (hypothesis of the first subordinate echelon)

3 4 5 (conclusion of the first subordinate echelon) 6 (conclusion of the main echelon) 7 (theorem) The echelon 2-5 is said to be subordinate to the echelon 1-6, while the latter is said to be the superior of the former. If there are more than two hypotheses in the proof, the procedure is precisely analogous. The second sub-(subordinate)-echelon begins one step number after the first, and to its right, generally terminating one step number prior to the termination of the first sub-echelon. If there are three or more sub-(subordinate)hypotheses, they are structured in the proof in exactly the same manner as is the first sub-echelon. The following is a skeleton of a typical proof structure having five hypotheses: Main hypothesis First sub-hypothesis Second sub-hypothesis

I

Third sub-hypothesis

I Fourth sub-hypothesis Fifth sub-hypothesis

Fifth sub-conclusion Fourth sub-conclusion Third sub-conclusion Second sub-conclusion First sub-conclusion Main conclusion

Theorem

134

SYSTEMS OF NATURAL DEDUCTION

In this case, the fifth sub-echelon is subordinate to all the hypothetical echelons to its left; a similar subordination-superiority relation obtains between the remaining echelons of the proof. In summary, we note that for every hypothesis of a proof in PND there is one and only one correlated vertical echelon. Hence if the proof has N hypotheses, the total number of echelons will be N + 1, i.e., the total number of hypotheses plus the theorem echelon. 4.54 Derivations in PND proceed in the same manner, differing from proofs of the system in that they lack the terminal deduction of the theorem: i.e., they terminate with the derivation of the main conclusion. Obviously any derivation in the system having only one hypothesis per echelon may be converted into a proof by simple recourse to adding the theorem line and asserting the main conclusion subject to the condition given in the main hypothesis. Hence if the main conclusion of the proof is some expression B and the main hypothesis A, the theorem will be A:::)B. (Derivations may possess more than one main hypothesis, while proofs possess one and only one.) 4.55 The formation of hypotheses in PND and the justification column. The determination of the hypotheses for the proof of a given theorem follows a simple procedural rule. This rule is called the rule for the analysis of the theorem. It will be stated here for implication, its application for other connectives being reserved for later. The rule is stated as follows: Given any expression to prove, the hypothesis from which the expression is to be proved is the antecedent of the expression.

Following this rule, the main hypothesis of a proof of a theorem in which implication is the main connective will always be everything to the left of the main implication. Hence, in a proof of the first syllogistic law,

the main hypothesis is q:::) r, since this is the expression to the left of the main implication, i.e., the antecedent of the theorem. The rule is then applied to the consequent of the law, namely, p:::)q:::) .p:::)r. The application of the rule to this expression gives the first sub-hypothesis, which is, of course, p:::) q, since this expression is the antecedent of the consequent of the theorem. Following the rule, it can be seen that to prove p:::) r, 135

SYSTEMS OF FORMAL LOGIC

p is to be hypothesized, since it is antecedent of the expression in question. When p has been assumed as the second sub-hypothesis the analysis of

the theorem is complete and the preliminary structure of the proof is determined. The preliminary structure will have the following appearance. iq:::>r Ip:::>q

II;

hyp hyp hyp

III·

The expression "hyp", abbreviating "hypothesis", is written in a column at the right of the proof, indicating that the expression to its left is an assumption of the sequence. This column is designated the "justification column" of the proof. No expression may be introduced into the proof unless it has one of the three following forms of entry in the justification column: (a) It is an hypothesis; hence it has the entry "hyp" in the justification column and occurs as the first step in a proof echelon. (b) It is an immediate consequence of one or more steps preceding it in the proofin accordance with the rules of the system presented below. In this case the step numbers from which the deduction is made are cited in the justification column along with citation of the rule of transformation by virtue of which the deduction is made. (c) It is a previously proved theorem of PND, in which case the number of the theorem in PND is cited in the justification column. (This procedure is introduced in order to abbreviate proofs. In each case of the introduction of a theorem as a step in a proof the full proof of that theorem could be presented, using none but primitive rules of PND; however, when a theorem is once proved, it is tiresome to prove it anew in a succeeding proof. If anyone finds this method of abbreviation objectionable he is free to construct all the proofs of this chapter using none but the primitive rules of the system. When the technique of introducing previously proved theorems is employed, however, it should be remembered that this is an abbreviative method, and that at this point 136

SYSTEMS OF NATURAL DEDUCTION

in the proof the entire proof of the theorem cited enters the deduction.) If there are no implication signs in the consequent (or in the consequent of the consequent, etc.) of the theorem to be proved then the sole hypothesis will be the antecedent. In the case of theorems like p':::J .q':::J .p&q the sequence of hypotheses terminates with the first occurrence of q, since the main connective in the consequent of the consequent of the theorem is not ':::J but &. In this and similar cases it is necessary in general- though there are important exceptions - to derive the remaining expression in which the main connective is not implication from those hypotheses, and only those, already available. Cases,in which the main connective of the theorem is not implication are treated in a special manner discussed below. 4.56 Concluding comments. The above technique of constructing proofs and derivations in PND has the characteristic of preserving the structure of the theorem to be proved through the order in which the hypotheses are assumed. (For reasons which will become evident later, there is one and only one order in which a sequence of hypotheses may be set out.) To the antecedent of the theorem corresponds the main hypothesis; to the antecedent of the consequent, the first sub-hypothesis; to the antecedent of the consequent of the consequent, the second sub-hypothesis, and so on. Deduction then takes place by the application of the rules of PND to the sequence of assumptions hypothesized. The reason for this procedure is clear. Implicative theorems are conditionals, asserting that if the antecedent then the consequent, the antecedent being given. This circumstance is reflected by the sequence of hypotheses in a proof in PND. It can be seen that the problem of proving a theorem in which implication is the main connective in PND reduces to the problem of finding a proof for its terminal consequent upon having applied the rule for the analysis of the theorem. 4.6

Rules of Transformation of PND

The transformation rules of the system may be conveniently divided into two classes, one of which may be further divided into two sub-classes: Class one: the primitive rules of the system; these are the basic underived rules by means of which expressions are derivable from expressions in the system. 137

SYSTEMS OF FORMAL LOGIC

Class two: derived rules; these rules are systematically inessential, since every result derived through them is derivable by recourse to primitive rules of the system without further supplementation: none the less, it will be seen that they playa useful role in shortening proofs. The primitive rules may be classified into rules of construction and rules of transportation: construction rules allow the derivation of a new expression from one or more expressions already present in the proof; transportation rules provide justification for the movement of the same expression from one point to another in the proof. It will be noted that another classification immediately suggests itself on investigating the list of sixteen construction rules. Eight of these rules allow the derivation of a shorter formula from a longer one, while eight allow the derivation of a longer formula from one or more shorter ones. Following Fitch, the former are called elimination rules, while the latter are named introduction rules. Corresponding to these in P+ and PLT are modus ponens, the sole elimination rule, and substitution, the sole introduction rule, of the systems. (As before, an element of a proof is any integral wff occurring as a step in the proof.) It will be useful, first, to present the rules of the system in toto and then to illustrate their employment by means of proofs in the system. Below are presented the two transportation rules (4.61) and the sixteen construction rules (4.62) of the system. The derived rules will be presented at those points at which the logical laws corresponding to them are proved. Following the rubric giving the name of the rule, a derivation-form illustrating the use of the rule is presented. As in all systems of logic, no proof is valid unless every step needed for the deduction of its terminal element is explicitly included in the proof.

4.61 The transportation rules of PND. 4.611 The rule of repetition, abbreviated "rep".

1 A

hyp

2 3 4

5 6 A

1, rep

An element in any echelon of a proof may be restated at any point in the same echelon. If an element has been assumed or derived in a proof

138

SYSTEMS OF NATURAL DEDUCTION

echelon no invalidity can occur if that element is repeated in its own echelon. 4.612 The rule of reiteration, abbreviated "reit". a

A

hyp hyp

a, reit

An element in a superior echelon may be reiterated into any echelon subordinate to it at any point in that subordinate echelon. The rule of reiteration makes hypotheses and steps deduced in superior proofs available for deductive operations in subordinate ones. It is based on the dictum that if one has explicitly assumed or deduced an element once, it may be validly restated. N.B.: the rule of reiteration allows an element in a superior echelon to be restated in a echelon subordinate to it, but not vice versa. If a rule allowing reiteration of elements from subordinate echelons to superior echelons were included among the rules of PND it would be possible to prove any expression whatever as a theorem of the system. 4.62 The construction rules of PND 4.621 Rules for implication. 1 (a) Implication elimination, symbolized - =>, (modus ponens) 1 A=>B 2 A

hyp hyp

3 B

1,2, - =>

From two wf elements A=>B and A, B may be inferred as an immediate consequence. As in P + and PLT, element 1 is called the major premise; element 2, the minor premise; element 3, the conclusion. The variables 1

The names for the rules are drawn, for the most part, from F. B. Fitch.

139

SYSTEMS OF FORMAL LOGIC

are variables of the syntactical meta-language here, as in all the rules, for reasons identical with those given under the discussion of the same rule in Chapter 1. N.B.: the premises from which the conclusion is derived by modus ponens must be elements of the same echelon. Hence the following deduction is invalid: 1 p

hyp

2 ip::>q

hyp

3

1--;-

1, 2, - ::>

The transportation rule of reiteration, of course, makes it possible to avoid this invalidity. The restriction set out above applies to all the remaining rules unless otherwise noted (cf. the rule of disjunction elimination). (b) Implication introduction, symbolized +::> 1

A

2 3 4 5

B

61 A::>B

hyp

1-5, + ::>

From an echelon with the hypothesis A and the conclusion B, the expression A::>B may be inferred as resultant. (We assume, here, that A has been so chosen that from it B is derivable.) The term "resultant" is reserved for elements having the following characteristics: the resultant is derived by operations upon elements in one echelon, but is expressed immediately below the terminus of the vertical line ranging over that echelon in such a way that it is an element of the echelon immediately superior to the echelon in which the elements from which it is deduced occur. Only two rules in the system allow the deduction of a resultant, as contrasted with a conclusion: they are implication introduction and negation introduction (see below 4.625c). Here, as in the remaining rules, a dash between step numbers in the justification column is to be read "through", e.g., steps 1 through 5, indicating that every step cited from first to fifth is necessary for the justification; a comma between step numbers, read "and", is understood to mean that only the steps having the number or numbers 140

SYSTEMS OF NATURAL DEDUCTION

cited are required as justification. It will be seen in the examples of deductions following the full statement of the rules that in an echelon possessing no redundant elements, each and every element is necessary for the derivation of B from A, and hence for the resultant A::JB by implication introduction. 4.622 Rules for conjunction. (a) Conjunction elimination, symbolized, -&. This rule has two forms, represented by the following two derivation forms.

I. 1 I A&B 21A

II. 1 A&B 2 B

hyp

1, -& hyp

1, -&

(I) From an element of the form A&B, A may be iriferred as conclusion. (II) From an element of the form A&B, B may be iriferred as conclusion. To these two forms of the rule of conjunction elimination correspond axioms 2A4 and 2A5 of P +. (In strictness, these two forms should be regarded as separate rules, but it will be convenient to consider them sub-cases of the dictum that if a conjunction is asserted, both of its conjuncts may be asserted separately.) (b) Conjunction introduction, symbolized + &. 1 A 2 B

hyp hyp

3 A&B

1-2, +&

From an element A and an element B, an element of the form A&B may be inferred as conclusion. The rule of conjunction introduction exhibits some correspondence with the axiom 2A6 of P +, though not a one-one correspondence, since the axiom would be represented not by the derivation form 4.622b but by the following derivation form:

1 A

hyp

2

hyp

B

3 A 4 A&B 5 B::J .A&B

1, reit 3,2, +& 2-4, +::J 141

SYSTEMS OF FORMAL LOGIC

4.623 Rules for disjunction (a) Disjunction elimination, - v 1 AvB

hyp

2

hyp

A

3 4

5 6

C B

hyp

7

: Ie

10 C

1, 2-5, 6-9, - v

The conclusion C is a consequence of the expression A v B and two subechelons placed one above the other such that A is the hypothesis and C the conclusion of the first, and B is the hypothesis and C the conclusion of the second. An implication between a disjunction A v B and an expression C is asserted if and only if C is a consequence of both sides of the disjunction. (b) Disjunction introduction, + v. This rule has two forms.

I. 1 I A

21AvB

II. 1 I B

21AvB

hyp 1,

+v

hyp 1,

+v

(I) From an expression A, A v B may be inferred as conclusion. (m From an expression B, A v B may be inferred as conclusion. If an expression A has been either assumed or derived any expression B may be adjoined to it where the connective between A and B is v. 4.624 Rules for equivalence (a) Equivalence elimination, - =. This rule has two forms.

I. 1 A=B 2 A:::>B

I

II. 1 A=B 2 B:::>A 142

hyp

1,

-=

hyp 1,

-=

SYSTEMS OF NATURAL DEDUCTION

(I) From the expression A == B, A:::> B may' be inferred as conclusion. (II) From the expression A == B, B:::> A may be inferred as conclusion. (b) Equivalence introduction, + ==

I~

hyp

I-a, +:::> hyp

IA

d e B:::>A f A:::>B&.B:::>A g A==B

c-a, +:::> b,e, +& f, +==

The rule may be stated briefly as follows: From the expression A:::>B&. B:::>A, A==B may be inferred as conclusion. (The derivation form above indicates in schematic form the manner in which such a circumstance would obtain.) A system possessing only the rules for articulating proofs, plus 4.611, 4.612, 4.621, 4.622, 4.623, and 4.624 is first developed below; then the rules following in 4.625 are adjoined, forming a more extensive system. 4.625 Rules for negation

A. Double negation (a) Double negation elimination, -,...,,..., hyp 1, -,...,,...,

From an expression ,...,,..., A, A may be inferred as conclusion. (b) Double negation introduction, +,...,,..., hyp 1, +,...,,..., 143

SYSTEMS OF FORMAL LOGIC

Froman expression A, '" ",.A may be inferred as conclusion. B. Single negation

(c) Negation elimination, - '" hyp hyp

1,2, - '" From two expressions A and '" A, and expression B may be inferred as conclusion. This rule allows the derivation of any expression from a contradiction. (Considered in the light of the principal interpretation of the system, it may be taken as meaning that if contradictory propositions are true then anything is true.) (d) Negation introduction, 1

+ '"

A

hyp

I-b,

+ '"

From an echelon having the hypothesis A and the two expressions Band", B as elements, '" A may be inferred as resultant. (In terms of the principal interpretation of PND, this rule may be taken to embody the principle that any proposition which leads to logically absurd or contradictory consequences is false.) C. Rules for the Law of De Morgan (e) Negated disjunction elimination, - '" v hyp

1, - '" v

From an expression "'. A v B, '" A & '" B may be inferred as conclusion. 144

SYSTEMS OF NATURAL DEDUCTION

(f) Negated disjunction introduction, + '" v hyp

From an expression ",A&",B, "'.A vB may be inferred as conclusion. (g) Negated conjunction elimination, - '" &

hyp

From an expression", .A&B, ",A v ",B may be iriferred as conclusion. (h) Negated conjunction introduction, + '" &

From an expression ",A v ",B, '" .A&B may be iriferred as conclusion. (The four rules for the Law of De Morgan are, strictly speaking, derivative, though formulated here as primitive.) The reader will note the correlation between the rules (a)-(h) above with the corresponding theorems of PLT. This completes the exposition of the primitive rules of PND. No expression may be an element in a valid deduction unless it is introduced by hypothesis or by application of one of the transformation rules of the system. From this it may be seen that, in an analogous sense of effectiveness, the rules of inference and proofs of PND are effective. Proof of this is left to the reader. Likewise, since the rules of formation for PND are otherwise identical with those of PLT, the specification of primitive symbols and the definition of a wff is effective. In this section we will prove the axioms and a number of the theorems of the axiomatic systems previously introduced, as well as some standard logical laws not previously deduced.

4.7 Proofs and Theorems of the System PND (The method of numbering of the theorems and proofs of the system is analogous with that used in preceding chapters.) 4.71 Theorems of PND in implication. 145

SYSTEMS OF FORMAL LOGIC

p-:::J .q-:::Jp

4T1 1

hyp

P

2

hyp

q

3 4

p q-:::Jp 5 p-:::J.q-:::Jp

1, reit 2-3, +-:::J 1-4, +-:::J

p-:::J .q-:::Jr-:::J:p-:::Jq-:::J .p-:::Jr

4T2

p-:::J .q-:::Jr

hyp

2

p-:::Jq

hyp

3

p

hyp

4

5 6

7 8 9 10 11

p-:::J .q-:::Jr q-:::Jr p-:::Jq q r p-:::Jr p-:::Jq-:::J .p-:::Jr p-:::J .q-:::Jr-:::J:p-:::Jq-:::J.p-:::Jr

5, 7, --:::J

3-8, +-:::J 2-9, +-:::J 1-10, +-:::J

p-:::Jp

4T3 1

1, reit 4,3, --:::J 2, reit 6,3, --:::J

hyp

p

1, rep 1-2, +-:::J

2 p 3 p-:::Jp 4T4

q-:::Jr-:::J .p-:::Jq-:::J .p-:::Jr

(Here and hereafter we employ the previously introduced convention of writing TH instead of detailing the theorem once it has been demonstrated.) 146

SYSTEMS OF NATURAL DEDUCTION

hyp

q-=:1r

2

p-=:1q

hyp

3

p

hyp

p-=:1q 4 5 q q-=:1r 6 r 7 8 p-=:1r 9 p-=:1q-=:1 .p-=:1r 10 TH

2, reit 4,3, --=:1 1, reit 6,5, --=:1 3-7, +-=:1 2-8, +-=:1 1-9, +-=:1

(Note that in the above proofs each vertical line defining an echelon terminates with the step needed for the derivation of the resultant, the resultant being placed immediately below the vertical line to the right of which the deductions are carried out by virtue of which the resultant is inferred. Note also that in the case of proofs of 4T1, 4T2, and 4T4 the resultant of each sub-echelon is the conclusion of the echelon immediately superior to it. In an extended sense, each subordinate echelon is an element of the proof superior to it. Thus, for instance, the second subordinate echelon in the proof of 4T4 is an element, in this extended sense, of the first subordinate echelon; thus the resultant of the former is the conclusion of the latter, the latter, likewise, being an element of the main proof.) 4T5 1

p-=:1q-=:1 •q-=:1 r-=:1 .p-=:1r p-=:1q

hyp

2

q-=:1r

hyp

3

p

hyp

4

5 6 7 8 9

p-=:1q q q-=:1r r p-=:1r q-=:1r-=:1 .p-=:1r

10 TH

1, reit 4,3, --=:1 2, reit 6,5, --=:1 3-7, +-=:1 2-8, +-=:1 1-9, +-=:1 147

SYSTEMS OF FORMAL LOGIC

4T6 1

p~ .q~r~:q~ .p~r p~ .q~r

hyp

q

2 3

hyp hyp

p

4 p~ .q~r q~r 5 6 q 7 r p~r 8 9 q~ .p~r 10 I TH 4T7

1, reit 4,3, - ~ 2, reit 5, 6,-~ 3-7, + ~ 2-8, + ~ 1-9, + ~

p~ .p~q~q

1

P

hyp

2

Ip~q

hyp

3 4 5 p~q~q 6 TH

I~

4T8

p~q~r~s~t~ .p~ .q~ .r~ .s~t p~q~r~s~t

2 3 4 5

hyp hyp

P

hyp

q

hyp

r

hyp

s

6 7 8 9 10 11 12 13 14 15

1, reit 2,3, - ~ 2-4, + ~ 1-5, + ~

p~q~r

s p~q~r~s

p~q~r~s~t

t s~t r~ .s~t q~ .r~ .s~t p~ .q~ .r~ .s~t

TH

hyp 5, reit 6-7, + ~ 1, reit 9,8, - ~ 5-10, + ~ 4-11, + ~ 3-12, + ~ 2-13, + ~ 1-14, + ~

SYSTEMS OF NATURAL DEDUCTION

The proof of 4T8 is somewhat more complex than those of previous theorems. The crucial steps are 6 and 7. For reasons which should be clear when bracketing is restored in the main hypothesis, viz.,

implication elimination cannot be validly carried out by sequential application of the hypotheses 2, 3, and 4 reiterated into the fourth subechelon. This is because the main hypothesis states that its entire antecedent [[[p::Jq]::Jr]::Js] implies t. Hence the problem of the proof reduces to finding a method whereby the antecedent of the main hypothesis may be inferred. The solution to this problem is found by reference to a procedure similar to that of 4T1. While not strictly essential it will be useful to have a derived rule in the system for this purpose. This rule will be indexed 4*1, the asterisk indicating, by convention, that the citation is of a rule, the numbers indicating the theorem of PND from which the derived rule is derived. The derivation-form of 4* 1 is as follows: hyp 1,4*1 Two derivations using 4*1 will now be presented. (We here adopt the convention that, in numbering derivations, the letter 'D', for derivation, will intervene between the section number and the derivation number. Thus 4Dl will index the first derivation, 4D2 the second derivation, etc., in Chapter 4.) 4Dl

4D2

1 P 2 q::JS::JP::J .p::Jr

hyp hyp

3 q::JS::JP 4 p::Jr

5 r

1,4*1 2,3, -::J 4, 1, - ::J

1 r

hyp

2 p::Jq::Jr 3 S::Jt::J .p::Jq::Jr

1,4*1 2,4*1

It should be noted that, in its present form, 4Dl cannot be transformed into a proof. The reason for this circumstance is that no rules of the 149

SYSTEMS OF FORMAL LOGIC

system in which there are two main hypotheses give rise to a resultant; but a resultant is exactly what is required if 4Dl is to be transformed into a proof. (This situation, however, may be rectified, if the two hypotheses of 4Dl are conjoined into the single hypothesis p&.q-=:Js-=:Jp-=:J .p-=:Jr, an expression different from but equivalent with the hypotheses of 4Dl.) Some further implication theorems of PND are: 4T9

p-=:Jr-=:J .p-=:J .q-=:Jr

4T10

p-=:J .p-=:Jq-=:J:p-=:Jq

4Tll

p-=:J .q-=:J .r-=:Js-=:J:q-=:J:r-=:Jp-=:J .r-=:Js

4T12

p-=:J .q-=:J .r-=:J .S-=:Jt-=:J:S-=:J .r-=:J .q-=:J .p-=:Jt

4T13

p-=:Jq-=:J .r-=:Js-=:J .q-=:Jr-=:J .p-=:JS

4T14

p-=:Jr-=:Jr-=:J . q-=:J r-=:J .p-=:Jq-=:Jr

4T15

p-=:J . q-=:J r-=:J .p-=:Jq-=:Jr

4.72 Theorems with conjunction and disjunction. Conjunction theorems with implication: 4T16

p&q-=:Jp

~ 11;&q

hyp 1, -& (first form) 1-2, +-=:J

31TH 4T17

p&q-=:Jq

1 Ip&q

hyp

2

1, -& (second form) 1-2, +-=:J

1-;-

3 TH 4T18 2

3 4

5

p&.q&r-=:J:p&q&r

p&.q&r

hyp

p q&r q r p&q p&q&r

1, -& (first form) 1, -& (second form) 3, - & (first form) 3, - & (second form) 2,4, +& 6,5, +& 1-7, +-=:J

6 7 8 TH 150

SYSTEMS OF NATURAL DEDUCTION

4T19 1

p&q&r:::J .p&.q&r

hyp

p&q&r

-

2 p&q 3 p 4 q 5 r 6 q&r 7 p&.q&r 8 TH

1, - & (first form) 2, - & (first form) 2, - & (second form) 1, -& (second form) 4,5, +& 3,6, +& 1-7, +:::>

4T18 and 4T19 show that p&.q&r and p&q&r, though distinct expressions, imply and are implied by each other. (Hereafter the first and second forms of conjunction elimination will not be explicitly distinguished, since it is always possible, effectively, to distinguish them, when necessary, by inspection.) 4T20

p:::> .q:::>r:::>:p&q:::>r

p:::> .q:::>r

2

hyp

p&q

3 p p:::> .q:::>r 4 q:::>r 5 6 q r 7 8 p&q:::>r 9 TH 4T21 1 2 3

hyp 2, -& 1, reit 4,3, -:::> 2, -& 5,6, -:::> 2-7, +:::> 1-8, +:::>

p&q:::>r:::> .p:::> .q:::>r p&q:::>r

hyp hyp

p q

4 p p&q 5 p&q:::>r 6 r 7 q:::>r 8 9 p:::> .q:::>r 10 TH

hyp 2, reit 4,3, +& 1, reit 6,5, '-:::> 3-7, +:::> 2-8, +:::> 1-9, +:::>

SYSTEMS OF FORMAL LOGIC

The proofs of 4T20 and 4T21 show that p&q:::Jr and p:::J .q:::Jr are mutually implicative. 4T22

p&q:::J .q&p

1 Ip&q

hyp

2 p 3 q 4 q&p 5 TH

1, -& 1, -& 3,2, +& 1-4, +:::J

4T23 1

q&p:::J .p&q

hyp

q&p

1, -& 1, -& 3,2, +& 1-4, +:::J

2 q 3 p 4 p&q 5 TH 4T24 1

p:::J .q:::J .p&q hyp

p

2

hyp

q

3 p 4 p&q 5 q:::J.p&q 6 TH 4T25 1

p&.p:::Jq:::J:q

p&.p:::Jq

2 p 3 p:::Jq 4 q 5 TH 152

1. reit 3,2, +& 2-4, +:::J 1-5, +:::J

hyp 1, -& 1, -& 3,2, -:::J 1-4, +:::J

SYSTEMS OF NATURAL DEDUCTION

4T26

p::::>q&.p::::>r::::>:p::::> .q&r p::::>q&.p::::>r

2

hyp

p

p::::>q&.p::::>r 3 p::::>q 4 5 q p::::>r 6 r 7 8 q&r 9 p::::>.q&r 10 TH

4T27

1, reit 3, -& 4,2, -::::> 3, -& 6,2, -::::> 5,7, +& 2-8, +::::> 1-9, +::::>

p::::>q::::> .p&r::::> .q&r

hyp

p::::>q

2

hyp

hyp

p&r

p 3 p::::>q 4 5 q 6 r q&r 7 8 p&r::::> .q&r 9 TH

2, -& 1, reit 4,3, -::::> 2, -& 5,6, +& 2-7, +::::> 1-8, +::::l

Further theorems of PND in conjunction and implication: 4T28

p::::>r::::> .p&q::::>r

4T29

p&q::::> .p&q::::>r::::> .q&r

4T30

p::::>r&.q::::>s::::>:p&q::::> .r&s

4T31

p::::> .q&r::::>:p::::>q&.p::::>r

4T32

p&q&.p::::>r&.q::::>s::::>:r&s

4T33,4T33' p::::> .p&p; p&p::::>p 4T34

p::::>q::::> .r::::>s::::>:p&r::::> .q::::>s

4T35

p::::> .q::::> .r::::> .s::::>t::::> .p&q&r&s::::>t

153

SYSTEMS OF FORMAL LOGIC

Disjunction theorems with implication and conjunction: 4T36

p=>.pvq

Ip

hyp

I--;vq

1, + v (first form) 1-2, +=>

2 3 TH

4T37 1

q=>.pvq

II q

hyp 1, + v (second form) 1-2, + =>

2 1p v q 3 TH 1

4T38

p=>r=> . q=>r=> .p v q=>r p=>r

hyp

2

q=>r

hyp

3

I~q

hyp

4

hyp

I;=>r

5 6

Ir

7 8 9 10

11

1, reit 5,8, - => hyp

I:=>r Ir r pvq=>r q=>r=> .pvq=>r

12 13 TH

154

2, reit 8,7- => 3, 4-6, 7-9, - v 2-10, + => 2-11, + => 1-12, + =>

SYSTEMS OF NATURAL DEDUCTION

4T39 1

p:::Jr&.q:::Jr:::J:pvq:::Jr p:::Jr&.q:::Jr

hyp

2

pvq

hyp

3

p

hyp

4 5 6

p:::Jr&.q:::Jr p:::Jr

1, reit 4, -& 5,3, -:::J

q

hyp

Ir

7

p:::Jr&.q:::Jr 8 q:::Jr 9 10 r 11 12 pvq:::Jr 13 TH

Ir

1, reit 8, -& 9,7, -:::J 2,3-6,7-10, - v 2-11, +:::J 1-12, +:::J

The relation of 4T38 and 4T39 is analogous to that between antecedent and consequent in the law of exportation (4T20). The converse of 4T38 and that of 4T39 are, of course, also theorems of PND. 4T40

pv.qvr:::J:pvqvr

1

pv.qvr

hyp

2

I~

hyp

3 4

Ipv q pvqvr

2, 3,

+ v (first form) + v (first form)

5

qvr

hyp

6

I~

hyp

7 8 9 10

11

Ipv q pvqvr r pvqvr pvqvr pvqvr

12 13 TH

6, 7,

+ v (second form) + v (first form)

hyp 9, + v (second form) 5,6-8,9-10, - v 1,2-4,5-11, - v 1-12, +:::J 155

SYSTEMS OF FORMAL LOGIC

4T41

pvqvr:::;.pv.qvr

1

pvqvr

hyp

2

pvq

hyp

3

P

hyp

4

pv.qvr

3,

5

q

hyp

6 7 8

qvr

I

5, + v (first form) 6, + v (second form) 2, 3-4, 5-7, - v

9

I~

pv.qvr pv.qvr

+ v (first form)

hyp

10 Iqvr 11 pv .qv r 12 pv.qvr 13 TH

9, + v (second form) 10, + v (second form) 1,2-8,9-11, - v 1-12, -:::;

Thus p v . q v rand p v q v r imply each other. pvq:::; .qvp 4T42 1

pvq

hyp

2

p

hyp

3

qvp

2,

4

q

hyp

5

4, + v (first form) 1,2-3,4-5, - v 1-6, +:::;

qvp qvp

6

7 TH 4T43

+ v (second form)

1

qvp:::; .pvq

1

qvp

hyp

2

q

hyp

3

pvq

2,

4

p

hyp

5

pvq pvq

6

7 TH 1

156

+ v (second form)

4, + v (first form) 1, 2-3, 4-5, - v 1-6, +:::;

SYSTEMS OF NATURAL DEDUCTION

Hereafter, the distinction between the first and second forms of disjunction introduction will be ignored.

p&qv .p&r=:l:p&.qv r

4T44 1

p&qv.p&r

hyp

2

Ip&q p q qvr p&.qvr

hyp

3 4 5 6

!p&r p r qvr p&.qvr 121 p&.qv r 7 8 9 10 11

13 TH The converse of 4T44 is also provable in

3 4 5 6

hyp 7, -& 7, -& 9, +v 8,10, +& 1,2-6, 7-11, - v 1-12, +=:l PND.

pv.q&r=:l:pvq&.pvr

4T45 2

2, -& 2, -& 4, +v 3,5, +&

pv.q&r

hyp

I~

hyp

pvq pvr pvq&.pvr

2, + v 2, +v 2,3, +&

q&r

hyp

q 7 pvq 8 9 r pvr 10 11 pvq&.pvr 12 pvq&.pvr 13 TH

6, -& 7, +v 6, -& 9, + v 8,10, +& 1,2-5,6-11, - v 1-12, + =:l

157

SYSTEMS OF FORMAL LOGIC

The converse is proved as follows: 4T46

2 3 4

pvq&.pvr-=:>:pv.q&r pvq&.pvr

hyp

pvq pvr p

1, -& 1, -& hyp

5

pv.q&r

4, + v

6

q

hyp

7 8

pvr

3, reit hyp

9

\pv.q&r

I~ r

10

11 q 12 q&r pv.q&r 13 pv.q&r 14 15 pv.q&r 16 TH

8, + v hyp 6, reit 10,11, +& 12, + v 7,8-9, 10-13, - v 2,4-5,6-14, - v 1-15, +-=:>

The structure of the above proof reflects the fact that the consequent of the theorem does not follow from p v q or p v r separately, but from the conjunction of both.

158

SYSTEMS OF NATURAL DEDUCTION

pv qv rv S&t~ .p&tv .q&tv :r&tv :.S&t

4T47

pvqvrvs&t

hyp

2 3 4

pvqvrvs t pvqvr

1, -& 1, -&

hyp

5

pvq

hyp

6

p

hyp

7 8 9 10 11

t p&t p&tv.q&t p&tv .q&tv :r&t p&tv :q&tv: .r&tv :.s&t

3, reit 6,7, +& 8, + v 9, + v 10, + v

12

q

hyp

13 14 15 16 17 18

t q&t p&tv.q&t p&tv.q&tv:r&t p&tv.q&tv:r&tv:.s&t p&tv.q&tv:r&tv:.s&t

3, reit 12, 13, +& 14, + v 15, + v 16, + v 5,6-11,12-17, - v

19

r

hyp

20 21 22 23 24 25 26

r&t q&tv.r&t p&tv .q&tv :r&t : p&tv .q&tv :r&tv: .s&t Ip&tv.q&tv:r&tv:.s&t s

27 28 s&t 29 p&tv .q&tv :r&tv: .s&t 30 p&tv .q&tv :r&tv: .s&t 31 TH

3, reit 19,20, +& 21, + v 22, + v 23, + v 4,5-18, 19-24, - v

hyp 3, reit 26,27, +& 28, + v 2, 4-25, 26-29, - v 1-30, + ~

159

SYSTEMS OF FORMAL LOGIC

It will be noted that in the demonstrations of 4T46 and 4T47 the rule of c(mjunction elimination is applied in the main echelon. It will be a valid procedure to apply this rule to any elements at any point in a proof in PND.

4T48

p p

2 3 4 5 6

vq&.p=>r&.q=>s=> :rv s

vq&.p=>r&.q=>s

pvq p=>r&.q=>s p=>r q=>s I p

7 8 9

1, 1, 3, 3,

-& -& -& -& hyp

rvs

4, reit 7,6, - => 8, +

q

hyp

p=>r r

10

hyp

v

11 q=>s 12 s 13 I rvs 14 rvs 15 TH 4T49

5, reit 11, 10, - => 12, + 2,6-9, 10-13, 1-14, + =>

v

r&pv.q&s=>:rvs

1

r&pv.q&s

hyp

2

I~p

hyp

3 4 5

I;vs I;&s Irvs

6 7 8 rvs 9 TH 160

2, -& 3, +

v

hyp 5, -& 6, + 1,2-4, 5-7, 1-8, + =>

v

v

v

SYSTEMS OF NATURAL DEDUCTION

2 3 4

p&: q V .p::Jq V .p::J .p::Jq

hyp

p qv .p::Jqv:p::J .p::Jq qV.p::Jq

1, -& 1, -& hyp

5

q

hyp

6 7

p q&p

2, reit 5,6, +&

8

p::Jq

hyp

p

9 10 11 12

I q&p q&p

2, reit 8, 9, - ::J 10,9, +& 4,5-7,8-11, -

13

p::J .p::Jq

hyp

q

V

2, reit 13, 14, - ::J 15, 14, - ::J 16,14, +& 3,4-12,13-17, 1-16, + ::J

14 p 15 p::Jq 16 q 17 q&p 18 q&p 19 TH

V

Further theorems in disjunction: 4T51 4T52 4T53 4T54 4T55 4T56

The The The The The

converse converse converse converse converse

of 4T38 of 4T39 of 4T44 of 4T47 of 4T48

p&q::Jr&.p::J .qvr::J:p::Jq

4T57

p&qv .q&rv .r&p::J:pvq&.qvr&rvp

4T58

The converse of 4T57

4.73 Theorems with equivalence. An equivalence holds between two expressions A and B, when A::JB and B::JA. Thus the following equivalence is provable. 161

SYSTEMS OF FORMAL LOGIC

4T59

p= .p&qv p

hyp

p

2 p&qvp 3 P:::J·p&qvp 4 p&qvp

1, + v 1-2, +:::J hyp

5

p&q

hyp

6

p

5,

7

p

hyp

-

p 8 9 p 10 p&qvp:::Jp 11 p:::J .p&qvp&:p&qvp:::Jp 12 p=.p&qvp

-&

7, rep 4, 5-6, 7-8, - v 4-9, +:::J 3,10, +& 11, +=

The conjunction of step 3 and step lOin step 11 is valid because both steps are elements of the same echelon.

162

SYSTEMS OF NATURAL DEDUCTION

4T60 1

q::Jr-= .qv r-=r

hyp

q::Jr

2

qvr

hyp

3

I~

hyp

4 5

1;::J

6

I!-

7 8 9

r

1, reit 4,3, -::J hyp

10

Ir

6, rep 2, 3-5, 6-7, - v 2-8, +::J hyp

11

I-;-v r r::J.qvr qv r::Jr&.r::J .qv r

10, + v 10-11, +::J 9, 12, +&

12

13

Ir r qvr::Jr

14 qvr-=r 15 q::Jr::J.qvr-=r 16 qvr-=r

hyp

17

q

hyp

18 19 20 21

qvr qvr-=r qvr::Jr r

17, + v 16, reit 19, --= 20, 18, - ::J 17-21, +::J 16-22, + ::J 15,23, +& 24, +-=

221 23 241 25

q::Jr q v r-=r::J . q::J r q::Jr~ . q v r~r&:q v r-=r::J .q::Jr q::Jr= .qv r=r

13. +-= 1-14, +::J

The proofs of 4T59 and 4T60 show how operations are carried out in the theorem echelon. 4T60 illustrates the manner in which the rule of equivalence elimination functions and the procedure used when equivalence occurs as a subordinate functor in a theorem. Since A -= B is a theorem whenever A::JB and B::JA are theorems, it is clear that we can prove the equivalence p v q -= . q v p. The proof would be as follows: 163

SYSTEMS OF FORMAL LOGIC

4T61 1 2 3 4

pvq=..qvp

pvq=>.qvp qvp=>.pvq pvq=> .qv p&.qv p=>.p v q pvq=..qvp

4T42 4T43 1,2, +& 3, + =.

It should be noted that the proof of 4T61 takes place entirely within the theorem echelon. Hereafter, proofs of this form will often be used. This procedure is plainly valid, but it should be noted that in every case when a theorem number is used in the justification column as the reason for a step (whether in the theorem echelon or at any other point in a proof) the entire proof of that theorem enters into the proof at that point. Thus the proof of 4T61 may be regarded as an abbreviation for the following unabbreviated proof: 1

pvq

hyp

2 3

I:vp

hyp 2,

4

Iq

hyp

+v

5 I-;-vp 6 qvp 7 pvq=>.qvp 8 qvp

4, + v 1,2-3,4-5, - v 1-6, + => hyp

9

q

hyp

10

pvq

9,

11

p

hyp

12 pvq 13 pvq 14 qvp=> .pvq 15 pvq=> .qv p&:qv p=> .pvq 16 pvq=..qvp 4T62

+v

11, + v 8,9-10, 11-12, - v 8-13, + => 7,14, +& 15, + =.

p&q=..q&p

p&q=> .q&p

2 q&p=> .p&q 3 p&q=> .q&p&:q&p=> .p&q 4 TH

4T22 4T23 1,2, +& 3, + =.

SYSTEMS OF NATURAL DEDUCTION

Whenever abbreviated proofs using theorem numbers are presented, there is always an effective method for procuring them in unabbreviated form, merely by referring back to the original proofs of the theorems used and introducing them entire as elements in the abbreviated proof as was done above. In the following proofs of equivalences we will abbreviate even more. The abbreviation:

4Tn, 4Tn+ 1, +& is to be understood to mean, "the conclusion derived by application of the rule of conjunction introduction to theorems 4Tn and 4Tn+ I." The following theorems in which equiValence is the main connective are proved using this device. p&q&r=. .p&.q&r 4T63 1 p&q&r:;) .p&.q&r&:p&.q&r:;): p&q&r 4T19, 4T18, +& 2 TH 1, +=. 4T64 pvqvr=. .pv .qvr 1 pvqvr:;) .pv .qvr&:pv .qvr:;): p v qv r 2 TH

4T40,4T4I, +& 1, + =.

By virtue of the equivalences 4T63, 4T64, it will be regarded as a valid procedure in the application of disjunction elimination, to hypothesize each of the disjuncts separately, whatever the length of the disjunction, and, in the application of conjunction elimination to eliminate the conjunction functor at any point. The justification for this practice lies in the fact that, due to the above equivalences it is always possible, in a finite number of steps, to isolate every member of a complex disjunction or conjunction. Thus the order of the brackets in a pure disjunctive or a pure conjunctive expression is irrelevant with respect to proof, (though in strictness expressions of the form p v . q v rand p v q v r and of the form p&.q&r and p&q&r are respectively different expressions, expressing distinct propositions.)

p:;) .q:;)r=.:p&q:;)r 1 p:;) .q:;)r:;):p&q:;)r&:. p&q:;)r:;) .p:;) .q:;)r 2 TH

4T65

4T20,4T2I, +& 1, +=. 165

SYSTEMS OF FORMAL LOGIC

4T66

p=.p

1 Ip=>p&p=>p 2 TH

4T3,4T3, +& 1, +=.

It is easy to find further equivalence theorems, provable in the same manner, by inspecting those laws of PND in implication whose converses have also been proved as theorems of PND in foregoing sections. The remainder of the section will be devoted to proving, at greater length, some useful equivalences not already established.

4T67 1 2 I

3' 4

pvqvr=. .qvpvr pvqvr

hyp

l~

hyp (see note following 4T64)

I qvp

+v +v

qvpvr

2, 3,

5

q

hyp

6

qvpvr

5,

7

Ir

+v

hyp

I-;-v pvr qvpvr 10 pvqvr=>.qvpvr 11 qvpvr

7, + v 1, 2-4, 5-6, 7-8, - v 1-9, + => hyp

12

q

hyp

13 14

pvq pvqvr

12, 13,

15

p

hyp

16

pvqvr

15,

17

r

hyp

8 9

pvqvr 18 19 pvqvr 20 qvpvr=> .pvqvr 21 p vqv r=> .qv pv r&: qvpvr=> .pvqvr

22 TH 166

+v +v +v

17, + v 11,12-14,15-16, 17-18, - v 11-19, + => 10,20, +& 21, + =.

4T68

q~ .p~r==.:p~ .q~r

hyp

q~ .p~r

2 3 4 5 6 7 8 9 10 11 12

hyp

p

hyp

q

1, reit 4, 3,-~ 2, reit 5,6, - ~ 3-7, + ~ 2-8, + ~ 1-9, + ~ 4T6

q~ .p~r p~r

p r q~r p~ .q~r q~ .p~r~:p~ .q~r p~ .q~r~:q~ .p~r

q~ .p~r~:p~ .q~r&:.

10,11, +& 12, + ==.

p~ .q~r~:q~ .p~r

13 TH 4T69

p==.q~ .q==.r~

.p==.r

hyp

p==.q

2

q==.r

hyp

3

p

hyp

4 5 6

p==.q

7

q=r

1, reit 4, -==. 5,3, 2, reit

q~r

7, -

8 9 10 11

12 13 14 15 16 17 18 19 20 21

p~q

q

==.

8, 6,-~ 3-10, + ~

r p~r

r

hyp

q==.r

2, reit 12, -==.

r~q

q p==.q q~p

p r~p

p~r&.r~p

p==.r q==.r~

22 TH

~

.p==.r

13,11,-~

1, reit 15, - ==. 16,14, - ~ 11-17, + ~ 10,18, +& 19, +==. 2-20, + ~ 1-21, + ~

SYSTEMS OF FORMAL LOGIC

4T70

pvp=p

pvp 1I 2 Ip

---

hyp hyp

3

17

4

p

hyp

p

4, rep 1, 2-3, 4-5, - v 1-6, + ~ hyp

5 6 7 8

2, rep

p pvp~p

p

8, + v 8-9, + ~

pvp

9 10

p~.pvp

11

pvp~p&.p~

.pvp

12 TH 4T71

7,10, +& 11, +=

p&p=p

1 Ip&p

hyp

2 3 p&p~p

1, -& 1-2, + ~ hyp

17

4 I~ 5 Ip 6 Ip&p 7 p~ .p&p 8 p&p~p&.p2J .p&p 9 p&p=p

4, rep 4,5, +& 4-6, + ~ 3,7, +& 8, +=

Further theorems with equivalence provable in the system: p=q~ .p~q 4T72 4T73 p=q~ .q~p 4T74 p~q~ .q~p~ .p=q 4T75 p&qv .p&r=:p&.qv r 4T76 pvq&.pvr=:pv.q&r 4T77 p~q&.p~r=:p~ .q&r 4T78 p~r&.q~r=:pvq~r 4T79 pvqvrvs&t= .p&tv .q&tv .r&tv .s&t 4T80 p&.qv .p~qv .p~ .p~q=:q&p 168

SYSTEMS OF NATURAL DEDUCTION

4.74 Concluding comments on this system. All the theorems of the above sections, 4T1-4T80, are provable in P +. Indeed, it can be shown that all the theorems of P + are provable in this system employing only the transportation rules and the rules for implication, conjunction, disjunction, and equivalence, i.e., rules 4.611, 4.612, 4.621, 4.622, 4.623, 4.624, plus the procedural rules for the articulation of proof structures. Likewise it can be shown that all the theorems of the system, thus restricted, are theorems of P +. This circumstance suggests that the system so far constructed, devoid of the rules for negation, 4.625, may be considered in itself a system of positive logic equivalent with the axiom system P +. This partial system will be called PND+. When the negation rules 4.625 are added, it can be shown that the full system PND stands to PND+ in a relation analogous with that sustained by PLT to P +, i.e., the latter is a partial system of the former.

4.8 Theorems of the Full System PND For convenience, the serial numbering of theorems which began in section 4.7 will be continued here. It will be understood by stipulation that any theorem, the number of which is greater than 4T81, is a theorem of PND and not a theorem of PND+ except under the following circumstance: if the theorem number is followed by a dagger, e.g., 4T81 +nt, it is understood that the expression staled to its right is a theorem of both PND+ and PND. However, while not all theorems of PND are theorems of PND+ it follows that every theorem of PND+ is a theorem of PND, since every rule of PND+ is a rule of PND. Hence any theorem proved in PND+ is utilizable in proofs of theorems of PND. It may be thought a defect of rigor that theorems from PND+ are used without being proved again in PND. Any persons for whom this defect is intolerable are invited to rewrite this chapter by duplicating at this point all the proofs of 4T1 through 4T80 given in 4.7. It will be shown, further, that for every theorem of PND+, there is at least one method of demonstrating it in PND which is not available in PND+. In fact, it will be shown that for every theorem of PND there exist at least two, sometimes four, methods for its demonstration. Deduction will begin, as before, by presentation of proofs for the axioms and some of the theorems ofPLT. We note that to the axioms 3Al through 3All correspond respectively the following theorems ofPND:

169

SYSTEMS OF FORMAL LOGIC

3AI to 3A2 to 3A3 to 3A4 to 3A5 to 3A6 to 3A7 to 3A8 to 3A9 to 3AIO to 3All to

4T1 4T2 4T16 4T17 4T24 4T36 4T37 4T38 4T72 4T73 4T74

We will now show that the law of transposition is also a theorem; hence there is a correspondence of 3Al2 with 4T81 4T81 I

"'p~ "'q~ .q~p

"'p~

"'q

q

2 3

hyp hyp

"'p

4 "'p~ "'q 5 "'q q 6 7 '" '" p p 8 9 q~p 10 TH

hyp I, reit 4,3, - ~ 2, reit 3-6, + '" 7, - '" '" 2-8, + ~ 1-9, + ~

In the above proof the hypothesis, '" p, of the second subordinate echelon is assumed in order to show that it is incompatible or inconsistent with the hypotheses, '" p ~ '" q and q, of the echelons superior to it. It may be remarked that no system of natural deduction not possessing both the rule of negation introduction and the rule of double negation elimination (or an equivalent or equivalents) will allow the proof of this theorem or theorem 4T82. The correctness of this remark will be demonstrated in Chapter 6, in connection with a number of systems weaker than standard systems. Just as the presence of the two laws

170

SYSTEMS OF NATURAL DEDUCTION

'" p::::; '" q::::; . q::::; p and p v - p may be regarded as indicators that the axiomatic system in which they occur is standard, so these two rules are indicators that a system possessing them is standard (or may be made so by the introduction of definitions). The provability of 4T81 alone indicates that PND possesses all the theorems of PLT, since PND possesses modus ponens and, through its remaining rules, the equivalent of substitution. None the less, it will be useful to present a number offurther deductions in detail to familiarize the reader with the application of the negation rules of PND. 4T82 2

-

pv-p

"'.pv-p

- p & '" - p - '" p 4 p 5 I '" p 6 '" - .p v '" p 7 TH

3

hyp I, - - v 2, -&

3, - - 2, -& 1-5,

6,

+-

-1"ttJ I"ttJ

The reader will note the use made of indirect proof technique (negation introduction) in the demonstration of the law of excluded middle. The technique consists of showing that the denial of the theorem leads to nonsense, thence inferring the denial of the denial and thence, by the rule of double negation elimination, inferring the theorem itself. Thus the method of deduction is described as indirect. A proof is direct if, in it, there is no use in it of negation introduction. A number of influential systems of logic do not countinence, in general, the use of indirect proofs, employing only direct proofs, in the sense defined above. Chapter 6 is devoted to the exposition of some such systems. However, in the present system, indirect proofs are freely used. (The consequent restrictions which must be placed on this and other standard systems in order to avoid inconsistency at the level of functional logic is the subject of a later chapter.) For every theorem proved by direct methods in PND there is a proof by indirect techniques; the converse, however, does not hold in general, (e.g., there is no direct proof of LEM in PND.) This will be illustrated following the introduction of some useful derived rules.

171

SYSTEMS OF FORMAL LOGIC

4T83

"'pvq=..p=>q "'pvq

hyp

2

p

hyp

3 4

"'pvq "'p

1, reit hyp

5. 6

p q

2, reit 5,4, - '"

7

q

hyp

8 q 9 q 10 p=>q 11 "'pvq=> .p=>q 12 p=>q

7, rep 3, 4-6, 7-8, - v 2-9, + => 1-10, + => hyp

13

hyp

"'."'pvq

14 '" '" p &'" q 15 '" '" p 16 p 17 p=>q 18 q 19 I "'q 20 '" "'. "'p v q 21 "'pvq 22 p=>q=>."'pvq 23 '"p v q=> .p=> q&:p=>q=> "'pvq 24 TH

13, - '" v 14, -& 15, - '" '" 12, reit 17,16, - => 14, -& 13-19, + '" 20, - '" '" 12-21, + => 11,22, +& 23, +=.

By virtue of the equivalence established in the proof of 4T83 the following derived rules are introduced:

4*83

Rules for the definition of material implication.

",A vB

hyp

A=>B

4*83 (first form)

From the element '" A v B, A => ~ may be inferred as conclusion. 172

SYSTEMS OF NATURAL DEDUCTION

hyp 4*83 (second form)

From the element A=> B, '" A v B may be iriferred as conclusion. 4T84 2 3 4 5 6

"'p=>q=. .pvq "'p=>q

hyp

'" "'pvq '" '" p

1,4*83 (second form) hyp

p pvq

3, - '" '" 4, + v hyp

Iq

7 17vq 8 pvq 9 "'p=>q=> .pvq 10 pvq

6, + v 2,3-5,6-7, - v 1-8, + => hyp

11

p

hyp

12 13

'" '" p "''''pvq

12, + v

14

hyp

Iq

1-=

11, + '" '"

15 "'pvq 16 '" "'pvq 17 "'p=>q 18 pvq=>."'p=>q 19 '" p=>q=> .p v q&:p v q=> '" p=>q 20 "'p=>q=. .pvq

14, + v 10, 11-13, 14-15, - v 16,4*83 (first form) 10-17, + => 9,18, +& 19, + =.

By virtue of the proof of 4T84 the following derived rules are introduced: 4*84 ",A=>B

AvB

Rules for the definition of material implication. hyp I, 4*84 (first form)

From the element '" A => B, A v B may be inferred as conclusion. AvB

h~

1,4*84 (second form)

From the element A v B, '" A => B may be inferred as conclusion. 173

SYSTEMS OF FORMAL LOGIC

It is clear that no results not derivable by the primitive rules of PND are derivable by 4*83 and 4*84, while these rules will serve to shorted proofs considerably. Though the first and second forms of 4*83 and the first and second forms of 4*84 are distinct rules, it will be convenient to consider them as a single class of rules. Thus the reason to be given for the application of anyone of them in the justification column of a proof will be the citation of the step from which the inference is made and the abbreviation dmi, (abbreviating "definition of material implication" the name given to the rule of passage in PM allowing the inference of p;:) q from '" p v q, and '" p v q from p;:) q. The meaning of this expression is here enlarged, of course.) The acute reader will notice, also, that in the proofs of theorems 4T81 and 4T82 special use is made of the rules for the Law of De Morgan. Inspection of the derivation forms for these rules will disclose that the application there performed is valid. None the less, it is useful to have the full Law of De Morgan explicitly demonstrated in the system. Two derivative forms will be proved, leaving the proof of the remaining forms to be worked out by the reader.

4T85

'" . p v '" q=.E=: '" p&q "'.p v "'q

2 '" p & '" '" q 3 "'p 4 '" '" q 5 q 6 "'p&q 7 '" .pv "'q;:):"'p&q 8 "'p&q 9 "'p 10 q 11 '" '" q 12 '" p & '" '" q 13 '" .p v "'q 14 '" p&q;:) . '" .p v '" q 15 '" .P v '" q;:) : '" p &q &: . '" p&q;:) . '" .p v '" q 16 TH

174

hyp 1, - '" v 2, -& 2, -& 4, -,.....,,.....,

3,5, +& 1-6, +;:) hyp 8, -& 8, -& 10, + '" '" 9,11, +& 12, + '" v 8-13, +;:) 7,14, +&

15, + ==

SYSTEMS OF NATURAL DEDUCTION

4T86 1

"'. "'p&"'q:=.:pvq

I~' "'p&"'q

2 3

'" '" p v '" '" q '" '" p

hyp

t, - "'& hyp

4 5

p pvq

3, - '" '" 4, + v

6

'" '" q

hyp

7 q 8 pvq 9 pvq 10 "'. "'p&"'q~:pvq 11 pvq

6, - '" '" 7, + v 2, 3-5, 6-8, - v 1-9, + ~ hyp

12

p

hyp

13 14

'" '" p '" '" p v '" '" q

13, + v

15

q

hyp

16 '" '" q 17 '" '" p v '" '" q 18 '" '" p v '" '" q 19 "'. "'p&"'q 20 pvq~.",."'p&"'q 21 "'. "'p&",q~:pvq&:. pvq~. "'. "'p&"'q 22 TH

12,

+ '" '"

15, + '" '" 16, + v 11,12-14,15-17, - v 18, + "'& 11-19, + ~ 10,20, +& 21, +:=.

Using the same procedures, the following may be established as theorems ofPND: 4T87

",.pvq:=.:"'p&",q

4T88

"'. "'pvq:=.:p&"'q

4T89

"'. "'pv "'q:=.:p&q

4T90

'" .p&q:=.:"'pv "'q

4T91

"'. "'p&q:=.:pv "'q

4T92

'" .p&",q:=.:"'pvq 175

SYSTEMS OF FORMAL LOGIC

The theorems 4T85-4T92 will be jointly referred to as the Law of De

Morgan. Hereafter, any use of the rules and theorems associated with the Lawof De Morgan will be referred to in the justification column by the abbreviation "dm", abbreviating "DeMorgan". The derived rules dm and dmi are applicable to parts of expressions as well as to integral expressions.

EXERCISE

Prove all the laws of De Morgan without using any of the four special rules for the Law of De Morgan, (the negation rules (e), (f), (g), (h) of 4.625C), thus showing that these rules are not essential for the construction of PND. We now return to the project specified at the end of the paragraph following the proof of 4T82. It will be shown that there is a method by means of which every theorem proved by direct techniques can also be proved by means of indirect techniques. As a sample case we will prove, by indirect methods 4T7t

p:::> .p:::>q:::>q

already proved directly. hyp

p

A. 2

"'.p:::>q:::>q

hyp

3

"':"'.p:::>qv:q p:::>q&"'q p:::>q p q

2, dmi 3, dm 4, -& 1, reit 5,6, -:::> 4, -& 2-8, + '" 9, - '" 1-11, +:::>

4 5 6 7 8 9 10

"'q '" '" .p:::>q:::>q p:::>q:::>q 11 p:::>.p:::>q:::>q

another proof 176

I"V

SYSTEMS OF NATURAL DEDUCTION

B.

1 I~'p~ .p~q~q 2 "'. "'pv .p~q~q 3 p&'" .p~q~q 4 '" .p~q~q 5 "':'" .p~qv:q 6 p~q&",q 7 p~q 8 p 9 q 10 "'q 11 '" '" .p~ .p~q~q 12 p~ .p~q~q

hyp 1, dmi 2, dm 3, -& 4, dmi 5, dm 6, -& 3, -& 7,8, - ~ 6, -& 1-10, + '" 11, - '" '"

The reader may be puzzled at these proofs. He may reason, "A curious procedure! The theorem is provable, directly, using only three primitive rules, in six steps. Here we are confronted with very much more awkward and complex proofs, one of which takes twelve steps and uses six rules, some of them derived; the other of which requires recourse to eight rules, two of which are derived. If the proofs were written out in full, using only primitive rules, it is clear that they would be considerably longer. A curious procedure!" The reader is correct. An unabbreviated proof using the technique of proof A requires a total of 22 steps, while a similar proof using the technique of B needs not less than 40! These methods, then, when they are employed using only primitive rules, are hardly abbreviative. None the less, there is a very good reason for desiring that every result provable by direct techniques be provable by indirect ones, since the indirect technique provides a decision procedure or algorithm for the system having a degree of simplicity unavailable were this not the case. This procedure will be outlined briefly below. The indirect technique used in proof A is called the method of denying the consequent, the technique of B, the method of total denial. The question: "Why is there always an indirect proof for every theorem of the system?" and its correlate, "Why is it not the case that there is a direct proof for every theorem of the system?" are recommended to the reader's consideration as disclosing a number of pertinent characteristics of PND and S-systems in general. One further indirect proof of a theorem provable directly follows:

177

SYSTEMS OF FORMAL LOGIC

1

p=>q&q=>r

hyp

2 3 4

p=>q q=>r ,.., .p=>r

1, -& 1, -& hyp

5 "".""pvr p&,..,r 6 p 7 p=>q 8 q 9 q=>r 10 11 r ,..,r 12 ,..,,.., .p=>r 13 14 p=>r 15 TH

4,dmi 5,dm 6, -& 2, reit 8,7, - => 3, reit 10,9, - => 6, -& 4-12, +,.., 13, -,..,,.., 1-14, + =>

1

q=>p

hyp

2

""p

hyp

3

q

hyp

q=>p 4 5 P 6 ""p ,..,q 7 8 ,..,p=> ,..,q 9 TH

1, reit 4,3, - => 2, reit 3-6, +,.., 2-7, + => 1-8, + =>

Thus we have

by 4T81, 4T93, conjunction introduction and equivalence introduction. Likewise, the stronger law 178

SYSTEMS OF NATURAL DEDUCTION

is a theorem of the system.

I~

'"

.p&"'p

2 IP&"'P 3 4 P"'P

I

5 '" '" '" .P& '" P 6 TH

hyp 2, -& 2, -&

1-4, 5, -

+ '"

f"O..;

'"

Another proof of LNC, using LEM 1 pv",p

4T82

2

P

hyp

3 4

'" '" P '"P v '" '" P '" .p&"'p

2, + '" 3, + v 4,dm

5 6

7 8

I~p I "'P V f"OoJ

9 TH

f"OoJ

hyp

"'P .p&"'p f"OoJ

+v 7,dm 1,2-5,6-8, - v 6,

In Chapter 3 the rather curious character of Peirce's Law and its correlates was noted. It was there remarked that, despite the absence of occurrences of negation in the expression of the law, it could not be proved in a system except from theorems or axioms in which negation does occur, (or from some correlate of the law, e.g., p-=:Jr-=:J .p-=:Jq-=:Jr-=:Jr). This characteristic of the law emerges very clearly in its proof in PND. An unabbreviated proof will be presented. (See next page.)

179

SYSTEMS OF FORMAL LOGIC

4T98

p:;:)q:;:)p:;:)p p:;:)q:;:)p

2

hyp hyp

"'p

3

p:;:)q

hyp

4 5 6 7 8 9

p:;:)q:;:)p

1, reit 4,3, -:;:) 2, reit 3-6, + '" 2-7, +:;:) hyp

10 11 12

p

I '" p '" .p:;:)q "'p:;:) '" .p:;:)q "'p "'p:;:)"'·p:;:)q '" .p:;:)q "'pvq

13

p

14 15

"'pvq

8, reit 10,9, -:;:) hyp hyp

"'p

12, reit hyp

16 17

p q

13, reit 15, 16, -:;:)

18

q

hyp

19 q 20 q p:;:)q 21 22 '" .p:;:)q 23 "'."'pvq 24 '" '" p & '" q 25 '" '" p 26 p 27 "'P 28 '" '" p 29 p 30 TH

18, rep 14, 15-17, 18-19, - v 13-20, +:;:) 11, reit 12-33, + '" 23, - '" v 24, -& 25, - '" '" 9, rep 9-27, + '" 28, - '" '" 1-29, + :;:)

In the above unabbreviated proof, steps 1-8 constitute a lemma, proving a result needed at step 10. The reader will note that use is made of three 180

SYSTEMS OF NATURAL DEDUCTION

primitive negation rules of the system. An abbreviated proof of the same theorem follows. p-:::Jq-:::Jp

hyp

2

"'p

hyp

3 4 5

p-:::Jq-:::Jp "'p-:::Jqv:p

1, reit 3,dmi hyp

6 7 8

""."'pvq p&"'q p

5, dmi 6,dm 7, -&

9

p

hyp

I_,:·p-:::J q

10 P 11 p 12 '" '" P 13 p 14 TH 4T99 1

9, rep 4, 5-8,9-10, - v 2-11, + '" 12, - '" '" 1-13, +-:::J p-:::Jr-:::J .p-:::Jq-:::Jr-:::Jr

p-:::Jr

hyp

2

p-:::Jq-:::Jr

hyp

3

I_",-r

hyp

4 5 6

p-:::Jq-:::Jr '" .p-:::Jqv:r '" .p-:::Jq

7 8 9 10 11

"'."'pvq p&"'q p p-:::Jr r

6, dmi 7,dm 8, -& 1, reit 10,9, --:::J

12

r

hyp

r

12, rep 5,6-11, 12-13, - v 3-14, + '" 15, - '" '" 2-16, +-:::J 1-17, +-:::J

13 14 15 16 17 18

r '" '" r r p-:::Jq-:::Jr-:::Jr

TH

2, reit 4, dmi 5,hyp

SYSTEMS OF FORMAL LOGIC

4T99 is an analogue of Peirce's law; innumerable others can be found. Further theorems of the system.

4T100 4T101

4T102 4T103

4T104 4T105 4T106

p~-p~-p -.p~q~:- .p~rv:-.r~q

-. -s~ -qv. -. -r~ -pv. -p&-q -. -p&. - .qv - .ra -:-p&-qv . -p&-r -r&-s~.

- - - -pv - - - -qv - ---ra. - - - - .p&q&r

4.9 A Decision Procedure for the System PND The decision problem for a logical system is to discover an effective method, an algorithm - decision procedure - by which, for any arbitrarily selected wfJ, it is possible to determine whether or not it is a theorem, and if it is a theorem, to secure a proof for it. For any well formed formula of a standard system, S, there are three and only three alternatives with respect to its truth value. (1) It is L-True, in which case it is true for any consistent substitutions for its variables. (2) It is L-False, in which case it is false for all consistent substitutions on its variables. (3) It is L-Indeterminate, in which case it is true for some consistent substitutions and false for others. In Chapter 1 truth-tables were presented which allow us to determine, for any wfJ, whether or not it is a law of standard logic (cf., also, Chapter 5). In the present section a different method is outlined, a procedure which uses the technique of natural deduction. The following algorithm gives an effective test for determining theoremhood or non-theoremhood with respect to the standard propositional calculus, PND. Given any arbitrary well formed formula B: A. If the denial ofB, -B, implies a contradiction, then -B is L-False and its contradictory, B, is L-True. The same method demonstrates also 182

SYSTEMS OF NATURAL DEDUCTION

that B is a theorem, since it constitutes an indirect proof of B in PND since PND possesses the full law of double negation. B. If '" B does not imply a contradiction, then B is either (I) L-False or (II) L-Indeterminate. If B is L-False, then B can be shown to imply a contradiction, hence we have an indirect proof of the theoremhood of '" B, since PND possesses the full law of reductio ad absurdum. C. If B does not imply a contradiction, then B is either L-true or Lindeterminate. If B is L-true, then '" B can be shown to imply a contradiction and there is an indirect proof of B by + '" and - '" '" . D. If neither B nor'" B imply a contradiction, then B (and of course '" B) is L-Indeterminate and neither B nor '" B are theorems of PND. In order to establish the effectiveness of this algorithm, an effective test must be given for determining whether or not a given formula does or does not imply a contradiction. 1. B is said to be an open formula if and only if its main connective is either & or v. It is said to be closed otherwise. Thus formulae having "', ==, ::::> as their respective main connectives are said to be closed. A single variable-formula is also, trivially, said to be a closed formula. 2. Any complex wff (a wffhaving two or more occurrences of variables), is either an open formula or can be converted into an open formula by the rules of PND. Thus the closed formula p::::>q

may be converted into an open one by dmi: "'pvq

The closed formula '" [p&q]

may be "opened" by dm: "'pv"'q

The closed formula '" [p::::> '" q]

can be opened by (1) application of dmi to give "'[ "'pv "'q] 183

SYSTEMS OF FORMAL LOGIC

and (2) dm, to give the open formula

p&q etc. 3. The procedure uses, besides rep and reit, the following rules, to be called "decision-rules":

(1) (3) (4)

dmi dm -& - v

(5)

---

(2)

Plus two derived rules of PND, namely, (6a)

(6b)

1 AvB 2 -A

hyp hyp

3 B

1,2, mtp

1 AvB 2 -B

hyp hyp

3 A

1,2, mtp

(These coordinate rules are referred to jointly as modus tollendo ponens, (mtp)) (7)

1 -[A~B]

hyp 1, --~

(This rule will be referred to as "negated implication elimination" and will be abbreviated as above.) This rule converts a negated implication into an open formula. Brackets will be used to avoid confusion. In the test deductions themselves, none but the decision-rules specified above may be used (though in the moves whereby a theorem is deduced in the theorem echelon from a contradiction obtained in the main echelon (the test deduction itself) + - is required). 4. Now take some formula as a test case. Let us select

184

SYSTEMS OF NATURAL DEDUCTION

This wff is either an L-truth, an L-falsehood, or an L-indeterminacy in PND. To decide, the formula must be "opened": hyp 1,-"'=>

Step 2 "opens" the formula, But from step 2, both p and", p are deducible. Hence, step 1 is L-False, and its denial

'" '" [p => p] is L-true. And so, by - '" "', is

p=>p The procedure works, not by attempting to deduce a theorem of PND, but by systematically reducing formulae, whose logical status we desire to discover, to their least elements (either bare variables or variables whose only associated constant is negation). If the formula we are testing entails the contradictories, p and '" p (or an equivalent pair, e.g., q and "'q) then it is L-false and its denial is L-true. Below is another test deduction illustrating the procedure. I I-::J[[p=>q]=>p]=>p]

hyp

2 '" [ '" [[p => q] => p] v p] 3 [[P=>q] => p] &'" p 4 "'p 5 [p=>q] =>p 6 "'[p=>q] v p 7 I~[p=>q]

1, dmi 2, dm 3, -& 3, -& 5, dmi hyp

11

"'["'pvq] p&"'q "'q p

7,dmi 8, dm 9, -& 9, -&

12

p

hyp

8 9 10

13 p 14 p 15 "'p

12, rep 6, 7-11, 12-13, - v 4,rep

Again, since 1 implies the contradictory formulae p and '" p, the above proof shows that this formula is L-false. 185

SYSTEMS OF FORMAL LOGIC

By application of mtp this deduction may be abbreviated: 1 "" [[[P:::lq]:::lp]:::lp]

hyp

2 ""[ ""[[P:::lq]:::lp] v p] 3 [[P:::lq]:::lp]&""p 4 ""p 5 [p:::lq]:::lP 6 ",,[p:::lq]vp

1, dmi 2,dm 3, -& 3, -& 5, dmi 6,4, mtp 7, dmi 8,dm 9, -& 9, -& 4, rep

7 ",,[p:::lq] 8 ""[ ""pvq] 9 p&""q 10 ""q 11 p 12 ""p

In subsequent deductions, mtp will be used whenever possible. What of the case when the formula to be tested is L-true or L-indeterminate? In both such cases the search procedure instituted in order to discover a contradiction will break down or abort, since neither L-true or L-indeterminate formulae are equivalent with, or imply, a contradiction. A decision-procedure-deduction (or test-deduction) is said to 'abort' if and only if (a) the deduction cannot be carried on in such a way that, after exhaustive application of the decision-rules to each element, its terminal element is in the main echelon (see deduction below), or (b) the deduction terminates with items in the main echelon none of which stands to any other in a relation of the form A and "" A: i.e., all elements in the main echelon are consistent with one another. To illustrate, we select the formula, P:::l[q:::lp]

186

SYSTEMS OF NATURAL DEDUCTION

1 p=>[q:::)p]

hyp

2 "'pv[q:::)p] 3 "'P 4 "'P

1, dmi hyp

-

3, rep

5

q:::)p

hyp

6 7

"'qvp

I "'q

5, dmi hyp

8

I---:q

7,rep

9

Ip

hyp

10

1-;-

9, rep

The deduction aborts after step 10. No further application of the decisionrules will give lesser elements than those already obtained. And the attempt to apply - v to the elements already available is frustrated. When a deduction aborts in this way - when all the specified decision-rules have been exhausted with respect to each step - the formula with which the sequence of steps began is either an L-true or an L-indeterminate one. (The deduction must always be checked to ascertain that no step is further reducible by the application of the decision-rules. This can, in fact, be done by an appropriately programmed machine.) How is it possible, using this procedure, to discriminate L-truths from L-indeterminacies? If a formula, S, is L-true, then its denial is L-false, a contradiction. Hence, to check the logical status of a formula, the test-deductionprocedure for which has aborted, - as above - one checks the denial of S, '" S, using the same technique. If, using the procedure, '" S is shown to imply a contradiction, then '" '" S is L-true, and so is S. If, the deduction from ",S also aborts (as by assumption did the deduction from S) then S is L-indeterminate. (For illustrations see next page.)

187

SYSTEMS OF FORMAL LOGIC

I.

1 I---=:.[p:::>[q:::>p]]

hyp

2 3 4 5 6

1, - '" :::> 2, -& 2, -& 4, - "':::> 5, -& 5, -& 3, rep

p&"'[q:::>p] p "'[q:::>p] q&"'p q 7 "'p 8 p

(Thus, while the decision-procedure-deduction from p:::> [q:::> p] showed only that this formula is either an L-indeterminacy or an L-truth, the fact, shown above, that its denial implies contradictory formulae demonstrates that the original formula, p:::>[q:::>p], is L-true, not L-indeterminate.) L-indeterminacy

II.

1 [p:::>q]:::> [[r:::>q]:::> [p:::>r]]

hyp

2 "'[p:::>q]

1, dmi hyp

-

V

[[r:::>q]:::> [p:::>r]]

3 I---=:.[p:::> q] 4 p&"'q 5 p 6 "'q 7

[r:::>q]:::> [p:::>r]

hyp

8

"'[r:::>q] v [p:::>r] I---=:. [r:::>q]

7, dmi hyp

9 10

11 12

13

188

3, - '" :::> 4, -& 4, -&

r&"'q r "'q Ip:::>r

9, - "':::> 10, -& 10,-&

hyp

14 15

"'pvr

I '"p

13, dmi hyp

16

I-:p

15, rep

17

Ir

hyp

18

I~

17, rep

SYSTEMS OF NATURAL DEDUCTION

The deduction aborts at Step 18. III.

1 '" [[p::> q] ::> [[r::> q] ::> [p::> r ]]] hyp 2 3 4 5 6 7 8 9 10 11 12 13 14 15

[p::> q] & '" [[r::> q] ::> [p::> r]] p::>q "'pvq '" [[r::> q] ::> [p::> r]] [r::> q] &'" [p::> r] r::>q -rvq '" [p::> r] p&"'r p "'r q "'rvq -r

16

"'r

Iq

17

1-;-

18

1, -"'::> 2, -& 3, dmi 2, -& 5, - '" ::> 6, -& 7,dmi 6, -& 9, -&::> 10,-& 10, -& 4,11, mtp 8, rep hyp 15, rep hyp 17, rep

Thus deduction III, which assumes the denial of step 1 of deduction II, also aborts: thus this formula, and, of course, its denial, are L-indeterminate. IV.

1 "'[[ "'p::>q] v "'[p&"'q]]

hyp

2 3 4 5 6 7 8

l,dm 2, -& 3, dmi 2, -& 5, -& 5, -& 4,6, mtp

-

[p::> q] &[p& '" q] p::>q ""'pvq p&-q p -q q

Thus, the lead-formula in IV is a contradiction, L-false. The addition of a theorem echelon and two more steps would obtain a law of PND, 189

SYSTEMS OF FORMAL LOGIC

namely, ["'P:::lq] v ["'[P&"'q]], (though such an extension of the deduction would use rules of PND other than those specified for testdeduction). Clearly, the decision-procedure is applicable to any wi expression in PND.

4.10 A Reduction of PND

The decision-procedure outlined above suggests a number of reductions in the rules of PND, reductions analogous with those by means of which PLT was reduced to PLT', In the present section, however, the radical reduction by recourse to definitions thus suggested is bypassed. The primitive rules given for PND+ will be retained. The reduction is restricted to the rules of negation. PND has, in the formulation given in 4.62 (cf. 4.625), not less than eight primitive rules for negation. In an exercise, the student was instructed to reduce this number by four. It is possible to reduce the remaining rules to two by modifying the rule of negation introduction. The reduction accomplished in the exercise shows that the four special rules for the Law of De Morgan can be removed without loss. The present reduction shows how to eliminate - '" '" and + '" '" . Let us agree to call this latter system PND(Red). PND(Red) has just two rules for negation: A

2 ",A

hyp hyp

3 B

1, 2, - '"

A

Rule A is identical with the rule of negation elimination of PND. A

hyp

I-b, un red (first form)

190

SYSTEMS OF NATURAL DEDUCTION

",A

a b

hyp

B ",B

lob, un red, (second form)

c A

(The abbreviation "un red" means "by the unrestricted rule of reductio ad absurdum") Un red (first form) is identical with + "'. But un red (second form) has no correlate among the primitive rules of PND. Even so, as one would expect, it is possible to prove a theorem in PND corresponding to it. namely:

[ "'P::J[q&"'q]]::JP The

PND

proof follows:

1

"'P::J [q&"'q]

hyp

2

"'P

hyp

3

'" p::J [q & '" q] q&"'q q "'q

1, reit 3,2, -::J

4 5 6

7 '" '" p 8 P 9 TH

4, -& 4, -& 2-6, + '" 7, - '" '" 1-8, +::J

The presence in PND(Red) of negation elimination and the two forms of unrestricted reductio allow, without further negation rules, the deduction of all theorems of PND. The following two proofs in PND(Red) show how the laws of double negation are provable in this system.

1.

1

2

P

hyp

"'P

3 P 4 "'P 5 '" '" P 6 p::J '" '" P

hyp I, reit 2, rep 2-4, un red (first form) 1-5, +::J 191

SYSTEMS OF FORMAL LOGIC

II. 2

'" '" p

hyp

"'p

hyp 1, reit 2-3, un red (second form) 1-4, +::>

3 '" '" p 4 p 5 '" '" p ::> p

(Note in II that just as p and", p are inconsistent, so are'" p and", '" p.) Hence the full Law of double negation, as well as the full Law of De Morgan, are provable in PND(Red). EXERCISES

1. Prove the theorems left unproved for PND. 2. Show that the following are L-false by the method of the decisionprocedure. (i) (ii) (iii) (iv) (v)

(vi)

p&"'p '" .pv "'p p::> q &. '" . '" q::> '" p

'" .pvq::>. "'p::>q "':p::> .q::>r::>:"'qv. '" .p&",r "':"'pv. '" .q&",r::>:"'pvqv. '" .p&"'r

3. Evaluate: (which are L-true, which L-false, which L-indeterminate?) (i)

p::> . s::> • r::> . t ::> p

(ii)

"'p::>. '" .p&"'q '" .p v q::> : '" q

(iii) (iv)

p::> "'s::> .q::> "'s::>. '" "'p::> "'s::> "'p

4. Prove the following in (i)

(ii) (iii) (iv) (v)

192

PND(Red)

p::> q::> . '" q::> '" p '" p::> q::> . '" q::> p

p v '"P

'" .p&.q&r::>:"'pv. "'qv "'r p::>q::>p::>p

CHAPTER 5

THE CONSISTENCY AND COMPLETENESS OF FORMAL SYSTEMSl

(The present chapter is more difficult than its predecessors. Consequently it may be passed over by those using the work as an elementary text book in symbolic logic.) 5.1

Summary

Of every formal system it is desirable to show that the system is (1) consistent, and (2) complete. 2 This is shown for the system PLT' in this chapter. A decision procedure is first developed for PLT', and by means of this, the desired results are demonstrated. It will be shown that (a) every theorem of PLT' is a tautology and that every tautology of propositionallogic is a theorem of PLT'. The results thus derived will be extended to the system PLT. Of P + it will be shown that while the system is consistent (in an appropriate sense) it is not complete. In the Introduction it was shown that the primitive rules of the system PLT', substitution and modus ponens, were L-truth-preserving, i.e., that if the premises of a proof are L-true then the conclusion is L-true. (In this chapter the term "L-true" is used interchangeably with the term "tautology".) This being the case, it is possible to establish the following metatheorem concerning PLT" If ~ B in PLT' then B is a tautology, i.e., every theorem is a tautology.

Metatheorem 5.1

of PLT'

The proof is as follows. The metalogical proofs here presented employ methods due to Post, Kalmar and Church. 2 In S-systems substantially more comprehensive than the propositional calculus (e.g., LFLT' of Ch. 7), these desiderata cannot both be obtained.

1

193

SYSTEMS OF FORMAL LOGIC

(1) The three axioms of PLT' are L-true. This can be shown by resort to truth-tables of the kind illustrated in Chapter 1. P T T F F

=>. q => P

T T T T

T F T F

T T F T

"'p => "'q

T T F F

F F T T

T T F T

F T F T

=>

q => P

T T T T

T F T F

T T F T

T T F F

p =>. q => r =>: p => q =>. P => r

T T T T F F F F

T F T T T T T T

T T F F T T F F

T F T T T F T T

T F T F T F T F

T T T T T T T T

T T T T F F F F

T T F F T T T T

T T F F T T F F

T F T T T T T T

T T T T F F F F

T F T F T T T T

T F T F T F T F

Thus the axioms of PLT' are verified as L-true. (2) The rules of PLT' preserve tautologies (L-truths). (3) Hence the metatheorem follows. We will now present a decision procedure for PLT' through which it will be possible to prove the consistency and completeness of the system. The demonstration of the metatheorems from which the decision procedure results require the application of the two primitive rules of the system, the axioms and the deduction theorem, which is a metatheorem of PLT' as for PLT. In addition, the following theorems of PLT' are required: PLT,T12

p=>p

PLT,T13

"'p=> .p=>q

PLT,T14

p=>. "'q=> '" .p=>q

PLT,T15

"'p=>q=> .p=>q=>q

The proof follows the methods of Kalmar and Church. Metatheorem 5.2 Let B be a wff of P LT ,. The variables bl, b2, . '" bn constitute an exhaustive list of the distinct variables which have occurrence in B. PI, P2, ... Pn are truth values, T or F. Let the hypothesis Ai

194

CONSISTENCY AND COMPLETENESS OF FORMAL SYSTEMS

(where i is the step number of the hypothesis) be bi if the truth value Pi of bi is T; and let the hypothesis Ai be "-' bi if the truth value, pi, of bi is F. B' is B if the value of B for the values Pi, P2, ... Pn of its variables bl, b2, ... , bn, is T. B' is "-' B if the value of B for the values PI, P2, ... Pn of its variables bl, b2, ... , bn is F. We will show how to prove (I)

AI, A2, ... ,

An~B'

The proof is by induction on the occurrences of ~ in B. There are three cases, (a) there is no occurrences of ~ in B, (b) there is one occurrence of ~ in B, (c) there is more than one occurrence of ~ in B. It will be shown how to reduce the solution of case (b) to case (a) and how to reduce the solution of case (c) to case (b). (a) If ~ does not occur in B, then B is either a variable, bi, or a negated variable, "-' bi, standing alone. (This follows from the formation rules of the system PLT', no expression of the system of a length greater than 1 being wi in this system without the occurrence of ~.) If B is bi then B' is the same wff as one of the hypotheses Ai. In that case the derivation of B' from the hypotheses AI, A2, ... , An would be as follows.

1. Al 2. A2

hyp hyp

hyp

n. An o. Ai~B' p. B'

hyp PLT' TI2, 0, i, - ~

+ S .,.

The step number 0 is justified by the fact that Ai and B' are the same wjJ, hence, 0 is a variant of PLT,TI2 (see above). B' then follows by modus ponens. Hence, we have a derivation justifying

AI, A2, ... , An~B' where B' is the same as Ai. Hence we consider the case when B is a negated 195

SYSTEMS OF FORMAL LOGIC

variable, '" hi. When B is '" hi, then B' is the same as one of the hypotheses Ai, where Ai is the same as ",hi, and the same argument applies. Thus in both cases we can establish I, where there is no occurrence of ::::>. (b) If there is one occurrence of ::::> in B then we have that B is BI::::> B2, where there are no signs'::::>' in BI and B2. In that case, by induction, we have that: (II) (III)

AI, A2, ... , Anl-B'l AI, A2, ... , A n l-B'2

where B'l is BI or '" BI in accordance with whether the value of BI for the several values of its variable is T or F, and where B' 2 is B2 or '" B2 under the same circumstances. There are three sub-cases: Sub-case 1. B'l is '" BI. In this circumstance we can prove I by II, as follows. Substitute on the theorem PLT,T13, '" p::::> p::::> q. Thus by modus ponens we obtain BI::::> B2. In this case B' is BI::::> B2, i.e., the same as B. Sub-case 2. B' 2 is B2. In this case, again, B' is BI::::> B2. This may be shown by substitution on an appropriate variant of PLT,Al, p::::> .q::::>p, giving B2::::> .BI::::>B2. Then from this, B2, and modus ponens, we obtain BI ::::> B2. In this circumstance, I is proved from III. Sub-case 3. B'l is BI and B' 2 is '" B2. In this case B' is "'. B1 ::::> B2• Thus we obtain B'l from II and B' 2 from III (where B' 2 is '" B2), and a substitution on PLT,T14, giving BI::::>. ",B2::::> '" .BI::::>B2. We obtain B' by two applications of modus ponens upon the result of the substitution. Hence, by applying the same procedure used in proving I for expressions with no occurrence of ::::> to the antecedent and consequent of an expression with one occurrence of ::::>, and then applying the rules of the system plus three laws ofPLT' to the resultant, we prove the metatheorem for cases in which B' has one implication sign. (c) Now it is clear by induction that for any B containing two or more implication signs, the procedure is exactly analogous. If B has occurrences of ::::> the procedure above provides a method of reducing the problem of finding a derivation of B' on the hypotheses AI, A2, ... , An to the problem of finding a derivation of B'l from AI, A2, ... , An, and to the problem of finding a derivation of B' 2 from AI, A2, ... , An, the solutions to which are immediate if there are no occurrences of ::::> in B'l or in B'2. If B'l or B' 2 have occurrences of ::::> then the same procedure is carried out with respect to the wi parts of B'l or B' 2 until the terminal 196

CONSISTENCY AND COMPLETENESS OF FORMAL SYSTEMS

reduction reaches expressions having no occurrences of =>, in which case the same procedure is carried out as was performed in the case of B with one occurrence of =>. Thus we have an effective method of derivation of any B' from hypotheses AI, A2, ... , An. We will use metatheorem 5.2 as a lemma in the proof of the following metatheorem. Metatheorem 5.3 If B is L-true, then rB in PLT'. This metatheorem means that all L-truths (of propositional logic) are theorems of PLT" This is the converse of metatheorem 5.1. We prove this as follows: bI, b2, ... , b n are all the variables of B. For any system of values PI, P2, ... , Pn of the variables of B, the hypotheses Al, A2, ... , An are subject to the same conditions as in Metatheorem 5.2. If B is L-true, then the B' correlated to B of Metatheorem 5.2 is always B and never", B, since an L-truth is never false for any system of values for its variables. Hence, if B is L-true we have the following special form of Metatheorem 5.2: (I)

AI, A2, ... , AnrB

This holds regardless of the value Pn of b n, whether Pn be T or Pn be F. Hence we have both

(II)

AI, A2, ... , An-l, ",bnrB

and

(III)

AI, A2, ... , An-I, bnrB

By (II) and the deduction theorem we have (IV)

AI, A2, ... , An-Ir",bn=>B

and by (III) and the deduction theorem we have,

(V)

.AI, A2, ... , An-Irbn=>B

Thus we can set up the following proof by the method of the deduction theorem starting with (IV) above. We use a variant of PLT,T15 at step 3. 1. Al, A2, ... , An-Ir",bn=>B 2. Al, A2, ... , An-Irbn=>B

3. AI, A2, ... , An-Ir",bn=>B=> .bn=>B=>B 4. AI, A2, ... , An-Irbn=>B=>B 197

SYSTEMS OF FORMAL LOGIC

5. AI, A2, ... , An-II-B 6. AI, A2, ... , An-21-",b n-l=>B 7. AI, A2, ... , An-2I-bn-I=>B etc. Clearly, by the successive application of the above procedure it is possible to eleminate all the hypotheses, thus giving I- B. Hence, if B is L-true then there is a proof ofB in PLT', i.e., B is a theorem ofPLT', which was to be proved.

Summary (1) The method of truth-tables gives an algorithm for determining for any wffB of PLT', whether B is L-true. (2) The proof of Metatheorem 5.1 demonstrates that if I-B in PLT' then B is L-true. (3) The proof of Metatheorem 5.3 demonstrates that if B is L-true then I-B in PLT" Thus, since the decision problem for a system S is to discover a testprocedure by which it is possible to determine, for any arbitrary wff B, whether B is a theorem, and to discover a proof-procedure by which it is possible to obtain a proof for any arbitrarily selected theorem, T, of S, the decision problem for PLT' is solved.

5.2 The Consistency of PLT' A system, S, may be consistent in a number of ways: there are four especially important forms of consistency, all of which will be shown to be properties of PLT" The need for a proof of consistency for an interpreted system, S, is motivated by the desire that nothing logically absurd or contradictory be provable in S. Consistency may be expressed syntactically in the following four ways. 5.21 Absolute consistency. A propositional calculus S is absolutely consistent if not all wffs of S are theorems of S. (This definition links inconsistency with all inclusiveness, thus relating it with the theorem p=>. "'p=>q, or its correlate, p&""'p=>q.) 5.22 Aristotle consistency. A propositional calculus S is Aristotle consistent if, given two wffs of S, A and A', not both I-A and I- A' in S. (This

198

CONSISTENCY AND COMPLETENESS OF FORMAL SYSTEMS

definition links consistency with the propositional law '" .p&"'p.) 5.23 Hilbert consistency. A propositional calculus S is Hilbert consistent, if there is a specially selected wff A such that A is not a theorem of S. 5.24 Post consistency. A propositional calculus is Post consistent if no wff consisting of a propositional variable standing alone is a theorem.

Metatheorem 5.4 PLT' is absolutely consistent. The wff p ~ q is not a logical truth. Hence, since by Metatheorem 5.1 none but logical truths are theorems of PLT', p~q is not a theorem of PLT'.

Metatheorem 5.5 PLT' is Aristotle consistent. Given two wffs ofPLT', A and "'A, it is not the case that both ~A and ~ '" A. By the definition of a logical-truth and the truth-tables for ~ and "', not both A and '" A can be L-true. We know further that if A is L-true then '" A is L-false and if '" A is L-true then A is L-false. Hence, by Metatheorem 5.1, it cannot be the case that both ~ A and ~ '" A in PLT'.

Metatheorem 5.6 PLT' is Hilbert consistent. We select as the special wffthe simplest contradictory formula of PLT', namely, "'.p ~ p. PLT' is Hilbert consistent, since it is not the case that ~ '" .p ~ p. This is demonstrated by the fact that p ~ p is a theorem of the system, PLTTl2; since ~p~p in PLT', p~p is an L-truth by Metatheorem 5.1. But if p ~ p is an L-truth, then, by the same reasoning as in Metatheorem 5.5, '" .p ~ p is not an L-truth, in fact it is a contradiction. Hence, by Metatheorem 5.1, '" .p~p is not a theorem ofPLT"

Metatheorem 5.7 PLT' is Post consistent. A wff of PLT' consisting of a propositional variable standing alone is not a theorem of PLT' since it is never L-true, its value being F when the value of the variable is F. Hence, by the definition of L-truth and metatheorem 5.1, the bare variable p is not a theorem of PLT" PLT' is also consistent in an extended sense of Post consistency. A system is consistent in this extended sense if no wff consisting of a single variable as the argument of a negation sign is a theorem of the system. The proof is analogous to that given for simple Post consistency. The expression'" p 199

SYSTEMS OF FORMAL LOGIC

has the value F when the value of the variable is T. Hence, by the definition of an L-truth and Metatheorem 5.1, it is not the case that ~ "'pin PLT'. Thus PLT' is consistent in all the above senses. A similar proof of the consistency of PLT may be constructed. One first proves an analogue of Metatheorem 5.1 by showing (a) that all the axioms of PLT are L-truths, and (b) that the rules of PLT, primitive and derived, preserve L-truths. Then the arguments of Metatheorems 5.4-5.7 may be applied to PLT. (The simplest way to carry out this procedure is to reduce the present system PLT to a system with only the primitive rules, modus ponens and substitution, and the single derived rule of substitutivity of equivalences at any point: then to prove as a metatheorem, along the lines of the argument on page 52, that this derived rule preserves L-truths.) 5.3 The Completeness of PLT'

For any formal system, S, it is desirable that S be such that for every wff B of S, either ~ B or the addition of B to S as an axiom of S would make S inconsistent. This definition of completeness is due to Post. The motivation for the desire that a system be complete is to be found in the desire that a system be capable of proving every result consistent with its main interpretation. Thus, given the main interpretation of the propositional calculus, it is desirable that a propositional logic, S, possess as theorems all propositions true solely by virtue of the structural relations of their propositional connectives and variables. A system, S, is incomplete, then, if there is at least one wff of S, (a) which is not a theorem of S, and (b) which, if added to the axioms of S, does not make S inconsistent. There are four major definitions of completeness. These parallel those of consistency; hence the same qualifying expressions will be used. 5.31 Absolute completeness. A system S is absolutely cbmplete if, for any wffB, either ~B or the addition ofB to the axioms ofS would make S absolutely inconsistent. 5.32 Aristotle completeness. A system S is Aristotle complete if, for every wff B, either ~ B or the addition of B to the axioms of S would make S Aristotle inconsistent. 5.33 Hilbert completeness. A system S is Hilbert complete if, for every

200

CONSISTENCY AND COMPLETENESS OF FORMAL SYSTEMS

wff B, either ~ B or the addition of B to the axioms of S would make S Hilbert inconsistent. 5.34 Post completeness. A system S is Post complete if, for every wffB, either ~ B or the addition of B to the axioms of S would make S Post inconsistent. PLT' is complete in all four of these senses. The proof is as follows: By Metatheorem 5.3, if B is L-true then B is a theorem (or axiom) of PLT'. Hence, by transposition, if B is not a theorem of PLT', then B is not L-true. If B is not L-true, then by the truth-value metatheorem of Chapter 1, there are two cases. In the following it is always assumed that B is a wff of PLT'. Case 1. B is L-false. If B is L-false, then B has the value F for all substitutions on its variables. Likewise, by the definition of L-truth and, the truth-table for negation, if B is L-false then "" B is L-true and by Metatheorem 5.1, if ""B is L-true then ""B is a theorem ofPLT" Sub-case (a). If B is L-false and if B is added as an axiom to PLT' then we have both ~ B and ~ "" B in the system, since "" B is a theorem of PLT' by the argument above. Hence, the addition of B to the axioms of PLT' would make PLT' Aristotle inconsistent. Sub-case (b). If B is L-false and if B is added as an axiom to PLT', then since ~""B is in PLT', both ~B and ~""B. Substitution on a variant of PLT,T13 gives ""B::::l .B::::lq. Hence q is provable as a theorem from this formula and two applications of modus ponens. Hence, since any wff may be substituted for q, every wff is a theorem of PLT', with B added as an axiom. Hence the addition of B to the axioms of PLT' makes PLT' absolutely inconsistent. Sub-case (c). If B is L-false and if B is added as an axiom of PLT', then we have both I- B and I- "" B in PLT'. Substitution on a variant of P LT ,T14 gives B::::l. ""B::::l. "" .B::::lB. Hence 1-"" .B::::lB is derivable from this formula and two applications of modus ponens. Thus the addition of B as a axiom of PLT' makes PLT' Hilbert inconsistent. Sub-case (d). If B is L-false and if B is added to the axioms of PLT' then we have both ~B and ~""B. By substitution on a variant ofPLT,T13 we have ""B::::l .B::::lq. Thus by this formula and two applications of modus ponens we prove the bare variable q as a theorem ofPLT" Hence the addition ofB makes PLT' Post inconsistent. PLT' could be shown Post 201

SYSTEMS OF FORMAL LOGIC

inconsistent in the extended sense under the same circumstances, by substituting "'q for q in the variant of PLT,T13, and applying modus ponens twice. Case 2: B is L-contingent. If B is L-contingent (or L-indeterminate), then by the definition of L-contingency, there is at least one system of values of B such that B is F, and at least one system of values of B such that B is T. If B is L-contingent and B is added to the axioms of PLT', then any immediate consequence of B derived by the rule of substitution is a theorem of PLT'. Similarly if B is L-contingent then there is at least one theorem, C, of PLT' derivable by substitution on B such that C is L-false. This follows by induction by virtue of the fact that there is at least one system of values of B such that B is F. (The proof of this result by induction is left to the reader.) Thus if B is p-::Jq then C may be P-::JP-::J. '" .P-::JP, substituting p-::Jp for p and", .p-::Jp for q. If B is '" .p-::Jq then C may be '" .P-::JP, substitutingp for q. Thus we have the following sub-cases: Sub-case (a). If B is L-contingent (or L-indeterminate) and B is added as an axiom of PLT' then B has as consequence, by substitution, wff C which is then a theorem of PLT' and such that C is L-false. Hence by the truth-value metatheorem, the wff '" C is L-true. Hence by Metatheorem 5.3 it follows that ~"'C is in P LT ,. Thus both ~C and ~"'C are in PLT' with B added as an axiom. Hence, the addition of B as an axiom makes P LT , Aristotle inconsistent. Sub-case (b). Under the same circumstances as in sub-case (a) we have thanC and ~ ",C are in PLT" Thus by substitution on a variant of PLT,Tl3 we have ",C-::J .C-::Jq. Thus q is a theorem by two applications of modus ponens. Since any wff of PLT' may be substituted for q, every wff is a theorem of PLT' with B added as an axiom, and PLT' with B added as an axiom is absolutely inconsistent. Sub-case (c). Under the same circumstances as in sub-case (a) we have that ~C and ~"'C are in PLT'. Thus by substitution on a variant of PLT,Tl4 we have C-::J. ",C-::J '" .C-::JC, and by two applications of modus ponens, "'. C -::J C. Thus the addition of B to the axioms of PLT' makes PLT' Hilbert inconsistent. Sub-case (d). Under the same circumstances as in subcase (a) we have that ~C and ~"'C are in PLT'. Thus by two substitutions on a variant

202

CONSISTENCY AND COMPLETENESS OF FORMAL SYSTEMS

of PLT,T13 we have ",C=> .C=>q, and ",C=>C=> "'q. Hence from each of these plus two applications of modus ponens, it can be seen that the addition of B to the axioms of PLT' makes PLT' Post inconsistent, and Post inconsistent in the extended sense since the bare variable q is provable and the negated variable '" q is provable. Hence we can prove the following metatheorems for PLT'. Metatheorem 5.8 PLT' is absolutely complete. Proof: by Case 1, sub-case (b); and Case 2, sub-case (b). Metatheorem 5.9 PLT' is Aristotle complete. Proof: by Case 1, sub-case (a); and Case 2, sub-case (a). Metatheorem 5.10 PLT' is Hilbert complete. Proof: by Case 1, sub-case (c); and Case 2, sub-case (c). Metatheorem 5.11 PLT' is Post complete. Proof: by Case 1, sub-case (d); and Case 2, sub-case (d). Metatheorem 5.11a PLT' is Post complete in the extended sense. Proof: by Case 1, sub-case (d) and Case 2, sub-case (d). Hence PLT' is complete according to all the definitions specified.

Summary of results concerning PLT' (1) The primitive functors of PLT' are independent. (2) PLT' is consistent. (3) PLT' is complete.

Another result which has not been proved, but which is provable is: (4) The axioms of PLT' are independent. 5.4 Metatheorems on P + A number of metatheorems will now be proved for P + Metatheorem 5.12

If f-B in P + then B is L-true.

Proof: the axioms of P + are L-true and the primitive rules of P + preserve L-truths. It will be left to the reader to verify the eleven axioms of P + according to the method of truth-tables. Metatheorem 5.13 Some L-truths are not theorems of P +' 203

SYSTEMS OF FORMAL LOGIC

Proof: P-:::Jq-:::J. "'q-:::J "'P is L-true, as is shown by the following truthtable.

P T T F F

-:::J

q

-:::J

"'q

-:::J

T F T T

T F T F

T T T T

F T F T

T F T T

"'P F F T T

But P -:::J q -:::J • '" q -:::J "'pis not a theorem of P +, since it is not a wff according to the formation rules of P +, and no expression which is not wff in a system S can be a theorem of S. In fact we have a further metatheorem concerning L-true statements wf in P +.

Metatheorem 5.14 There are some L-truths wf in P + which are not theorems of P +. Proof: Peirce's law is L-true. p

-:::J

q

-:::J

T T F F

T F T T

T F T F

T T F F

P T T F F

-:::J

T T T T

P T T F F

It is also wfby the formation rules of P +. But it is not a theorem of P +. To prove this, we construct the following three-valued tables for the functors of P +: p

-:::J

q

1 1 1 2 2 2 3 3 3

1 2 3

1 2 3 1 2 3 1 2 3

1 3 1 1

P & 1 1 1 2 3 2 2 2 2 2 3 3 3 3 3 3 3

q

p

v

q

p

1 2 3

1 1 1 2 2 2 3 3 3

1

I 2 3

I 1 1 2 2 2 3 3 3

2 3 2 3

1 1 2 2 1 2 3

2 3 2 3

I 2 3 2 1 3 3 3

q

1 2 3 2 3 2 3

(We could interpret the "truth-values" of the above tables as follows: 1 = true, 2 = "indeterminate", 3 = false.) Using the above tables, all

204

CONSISTENCY AND COMPLETENESS OF FORMAL SYSTEMS

the axioms ofP+ can be verified as tautologies of this three-valued system. The following show how the three-valued tables verify IAl, lA3, and lA6:

P 1 1 1 2 2 2 3 3 3

:::J.

1 1 1 1 1 1

p & q :::J P 1 1 1 1 221 133 1 2 2 2 222 2 2 233 331 3 3 3 3

q :::J P 1 1 1 2 1 1 3 1 122 2 2 312 133 233 3 3

3 3

~l

3

P 1 1 1 2 2 2 3 3 3

:::J.

1 1

P

V

q

1 1 1 2 1 3 2 1 222 223 3 1 322 333

It is easy to show by similar tables that the remaining eight axioms of P + are similarly verified. (Note: the table for a formula with one distinct variable has 31 = 3 rows; that for a formula with two distinct variables, like those tested above, has 32 = 9 rows; those for formulae, like 2T2 and 2T8, with three distinct variables have 33 = 27 rows; etc.) By virtue of the characteristics of the rules of substitution and modus ponens, it follows that, if the axioms of P + are verified by the three-valued tables, then all theorems of P + are also verified by them. (Why does this follow?) Hence, if a formula is a theorem of P + then it is verifiable by the tables. If it is not verifiable by the three-valued tables, it is not a theorem of P +. The following shows that Peirce's law is not verified by the three-valued tables, since it has the value 2 in the index column at row 6.

1 1 I 2 2 2 3 3 3

1 2 3 I I 3 1 I 1

1 2 3 I 2 3 1 2 3

1 1 I 2 2 I 3 3 3

1 1 1 2 2 2 3 3 3

1 1 I I I 2 1 I 1

2

2 2 3

3 3 205

SYSTEMS OF FORMAL LOGIC

Thus the Metatheorem follows. (A procedure similar to this is used in determining the independence of axioms of a system.) Metatheorem 5.15 P + is not complete.

This follows from Metatheorem 5.l4. There is at least one wff of P + which could be added to the axioms of P + and which is not a theorem of P +, and P + would remain consistent in every sense. That wff is Peirce's law (or any correlate). Since the primitive rules of P + preserve L-truths, and Peirce's law is L-true, it follows that every theorem deduced by application of the primitive rules to Peirce's law is L-true. Hence, P + is incomplete. Metatheorem 5.16 P + is consistent.

Since Aristotle consistency and Hilbert consistency are syntactic definitions which are relevant only to systems having ,..., or an equivalent functor as primitive functors, it will be sufficient to show that P + is absolutely consistent and Post consistent. Metatheorem 5.16a P + is absolutely consistent.

The wff p => q is not a logical truth. Hence it is not a theorem of P +, by Metatheorem 5.12. Metatheorem 5.16b P + is Post consistent.

A wff consisting of a propositional variable standing alone is not a theorem of P +, since it is never L-true, its value being F when the value of the variable is F. Hence, by the definition of L-truth and the Metatheorem 5.12, a wff B consisting of a variable is not a theorem of P +. Thus P + is consistent. The proofs of the following metatheorems are left to the reader. Metatheorem 5.17

PLT

is complete and consistent.

Metatheorem 5.18

PPM

is complete and consistent.

Metatheorem 5.19

PLTF

Metatheorem 5.20

PND

is complete and consistent. is complete and consistent.

(Hint: In the case of Metatheorems 5.17, 5.18, and 5.19, show that PLT, PPM, and P LTF are equivalent with PLT" In the case of Metatheorem 5.20 use the decision procedure of Chapter IV.) 206

CHAPTER 6

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

6.1

Summary

In this chapter some non-standard propositional calculi are considered and the motives for their construction are discussed. The main systems studied are: (1) a version of the intuitionistic propositional calculus, (2) the propositional calculus of F. B. Fitch, and (3) a calculus due to A. Kolmogoroff and I. Johansson, the minimum calculus. It is shown how each of these systems departs from S-systems like PLT' and PPM. Each system is set up both as an axiomatic system, using the system P + as basis, and as a system of natural deduction. It is shown how each nonstandard system can be made into a standard one. 6.2 What is a Non-Standard System? A logical system, L, is defined as non-standard if some law or laws characteristic of standard systems is not a theorem of L. The following are offered as among the characteristic laws of standard systems, (S systems). Cl. C2. C3. C4. C5.

pv"'p '" '" p=> P "'p=> "'q=> .q=>p p=>q=>."'pvq '" .p&"'p

There are other such laws, but those given above suffice for the present purposes. It is now possible to reformulate the definition given above:

207

SYSTEMS OF FORMAL LOGIC

L is a non-standard system = df CI or C2 or C3 or C4 or C5 is not a theorem of L.1 The disjunction employed in the definition should be interpreted inclusively. Since none of the classical negation laws, CI-C5, is a theorem of P +, P + is, in accordance with this definition, a non-standard (Non-S) system. P +, however, was constructed in the interest (among others) of serving as a basis for the construction of more extensive propositional systems, among which is PLT, an S-system. Hence, P + may be regarded as a partial system for the full propositional calculus. This motive, however, does not feature in the construction of some of the other Non-Ssystems to be studied in this chapter. They are not constructed to serve as bases for the development of more extended propositional logics. They are designed as systems in their own right, in some sense competitive with S-systems. 6.3 The Intuitionistic System and the Fitch Calculus (PI and PF) The first two Non-S-systems considered in this chapter are an intuitionistic propositional calculus, called PI, and the propositional calculus of Fitch, called PF. The axiom systems for both are based on P +. 6.31 It will be useful to consider both these systems in relation to a classical system, PLT. The most notable feature shared by both PI and PF, but not by PLT, is the suspension of the principle of excluded middle as a general law. Neither system has

pv"'p as a theorem. 6.32 Both systems employ a concept of negation different from that of PLT, which accounts for the absence of LEM in the systems. The systems PI and PF also differ from each other in their negation concepts. The definiens is more precisely stated as follows: It is not the case that CI is a theorem of L or it is not the case that C2 is a theorem of L or it is not the case that C3 is a theorem of L or it is not the case that C4 is a theorem of L or it is not the case that C5 is a theorem of L. Given this formulation it can be seen that the statement is true if all the disjuncts are true (as in the case of P+), if more than one but not all are true (as in the Fitch and Heyting systems), and if one and only one disjunct is true. Otherwise it is false. 1

208

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

6.33 These revisions of classical negation lead to restrictions on forms of inference conceived of as valid in standard systems. These restrictions have a number of important consequences, the most obvious of which is the fact that many of the elementary theorems of S-systems are not theorems of PF or are not theorems of PI. A few outstanding examples are the following theorems of PLT, PLT', PLTF and PPM.

. "'q~ "'p '" .p&"'p p~ ",p~ "'p

p~q~

These tautologies are not theorems ofPF, though they are theorems of PI. The S-logical theorems

'" ",p~p '" .p&q~:"'pv "'q "'. "'pv "'q~:p&q are not theorems of PI, though they are theorems of PF. It should not be thought that the two Non-S-systems differ from S-systems only with respect to wffs in which negation is an ingredient functor. The classical theorems p~q~p~p

(Peirce's law)

p~qv .q~p

are theorems of neither of the present systems, though, of course, they are provable in all of the four S-systems considered. We begin with a formulation of PI as an axiom system, then, after some illustrative deductions, formulate it as a system of natural deduction, PNDI. 1

6.4 Rules of Formation of PI 6.41 The system possesses seven improper symbols as primitive functors. (The names of these functors are given to their right, in parentheses.) (1)

[

(2)

]

1 A system resembling them by A. Church.

PI

was developed by H. Scholz and K. Schroter, attributed to

209

SYSTEMS OF FORMAL LOGIC

(3)

(implication) v (disjunction) A (conjunction) +-+ (equivalence) -, (negation) ~

(4) (5) (6) (7)

6.42 The system possesses an infinite list of propositional variables as proper symbols. p, q, r, s, t, PI, q1, ...

6.43 An expression is a wff of PI only under the following circumstances: 6.431 A variable standing alone is a wjJ. 6.432 If A and Bare wf then,

-,A

(a) (b)

[A~B]

(c) (d)

[AvB]

(e)

[A+-+B]

[AAB]

are WffS.1 6.433 Brackets are abbreviated by points as in previous systems. 6.5

Rules of Transformation of PI

6.51 Modus ponens. 6.52 Substitution. We will abbreviate the rubric noting the application of modus ponens thus, - ~. The abbreviation + S is retained. 6.6 Axioms of PI IAI IA2 IA3

~p~.q~p ~p~.q~r~:p~q~.p~r ~pAq~p

1 Any formula formulated in terms of PI constants can be formulated in the symbolism used for PLT; it should not, however, be assumed that the functors used in PI have the same interpretation as those used in PLT and its correlate standard systems (cf6.82).

210

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

IA4 IA5 IA6 IA7 lA8 lA9 lAW lAll IAl2 IA13

f-P/\q~q f-p~ .q~

.P/\q

f-p~.pvq f-q~

.pv q

f-p~r~. q~r~.p V q~r f-p~q~.p~q f-p~q~.q~p

f-p~q~ .q~p~ .p~q f-p~-,p~-,p

f--'p~ .p~q

6.7 Definitions of PI None but the definitions of 1-, A, and TH, introduced in P+, are needed. 6.8 Deductions in PI The theorems and proofs of P + become theorems and proofs of PI when the symbolism is revised to meet the requirements of the formation rules. These proofs will not be repeated, but theorems 1-90 of P + will be taken as theorems of PI, revising the symbolism. To 2T! will correspond IT!; to 2T2, IT2, etc. We will restrict ourselves to proving only the most important theorems of the system while pointing out why some classical theorems are unprovable. Some remarks on the principal interpretation of PI will follow after some deductions. By virtue of IAl, IA2, and IT! the deduction theorem is provable for PI. It will be used when convenient. IT91 D91

p~q~. -'q~-'p

1.

p~q

2. -,q 3. -'r~ .r~s 4. -'q~ .q~-'p 5. q~-'p 6. P~-'P 7. P~-'P~-'P 8. -,p

(I-Law of transposition) hyp hyp Variant ofIAl3 2, + S, q/r, -,p/s 4,2, -~ 1,5, +lS IAl2 7,6, -~ 211

SYSTEMS OF FORMAL LOGIC

1. p-+q, -,ql--,p 2. p-+ql--,q-+-,p 3. I-p-+q-+. -'q-+-,p

D91 I,DT 2,DT

The converse I-law of transposition (the correlate of 3AI2) is not a theorem of PI nor is the wff corresponding to 3T97. IT92

p-+q-+ .p-+-'q-+-'p (I-Law ofreductio ad absurdum)

092

1. p-+q 2. p-+-'q 3. p-+q-+. -, q-+-'p 4. -'q-+-,p 5. P-+-'P 6. P-+-'P-+-'P 7. -,p

1. p-+q,p-+-,ql--,p 2. p-+ql-p-+-,q-+-,p 3. I-p-+q-+ .p-+-'q-+-'p IT93

p-+ -, -,p

093

1.p 2. -,p 3. -'P-+ .p-+ -, -,p 4. P-+-'-'P

5. -,-,p

1. p, -,pl- -, -,p 2. pl--,p-+-,-,p 3. pl--,p-+-, -'P-+-'-'P 4. pl--,-,p

5.I-TH

hyp hyp IT91 3, I, --+ 2,4, +IS IAI2 6,5, --+ 092 I,DT 2,DT

hyp hyp IA13, + S, -, -,p/q 3,2, --+ 4, 1, --+ 093 I,OT IAI2, + S, -,p/p 3,2, --+ 4,DT

The converse is not a theorem of PI. IT94

p-+q-+ . -, -'P-+ -, -, q

1. I-p-+q-+. -'q-+-,p 2. I--,q-+-,p-+. -, -'p-+-,-,q 3.I-TH 212

IT91 IT91, +S, -,q/p, -,p/q 1,2, +IS

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

1T95 1. f-TH The converse also holds. 1T96 -, -, -'P-+ -,p

+S, -,p/p

1T93,

1T91, + S, -, -,p/q 1T93 1,2, --+ Thus, though we are not permitted to infer from the intuitionistic double negation of a wff -, -, A to A, inference is permitted from -, -, -, A to -,A. Hence, any wff of PI which is qualified by three (or more) signs of negation may be simplified, using the above law.

1. f-p-+-, -'p-+. -, -, -'P-+-'P 2. f-p-+ -,-,p 3. f-TH

1T97

-, .pvq-+:-'P/\ -,q

1. f-p-+.p v q 2. f-p-+.pvq-+:-, .pvq-+:-'p 3. f--, .pvq-+:-'p 4. f-q-+.p v q 5. f-q-+.pvq-+:-, .pvq-+:-'q 6. f--, .pvq-+:-'q 7. f--, .pvq-+:-'P/\:. -, .pvq-+:-'q 8. f--,.pvq-+:-'P/\:.-,.pvq-+: -,q-+::-, .pvq-+:-'P/\-'q

IA6 1T91, +S,pvq/q 2,1, --+ IA7 1T91, + S, q/p, p v q/q 5,4, --+ 3,6, *IA5 1T32,

+S, -, .p v q/p,

-,p/q, -, q/r

8,7,

9. f-TH

--+

The converse

1T98

-,p /\ -'q-+. -, .p v q

holds also. Previous to the proof of this theorem a lemma will be proved. DV 1. -,p hyp 2. -'P-+ .p-+q IA13 3. p-+q 2,1, --+ 4. q-+q ITl 5. p-+q-+. q-+q-+.p v q-+q IA8 6. q-+q-+.p v q-+q 5, 3, --+ 7.pvq-+q 6,4,--+ V 1. -,pf-pvq-+q DL' 2. f--,p-+.p V q-+q DT

213

SYSTEMS OF FORMAL LOGIC

D98

1. 2. 3. 4.

-,p A -, q -,p A -, q-+ . -,p -,p -'p-+.p v q-+q pvq-+q r-+s-+.-,s-+-,r pvq-+q-+.-'q-+-'.pvq -'q-+-'.pvq -'pA -'q-+-,q

5. 6. 7. 8. 9. 10. -,q

11. -'.p v q 1. -,pA-,ql--,.pvq 2.I-TH

hyp IA3 2,1, --+

V 4,3, --+ VariantofIT91 6, +S,pvq/r,q/s 7,5,--+ IA4 9,1, --+ 8, 10, --+ D98 DT

The wff -,p v -'q-+-' .p A q is also a theorem, but the converse does not hold. In general, for the Law of De Morgan in PI, disjunctions imply negated conjunctions and conjunctions imply negated disjunctions, but the converses do not hold, except in the unique case of IT97. The I-law of contradiction is a theorem of PI. IT99

-, .p A -,p

1. I-PA -'p-+.p-+:PA -'P-+ -'P-+. -, .p A -,p 2. I-p A -'P-+P 3. I-p A -'P-+-'P-+. -,.p A -,p 4. I-PA -'P-+-'P

5. I-TH

IT92, +S,PA -,p/p IA3, + S, -,piq 1,2, --+ IA4, +S, -,piq 3,4, --+

As mentioned earlier, the intuitionistic law of excluded middle is not a theorem of PI, but its double negation is a theorem. IT 100

-, -'.p v-,p

1. I--,.pv-'P-+:-'PA-'-'P 2. I- -, .p v -,p-+:-,p A -, -'P-+:. -, . -,p A

-,

-,p-+:-, -, .p V -,p

3. I- -, . -,p A -, -'P-+ -, -, :p v -,p 4. 1--,. ip A -'-'P 5. I-TH 214

IT97,

+S, -,p/p

IT91, +S, -, .pv -,p/p, -'p A -, -,piq 2,1, --+ IT99, + S, -,pip 3,4, --+

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

6.81 An Intuitionistic System of Natural Deduction. It is also easy to devise an intuitionistic system of natural deduction on the basis of PND+ having the same theorems as PI. This system will be called PNDI. It is constructed by adding to PND+ the rule of double negation introduction, (+ . . , ""'), the rule of negation elimination, ( - ""'), and the rule of negation introduction, (+....,). The remaining rules, double negation elimination, ( - ...., ""'), and all of the four rules for the Law of De Morgan, are not included in PNDI.1 Five illustrative deductions are made below. PNDIT91

p--.q--. . ...., q--.""'p

p--.q

hyp

I....,q

hyp

3

p

hyp

4

p--.q q ....,q

I, reit 4, 3,-~ 2, reit 3-6, +...., 2-7, +--. 1-8, +~

-

2

5

6 7 ""'p 8 ""'q--''''''p 9 TH

But the following deduction of the converse is invalid. hyp

....,p~....,q

2 3 4 5 6 7 8 9 10

q

hyp

""'p

hyp

""'p--''''''q ....,q q

1, reit 4, 3, ---. 2, reit 3-6, +...., 7, -....,...., (invalid) 2-8, +--. 1-9, +~

""''''''p p q~p ....,p~....,q~

.q--'p

A system, PNDI', which lacks +11, can be formulated. In fact, PNDI and PNDI' have exactly the same theorems, +I I has been retained in PNDI for deductive convenience.

1

215

SYSTEMS OF FORMAL LOGIC

PNDITlOO -,-,.pv-,p -, .pv-,p

1 2

p

hyp

pv-,p 3 -,.pv-,p 4 5 -,p 6 pv-,p 7 TH

PNDIT99

2, +v 1, reit

2-4, +-, 5, + v 1-6,

+-,

-'.pA-'p

pA -,p

1

hyp

hyp

2

p -,p 4 TH

1, 1, 1-3,

3

A

A

+.

In order to secure as theorems those parts of the Law of De Morgan which are regarded as intuitionistically valid, PNDI works as follows. PNDITlOI

1

-,p v -'q-+. -'.p Aq

-,pv -,q

hyp

2

pAq

hyp

3 4

p q -,p v-,q -,p

2, -A 2, -A 1, reit hyp

5 6 7 8

p -,q

3, reit 7,6, --,

9

-,q

hyp

-,q -,q 12 q 13 -, .pAq 14 TH

10

11

216

rep 5,6-8,9-10, - v 4,rep 2-13, +-, 1-13, +-+

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

The converse of this theorem is not provable. 1 PNDIT98

-,p /\ -, q~ . -, .p v q

1

-'P/\-,q

hyp

2 3 4

-,p -,q pvq

1, - /\ 1, - /\ hyp

5

p

hyp

6 7

-,p q

2, reit 5,6, --,

8

q

hyp

9 10 11

12

q q -,q -, .pvq

13 TH

8, rep 4,5-7,8-9, - v 3, reit 4-11, +-, 1-12, +~

The converse of this theorem is the one exceptional case, the negated formula, -, .p v q, implying -,p /\ -, q. This is proved as follows.

In his book, Filosofskie problemy mnogoznacnoj logiki ('Philosophical Problems of Many-Valued Logic', English translation by G. Kling and D. D. Corney, published by D. Reidel, Dordrecht-Holland), the Soviet logician A. A. Zinov'ev attempted to articulate a system equivalent with that of intuitionism on the basis of three-valued truth-tables. However, it has been shown that this attempt is unsuccessful, since, according to his tables, 1

I.p /I. q-+:Ip V Iq

is a tautology of intuitionist logic, which is not the case. For a proof of the failure of Zinov'ev's system see L. H. Hackstaff and J. M. Bochenski, 'A Study in Many-Valued Logic', Studies in Soviet Thought, 1962. (Added April 25, 1966:) D. D. Corney has informed the author that Zinov'ev's work asserts, not that his three-valued tables give a system equivalent with the intuitionist logic, but that the tables satisfy the axioms and theorems of this logic; i.e. all the axioms and theorems are tautologies when evaluated by the tables. This contention is correct. The Kling-Corney translation (1963) which embodies a thorough revision of the original 1960 Russian manuscript by Zinov'ev himself makes the distinction clear.

217

SYSTEMS OF FORMAL LOGIC

PNDIT97

-, .pvq--+: -'P/\-,q

-, .pvq

2

p

3 4 5

pvq -, .p v q -,p

6

I

7 8 9 10

hyp hyp 2, + v 1, reit 2-4, +-, hyp

I ;Vq

6, + v

I-,·pvq -,q

1, reit 6-8, +-, 5,9, + /\ 1-10, +--+

-'P/\ -,q

11 TH

It is clear that a similar process will not give the intuitionistically invalid parts of the De Morgan Law. For instance, -,.-,pvq

1 2

I-'p -'pvq -'pvq -, -,p

3 4 5

1-,.

6

p q

7

8

-'pvq 1-'. -'pvq -,q 11 P/\-,q 12 -,. -,p v q--+:P/\ -,q

9 10

hyp hyp

2, + v 1, reit 2-4, +-, 5, - -, -, (invalid) hyp 7, + v 1, reit 7-9, +-,

6, 11, + /\ 1-11,

+--+

It is clear that in order to infer step 12, step 11 must be available through conjunction introduction, but this step is available only if steps 6 and 10 are available, and step 6 cannot be had except by the intuitionistically invalid inference from -, -,p to p. 6.82 Remarks on Intuitionistic Logic. So far PI has been developed as a purely formal system. From that point of view PI and PNDI may be 218

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

regarded as, like P +, partial systems of the full propositional calculus. It is easy to see how the calculus PI could be revised so as to give a system equivalent with S-systems of propositional logics like PLT and PLTF. The most obvious recourse is merely to add, as a fourteenth axiom, the wff ,p--+, q--+ . q--+p. If this were added the system would become an S-system with two redundant axioms, namely IAI2 and IA13. This system is called PILT. Another way is to add "p--+p as a fourteenth axiom; this would allow the proof of ,p--+, q--+ . q--+p as a theorem. This system is called PIDN. Still a third way is to add LEM, p v ,p, as an axiom to the basic set of PI. Again, we can derive ,p--+,q--+ .q--+p, and hence have a system which is standard. This system is called PILEM. A fourth way is to add as axiom , . ,p A ,q--+: p v q. This allows the deduction of pv ,p and, thence, ,p--+,q--+ .q--+p as theorems. This system is called PIDM. Thus PI is easily transformable into PILT, PIDN, PILEM and PIDM, all of which are S-systems. But the main interest in PI is not as a mere calculus or as a partial system of the full propositional logic. The main interest rests in the interpretation of the system, i.e., as a logic related to the mathematical theories of intuitionism. It will be useful, then to consider this interpretation. The four functors, A, v, --+, +-+, give little trouble. An expression A is assertable if and only if there is a proof of A. (1) An expression A A B is assertable if and only if both A and Bare assertable. (2) An expression A v B is assertable if and only if at least one of the disjuncts is assertable. (3) An expression A--+B is assertable if and only if from a proof of A there is a method or construction, C, such that A conjoined with C gives a proof of B. In other words, A--+B is a theorem if and only if there is some valid deductive sequence leading from A to B. This definition differs in no essential respect from that which has been used in every system beginning with Chapter 2. (4) A+-+B is assertable if and only if A--+B is assertable and B--+A is assertable. The system derives its interest from the interpretation placed on negation (,). (5) An expression ,A is assertable if and only if there is a proof that 219

SYSTEMS OF FORMAL LOGIC

the assertion or proof of A leads to a contradiction. In the usual interpretation contradiction is taken as a primitive notion. Now the intuitionists are first and foremost concerned with mathematics and mathematical constructions. To say, for them, that a proposition in mathematics, p, is true or assertable means no more and no less than this: there is a proof or "construction" of p, or there is a method for proving or constructing p. To say that a mathematical proposition, p, is false is to say: there is a construction or proof which shows that p leads to a contradiction. Hence for "true" we may substitute the term "constructable"; for "false", the term "inconstructable", where a proposition is inconstructable if the attempt to construct a proof of p, (or the assumption that such a construction were carried out), leads to a contradiction. Hence, if IP is asserted, p is inconstructable. Now take the expression l i P . This means that the assumption that p is contradictory leads itself to a contradiction. In the light of these remarks, we may consider the suspended intuitionistic forms of some standard logical laws. (I)

pv IP

Taking 'I' to mean provably "contradictory" or "absurd", this form of LEM would assert that for every mathematical proposition p, either there is a proof of p or a proof that p is contradictory. This principle, then, asserts that there is a general decision procedure for all mathematical propositions, such that p is provable or provably contradictory. Such a general decision procedure is not available. Hence we have no right to assert the principle in this form.

"If the assumption that p is inconstructable leads to a contradiction, then p is true." This form of the law of double negation must also be suspended. One does not necessarily have a proof of p if he has a proof showing the assumption that p is contradictory to be, itself, contradictory. A proof that the intuitionistic denial of p is contradictory is one thing. A proof of p itself is another. The second does not immediately follow from the first. Similar remarks apply to the forms Ip--+lq--+ .q--+p and I.P 1\ q--+: Ipvlq.

220

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

Formulating LEM in intuitionistic attire it becomes pv-,p

Now is this the same law as pv"'p,

classically interpreted? Evidently not. The first formula which will be called "LEM-Int". has, as noted above, the consequence that, for every proposition p there is either a method by means of which p may be proved a theorem or that a method is available which shows that a formal contradiction results from the assumption that p is a theorem. Roughly, it means that every problem is solvable. LEM has no such consequences. It asserts that p is true or "'pis true, i.e., that p is either true or false; it is entirely consistent with the possibility that there exists some p such that we cannot prove either p or '" p "till the sun grows cold". Clearly, then, the law which is suspended in I-logics (namely LEM-Int.) is not the same law as the classical LEM. Consequently, there is an important sense in which the assertion 'LEM is suspended in the intuitionistic system' is false. Indeed, it may be noted that the classical LEM cannot even be formulated within the resources of PI. The reader is urged to use special care in translating classical laws into intuitionist laws. Considerable misunderstanding has resulted in this century from the ignoring of this simple admonition. If the interpretation of the primitive constants of PI is kept in mind, it will not be puzzling that while pv -,p

is not a theorem of the system, yet a weaker version of LEM-Int -, -, .pv-,p

is a theorem; and that while p-+-' -,p

is a theorem, -, -'p-+p

is not a theorem.

221

SYSTEMS OF FORMAL LOGIC

In connection with this discussion of the interpretation of PI, a brief presentation of an interpretation of I-logics due to A. Kolmogoroff will be given. The intuitionist systems appear to be very natural if they are interpreted as solution calculi. In this interpretation the variables of the system are taken to designate mathematical problems. The system PI then formulates a set of methods for the solution of problems in mathematics. The derivation of the expression 'p--+q' is thus interpreted as a problem in itself, namely, the problem of deriving the solution of a problem q from the solution of p. The direction, "prove q on the hypothesis p," is interpreted to mean, "given the solution of p, find the solution for q". The expression p 1\ q is taken to mean, "give the solution for both p and q". Likewise ip is interpreted, "assuming a certain solution of p is given, derive a contradiction from that solution". The expression ~p is interpreted as, "give a solution of the problem p for all values of p involved." The axioms of PI are taken as "solved" or as "assumed solved" and the solutions of other problems are to be given on their basis, e.g., if ~ A 1\ B 1\ C is considered solved - on the basis of the axioms or previously solved problems (theorems) - then ~C may be regarded as solved. The following is a statement of the further development of "the solution calculus" and its relation to I-systems and intuitionism given by Wilder, "We can ... following Kolmogoroff ... , approach the matter entirely independently, without any intuitionist presuppositions, purely as a question of formulating methods of solution of mathematical problems (such as, for example, problems in geometric construction) ... The development of the "solution calculus" turns out to coincide, symbolically, with the intuitionist propositional calculus. For instance, we will not expect to find ~. a v '" a - the Law of the Excluded Middle in the 'solution calculus', since it would be tantamount to having established, for all problems a, either a method of solving a or a method for showing that the assumption of a method of solution for a leads to a contradiction. As Brouwer asserts, the assumption of the universal validity of the Law of Excluded Middle would be equivalent to assuming that every problem is solvable."l 1

Raymond Wilder, Introduction to the Foundations of Mathematics, p. 246f.

222

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

It should be remembered that ---, ---, .p v ---'p, the weak law of excluded middle, is a theorem of the systems I, but this does not mean that it is false that the classical law is false. It may be roughly interpreted to mean that there are no provably unsolvable problems. There are unsolved problems, and may be "till the sun grows cold", but this does not mean that, in principle, some problem p must of necessity resist every attempt at solution. The axiom system PI is not that proposed by the celebrated intuitionist logician, Heyting, in his original paper of 1930, but one similar to a system devised by Scholz and Schroter, equivalent with Heyting's. Heyting's own axiom set has the same formation and transformation rules as PI. It will be called PIH. Its eleven axioms are:

HAl HA2 HA3 HA4 HA5 HA6 HA7 HAS HA9 HAlO HAll

p-+.pAp pAq-+ .qAp p-+q-+:p A r-+q A r p-+q A .q-+r-+:p-+r q-+ .p-+q pA .p-+q-+:q p-+ .pv q pvq-+ .qv p p-+r A .q-+r-+:p v q-+r ---'P-+ .p-+q p-+q A .P-+iq-+:iP

6.9 The Propositional Logic of F. B. Fitch The Fitch system, PF, may be axiomatized thus: It has the same rules of formation, transformation and definitions as does PLT. Its axioms are the axioms of P +, numbered F AI-FAll, plus the four following axioms. FAl2 FA 13 FA14 FAl5

",p~ .p~q

p= '" "'p '" .pvq=:"'p&"'q '" .p&q=:"'pv "'q

The system was originally formulated as a system of natural deduction. The illustrative proofs will, therefore, be done in this style. 223

SYSTEMS OF FORMAL LOGIC

The Fitch system of natural deduction, PNDF, is similar to PND. It differs from PND in only one rule, suspending the full rule of negation introduction and replacing it with the following derived rule. 1

'" A may be inferred as conclusion from A v '" A and a sub-echelon having A as hypothesis and the two elements Band", B as conclusions. 1 Av",A

hyp

A

hyp

2

a b c d B e ",B f ",A

1,2-e,

+ "'(res)

(The reader will notice that this rule resembles negation introduction, but differs from it in explicitly requiring the assumption of LEM at every point at which it is used. Following Fitch, we will entitle this rule the restricted rule of negation introduction, and symbolize it by + "'(res).) Five sample deductions will now be given. PNDFTl

p-::::;q&.pv "'p-::::;:"'q-::::; "'p

1

p-::::;q&.pv"'p

hyp

2 3 4

p-::::;q pv"'p "'q

1, -& 1, -&

5 6

pv"'p

7 8 9 10

p-::::;q q "'q I '" p "'q-::::; "'p

11

I~

121 TH

hyp 3, reit hyp 2, reit 7,6, --::::; 4, reit 5, 6-9, + '" (res) 4-10, +-::::;

1-11, +-::::;

It should be noted, however, that while the Fitch system requires +"'''', -"'''', -'" v, +'" &, and -'" & as primitive rules, a system equivalent with PND can be constructed in which all these rules are derivative, (cf. 4.10). 1

+'" v, 224

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

PNDFT2

1

'" .pv "'q~:"'p&q '" .pv-q

2 -p&", "'q 3 "'p 4 '" -q 5 q 6 -p&q 7 TH PNDFT3 1 2

-p&q~:",

hyp 1,-"'v 2, -& 2. -& 4, - '" '" 3,5, +& 1-6, + ~

.pv "'q

-p&q

hyp

"'p

1, -& 1, -& 3, +-2,4, +& 5, + '" v 1-6, + ~

3 q 4 '" -q 5 -p&", "'q 6 "'.pv"'q 7 TH PNDFT4

-.pv.q&r~.",.pvq&.pvr

1 1~'pV .q&r 2 -p&. '" .q&r

hyp

3 4

2, -& 2, -&

5 6

"'p

'" .q&r "'qv"'r "'q

1, - '" v

4, - "'&

hyp

7 8 9 10 11

"'p "'p&"'q "'.pvq "'.pvqv:"'.pvr "':pvq&.pvr

3, reit 6,7, +& 8, + '" v 9, + v 10, + "'&

12

-r

hyp

13 14 15 16 17 18

""p -p&"'r "'.pvr '" .p v q v: '" p v r I ",.pvq&.pvr ",.pvq&.pvr

19 TH

3, reit 12,13, +& 14, + - v 15, + v 16, + "'& 5,6-11,12-17, - v 1-18, +::>

SYSTEMS OF FORMAL LOGIC

The converse is also a theorem of PNDF. The following proof is invalid in PNDF (and PNDI), though valid in PND.

1

2 3 4

-

"'.pv"'p

hyp

'" p & '" '" p "'p

1, - '" v 2, -& 2, -& 4, - - - (invalid in PNDI) 1-5, + '" (invalid in PNDF) 6, - - '" (invalid in PNDI)

'" - p

5 ip 6 '" '" .p v - p 7 pv-p

Concerning the interpretation of the system, Fitch has written: "We shall assume that there are some propositions which are not to be asserted as true or false. Examples of these will be given later. They will be called 'indefinite' propositions. Propositions which are true or false will be called 'definite' propositions. The 'principle of excluded middle' asserts that all propositions are true or false. This principle will not be asserted here except in the limited sense of applying to definite propositions. "The classification of propositions as so far given may be outlined as follows: (A) Definite. 1. True. (a) Necessarily true. (b) Contingently true. 2. False. (a) Necessarily false. (b) Contingently false. (B) Indefinite. "An example of an indefinite proposition is the proposition expressed by the sentence. 'This proposition itself is false'. The latter proposition cannot be assumed true without also being assumed false, nor can it be assumed false without also being assumed true. If such a proposition is regarded as satisfying the principle of excluded middle, then it must be

226

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

treated as either true or false, and hence as both true and false. So we do not assert that it satisfies the principle of excluded middle."l The manner in which this question has been dealt with by most classical logicians is different. They have argued that such expressions as Fitch's example are not indefinite but meaningless. It has been noted that such expressions are self-referential, i.e., they are statements which refer to themselves. Classical logicians, then, have tended to class all such expressions as meaningless. Fitch maintains, on the contrary, that not only are not all self-referential expressions meaningless, but that some are true and important in philosophy and mathematics. He claims that the classical logic of excluded middle excludes such true and important propositions. Thus: "The problem is to find a logic which eliminates the 'vicious' sorts of self-reference that lead to the mathematical and semantical paradoxes but not those sorts of self-reference that seem to be such an important part of philosophical logic, or are required in developing the theory of numbers. The system of logic of this book seems to satisfy these demands. On the other hand, Russell's theory of types 2 , in its various forms, excludes the sort of self-reference that is essential to philosophy."3 Some remarks will be made here about the Fitch system, and LEM. The reader already knows that LEM is not asserted as a theorem in PNDF, though it holds for all theorems of the Fitch System, i.e., the theorems of Fitch are logically true, hence they are true, and hence, trivially, they are either true or false. In Fitch's classification of propositions, which is assumed to be exhaustive, he calls propositions which are either true or false (i.e., propositions which conform to the condition of LEM) definite propositions, while those which cannot be said to be either true or false are said to be indefinite. For indefinite propositions LEM does not hold. If it did hold for these propositions, then a contradiction would ensue. Fitch, in the interest of preserving the meaningfulness of some selfreferential statements, stipulates that some of these belong to the special class of "indefinite propositions". Hence he appears to classify all Fredric Fitch, Symbolic Logic, p. 8. The "theory of types" is the best known of the standard-logical techniques for avoiding the problems to which Fitch alludes in the paragraph quoted. 3 Fitch, op. cit., p. 225. 1

2

227

SYSTEMS OF FORMAL LOGIC

statements as definite or indefinite. Let us attempt to make somewhat more precise the statement "The system F presupposes that every proposition is either definite or indefinite." Our first aim must be to clarify the notion of presupposition involved in the statement. There are several possible meanings. Perhaps the clearest ofthese is that, for every propositionp in the Fitch System either it is a theorem that p is definite, or it is a theorem that p is indefinite. Now this interpretation, though it sounds plausible, is completely untenable. Why? Because it must never be a theorem that a particular p is indefinite: if there is such a theorem, then the system is an inconsistent one. Suppose there is such a theorem. It would be just this:

"'.pv"'p and this can never be a theorem in the system, since, by De Morgan's Law, this expression is equivalent with

a contradiction. Since this is the case, if the system F presupposes that every p is definite or indefinite in the sense defined, and if it is consistent, then the system presupposes that the definitiness of every p is a theorem. The reader will see that this argument uses the law of modus tollendo ponens a law which is valid in F.l Now what this appears to mean is that if it is true to say that}? presupposes every p to be either definite or indefinite then LEM is a theorem of F, since to say that all propositions are definite means just this. Apparently, then, the statement in question interpreted as above, must be false, since LEM is not a theorem of PNDF. The following valid proof of a theorem in the system PNDF is illuminating in that it presents in a formal way the results just presented. (See next page.)

1

See Fitch, op. cit., Section 10.11. Modus tol/endo ponens is the law p v q&'" p:::Jq.

228

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

Ip P I£! -p V -

2 3

4 5 6 7 8 9 10

V -

.p V

-

P

Ipv-p -.pv""p ""p&- ""p -p pv""p pv-p pv -pv "" .pv -p:::J:pV-p pv-p

11

pv ""pv "" .pv ""p 12 pv -P:::J .pv -pv "" .pv-p 13 pv ""pv - .pv -p==.:pv-p

hyp hyp 2, rep hyp

4, - 5, -& 6, + V

V

1,2-3,4-7, 1-8, +:::J hyp

V

10, + V 10-11, +:::J 9,12, + ==.

The theorem stated in step 13 states, in effect, that if a proposition is definite or indefinite then it is definite. (The observations set out above do not prove that insuperable difficulties are involved in every interpretation of the dichotomy definite/ indefinite, but they do show in general that this dichotomy is not devoid of problems of interpretation and in particular that it is by no means easy to relate the propositional dichotomy of the Fitch system to the explicit formal development of the system.) PNDF can be converted into a classical or standard system simply by adding + "" to its rules. 6.10 The Johansson Minimum Calculus The minimum calculus places upon negation even more severe restrictions than does PI, though it is more closely related to that system in its restrictions than to PF. We will use the same functors as in PI in formulating this system. The minimum calculus will be called PMIN. It was originally devised by removing a single axiom from Heyting's set, namely HAlO, and leaving the rest of the set in tact. PMIN may be obtained from P + in two ways. The first, P1MIN, closer to Johansson's original formulation, is obtained by adding the wfJ p~q~ .p~--'q~--'p

229

SYSTEMS OF FORMAL LOGIC

as a twelfth axiom to adding the two wffs MAl2 and MA13

P

+. A second formulation,

p2MIN,

is obtained by

'.pA ,p

as axioms. In the present section the second formulation will be employed since it has the interesting feature of using a form of the law of contradiction as an axiom; (some logicians have stated that this law could not function as an axiom in a logical system). The axioms of p2 MIN are those of P + in intuitionistic attire, plus the two axioms stated above. The axioms will be numbered in accordance with P MAl-MAll, with the above form of the law of transposition as MAI2 and the law of contradiction as MA13. The system is identical with PI with respect to rules of formation and transformation. The deduction theorem is, of course, provable for PMIN, pI MIN, and P2MIN• Abbreviative techniques introduced for P + will be employed freely. p2 MIN has all the theorems of P + in intuitionistic form. (Thus, e.g., 2T32 corresponds to MT32 when the symbolism has been appropriately changed.) 6.101 Deductions in P2 MIN . The key laws of the system are, p .... "p; p .... ,p .... ,p; and p .... q..... p .... ,q.... ,p. However, ,p..... p .... q is not a theorem of the system.

+,

MT91

p .... , ' p

4. f-,.pA ,p .... :p .... " p

MA5, + s, ,p/q MAI2, + s, ,pip, p A ,p/q 1,2, +IS 3, MT4, + S, ' .p A ,p/q,

5. f-,.pA ,p 6. f-TH

MA13 4,5, - ....

1. f-p ..... ,p..... pA ,p 2. f-,p ..... pA ,p .... :'.pA ,p .... :,'p 3. f-p .... :'.pA ,p.... :,'p( .... )

MT92

"p/r, -- ....

p .... ,p .... ,p

1. f-p .... ,p..... p ....p MTlI, +S,p .... ,p/q 2. f-p .... ,p..... p .... ,p MTl, +S,p .... ,p/p 3. f-p .... ,p..... p ....pA:p .... ,p..... p .... ,p 1,2, *MA5 230

SOME NON-STANDARD SYSTEMS OF PROPOSITIONAL LOGIC

4.

~p~ip~.p~p": p~iP~ .p~iP~:·

MT32,

p~ip~.p~p" .p~ip

+S,p~---,plp,

p~plq, p~ ---,plr

5.

~---,.p,,---,p

4, 3,-~ MT32, +S,plq, ---,plr MAI2, +S,p" ---,plq 5,6, 7, +IS 8, MT4, + S, p~ ---,plp, ---, .p" ---,plq, ---,plr, - ~ MA13

~TH

9, 10,

~p~ip~.p~p" .p~ip

6. ~P~P" .p~iP~:p~.p" ip

7.

~p~.p" ---'P~:---'

.p" ---'P~:---'P

8. ~P~---'P~:---' .p" ---'P~:---'P 9. ~---, .p" ---'P~:P~---'P~---'P 10. 11.

(~)

-~

The following proof uses the deduction theorem. MT93 D93

p~q~ .p~---'q~---'p

1. 2. 3. 4. 5.

p~q

6.

p~---'q~.---'q~---'p~.

p~---'q p~q~. ---'q~---'p

---'q~---'p r~s~.s~t~.r~t

P~---'P

7. ---'q~---'p~ .p~---'p 8. P~---'P 9. P~---'P~---'P 10. ---,p

1.

p~q,p~---,q~---,p

2.

p~q~p~---,q~---,p

3.

~TH

hyp hyp MA12

3, 1,-~ Variant of MT3

5, +S,plr, ---,qls, ---,plt 5,2, -~ 6,4, -~ MT92 8, 7,-~ D93 1, DT 2,DT

Up to this point p2 MIN has paralleled the development of PI. However, because of the absence of the formula: ---'P~ .p~q, the Law of De Morgan is more severely curtailed than in PI. Interestingly, the De Morgan principle MT94 is provable in P2 MIN . 231

SYSTEMS OF FORMAL LOGIC

1.

~p~.pvq

2.

~p--+.pvq~:-, .pvq~:-'p

2,1,

v q--+:-,p 4. ~q--+.p v q

3.

~-,.p

5. 6. 7.

~q~.pvq~:-,

8.

~TH

MA6 MAI2, +S,pvq/q

.pvq--+:-'q

~-, .pvq~:-'q

~-, .pvq~:-'pl\:.

-, .pvq--+:-,q(~)

---+

MA7 MAI2, + S, q/p, p v q/q 5,4, -~ 3,6, *MA5 7, MT32, +S, -, .p v q/p, -,p/q, -,q/r, -~

and thus, analogously with the proof of !TlOO, the weak intuitionist law of excluded middle is provable. But because the system does not possess -'p~ .p--+q, none of the remaining De Morgan principles is provable. Hence while PF possesses the full Law of De Morgan (while lacking the law of contradiction and the law of transposition in any form), and while PI possesses nine out of the sixteen implications which make up the Law, p2 MIN possesses only one of these implications, namely that proved as MT94. Doubts about the notion of contradiction as a primitive idea of intuitive mathematics and, particularly, about the dictum that a contradiction implies anything, (embodied in Al 13 of PI, and HAlO of PIH) motivated the construction of PMIN. All the systems of the minimum calculus become equivalent with PI and PIH when -'p--+ .p--+q is added to them as an axiom. Adding this axiom to p2 MIN gives PILNC. If we desire to develop a system of natural deduction embodying PMIN, this can be done by deleting the rule of negation elimination from the transformation rules of PNDI, and leaving the rest of the primitive structure of PNDI unchanged. This revision gives a system of natural deduction equivalent with PMIN : PNDMIN. Kolmogoroff developed a more limited system having only negation and implication as primitive functors. This system, PI-, ...., may be reproduced by employing the first two axioms of P +, p--+ .q~p p--+ .q~r~:p~q--+.p--+r,

and the two negation axioms of PI, p~ -,p~-,p

-'P~

232

.p--+q.

SOME NON -ST ANDARD SYSTEMS OF PROPOSITIONAL LOGIC

He discovered that if, for every theorem of the full calculus PLT', one substitutes I I a for every variable a, the result is a theorem of PII ...,. Thus, since

'" p ==> '" q ==> • q ==> P is a theorem of PLT' I I

IP~I

is a theorem of PII

I

Iq~.

I

IP~I

Iq

...,.

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CHAPTER 7

THE LOWER FUNCTIONAL CALCULUS

7.1 Summary and Remarks The lower functional calculus! forms the next step "up" in the logical hierarchy. Most formulations include the propositional calculus as a part and, hence, include axioms of propositional logic as among their axioms. Systems of the lower functional calculus always include individual variables (for which names of individuals may be substituted), predicate variables (for which names of properties of individuals may be substituted), and at least one quantifier (usually the so-called "universal quantifier"). A standard (S) system of the lower functional calculus may be built up out of the propositional system PLT by adding axioms and definitions, or out of PLT', by the same procedure. The system of functional logic developed in this Chapter and in Chapter 8 uses PLT' as its propositional basis. It is called LFLT,.2 The present chapter builts that part of LFLT' which possesses among its predicate signs, none but signs for simple or one-place predicates. The full system is outlined in Chapter 8. Chapter 5 gave a demonstration that the system PLT' is a complete propositional calculus, complete within the limitations of its own formalism. But, considering the main interpretation of PLT' as a logic of unanalyzed propositions, it is clear that there is another sense in which PLT' is not complete: it is not complete in the sense that it will not verify all valid inferences of formal logic, since some such inferences are not fully formulable within the limits of the formation rules and definitions 1

Often called "The functional calculus of the first order".

LFLT' resembles the lower functional calculus devised by Bertrand Russell and published in 1908 (see Bibliography). It differs from the Russell system in using the 2

propositional calculus PLT' instead of the Russell calculus PPM, and in formulating explicit rules of substitution missing in the Russell system. It is similar to the system Ftp of Alonzo Church (see his Introduction to Mathematical Logic, Chapter IV, par. 40), but unlike Ftp possesses individual constants and predicate constants.

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THE LOWER FUNCTIONAL CALCULUS

of PLT" A perfectly elementary valid inference is represented in the following conditional statement. (I)

If all males are mortal, then if some Americans are males, then some Americans are mortal.

It might be thought that this inference could be represented in the propositional calculus thus:

(II)

q=>r=> .p=>q=> .p=>r

making (I) a substitution on (II). Another such inference is, (III)

If Bertrand Russell is a man, then if anything is a man then it is mortal, then Bertrand Russell is mortal.

Which, it might be thought, could be regarded as a substitution instance of, (IV)

p=> .p=>q=>q

Since both (II) and (rV) are theorems ofPLT', PLT' is "complete enough" to validate them. However, (III) is not a correct substitution on (IV) and (I) is not a correct substitution on (II). It will be recalled that an application of the rule of substitution is correct if and only if carried out consistently. (III) is not, however, consistent substitution on (IV). For the first occurrence of p the statement "Bertrand Russell is a man", while for the second occurrence of p the statement "anything is a man" is substituted; likewise two distinct statements are substituted for q. Hence, if we were to represent (III) in the propositional calculus it would be, (V)

p=> .q=>r=>s

which is not a theorem of PLT" Likewise, if (I) is a substitution on an expression of propositional logic it must be expressed, (VI)

q=>r=> .p=>s=> .p=>r

which is not a theorem of PLT'. Hence if we wish to have a logic which verifies reasonings of these forms we must extend the propositional calculus. The lower functional calculus gives such an extension.

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SYSTEMS OF FORMAL LOGIC

7.2 Rules of Formation of LFLT' The lower functional calculus LFLT' which is constructed in this chapter has, besides all the elements of the primitive basis of PLT', the following primitive constituents. 7.21 As primitive functors of the system we have the following improper symbols, (together with the functors of PLT'), (1) ( (2) ) 7.22a An infinite list of variables from the end of the alphabet is employed, e.g.,

x, y,

Z, Xl, YI, Zl, X2, •.•

These are called individual variables, in accordance with the main interpretation of the system. (According to the main interpretation, names of individuals may be substituted for them. The expressions "name" and "individual" are, for the present, undefined. Individual variables are said to have individuals as values.) 7.22b An infinite list of variables, called singulary functional variables or predicate variables, taken from capitals of the Greek alphabet,

is employed. These are understood to be singularly or one-place variables, i.e., they take only one expression as argument in accordance with rule (2) under 7.23 stated below. (Singulary predicate variables are said to have one-place properties of individuals as values.) The full lower functional calculus includes not only singulary predicate variables and constants but also signs for many- or n-placed predicates. In the present chapter only a part of the full lower functional calculus LFLT', the so-called singulary or restricted calculus, is developed. The rules necessary for complete development of LFLT' for n-place as well as for one-place predicate signs are given at the beginning of Chapter 8. 7.22c A list, not necessarily infinite, of individual constants, abbreviated by small Greek letters from the beginning of the alphabet is employed: a,

fl,

y, 6, aI, ...

(These are, according to the main interpretation, abbreviations for names 236

THE LOWER FUNCTIONAL CALCULUS

of 'individuals', e.g., "Socrates", "Plato", "Thomas Blakeley", "I. M. Bochenski", "this table", "this house", "this atom", etc.) 7.22d A list, not necessarily infinite, of special individual constants,

a*, fJ*, y*, 15*,

al *,

...

is employed, the function of which is explained later. 7.22e A list, not necessarily infinite, of predicate (or functional) constants, abbreviated by capital letters from the middle of the Roman alphabet is employed, F, G, H, Fl, ... (These are, according to the main interpretation, abbreviations for names of properties of individuals, e.g., "white", "circular", etc.). 7.23 The necessary and sufficient conditions under which an expression of LFLT , is a wff of LFLT' are as follows. (1) A propositional variable standing alone is a wff. (2) If S is a predicate variable or a predicate constant, and a is an individual variable or an individual constant, then Sa is a wff. (3) If a formula A is a wff then ,...., A is a wff. (4) If A and Bare wffs then A::::lB is a wff. (5) If A is a wff and if a is an individual variable, then (a)A is a wff. The symbol (a), where a is an individual variable and where the parentheses (, ), are the improper symbols added to the system PLT' is called the universal quantifier. 7.3 Transformation of LFLT' See section 7.7 on page 248f. 7.4 Axioms of LFLT' See section 7.8 on page 255. (The transformation rules and axioms of axiomatic systems are usually stated concurrently with the rules of formation and definitions. In the present case, however, it was thought appropriate to reserve these rules for a later stage, after some explanation of the sense of the system has been presented.)

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SYSTEMS OF FORMAL LOGIC

Definitions of LFLT'

7.5

In addition to the definitions (D 1), (D2), and (D3) of PLT', the following definition is included in LFLT'. The expression on the left, below, is an abbreviation for the expression on its right: (D4)

(3a)A

where a is an individual variable and A is a wff. The symbol (3a), where a is an individual variable, is called the existential quantifier. This definition is called the definition of the existential quantifier. It may serve as a rule of inference, allowing the substitution of (3x) for any occurrence of '" (x) "', and of '" (x) '" for any occurrence of (3x). It is understood that this substitution need not be carried out for all occurrences of ",(x)", or (3x). Thus the following is valid: 1. f- ",(x)",. ",(x)", ",Gx]

read, "For all x, if x is F tnen x is not G", is true. Thus (D) and (G) are contradictories and (E) and (H) are contradictories. Both quantifiers are called operators. The matrices (or statements) to which they apply are called operands. In order to make clear the circumstances under which a matrix is converted into a statement by quantification, three concepts are introduced, (a) that of the bound variable, (b) that of the free variable, (c) that of the closed formula, along with some supplementary ideas. The range of a quantifier is determined as follows: To the right of every quantifier there is a left bracket: (x) [ ... (3x) [ ...

This bracket may be described as the associated left bracket of the 241

SYSTEMS OF FORMAL LOGIC

quantifier. By the rules of formation, to every left bracket there corresponds a right bracket which is its mate, as in previously developed systems. The right bracket which is the mate of the associated left bracket of the quantifier may be described as the associated right bracket of the quantifier: thus (x) [ ... ] (3x) [ ... ]

In some cases, e.g., (F) above, one or more negation signs may intervene between the quantifier and the first bracket to its right. In such a case, when full bracketing is restored, the associated left bracket will be to the immediate right of the quantifier and to the left of the initial negation sign following the quantifier, thus, (x) [-[ -[... ]]] The range of the quantifier is the entire expression falling between the

associated left bracket and the associated right bracket of the quantifier. Trivially, the quantifier also is said to be in its own range. Likewise, any expression which is not between the associated brackets of the quantifier, and is not the quantifier, is said to be outside the range of the quantifier. Thus the matrix [Fx::>Gx] of (D) is in the range of the quantifier (x), and the quantifier (3x) ranges over the entire matrix [Fx&Gx] in (E). The expression over which a quantifier or operator ranges is called its operand. An occurrence of a variable, a, in a wff B is said to be bound by the quantifier (a), (or (3a)), if (1) the variable a has the same alphabetic form as the variable of the quantifier (a), and if (2) the variable is in the range of the quantifier and if (3) the variable is not bound by an occurrence of another quantifier (a) subsequent to the initial occurrence of (a). Trivially, the variable of the quantifier itself is said to be bound by the quantifier. An occurrence of a variable, a, in a wff, B, which is not bound by any quantifier is said to have free occurrence in B, or, more simply, to be free in B. A variable bound by a quantifier in B is called a bound variable of B. A variable with free occurrence in B is called a free variable of B. Thus the two occurrences of x in (C) are free while all the variables of (D), (E), (F), (G), and (H) are bound by their respective quantifiers. A few more examples are presented to make these notions clearer. 242

THE LOWER FUNCTIONAL CALCULUS

(1)

[(x) [Fx::>Gx]::>Gx]

The first three occurrences of x, (including its occurrence in the quantifier) are bound, but the fourth occurrence is free, since it is not in the range of any quantifier of (I).

(J)

[(x) [[Fx::> (x) [Fx::>Gx]]::>Gx]]

In (J) all occurrences ofthe variables are bound. But only the first, second, and sixth occurrences of x are bound by the first quantifier. The third, fourth and fifth occurrences of x are bound by the second quantifier, but not by the first, even though they are in the range of the first quantifier. This analysis conforms to condition (3) of the definition of bondage given above. In this case, e.g., the fourth occurrence of the variable conforms with condition (1) and condition (2) of the definition of being bound by the initial quantifier, but it does not conform with condition (1) since the second quantifier takes precedence over the first with respect to binding the fourth occurrence of the variable, x. Condition (c) is imposed in order to secure that if a variable, a, is bound by a quantifier (a) or (3a) it is bound by one and only one quantifier. If we wish to avoid formulae resembling (J) this may be secured by the use of two variables as follows:

(K)

[(x) [[Fx::>(y) [Fy::> Gy]]::> Gx]]

A formula of this form allows the expression of all statements made by use of the form (J) with its repetition of the same quantifier. It should be obvious that formulae (J) and (K) are wi The consequent of (K), Gx, is wfby virtue of formation rule (2). The matrices Fy and Gy are wfby the same rule. Thus by formation rule (4) the matrix Fy::>Gy is wi By formation rule (5), the universal quantification (y) [Fy::> Gy] of the matrix Fy::> Gy is wi Fx is wf by formation rule (2). Thus by formation rule (4) [Fx::> (y) [Fy::> Gy]] is wi Again by (4) the matrix [[Fx::> (y) [Fy::> Gy]] ::> Gx] is wi Rule (5) then shows that the formula [(x)[[Fx::>(y)[Fy::>Gy]]::>Gx]] is wi The following is an example of a formula with a quantifier in which all variables (except the quantifier variable itself) are free.

(L)

[(x)[Fy::>Gy]::>Hx]

The operand of the quantifier in (L) is the matrix [Fy::> Gy]. The matrix 243

SYSTEMS OF FORMAL LOGIC

Hx is not part of the operand of (x). Thus all occurrences of variables (other than the quantifier variable) in (L) are free. (L) is a wff: Hx is wf by (2), Fy and Gy by the same rule. Thus [Fy::lGy] is wf by (4). (x)[Fx::lGy] is wfby (5); thus (L) is wfby (4). A quantifier which ranges over a matrix (or statement) none of whose variables it binds, is said to have vacuous occurrence, since the resulting formula is equivalent with a formula having the same form but without the quantifier. The formula (M)

(x)[Human Socrates]

read, "For all x, Socrates is human", (where we assume that 'human' is one of the predicate constants and 'Socrates' one of the individual constants of the system), is another instance of a quantifier with vacuous occurrence. Though formulae like (L) and (M) appear awkward, they are retained as wffs in order to allow full freedom of quantification over

wffs. (N)

[(x) [(y)[[Fx::lGy]::I [ ",GY::l "'Fx]]]].

(N) illustrates multiple quantification. A formula is said to be mUltiply quantified if more than one distinct quantifier appears in it. (N) is wf. The formula [[FX::l Gy] ::I [ '" Gy::l '" Fx]] is wf by rules (2), (3), and (4). The formula (y)[[Fx::lGY]::I[ ",Gy::l "'Fx]] is wfby rule (5). Thus (N) is wfby (5). When dealing with expressions having only universal quantification or only existential quantification, it is obvious that the order of the quantifiers is immaterial; hence formulae resembling (N) will be expressed as formulae resembling (N'). (N')

[(x)(y) [[Fx::l Gy]::I [ '" Gy::l '" Fx]]]

It is clear that both (0) and (P) are wf

(0)

[(3x) [Fx]::I [(y) [Fy::l Gy]]]

(P)

(x) [P::lFx]

where p is a propositional variable. It is now possible to define the circumstances under which a formula of LFLT' is said to be a quantificationally closed formula. A formula with no occurrences of free variables, i.e., having either no variables or only bound occurrences of variables is a quantificationally closed formula. IfB is a quantificationally closed formula ofLFLT' then B is a statement 244

THE LOWER FUNCTIONAL CALCULUS

of LFLT" A formula B with at least one free occurrence of a variable is said to be a quantificationally open formula. A quantificationally open formula with n occurrences of free variables is said to be an n-place q-open formula". If B is a quantificationally open formula then B is not a statement, but a matrix of LFLT" By the criteria presented here, it can be seen that (A), (B), (D), (E), (F), (G), (H), (J), (K), (M), (N) and (0) are quantificationally closed formulae (q-closed formulae) and are thus statements. (C), (I), (L) and (P) are q-open formulae, thus matrices. It can be seen that each of these matrices is convertable into a statement by substitution or by quantification, or both. In the case of q-closed formulae, the following should be observed with some care: the bound variables no longer serve as place markers for which individual names are to be substituted, but rather are to be regarded as points of reference to various positions of quantification. Thus (x)[Fx::JGx]

is not "about" x, but about the relation the functional constants F and G sustain to each other in the context of statements. In the language of Alonzo Church, a statement or a matrix, "which contains x as a bound variable only has a meaning which is independent of x - not in the sense of having the same value for every value of x, but in the sense that the assignment of a particular value to x is not a relevant procedure",! It should be noted that the initial predicate of a statement of the form

(here the predicate F), generally serves to specify the domain ofindividuals which the statement is "about". E.g., if F is the predicate 'human', then the statement asserts something about all humans. If the system were revised to become an applied functional calculus of the first order, this specification would be unnecessary since the domain of individuals the names of which are substitutable for individual variables would be fixed by the rules of formation of the system. For instance, if the formation rules specify that the domain of individuals of the system is the natural numbers, and if the predicate G names the property equal to or greater 1

A. Church, Introduction to Mathematical Logic, p. 40.

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SYSTEMS OF FORMAL LOGIC

than one', the qualifying antecedent, "If x is a natural number", is unnecessary, the formula being expressible as (x) [Gx]

The system LFLT', however, is fully general in this respect. Thus, the characterizing antecedent is required. The reader will note that functional variables are included in none of the formulae (A-P). However, formulae containing none but functional variables as functional signs will be dealt with in an exhaustive way below. As introductory to the presentation of the axioms and rules of LFLT' a formula possessing functional variables will be considered and shown to be L-true. The formula is (Q)

[(x)[(l)x]=>[(l)y]]

Observations on the formula (Q) 1. The formula (Q) is a purely logicalformula ofLFLT" In this it differs from all other members of the formula-sequence (A)-(Q). The formulae (A)-(P) all contained, besides logical signs, one or more non-logical or descriptive signs, e.g., the predicate constants. 2. (Q) is a q-open formula. The occurrences of the variable x are bound by the quantifier in (Q); but y has free occurrence and the functional variable (l) has free occurrence in (Q). By the formation rules of LFLT', while individual variables may be bound or free all occurrences of functional or predicate variables (and of propositional variables) are free occurrences. Since (Q) is open, it is a statement matrix not a statement. 3. In order to facilitate approach to the peculiarities of (Q) it is compared with two closely related formulae, both of which are also L-true. The first of these formulae is (Q')

[(x)[Fx]={Fa]]

4. (Q') is a q-closed formula since it contains no variables not bound by a quantifier. Hence (Q') is a statement. It asserts that if everything has the property F then the individual a has the property F; if everything is F then a is F. This is an L-true statement, as can be shown by the following considerations. There are only two relevant cases: 246

THE LOWER FUNCTIONAL CALCULUS

Case 1. Fa is true. If Fa is true then the formula is true, since a material implication is true when its consequent is true. Case 2. Fa is false. If Fa is false, the statement (Q') is false if its antecedent (x) [Fx] is true, and true in any other circumstance. But (Q/) cannot have the value false, since if Fa is false then (x) [Fx] must also be false. If Fa is false then the individual a does not have the property F. But if a is not F, then ",[(x)[Fx]], i.e., not everything has the property F, is true. But if this is true then (x) [Fx] is false. Hence, if the consequent of (Q/) is false then so is the antecedent. Thus (Q/) is true in every possible case, regardless of whatever are the facts about the individual a and the property F. 5. The second formula is (Q")

(x)[Fx]:::{Fy]

(Q"), unlike (Q/), is a q-open formula, since the variable y occurs free in (Q"); hence (Q") is a matrix. But it is L-true in the sense that it is true for all values of its free variables. (Q") contains only one occurrence of a free individual variable, namely the single occurrence of y; all the other symbols of (Q") being constants or bound individual variables. Now given any substitution of an individual constant for the free variable y in the consequent of (Q"), there are two cases: Case 1. Substitution on the matrix Fy gives a true statement. In that case, (Q") is true by the same considerations which showed (Q/) true when Fa is true. Case 2. A substitution on the matrix Fy gives a false statement. In that case (Q") is true, by the same considerations which showed (Q/) true when Fa is false. Thus, whatever the value of y, the matrix (Q") is true. Hence (Q") is L-true. In fact, (Q') is obtainable from (Q") simply by substituting the constant a for the free individual variable y in (Q"). Thus (Q/) is a substitution instance of (Q"). 6. The original formula, (Q), differs from (Q/) in that it has no occurrences of individual constants. It differs from (Q") in that it has no occurrences of functional constants. Hence, it is a purely logical formula, since it has no essential occurrences of non-logical constants (since it has no occurrences of non-logical constants at all). (Q) is a q-open formula and hence a matrix. Like (Q"), (Q) is an L-true 247

SYSTEMS OF FORMAL LOGIC

formula. The considerations given in number 5 showed that (Q") holds for any value of its free individual variable y. The present number shows that (Q) holds for any value of the predicate variable, ,p, which occurs free in (Q). Substitution of an arbitrarily selected predicate constant, say F, for the predicate variable, ,p, of the consequent of (Q) gives the matrix, Fy. This substitution forces the substitution of F for ,p in the antecedent, thus giving the statement (x) [Fx] as antecedent. By this substitution, we have the matrix

(x) [Fx] ::>[Fy] gained from (Q). This matrix is the same as (Q"). Hence the reasoning applied to (Q") applies to it. Now any arbitrary substitutions of individual constants for the free variable of (Q) and any arbitrary substitutions of predicate constants for the free predicate variable of (Q) give true statements, by reasoning exactly analogous with that given for (Q"). Case 1. Substitution on the individual variable and the predicate variable of the consequent of (Q) makes the consequent true. Then (Q) is true for that substitution by the same reasoning as before. Case 2. Substitution on the individual variable and the predicate variable of the consequent of (Q) makes the consequent false. Then by previous reasoning, the antecedent is false. Thus the implication between antecedent and consequent is true. Thus (Q) is true for any arbitrarily selected values of its free variables regardless of the facts about the individuals or the properties the names of which are substituted for the variables. Thus (Q) is L-true. We are now prepared to complete the statement of the primitive basis of the part of LFLT to be developed in this Chapter by specifying the rules of transformation and the axioms of the system. 7.7 Rules of Transformation of LFLT'

1. Rule of modus ponens From A and A=> B, B may be inferred. (The application of modus ponens is signalized in the justification column in the same way as in the propositional calculus, namely by the symbol

248

THE LOWER FUNCTIONAL CALCULUS

-::) preceded by the step numbers of the steps, in the order required to carry out the application of the rule.) 2. Rule of universalization From A, where A is a wff and a is an individual variable, (a) A may be inferred. (The application of the rule of universalization is indicated in the justification column by the use of the abbreviative symbol + UQ, read, "Universal quantifier introduction", preceded by the step number of the wffto which the rule has been applied.) 3. Rules of substitution

i. Rule of substitution for propositional variables. From the wff A with occurrences of the propositional variable a, the wff B may be inferred by substituting the wff r, for a at every occurrence of a in A - subject to the following restriction: the substitution of r for a is valid provided that no individual variable which occurs free in r becomes bound as a result ofr's substitution for a in A, and thus becomes bound in B. (The application of this rule is indicated in the justification column by the symbol' + S' preceded by the number of the formula from which the result is derived by substitution, and followed by the indications of substitutions made, e.g., ria, read, "Gamma substituted for a." Note: r need not be a formula from the propositional calculus, but may be a wff of functional logic, e.g., f/>x, or Fx, or Fa, so long as the restriction on the rule is observed. The rule is to be understood as formulated so as to allow simultaneous substitution. This holds also for the remaining substitution rules. The reason for the restriction upon this rule, and those upon the subsequent substitution rules, will be made clear following the full statement of the rules of transformation and the axioms of the system.) Application of the rule of substitution for propositional variables and of the rule of universalization allows the transformation of laws of the propositional calculus into laws of the restricted functional calculus of the first order. The procedure works this way:

249

SYSTEMS OF FORMAL LOGIC

1. r[p::J[q::Jp]] 2. r[Wx::J[l[Ix::JWx]] 3. r(x)[Wx::J[l[Ix::JWx]]

LFAI (see p. 255) 1, +S,Wxjp, l[Ixjq 2, +UQ

The function of + UQ in LFLT' is to allow generalization of laws of the propositional calculus so that they become functional laws, with quantification. ii. Rule for substitution on individual variables. From the formula A with occurrence of the individual variable a, the expression B may be inferred by substitution of b for a, where b is an individual variable or an individual constant, subject to the following restriction: the substitution of b for a must be such that, if b is an individual variable, b does not become bound at any of the positions at which it is substitutedfor a in A; likewise, in applying this rule substitution of an individual variable or individual constant b for an individual variable a is permissible only where a has free occurrence in A.

(The application of this rule is shown in the justification column by the symbol "+ SI", the same procedure of indexing and specifying substitutions being used as that given above for substitution on propositional variables. This similarity holds also in the following rules.) iii. Rules of substitution for predicate variables. (a) Rule of substitution of a predicate variable or a predicate constant for a predicate variable. (This rule is sometimes called the rule of simple substitution for predicate variables.) From a wff A having occurrences of a predicate variable, V, the wff B may be inferred by substituting W for V, where W is a predicate variable or a predicate constant, and where both Wand V are singulary.

(Since predicate variables of LFLT' always occur free in the wffs of the system, no restrictions need be imposed here. The application of this rule is shown in the justification column by the symbol" + SP.") (b) Rule of substitution of a wff for a predicate expression. (Sometimes

called formula-substitution or complex-predicate-substitution.) Prior to 250

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the general statement of this rule some discussion and illustration will be presented.1 In the discussion of the formula (Q) (Q)

[(x)[Wx]::J[Wy]]

it was shown that (Q) is L-true, thus true for all substitutions on its free variables (cf. p. 246f). Now not everyone-place property which can be expressed in LFLT' is designated by one-place predicates like F, G, H, etc. Inspection is sufficient to show that any matrix having an individual variable as its only free variable expresses a property of individuals. Thus (Q") (p. 247) states a property of all individuals. Consider the matrix (R)

FX::JGx

(R) expresses a property of individuals since the only free variable of(R) is the individual variable x. The property of individuals which is expressed by (R) is the property that if x is F then x is G. Now (Q) asserts that if any arbitrarily selected property W belongs to all things then any arbitrarily selected individual is W. Now, since (Q) is L-true, the property expressed by (R) must hold if (R) is substituted, according to a certain rule, for W. The result of such a substitution is

(S)

[(x) [Fx::JGx]::J[Fy::JGy]]

which states "If everything that is F is G then if y is F then y is G." (S), we wish to say, is a substitution instance of (Q), as legitimate as (Q') and (Q"). But it is not a case of the simple substitution of a predicate or a predicate variable for a predicate variable. An expression (R) has been substituted in (S), not merely for the predicate variable W of (Q) but for the entire statement matrix f/)x of (Q), and similarly for the matrix Wy. In a case of substitution of this kind we will call a matrix of the type (R) a substitution matrix for (Q). The substitution matrix is applied to the matrix on which substitution is made as follows. If the expression Wx is the first statement matrix having the predicate variable W within the formula upon which substitution is made, then the substitution matrix is applied by substituting this matrix for the expression Wx. This first statement matrix having the predicate variable W within the formula The author is indebted to Rudolph Carnap, Introduction to Symbolic Logic and its Applications, pp. 46ff, for the main lines of the present exposition.

1

251

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upon which substitution is made is called the initial matrix of the formula. In the case of (Q), [Fx:::> GxJ is substituted for rpx, where rpx is the initial matrix of the formula. To give a scheme for the substitution of the substitution matrix for the initial matrix we can write

[Fx:::> GxJ/rpx which means that Fx:::> Gx is what rpx becomes on formula substitution of the first for the second. Since (Q) also includes the matrix rpy as a constituent, consistent substitution on the predicate variable rp requires the application of the substitution matrix to rpy as well. Since the individual variable y occurs in this matrix, the rule of formula substitution requires that y be substituted for the free variable x is in the substitution matrix. Thus [Fy:::> GyJ is substituted in (Q) for rpy. The simultaneous substitutions of the substitution matrix to (Q) gives (S). The scheme showing that both substitutions have been carried out would be

[Fx:::> GxJ/rpx [Fy:::> GyJ/rpy If the formula upon which substitution is being carried out involves more than two occurrences of rp as predicate variable of other individual variables or individual constants, the procedure is precisely analogous. Suppose the individual variable z and the individual constant a are assigned the predicate variable rp in the formula upon which substitution is being carried out. Correct substitution is indicated as follows,

[Fx:::> GxJ/rpx [Fy:::> GyJ/rpy [Fz:::> GzJ/rpz [Fa:::> GaJ/rpa Note that once the correct formula substitution for the initial matrix of the formula has been determined by application to it of the substitution matrix, the determination of correct formula substitution for the subsequent matrices of the formula is strictly determined, (where, of course, the substitution matrix is applied only to subsequent matrices having the same predicate variable as the initial matrix). In the following formal statement of the rule of substitution of a wff for a predicate expression, the following abbreviations are used. 252

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S = df Statement matrix upon which substitution is made. SM=df Substitution matrix applied to S. 1= df The initial matrix of S. The rule of formula substitution may now be stated, following Carnap:

Given a matrix S with occurrence of a predicate variable, V, for which substitution is made having an initial matrix I, and given a matrix SM applied to S, formula substitution is carried out in accordance with the following rules:

A. I is made up of n individual variables where n ~ 1 and corresponding occurrences ofV.

B. SM of I is a statement matrix such that, (1) the variables of I do not have occurrence in the quantifiers of SM. If SM is quantified there is a free occurrence of the individual variables of I in SM. (2) The variables which have occurrence in S but do not have occurrence in I do not have occurrence in SM. Variables of SM which have occurrence in neither I nor S have arbitrary occurrence in SM. C. From the scheme

SM/I schemes for the remaining individual variables or individual constants which are arguments of V in S are obtained by substitution of those variables or constants for the variables of I.

D. The application of SM to an arbitrary formula, F, of S with occurrence of V is carried out as follows. Replace the formula F with the formula SM substituting the individual variables or individual constants which are arguments of V in F for the individual variable of SM in accordance with C above. (Thus if a subsequent matrix ofS is Vz, the SM for Vz is obtained by the same method used to obtain SM for I, substituting zfor the individual sign of I.) Thus: From a wff A the formula substitution of A, B, may be inferred by application of the four rules of formula substitution, A, B, C, D. The application of the rule of formula substitution is indicated in the justification column by the symbol + SF. This symbol is preceded by the

253

SYSTEMS OF FORMAL LOGIC

step number of the wff to which the rule of formula substitution has been applied and is followed by a set of formula substitution schemes

indicating, in serial order, all the formula substitutions made which give the formula being justified. Thus if Q is step n, and S is derived from Q at step n+ I by formula substitution, the proof configuration would be as follows:

n+1

f-[(x) [$x]::l [$y]] f-[(x) [Fx::l GX]::l [FY::lGy]]

n+2

f--

n

n, +SF[FX::l Gx]/$x, [FY::l Gy]/$y

iv. Rule for substitution on bound variables.

If the integral wff C occurs in A and if the individual variable, a, occurs bound in C, and if b is an individual variable which has no occurrence in C, then from A, B may be inferred by substituting b for a everywhere in

C. The application of this rule is indicated in the justification column by the symbol + SB.

v. Rule of substitutivity of equivalence. From the 'lfff A having occurrence of a wffC, the wffB may be inferred by substituting D for C at zero or more places in A (not necessarily for all occurrences of C) where C and D are materially equivalent; subject to the following restriction: no variable of D may become bound in B which was not bound in C of A. (The application of this rule is indicated by the symbol + SE followed by a specification of the formula which is substituted and of the formula replaced. This is done, as before, by the symbol D/C, where D represents the formula substituted and C represents the formula replaced and where C=D. This rule is not independent in LFLT'. It can be derived from the remaining rules. None the less it will be regarded as a primitive rule of the system. (Cfthe special definitional equivalence on page 322)). 254

THE LOWER FUNCTIONAL CALCULUS

7.8 Axioms of LFLT' LFAI LFA2 LFA3 LFA4

f-p::::l.q::::lp f-p::::l .q::::lr::::l:p::::lq::::l .p::::lr f-"'P::::l "'q::::l.q::::lp f-[(x)[P::::lqJx]::::l[P::::l(x)[qJx]]] (Law of quantifier distri-

bution) LFA5

f-[(x) [qJx]::::l [qJy]] (Law of specialization)

(Let it be agreed that the indices LFAI-LFA5 index the axioms of the particular system of the lower functional calculus here developed, LFLT" In future deductions and explanation the method of abbreviating brackets instituted at the beginning of the book will be employed for functional expressions, thus, and are abbreviations, by this convention, for LFA4 and for LFA5 respectively. 7.81

Explanation of the restrictions on substitution rules.

(1) The restriction on the rule of substitution for propositional variables

is imposed, like all the remaining restrictions, in order to prevent invalidity. That invalidity would occur if the restriction were violated is shown using the axiom LFA4: violation of the restriction would allow the substitution of the matrix qJx for p; x has, of course, free occurrence in the matrix qJx. The result of this substitution gives

(A')

(x).qJX::::lqJX::::l:qJX::::l(x)qJx

Now while the axiom LFA4 is L-true, this substitution, violating the restriction on substitution for propositional variables, is L-indeterminate. The antecedent of (A') is L-true; it is, in fact, the law of identity for the functional calculus of the first order, soon to be proved. Hence if (A') is L-true, its consequent must likewise be L-true. But it is not, as shown by a substitution instance, substituting the individual constant a for its only free individual variable and the constant "human" for the free occurring predicate variable qJ: "If a is human then everything is 255

SYSTEMS OF FORMAL LOGIC

human"; this is false when it is true that a is human, since the consequent is false; hence the consequent of (A') is not L-true. Hence (A') is not L-true. 1 Hence the rule of substitution for propositional variables without the requisite restriction will lead to invalidity. (2) Violation of the restriction on the rule of substitution for individual variables leads to the following invalidity: (B') is a valid substitution on LFA4: (B')

(x). WY:::J WX:::J: WY:::J (x) Wx

substituting Wy for p. Substituting x for y in B', disregarding the restriction, gives (e')

(x). WX:::J WX:::J: WX:::J (x)Wx

which is the same as the invalid A'. (3) No restrictions need be imposed upon the rule of simple substitution for predicate variables in LFLT', (In more inclusive systems, in which predicate variables may be bound by predicate quantifiers, e.g., in the functional calculus of the second order, restrictions must be imposed so that substitution is permitted only on free occurrences of predicate variables.) (4) The restrictions imposed in the B part of the rule of formula substitution are set out with the same motives as those imposed for forgoing rules, namely to prevent inadvertent binding or freeing of variables by substitution such that invalidity results. (5) The rule for substitution on bound variables is restricted to integral formulae. This restriction would be violated if in LFA4, i.e., (E')

l-(x).P:::JWx:::J:P:::J(x)Wx

we substituted y for x, since this would result in (F')

(x).P:::JWY:::J:P:::J(x)Wx

an obvious contingency; the quantifier has vacuous occurrence in the The reader should recall that the status of bound and free individual variables is quite distinct (cf p. 245). A free individual variable, regardless of its alphabetic shape, is available for substitution, either by another free individual variable or by an individual constant (as above). But bound variables are not subject to substitution by constants, even if these variables have the same alphabetic form as do the associated free variables in the same formula, which are subject to such substitution. 1

256

THE LOWER FUNCTIONAL CALCULUS

antecedent but essential occurrence in the consequent of (F'). Abiding by this restriction the correct substitution would be

(G') or

~(y).p':::)CPy-:::;):p':::)(x)CPx,

(H')

~(y).p':::)CPy':::):p':::)(y)CPy,

or (1')

(x).p':::) .(y)CPy,:::) IJ'x':::):p':::)(x).(y)CPy':::) IJ'x;

«I'), of course, also requires use of formula substitution.) The full substitution of y for x throughout (H') is unnecessary in order to abide by the restriction, since the antecedent of (E') is by itself an integral formula. In fact, (G') and (H') state exactly the same thing, the only difference being the alphabetic shape of the quantified variables in the antecedent and consequent in (G'). For reasons stipulated before in discussing quantification, this difference of alphabetic shape is irrelevant. Both (G') and (H'), irrespectively, can be read: "For anything, if p implies that it is cP then p implies that anything is CP." The rule of substitution for bound variables merely reflects the fact that, with respect to quantification and bondage, alphabetic changes of individual variables in integral formulae are irrelevant to the truth-value of the formulae. (6) If C is an integral formula constituting a part of the formula A and if a is a bound variable occurring in C it is forbidden to substitute b for a throughout C if it is the case that b has occurrence in C. Violation of this restriction would lead to invalidity. The formula (B')

(x). CPy ':::) CPx ':::): CPy ':::) (x) CPx

is, as seen above, a valid substitution on LFA4. Here the antecedent and consequent are both integral formulae. If, in violation of the restriction, y is substituted for the bound variable x at every point in the antecedent, the contingent formula (D')

(y). CPy ':::) CPy ':::): CPy':::) (x)CPx

results. This formula can be seen to be contingent on the same grounds as for (C'). (As an exercise the reader should show that the rules of substitution of LFLT' are L-truth preserving, using arguments analogous with those establishing the L-truth preserving character of the rule of substitution for the calculus of propositions.) 257

SYSTEMS OF FORMAL LOGIC

7.9 The Propositional Calculus and LFLT' From every theorem of PLT' it is possible to obtain a wf expression of LFLT' having occurrences of individual variables and predicate variables. Likewise, from every wf expression of LFLT' having occurrences of individual variables or constants and of predicate variables or constants it is possible to obtain a wfexpression ofPLT" Now since all the theorems of PLT' are included in LFLT' (all the rules of formation and transformation and all the definitions and axioms of PLT' being included in LFLT'), the two statements above can be reformulated to say: from every theorem of LFLT' having propositional variables as its only variables, an expression of LFLT' having occurrences of individual and predicate variables is obtainable; and from every wff of LFLT' having occurrences of individual variables or constants and predicate variables or constants, a wff of LFLT' having occurrences only of propositional variables as its only variables is obtainable. If a formula ofLFLT' having individual and functional signs is obtained from a formula ofLFLT' having only propositional variables and functors the formula thus obtained is called the F-correlate of the formula from which it is obtained. Thus, if the expression (A)

l})x ~ l})x

is obtained, by rules about to be specified, from the expression (B)

p~p

(A) is said to be the F-correlate of (B). If a formula of LFLT' having only propositional variables and functors is obtained from a formula ofLFLT' having individual and functional signs, the formula so obtained is called the P-correlate of the formula from which it is obtained. Thus if (B) above is obtained, by rules soon to be specified, from (A), (B) is said to be the P-correlate of (A). If a wff of LFLT' consists only of propositional variables and propositional functors it will be called a P-formula. If it consists of individual signs and functional or predicate signs (whether variables or constants), plus, possibly, some propositional signs, it will be called an F-formula. If a P-formula is a tautology of the full propositional calculus it will be called a P-truth. If an F-formula is an L-truth of LFLT' (or of any

258

THE LOWER FUNCTIONAL CALCULUS

functional system of the first or higher order) it is called an F-truth. As noted above, every theorem of PLT' is a theorem of LFLT'. Since PLT' is a complete and consistent propositional calculus, it follows that LFLT' contains a complete and consistent propositional calculus as a proper part. For terminological convenience the propositional calculus of LFLT' will continue to be called PLT', by virtue of its isomorphism with that system. We know from the decision procedure for PLT' plus the definition of tautology that if a wff A is a theorem of PLT', then A is L-true for all appropriate substitutions on its propositional variables. Thus, by the definition of a substitution instance, all substitution instances on a theorem of PLT' are theorems of PLT' and are L-true. Hence, if B is a substitution instance of a theorem A of PLT', then since A is a theorem of LFLT', then B is a theorem of LFLT" Now it is easy to show that every F-correlate of a theorem ofPLT' is a theorem ofLFLT'; (as before, every axiom ofPLT' is, trivially, a theorem of PLT')' Some examples are given below, then the appropriate derived rule upon which the examples are based, is formally stated. Consider the theorem of PLT' (C)

p=:J .q=:Jp

The F-correlate of (C) is (D)

t/>x=:J . 'Px=:Jt/>x

(D) is obtained from (C) by the substitution of t/>x for p and 'Px for q,

by the rule of substitution for propositional variables. Likewise (F) is the F-correlate of the theorem of PLT' given in (E). (E) (F)

p=:Jp 'Px=:J 'Px

(G) and (H) are also F-correlates of (E). (G) (H)

t/>x=:Jt/>x t/>x=:J . 'Px=:J eX=:J: t/>x=:J . 'Px=:J ex

(G) and (H) are valid substitutions on (E) by the same rule. (Note that it is unnecessary to use the rule of formula substitution in order to obtain (H) from (E), since the rule of substitution for propositional variables is sufficiently general to validate the inference.) If we take a more complex theorem of PLT',

259

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(I)

p::Jq::J .q::Jr::J .p::Jr

the following are among its substitution instances: (J) (K) (L) (M) (N)

tPx::J lJIx::J .lJIx::J@x::J .tPx::J@x tPx::J lJIy::J .lJIY::J@Y::J .tPx::J@Y tPx::J lJIy::J .lJIY::J@z::J .tPx::J@z tPx::J .lJIx::J@x::J:lJIX::J@X::JtPIX::J:tPX::JtPIX tPx::JtPY::J . tPy::JtPz::J . tPx::J tPz

These examples show that the substitution-instances of a theorem of PLT' may be such that the individual variables of its F-correlate are all occurrences of the same variable or of distinct variables, as long as substitution on propositional variables is carried out consistently. The following illustrate substitution instances of a formula, (0), of PLT'. with negation.

(0) (P) (Q) (R)

-P::J -q::J .q::Jp -tPx::J - lJIx::J .lJIx::JtPx - . tPx::J lJIx::J: '" .lJIx::J eX::J: .lJIx::J eX::J . tPx::J lJIx '" . tPx::J P::J: '" .p::J lJIx::J: .p::J lJIx::J . tPx::J p

The derived rule is now easily stated: QRI

If a wff A ofPLT' is a theorem ofPLT' and if a wffB ofLFLT' is obtained from A by application of the rule of substitution for propositional variables of LFLT'. then B is a theorem of LFLT'.

This result may be more briefly and generally stated as follows. QRI'

IfB is a substitution instance of a tautology ofPLT', then B is a theorem ofLFLT'.

By virtue ofQRI, we ha.ve an infinite set of theorems ofLFLT' in addition to the infinite class of theorems which are members of LFLT' merely by virtue of being theorems of PLT" This infinite set is made up of theorems of LF LT' derivable as substitution instances of the theorems of PLT' by virtue of the derived rule QRI. Now it should be clear that some proofs in LFLT' can be carried out solely by virtue of QRI. Thus the formula (S)

260

tPx::JtPx

THE LOWER FUNCTIONAL CALCULUS

is L-true in LFLT', This can be shown, as in the case of (G) above, by simple recourse to showing that S is a substitution instance of the theorem p~p of PLT' (and therefore of LFLT'), The law of identity for LFLT' though similar to (S) is not identical with it, since the law of identity is usually expressed with quantification. However, it is easy to see how this law is provable from (S). The proof is as follows. 1.

~(/>x~ (/>x

2.

~(x). (/>x ~ (/>x

S 1, +UQ

Thus we have a proof of (x). (/>x~ (/>x from (S) and the rule of universalization. Whenever a proof is constructed using a P-truth of PLT' as ground for its F-correlate as consequent, the theorem number of the requisite theorem of PLT' is given in the justification column and the rule of substitution for propositional variables cited as reason for the inference. Thus a proof of the law of identity for LFLT' would appear as follows: 1.

~(/>x~(/>x

2.

~(x), (/>x~(/>x

PLT,Tl, +S, (/>x/p 1, +UQ

This result may be regarded as the first proved F-truth of LFLT'. By the foregoing discussion it is proved that if the formula P is a tautology of PLT' and if the formula F is the F-correlate of P, then F is a theorem ofLFLT'. It is important to know whether a similar result can be proved when a formula P is a tautology of PLT' and P is a P-correlate of a formula F ofLFLT'. It is of particular importance to know whether, when P is a tautology and P is a P-correlate of F, (i.e., when P can be obtained from F by application of a rule about to be presented) F is L-true. If it can be established that any arbitrary formula of LFLT', F, is L-true whenever its P-correlate is a tautology, then it is possible to establish an isomorphism between F-truths and P-truths, such that for every F-formula, if its P-correlate is P-true, then the F-formula is F-true, and such that for every P-formula, if its F-correlate is F-true then that P-formula is P-true. If this isomorphism can be established then there is an effective decision procedure for LFLT'. This follows since if the isomorphism can be established then it is always possible to reduce any wff of LFLT' to a wff of PLT', And if an arbitrarily selected formula, A, of LFLT' has a P-correlate B such that the P-correlate B is a theorem of

261

SYSTEMS OF FORMAL LOGIC

PLT', then A is a theorem of LFLT'. If this can be carried out then there is an effective decision procedure for LFLT', since there is an effective decision procedure for PLT'. First it is necessary to specify a rule by virtue of which it is possible to reduce every formula of LFLT' to its P-correlate, i.e., to an expression containing only propositional variables and functors. The rule has two stages. Since quantifiers do not appear in the APC of PLT' the first half of the rule specifies that all quantifiers of the F-formula are to be removed. Since neither individual nor predicate variables occur in the APC of PLT', the second half of the rule indicates how expressions having such variables are reducible to expressions having only propositional variables. The first rule of reduction, then, states:

QR2.

If A is an F-formula, it may be first-stage reduced by the removal of all quantifiers, universal or existential, from A.

Thus, if A is the F-formula

(A)

(x)f/Jx=> . f/Jy

the resultant of applying the above rule is (B) If A is (C)

the resultant of applying the rule to (C) is, (D)

f/Jx=> Px=>. Px=>(i)x=> .f/Jx=>(i)x

And the resultant of applying the rule to (E)

(x)(y).f/Jx=>. Py=>f/Jx

IS

(F) If A is (G)

f/Jx=> . Px=> f/Jx

f/J x => •f/Jx => P x => f/Jx

then application of the rule gives,

(G')

f/Jx=> . f/Jx=> Px=>f/Jx

which is the same as (G), since (G) contains no occurrences of quantifiers. Thus (G) is its own reduction by the application of the first rule.

262

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The second reduction rule is as follows: QR3.

Wf parts of a formula, A, resulting from application of QR2 having the same predicate variable, V, are to be replaced by the same propositional variable, a. Wf parts of a formulae, A, resulting from application of QR2 having a different predicate variable from V, e.g., W, are to be replaced by a different propositional variable, b.

Application of the second rule upon (B) and (D) give (H) and (I)

p"::JP

p::::Jq::::J .q::::Jr::::J .p::::Jr

respectively. Thus there is always a P-formula corresponding to any F-formula by application of QR2 and QR3. The question remains as to whether the L-truth of an F-formula, the P-formula of which is derived by application of rules QR2 and QR3 is derivable from the L-truth of the P-formula so derived. The answer unfortunately is negative. (Cf. Section 9.5). Thus we cannot establish a decision procedure for LFLT' such that the determination of whether a formula F is a theorem of LFLT' may be reduced to the determination of whether the reduction formula P of F is a theorem of PLT'. None the less, it is always possible to establish an expression B as a theorem of LFLT' by showing that B is a substitutioninstance of a theorem, A, of PLT" As a matter of fact, it can be shown that, though the consistency of LFLT' follows from the consistency of PLT', there is no decision procedure adequate to the full lower functional logic, and, more particularly, there is no full proof of completeness for LFLT', though a proof of completeness for the present restricted system is available. (Since, by the proof of completeness and consistency, all tautologies are theorems of PLT', and since the five standard propositional functors of PLT are available either as primitive or by definition, the theorems 1-90 of P + are theorems of PLT' and the theorems 91-123 of PLT are theorems of PLT' along with derived rules. When citing theorems of PLT' we will simply write "T" followed by a number from 1-123. If the reader wishes to check the reference he will find the theorems Tl-T90 as theorems 263

SYSTEMS OF FORMAL LOGIC

2Tl-2T90 of Chapter 2: theorems T91-Tl23 correspond to theorems 3T91-3Tl23: the axioms ofPLT, 3A3-3All are also proved as theorems ofPLT' in Chapter 3. For convenience the numbers Tl24-Tl32 found in parentheses to the right of the theorems will index those results. Any citation consisting of a "T" followed by a number less than 1000 is a theorem (or an application of a theorem) of the propositional calculus PLT'. Any citation consisting of a "T" followed by a number equal to or greater than 1000 is a theorem ofLFLT' requiring for its proof application of some rule and/or axiom of LFLT' beyond those of its propositional calculus, though not, of course, excluding the application of the rules, axioms, and theorems of its propositional calculus.) 7.10 Deductions in LFLT' With these conventions in mind, the following theorems of LFLT' are established. TlOOO

(x). tPx::::J tPx (Law of identity)

1. I-tPx::::JtPx 2. l-(x).tPx::::JtPx (TH) T1001

Tl, +8, tPx/p 1, +UQ

tPY::::J(3x)tPx (Law of existential generalization)

1. l-(x)",tPx::::J. ",tPY(::::J) 2. I-tPY::::J. ",(x)",tPx 3. I-tPy::::J . (3x)tPx(TH)

LFA5, +8F, ",tPx/tPx, ",tPy/tPy I, T96, +8, (x)",tPx/p, tPy/q, -::::J 2, D4

It might be thought that step 1 of the proof of TlOOl could be obtained by application of the simple rule of substitution for predicate variables, substituting ",tP for tP. This is incorrect, since '" is a propositional functor, the application of which is restricted by the formation rules to propositions and propositional functions and matrices. Thus formula substitution is required. The second step is available by virtue of the fact that(x)",tPx::::J. ",tPyis a valid substitution onLFA5and the expression (x)",tPx::::J. ",tPY::::J:tPY::::J. "'(x) '" tPx

is a substitution-instance on T96. (Note: the wff 264

THE LOWER FUNCTIONAL CALCULUS

(x) '" (/)x => • '" (/)y => : (/)y => (X) '" '" (/)x

is not a substitution instance of T96 (and is not L-true) since the third occurrence of '" must have the integral formula (x)"'(/)x as argument, not the wi part '" (/)x.) Step 3 is obtained from 2 by the definition of the existential quantifier, D4. TlOO2

(x)", . (/)x&"'(/)x (Law of contradiction for the LFC)

1. 1-", . (/)x&"'(/)x 2.I-TH Tl003

(x).(/)xv "'(/)x (Law of excluded middle for the LFC)

1. I-(/)xv"'(/)x 2. I-TH Tl004

Tl18, +S, (/)x/p 1, +UQ

(x)(/)x=> . (3x)(/)x

1. I-(x) (/)x => . (/)x 2. I-(/)x=> (3x)(/)x 3.I-TH TlO05

Tl17, +S, (/)x/p 1,+UQ

LFA5, +SI, x/y Tl001, +SI, xjy 1,2, +IS

(x). Px=>8x=>:(3x). Px=>8x

Tl004, +SF, [Px=>ex]/(/)x, [Px => ex ]j(/)x

1. I-TH

Note that the second scheme in the justification column is exactly the same as the first, since the successor of the initial matrix of TlO04 is the same as the initial matrix. In this and similar cases the inclusion of the second scheme is redundant and, therefore, may be dropped. Tl006

1. 2. 3. 4. 5.

(x). (/)x=> Px=> :(x) (/)x => . Px (Second law of quantifier dis-

tribution) I-(x) (/)x => •(/)x l-(x).(/)x=>Px=>:(/)x=>Px I-(x)(/)x=> .(/)x=>:(/)x=> Px=>: (x)(/)x=> . Px I-(/)x=> Px=> . (x) (/)x => . Px I-TH

LFA5, +SI, x/y 1, + SF, [(/)x => Px]j(/)x T3,

+ S, (x)(/)xjp, (/)x/q, Pxjr

3, 1, -

=>

2,4, +IS

Note that the substitution in step three does not violate the restriction on the rule of substitution, since no individual variables not already 265

SYSTEMS OF FORMAL LOGIC

bound become bound in step three. Note also that in the expression (x)tJ>x:::J .tJ>x, step 1 in the proof of TlO06 (and step 1 in the proof of Tl004) only the first two occurrences of x are bound, while the third occurrence of the same variable is free. Thus (x)tJ>x:::J . tJ>x is a different formula than (x). tJ>x:::J tJ>x, the law of identity, in which all occurrences of individual variables are bound. As a lemma for Tl007 the following theorem is proved. TlOO7'

(x).P:::J . (x)tJ>x:::J . 'I'x:::J:P:::J(x).(x)tJ>x:::J. 'I'x

1. 'r(x).P:::JtJ>x:::J:P:::J(x)tJ>x 2. 'r(x).P:::J . (y)tJ>y:::J . 'I'x:::J: p:::J(x).(y)tJ>y:::J. 'I'x 3. 'r(x).P:::J. (x)tJ>x:::J . 'I'x:::J: P:::J(x).(x)tJ>x:::J. 'I'x

LFA4 1, +SF, [(y)tJ>y:::J. 'I'x]/tJ>x 2, +SB, x/y

Step 2 is derived from step 1 by formula substitution on step 1, substituting (y) tJ>y:::J . 'I'x for tJ>x at both occurrences of tJ>x. By the restriction on formula substitution, we could not have substituted the formula (x)tJ>x:::J 'I'x for tJ>x by this rule since in that case a variable ofI, namely x, would have occurrence in the quantifier (x) of SM, thus violating the restriction; hence we substitute (y)tJ>y:::J . 'I'x for tJ>x, since here y has arbitrary occurrence in SM, occurring neither in S, (step 1) nor in I, (tJ>x). Step 2 is transformed into step 3 by the rule of substitution on bound variables. Here we choose the integral formula (y)tJ>y as C. The restriction on the rule is not violated since the variable x has no occurrence in C. TlO07

(x).tJ>x:::J 'I'x:::J:(x)tJ>x:::J . (x)'I'x (Third law of quantifier

distribution) 1. 'rex): (x). tJ>x:::J 'I'x:::J: (x) tJ>x:::J . 'I'x 2. 'r(x).P:::J . (x)tJ>x:::J . 'I'x:::J: p:::J (x). (x)tJ>x:::J . 'I'x 3. 'rex): (x). tJ>x:::J 'I'x:::J: (x)tJ>x:::J . 'I'x:::J:. (x).tJ>x:::J 'I'x:::J(x).(x)tJ>x:::J. 'I'x 4. 'r(x).tJ>x:::J 'I'x:::J(x).(x)tJ>x:::J. 'I'x 5. 'r(x).P:::J 'I'x:::J:P:::J(x) 'I'x 6. 'r(x).(x)4>x:::J-. 'I'x:::J:(x)4>x:::J(x) 'I'x 7. 'rTH 266

Tl006, +UQ TlOO7'

2, +S, (x).tJ>x:::J 'I'x/p 3,1, -:::J LFA4, + Sp, 'I'/tJ> 5, +S, (x)4>x/p 4,6, +IS

THE LOWER FUNCTIONAL CALCULUS

Step I of the proof of Tl007 is available by universalization of TlOO6. Note that step I of the proof Tl007 is q-closed, not with respect to all its variables, but with respect to its individual variables, while Tl006 is q-open with respect to its individual variables at the sixth occurrence of the variable x. A formula q-closed with respect to its individual variable (though otherwise open) is said to be I-closed. If it is not I-closed, it is said to be I-open. Thus Tl006 is I-open, while step I of TlOO7 is I-closed. Step 2 of this proof is also I-closed. Step 3 of the same proof is a valid application of the rule of substitution for propositional variables, since no free individual variable of step 2 becomes bound by the substitution of (x).(/)x:::;) 'JIx for p. Step 3 is also I-closed. Step 4 is unproblematic. Step 5 merely applies the rule of simple substitution for predicate variables to LFA4. Step 6 is a valid application of the rule of substitution for propositional variables, since the substitution of (x)(/)x for p binds no individual variable of step 5, not already bound in step 5. Step 7 is unproblematic. The correlative law for existential quantification, Tl007'

(3x). (/)x& 'JIx:::;) : (3x)(/)x&(3x) 'JIx

is also a law of LFLT'. By application of the law of importation, T30, to TlO06, Tl007, and LFA4 respectively we have: TlO08

(x).(/)x:::;) lJIx&:(x)(/)x:::;):.lJIx

Tl009

(x).(/)x:::;) lJIx&:(x)(/)x:::;): . (x) lJIx

TlOlO

(x) .p:::;)(/)x&:p:::;):. (x)(/)x

It is useful at this point to state a derived rule of inference frequently used in the following pages: the rule, based on the establishment of Tl007 is as follows:

From a formula of the form (a)[A:::;)B], where A and B are arbitrary wffs and where a is an individual variable, a formula of the form (a) A:::;)(a)B is derivable. This rule serves to shorten proofs. It will be called the rule of universal quantifier distribution, and its application will be symbolized by the abbreviation +UQD. A proof using UQD is presented below. 267

SYSTEMS OF FORMAL LOGIC

TlOll

(x) JPx:::> Px:::>:(3x) qJx:::>(3x) Px

1. ~qJx:::> Px:::> . '" Px:::> ,.·"(/Jx 2. ~(x).qJx:::> Px:::>. '" Px:::> ",qJx(:::» 3. ~(x).qJx:::> Px:::>:(x). '" Px:::> ",qJx 4. ~(x). Px:::>qJx:::>:(x)Px:::>(x)qJx 5. ~(x). '" Px:::> ",qJx:::>:(x)", Px:::> (x) '" qJx 6. ~(x)", Px:::> (x) '" qJx:::> . ",(x)",qJx:::> ",(x)", Px

T95, + S, qJxjp, Pxjq 1, +UQ

2, +UQD Tl007, +Sp, PjqJ, qJjP

T95,

+ S, (x) '" Px/p,

(x) '" qJxjq

7.

Px:::>: ",(x) '" qJx:::> ",(x)", Px

~(x).qJx:::>

8.

~TH

3,5,6, +IS 7, D4

The derived rule, + UQD, is used in the deduction of step 3 from step 2. Step 4 is derived from Tl007 by simultaneous substitution of P for qJ, and qJ for P. The necessity for this step may be understood by observing that if '" Px were substituted for qJx in Tl007 without the substitution given in step 4, the second restriction on formula substitution would be violated. In future deductions, steps like 4 and 5 of the proof above will be compressed into a single step, identical with 5.

5'.

'" Px:::> ",qJx:::>: (x)", Px:::> (x) ",qJx ~(x).

Tl007, +Sp, PjqJ, qJjP; +SF, '" PxjPx, '" qJxjqJx

The justification column in this case stipulates that the step has been derived by formula substitution of '" Px for Px and '" qJx for qJx on the result of substituting P for qJ and qJ for Pin Tl007. By the law of importation and TlOII we have TlOl2

(x). qJx:::> Px&: (3x)qJx:::>:. (3x) Px

By application of T4 on Tl007 we have Tl013

(x)qJx:::>:(x).qJx:::> Px:::>:(x) Px

and by T4 and TlOll we have TlOl4

268

(3x)qJx:::> :(x). qJx:::> Px:::> : (3x) Px

THE LOWER FUNCTIONAL CALCULUS

as theorems of LF LT'. By application of other propositional laws, plus the rules of the system, the following are theorems: Tl015

(x). O O=>-A

T1033

{

272

A=> -0 -O=>A

THE LOWER FUNCTIONAL CALCULUS

Tl034

T1035 But some other inferences of Aristotelian and traditional logic do not hold when these are expressed in terms of the variables, functors, and operators of LFLT'. For instance, the implication

and the implication

are not theorems of LFLT', though they are valid in Aristotelian. It is possible to show the invalidity of these two formulae in the present system of logic by showing that there is at least one substitution-instance upon them which is false. This may be done by finding a case in which A is true but I is false. We choose as the value for (/> the term "perfect circle" and for P the term "centered," i.e., has a center. Now consider the matrix (/>x~ Px, i.e., "If x is a perfect circle then x is centered." We universalize this to give

(c)

(x)[(/>x~ Px]

i.e., "If anything is a perfect circle then it is centered." We represent this as the statement (d)

(x)[Px~ Cx]

where "P" stands for "perfect circle" and we accept

(e)

(x).Px~

"c"

for "centered". Now if

Cx

as true, then, by the implication

we have

(g)

(x).Px~Cx~:(3x).Px&Cx

273

SYSTEMS OF FORMAL LOGIC

but while (e) is true, (g) is not, since (g) is true only if there is some value of x such that x is a perfect circle. If there are no perfect circles then there is no case when the consequent of (g) is true. But then (g) is false when its antecedent is true and its consequent is false. While everyone would have to grant (e), by the definition of "perfect circle", the implication (g) commits us, by (e), (g) and modus ponens, to the assertion there exist perfect circles which are centered, and thus to the assertion that there are perfect circles. While this may be true on the assumption of certain forms of Platonistic philosophy, it is not a truth of logic. A similar case showing that (b) is not L-true is as follows. Substitute for f/>, in the matrix f/>x::J '" lJIx, the constant "perfect circle" and for lJI in the consequent'" lJIx, the constant "square." We abbreviate the constant "perfect circle" by the letter "C", and we abbreviate the constant "square" by the letter "S"; thus we convert the matrix f/>x::J '" lJIx

(h)

into the matrix (i)

CX::J '" Sx

Thus we obtain the true statement G)

(x).Cx::J"'Sx

But by the formula (b), E::JO, we obtain from (j). (k)

(x).Cx::J ",Sx::J:(3x).Cx&",Sx

which is false if the statement (1)

(3x)Cx

is false. Thus the formulae (a) and (b) are not theorems of LFLT'. (It is possible to construct symbolic systems in which the implications (a) and (b) are theorems. In such systems the formation rules are set out so as to provide that all universal statements have existential import, i.e., entail the existence of entities having the properties specified by their antecedents. Universal formulae in LFLT' do not have existential import. To assert that the antecedent of such a formula refers to existing things it is necessary to indicate this explicity. Thus, where H is an abbreviation for the property Human and M for the property mortal (x)[Hx::JMx]

274

THE LOWER FUNCTIONAL CALCULUS

is consistent with there being no human and no mortal things. If we wish to assert not merely the universal, but existentially agnostic, formula, "All humans (if there are any) are mortal", but also that there are in fact humans, the requisite formula would be (3x) [HxJ & (x) [Hx~ MxJ

A similar remark holds for the universal negative formula, which is also denied existential import in LFLT") The part of the lower functional calculus now being developed is closely related to the "Syllogistic" of Aristotle, i.e., that part of Aristotle's logic in which he develops his theory of the syllogism. In the propositional calculus, two forms of the syllogism having occurrences only of propositional variables and functors were proved, T2 and T3. By virtue of these propositional laws, plus some laws and rules of functional logic, a number of forms of Aristotle's syllogisms are provable, though not all of them are valid in LFLT'. The Aristotelian syllogisms which are invalid in LFLT' are those which depend for their validity upon the acceptance of the formulae (a) and

A~I

(b)

E~O

as valid. The syllogism is an L-true conditional formula having three terms (represented in LFLT' by three distinct predicate variables) and having a conjunction of two (complex) formulae as antecedent and one (complex) formula as consequent; in both the formulae of the antecedent a term, M, called the middle term, occurs. Besides the middle term there are two other terms, P and S, such that P and M occur in the left conjunct of the antecedent and such that Sand M occur in the right conjunct of the antecedent: Sand P both occur in the consequent and there is no occurrence in the consequent of M. The following is an example of a syllogism and a proof of it in LFLT"

Tl036 1.

(x): lJIx~ex&.t/lx~ lJIx~:t/lx~ ex

I-lJIx~ex~ .t/lx~ lJIx~ .t/lx~ex(~)

2. I-lJIx~ex&.t/lx~ lJIx~:t/lx~ex

3. I-TH

T2, +S, t/lx/p, lJIx/q, ex/r 1, T28, +S, t/lx/p, lJIx/q, ex/r, - ~ 2, + UQ 275

SYSTEMS OF FORMAL LOGIC

Here, '1' is M; e, P; and cP, S. Notice that the quantifier ranges over the entire statement. There are, given the basic structure of the syllogism articulated above, exactly four different sub-structures of the syllogism, three of which are Aristotelian. Note that Tl036 has the basic form

(A)

(Q)[[[M:::>P]&[S:::>M]]:::> [S:::>P]]

If we ignore the quantifier we have

(B)

[[[M:::>P]&[S:::>M]]:::> [S:::>P]]

If we give this form in vertical order we have

(I)

1. M:::>P 2. S:::>M 3. S:::>P

where 1 and 2 represent the antecedent of (B) and 3 represents its consequent. All syllogisms having the form represented by are said to be syllogisms of the first figure. Given this scheme there are four forms, one of which is represented by (I). We have a second form:

cn

(II)

P:::>M S:::>M S:::>P

the so-called second figure of the syllogism, and a third form:

(III)

M:::>P M:::>S S:::>P

the third figure of the syllogism. The first, second, and third figure of the syllogism are Aristotelian, but Aristotle did not include in his syllogistic the so-called fourth or "Galenian" figure, clearly valid under certain quantifications:

(IV)

P:::>M M:::>S S:::>P

The reasons for Aristotle's failure to include the fourth figure do not

276

THE LOWER FUNCTIONAL CALCULUS

concern us here. The interested reader is referred to the discussion in I. M. Bochenski's Ancient Formal Logic, Chapter III, section 8B. Whatever may have been Aristotle's motives for excluding (or ignoring) the fourth figure, it can be shown that there are some valid fourth figure syllogisms. The theorem Tl036 is given the name Barbara, the "Scholastic name" due to Peter of Spain.! The Scholastic name will be given in parentheses following the statement of the syllogistic theorem. (Most of the syllogistic theorems are left to the reader as exercises.)

e

Here and hereafter is used for the P- or predicate-term, l/J for the Sor subject-term, and 'P for the M- or middle-term.

Tl036

(x): 'Px~ ex&.l/Jx~ 'Px~ :l/Jx~ ex (Barbara)

The vowels in the scholastic names signify the quantity of the formulae of the syllogism. All the vowels in the name "Barbara" are "a": likewise all three complex formulae making up Barbara are (or are reducible) to A formulae. The full quantificational reduction of Tl036 would be (x). 'Px~ ex&:(x).l/Jx~ 'Px~: .(x).l/Jx~ex

We have as a theorem also the syllogism of the first figure: Tl037

(x). 'Px~ "" ex&: (x) .l/Jx~ 'Px~:. (x) .l/Jx~ "" ex (Ce1arent)

(We note that the left side of the antecedent is an E proposition, the right side being an A proposition and the consequent an E proposition, thus CEIArEnt.) 1. r'Px~ ""ex~ .l/Jx~ 'Px~ .l/Jx~ ""ex T2, +S, x/q, ""ex/r, l/Jx/p 2. rex). 'Px~ ""ex~ .l/Jx~ 'Px~. l/Jx~ ""ex 1, +UQ 3. r(x). 'Px~ ""ex~. (x).l/Jx~ 'Px~ .l/Jx~ ""ex 2, +UQD 4. r(x) .l/Jx~ 'Px~ .l/Jx~ "" ex~: (x)l/Jx~ 'Px~:(x).l/Jx~ ""ex 5. r(x)'Px~ ""ex~:(x)l/Jx~ 'Px~: (x)l/Jx~ ""ex

6. rTH 1

TlO07, +SF

2,4, +IS 5, T28

For the syllogistic names see J. M. Bochenski, A History of Formal Logic, p. 212ff.

277

SYSTEMS OF FORMAL LOGIC

The details of justification in this, and the following proofs, are left to the reader. The proof of Tl038 makes use of the definitional equivalence stipulated on p. 238. T1038 1.

(x) .lJIx~ 8x&:(3x)f/>x& lJIx~: . (3x) . f/>x&8x (Darii)

f-(x).lJIx~ 8x&:(3x). f/>x& lJIx~:.

(3x). f/>x& lJIx 2. f-(x).lJIx~8x&:(3x).f/>x&lJIx~:. f/>a* & lJIa* 3. f- f/>a* & lJIa* ~ f/>a* 4. f-f/>a* & lJIa* ~ lJIa* S. f-(x).lJIx~ 8x&:(3x). f/>x& lJIx~:. f/>a* 6. f-(x) .lJIx~ 8x&: (3x). f/>x& lJIx~:.

1, +SE,(cf.p.238) Tl24, +S Tl2S, +S 2,3, +IS Tl24, +S

(x).lJIx~8x

7.

Tl2S, +S

f-(x).lJIx~8x&:(3x).f/>x&lJIx~:.

lJIa* 8. f-(x).lJIx~8x&:(3x).f/>x&lJIx~:. (x).lJIx~8x&: lJIa* 9. f-(x).lJIx~8x&:lJIa*~:.8a* 10. f-(x).lJIx~8x&:(3x).C/Jx& lJIx~:. 8a* 11. f-(x) .lJIx~ 8x&:(3x). C/Jx& lJIx~:. f/>a*&8a* 12. f-f/>a*&8a*~ .(3x).f/>x&8x 13. f-TH Tl039

(x).lJIx~

2,4, +IS 6,7, *AS, T32, +S LFAS, + SF, + SI, T28 8,9, +IS S, 10, *AS, T32, + S T1001, +SI, +SF 11, 12, +IS

",8x&:(3x). f/>x& lJIx~:. (3x)f/>x&'" 8x (Ferio)

The proof of this theorem is exactly analogous with that of Tl038. There is another method allowing proof of syllogistic laws having both universal and particular formulae as antecedents. This method allows us to dispense with the starred variables used in the proof of Darii given above. The method rests upon the realization that, by the law of exportation, T36, the basic syllogistic form

is convertable into the pure implication 278

THE LOWER FUNCTIONAL CALCULUS

p=>[q:::lr]

and, by the law of importation, T28, vice versa. Thus Darii becomes, by application of the law of exportation, (x) .lJIX:::l@x:::l:(3x).fPx&lJIx:::l:(3x).fPx&@x

If this formula is provable, then so is Darii, by application of the law of importation. The advantage of the above formula, for the present purposes, is that it positions the two existential formulae in the consequent, while the sole integral formula of the antecedent is universal. That this is an advantage will become clear in the proof. The proof begins with a substitution on a theorem of PLT' which is easily proved: 1. 2.

3. 4.

5. 6.

7.

.r&q:::l:'" .r&p ~lJIX:::l@X:::l:'" .fPx&@X:::l: '" .fPx&lJIx ~(x): lJIX:::l@X:::l:'" . fPx&@X:::l: '" . fPx& lJIx ~(x).lJIX:::l@X:::l:(x):'" .fPx&@X:::l: ",.fPx&lJIx ~(x):'" .fPx&@X:::l:",.fPx&lJIX:::l:. (x)", .fPx&@X:::l:(x)'" .fPx& lJIx ~(x) lJIX:::l@X:::l:(x)", .fPx&@X:::l: (x)", .fPx&lJIx ~(x)", .(/)x&eX:::l:(x)", .(/)x&lJIX:::l:. ",(x)", .fPx&lJIX:::l:"'(x)", .fPx&@x ~p:::lq:::l:'"

1,

+S, lJIx/p, @x/q, fPx/r

2, +UQ 3, +UQD TlO07, +SF 4,5, +IS

T95, +S, (x)", .fPx&@x/p, (x)", .fPx&@x/q

lJIX:::l@X:::l:"'(x)", . fPx& lJIX:::l: ",(x)", .fPx&ex

8.

~(x).

9.

~(x).lJIX:::l@x:::l:(3x).fPx&lJIX:::l:

(3x).fPx&@x

6,7, +IS 8, D4

The crux of the above proof is at step 7 where, by a substitution on the converse law of transposition, the two main formulae of the consequent are reordered and negated, giving, through application of the implicative series, the formula "'(x) '" . fPx& lJIX:::l: ",(x) '" . fPx&@x

as consequent of step 8. This, in turn, can then be transformed into the

279

SYSTEMS OF FORMAL LOGIC

desired consequent by application ofD4. A similar proof of Ferio, Tl039, can be obtained beginning with the propositional theorem p~ -q~:- .r&-q~:-

.r&p

then substituting lJIx~ -ex~:- .rJ>x&-ex~:- .rJ>x& lJIx

Besides the four syllogisms proved above, the syllogistic part of LFLT' allows proof of a considerable number of others, all of which are valid according to the Syllogistic logics of Aristotle and/or his successors. Four of the syllogisms developed by Aristotle, however, are invalid in LFLT', since they have universal formulae as antecedents and particular formulae as consequents. None of the five Scholastic "subaltern syllogisms", e.g., Barbari, (x).lJIx~ex&.r/Jx~ lJIx~:(3x).rJ>x&ex,

are valid in LFLT' for the same reason.! The proof of one further syllogism is begun for the sake of illustration. Tl040

(x). ex~ lJIx&: (3x). rJ>x&- lJIx~: .(3x). rJ>x&- ex (Baroco)

1. I-(x). ex~ lJIx&:(3x). if>x&- lJIx~: . (x) . ex~ lJix 2. I-(x). ex~ lJIx~: (x). - lJIx~ - ex 3. I-(x).ex~ lJIx&:(3x).rJ>x&- lJIx~: .(x). - lJIx~ -ex

4. I-(x). ex ~ lJIx&: (3x). rJ>x&- lJIx~:. (3x)rJ>x&- lJIx 5. I-(x).ex~ lJIx&:(3x).rJ>x&- lJIx~: .rJ>a*&- lJIa* 6. l-rJ>a* &- lJIa* ~ rJ>a* 7. l-rJ>a* &- lJIa* ~ - lJIa*

The completion of the justification column is left to the reader, as is the completion of the proof; it proceeds just as does Tl038. Using the alternative method, one begins the proof of Baroco with the theorem

and then substitutes

1

For the 'subaltern moods' of the syllogism, op. cit., p. 215f.

280

THE LOWER FUNCTIONAL CALCULUS

There are some valid syllogisms having mixed quantification in the antecedent, the left hand side of which is not a universal but a particular formula, e.g., (3x) .lJIx&8x&: (x) .lJIx-:::J (/>X-:::J:. (3x). (/>x&8x (Disamis)

Though this circumstance would appear to prevent application of the alternative method, it does not in fact do so. Why? In fact, no use need be made of starred variables in LFLT'. EXERCISES

1. Metatheorems about distribution. (A predicate is said to be "distributed" by a formula if the formula refers the predicate to every individual which possesses it. Otherwise it is undistributed by the formula. Thus in "All men are mortal," a substitution on (x). (/>x-:::J lJIx, the formula distributes the first predicate "men" since it refers to all men, but "mortal", the second predicate, does not refer to all mortal beings in the formula. Hence the second predicate is undistributed by the formula. Put in another way, if one were to attempt empirical verification of the formula in question, he would have to examine every member of the class of men to determine whether each was mortal. But he would not have to examine every member of the class of mortal beings, but only those which were human. If one has to examine every member of the class named by the predicate in order to verify the proposition, the predicate is distributed by the proposition. If total examination is unnecessary for verification, the predicate is undistributed. Inspection of the four Aristotelian forms shows the following. A. In the A formula: the predicate of the left side of the conditional is distributed; that of the right side, undistributed. E. In the E formula: the predicates of both sides are distributed. I. In the I formula: neither predicate is distributed. O. In the 0 formula: the predicate ofthe left conjunct is undistributed; that of the right conjunct is distributed. Metatheorem I. (a). The predicate M (the so-called middle term) must be distributed at least once. (b). No syllogism is valid if there is a predicate distributed in the consequent which is undistributed in the antecedent. 281

SYSTEMS OF FORMAL LOGIC

(c). No syllogism is valid if a predicate distributed in the antecedent is undistributed in the consequent. Verify I(a)-(c) by showing how invalidity would result if syllogistic inference were premitted which violated their conditions. Metatheorem II. By applying the metatheorems I(a)-(c) verify the following further metatheorems. (a). No valid syllogism has two particular propositions as antecedent. (b). No valid syllogism has two negative propositions as antecedent. (c). No valid syllogism has a particular proposition as one of the conjuncts of its antecedent and a universal proposition as consequent. (d). If a syllogism has two universal propositions as antecedent then its consequent must be universal. (e). If one of the conjuncts of the antecedent is negative, then the consequent must be negative. 2. Using the results of exercise 1, determine which syllogistic forms of the three Aristotelian figures are valid syllogisms in LFLT'. 3. Give an example of a valid fourth figure syllogism by applying the metatheorems of exercise 1. Prove the result by use of the axioms and theorems of LFLT" 4. Taking account of all four figures, how many syllogisms are theorems ofLFLT'?

5. The deduction theorem is provable for LFLT" It has the same form as the deduction theorem for PLT' and is proved in an exactly analogous way. Derivations in LFLT' are set up in the same way as for PLT"

The deduction theorem If AI, A2, A3, ... An~B, then AI, A2, A3, ... the immediate consequence, If

A~B

An-I~An::::>B.

Which has

then A::::>B

Bn in the following is the same as B of the theorem. There are ten cases: Case 1: Bn is one of the hypothesis AI, A2, A3, ... An. (Use axiom LFA1) Case 2: Bn is the same as An. (Use theorem Tl.) Case 3: Bn is a variant of an axiom. (Use axiom LFAl.)

282

THE LOWER FUNCTIONAL CALCULUS

Case 4: Bn is inferred by the rule of modus ponens from earlier derived elements in the proof Bf and Bg. (Use axiom LFA2.) Case 5: Bn is inferred by the rule of universalization from an earlier element in the proof Bg. (Thus Bn is (a)Bg where a is an individual variable which does not occur free in the hypotheses, AI, A2, A3, ... An.) (Use axiom LFA4.) Case 6: Bn is inferred by the rule of substitution for propositional variables from the earlier element Bg (where the variable substituted for is not a free variable of the hypotheses AI, A2, A3, ... , An.) (Analogous with case 4 of the proof of the DT in PLT") Case 7: Bn is inferred by the rule of substitution for individual variables from an earlier element Bg (where the individual variable substituted for does not occur free in the hypotheses.) (Analogous with case 4 of the proof of the DT in PLT") Case 8: Bn is inferred by the rule of simple substitution from an earlier element Bg (where the predicate variable substituted for does not occur in the hypotheses.) (Analogous with case 4 of the proof of the DT in PLT") Case 9: Bn is inferred by the rule of formula-substitution from an earlier element Bg (where the predicate variables substituted for do not occur in the hypotheses.) (Analogous with case 4 of the proof of the DT for PLT") Case 10: Bn is inferred by the special rule of substitutivity of equivalence from an earlier element, Bg. (Analogous with case 4 of the proof of the DT for PLT") Using the cases and hints provided above, prove the deduction theorem for LFLT'. Derivations and proofs by the method of the deduction theorem can now be carried out in LFLT'. When derivations and proofs by the method of the deduction theorem are constructed care must always be exercised so that the restrictions on the rules for substitution are observed. In particular, in using substitution and the rule of generalization the variables substituted for or generalized upon must not have free occurrence in any of the hypotheses. This restriction parallels that of PLT' where it is specified that the propositional variable substituted for must not have occurrence in any of the hypotheses. 6. Prove theorems T1 036-T1 040 by the method of the deduction theorem. 283

CHAPTER 8

AN EXTENSION OF LFFT' AND SOME THEOREMS OF THE HIGHER FUNCTIONAL SYSTEM. THE CALCULUS OF CLASSES

8.1 Summary and Modification of the Formation Rules of LFLT' In this chapter some extensions of LFLT' will be outlined permitting formation of statements and matrices having predicates and predicate variables which are more complex than the monadic predicates of the system as developed in Chapter 7. This is done by adding signs for twoplace, three-place, four-place, ... , n-place predicates and predicate variables to the primitive basis of the system; adding rules for the determination of well-formedness with respect to formulae; and including rules of transformation for the new signs and wffs. In addition, a brief outline of a further extension of the system is set out and the basis of a Boolean algebra (calculus of classes) is presented. We add an infinite number of two-place predicate variables: (/)2, lJI2, 8 2, (/)21, ... and a corresponding, though not necessarily infinite number of two-place predicate constants: £2, G2, H2, ...

adding as many of such lists as desirable specifying n-place predicates and constants. With respect to formation rules we need modify only a single rule; we add,

If W is an n-place predicate variable or an n-place predicate constant, and if al, a2, a3, ... , an are individual variables or individual constants, then Wa1 a2 ... an is a wff. In the case of the transformation rules all that is necessary is to reword the present rules in such a way that they include reference to n-place predicates and constants only.

284

HIGHER FUNCTIONAL SYSTEM AND CALCULUS OF CLASSES

No new axioms need be added. By virtue of this part of LFLT', we can construct proofs about relations. For instance: Tl041 (X)@2XX::::J .@2yy.

LFA4 I, + SF, @2XX/WX,

1. f-(x)Wx::::J .Wy 2. f-(X)@2XX::::J .@2yy

@2yy/Wy

Step 2 is read, "If everything bears the two place relation (predicate) 61 2 to itself, then y bears the two place relation 61 2 to itself." If we substitute the two place relation "is identical with" for 61 2 we obtain the matrix, If, for all x, x is identical with x, then y is identical with y. or, If everything is self identical then y is self identical. Substituting the name "Socrates" for the free variable y, we obtain the statement, If everything is self identical then Socrates is self identical.

8.2 The Lower Functional Calculus with Identity (Tl041-10S2) If we add the two-place predicate constant and the axioms

LFA6 LFA7

X=X X=y::::J . WX::::J Wy

to the system we obtain the lower functional calculus with identity. The rule of formation governing the = symbol is, If a and b are individual variables or individual constants, a = b is a wff. This system is called Tl042

LFZ'T'

X=y::::J .y=X

1. f-x=Y::::J .Wx::::JWy 2. f-x= y::::J .X=X::::J .y=x(::::J)

LFA7 I, +SF, [x=x]/Wx, [y=x]/Wy

285

SYSTEMS OF FORMAL LOGIC

3. I-x=x::> .x=y::> .y=x 4.l-x=x 5. I-x= y::> .y=X

2, T4, +S LFA6 3,4, - ::>

It is easy to see that the formula Tl042'

X= y= .y=X

is also provable by substitution on Tl042 and application of the law establishing equivalence, PLT,Tl1, corresponding with 2Al1 of PLT. Tl043 D1043

x=y&.y=z::>:x=z

1. X= y&.y=z(::» 2. x=y 3. y=z 4. x=y::> JPx::>f/Jy 5. f/Jx::> f/Jy 6. y = z::> . f/Jy::> f/Jz 7. f/Jy::> f/Jz 8. f/Jx::> f/Jy&.f/Jy::> f/Jz(::> ) 9. f/Jx::> f/Jz 10. X=X::> .Z=X

11. Z=X::> .X=Z 12. X=X::> .X=Z 13. X=X 14. X=Z

1. x=y&.y=zl-x=z 2. I-x=y&.y=z::>:x=z

hyp 1, Tl24, +S, - ::> 1, Tl25, + S, - ::> LFA7 4,2, -::> LFA7, +SI, y/Xl, Z/Yl 6,3, -::> 5,7, *A5, +S 8, T30, + S, f/Jx/p, f/Jy/q, f/Jz/r, -::> 9, +SF, [x=x]/f/Jx, [z=x]/f/Jz Tl042, +SI, Z/Xl, X/Yl 10, 11, +IS LFA6 12, 13, -::> D1043 1,DT

The following observations throw light on the above proofs. In step 2 of the first proof, the predicate "is identical with x" is substituted by formula substitution for the predicate variable f/J. This is permissible by the expansion of the system to include n-place variables and predicate constants. Step 2 of the second proof is derived from step 1 by the theorem of PLT', p&q::> p; application of the rule of substitution for propositional variables; substituting X= y for p, and y=z for q; and modus ponens. Step 3 is derived from step 1 by the theorem p&q::>q, substitution, and 286

HIGHER FUNCTIONAL SYSTEM AND CALCULUS OF CLASSES

modus ponens. Step 4 is unproblematic, as is step 5. Step 6 is obtained by the variant ofLFA7, Xl = Yl:::> J/JX1:::> (/)Yl

and substitution of y for Xl, and z for Yl. The variant is employed in order to prevent violation of the restriction that in derivations no substitution may be made on variables which have free occurrence in the hypotheses. A similar remark applies to step 11, where a variant of theorem T1042 is indicated. (In an unabbreviated derivation the full demonstration of a variant of Tl042 would be included, but we use theorems demonstrated by previous proofs, here, as before, giving them the same status as axioms.) In steps 9 and 10 the theorem of PLT' p:::>q&.q:::>r:::>:p:::>r

substitution, and modus ponens are applied. It is clear that just as (/)X:::> 'Px&. 'Px:::>ex:::>:(/)x:::>ex

is a substitution on this theorem by the rule of substitution for propositional variables, so is (/)x:::> (/)y&. (/)y:::> (/)z:::>: (/)x:::> (/)z

by the same rule. In the expression (/)x the left member, (/), is said to have the right member, X as argument. When, however, an expression fPx is converted by formula substitution into, e.g., x=y, the left member, x, is the argument, and the right member, = y, has the left member, x, as argument. Thus in the case of the formula substitution of step 2 in the proof of Tl 042, y = X is the result of the application of the rule of formula substitution since y is the argument of (/) in step 1. A similar remark applies to step 10 of the proof of T1043. As a result of Tl042 and the rule of universalization we have the following theorems. Tl044 Tl045 Tl046

(x).x=y:::> .y=X (x)(y).x=y:::> .y=X (y)(x).x=y:::> .y=X

Similar results hold for Tl043, three of which are given below. 287

SYSTEMS OF FORMAL LOGIC

Tl047

(X):X=Y&·Y=Z~:X=Z

Tl048

(x) (y): X= y&.y=z~:x=z

Tl049

(x)(y)(z):x=y&.y=z~:x=z

By virtue of Tl 043, the law of exportation, T36 of PLT', substitution, and modus ponens we have Tl050

X=Y~ .Y=Z~

.X=Z

A similar operation on Tl 049 gives Tl051

(x)(y)(z).x=y~ .Y=Z~

.X=Z

Euclid's axiom: "Things equal to the same thing are equal to each other" is proved as Tl052. Tl052

X=Z&·Y=Z~:X=Y

DI052

1. 2. 3. 4. 5. 6.

7.

x=z&.y=z(~)

X=Z y=Z .Z=Y z=y x=z&.z=y

y=Z~

x=z&.z=y~:x=y

8. X=Y

hyp 1, Tl24, +S, - ~ 1, Tl25, +S, - ~ Tl042, + SI, Y/Xl, Z/Yl 4,3, -::::> 2,5, *A5 Tl043, + SI, X/Xl, Z/Yl, Y/Zl 7,6, -~

1. x=z&.y=z!-x= Y 2. !-x=z&.y=z~:x=Y

LT'

8.3 Quantification over Predicate Variables; the System IF

In order to express other truths concerning identity it would be useful to have as a theorem the result (A)

tPx~tPy::::>

.X= Y

so that we could have the formula (B)

x=y=

.tPx~tPy

as a theorem, not just X=Y~ 288

.tPx~tPy.

But (A) is not L-true, since some

HIGHER FUNCTIONAL SYSTEM AND CALCULUS OF CLASSES

substitutions on (A) are false, (for instance when "human" is substituted for fP, and the names of two human beings, e.g., Bertrand Russell and A. N. Whitehead, for x and y respectively. In such a case we get, "If Russell is human then Whitehead is human, then Russell is identical with Whitehead".) Thus (B) is not L-true. However, consider the expression (C)

(fP).fPx:=JfPY:=J:x=y

(C) is not a theorem of LFLT' even in its extended form. The reason that this is the case is found in the fact that (C) is not a wff of LFLT', since LFLT' has quantification only over individual variables, and fP is a predicate variable quantified in (C). But (C) is L-true. In fact, (C) is a theorem of the higher functional system HFLT', specifically, of its second order predicate calculus with identity in which not only individual variables but predicate variables are bind able by quantification. Thus while in LFLT' all predicate variables have free occurrence, in the higher system of the second order predicate calculus, there are theorems in which predicate variables may have occurrence as bound variables. We will not present a full articulation of second and higher order predicate calculi in this book, but only outline some results provable in the second order functional calculus with identity which do not require much elaboration beyond that of the lower functional calculus. 8.31 Axioms and Deductions in 2Fi:'T' (Tl053-1061). It was claimed that (C) above is L-true. This can be shown intuitively as follows: if the antecedent of (C) holds then every predicate of x is a predicate of y, i.e., there is no property of x which is not a property of y. Now by axiom LFA6 of LFi:'T" the property of being identical with x is a property of x. But if being identical with x is a property of x, and y has all the properties of x, then being identical with x is a property of y. Assuming now that we have articulated a suitable primative basis for 2Fi:'T" the second order functional calculus with identity, adding formation rules allowing for quantification over predicate variables, specification of conditions for wffs analogous with LFLT', a rule of universalization for predicate variables, and the axioms, 2FA8 2FA9

X=y:=J .(fP).fPx:=JfPy (fP)fPx:=J .fPy

we can prove (C) formally in the following way, (continuing the numbering

289

SYSTEMS OF FORMAL LOGIC

of theorems for LFLT' but prefacing each new number, n, with the index 2FT.) As an immediate consequence of 2FA9 we have 2FTl053

(tP)tPx=> .tPx

And the next theorem follows by formula substitution on 2FTl053, 2FT 1054

(tP).x=x=> .Y=x=>:x=x=> .y=X

The quantifier of 2FTl054 occurs vacuously. 2FT 1055

(tP).tPx =>tPy=>:x=y

D2FTl055 1. (tP).tPx=>tPy 2. (tP).x=x=> .y=X 3. (tP).x=x=> .y=X=>: X=X=> .y=X 4. X=X=> .y=X 5. X=X 6. y=X (=» 7. X=y

hyp 1, + SF, [x=x]/tPx, [y=x]/tPy

2FT1054, + SI, X/Xl, y/YI 3,2, - => LFA6 4,5, -=> 6, LFTI042, + SI, Y/XI, X/YI

1. l-(tP).tPx=>4>yl-x=y 2. l-(tP).tPx=>tPy=>:x=y

Thus we have as a theorem of 2FLT', 2FT1056

X=y= . (4)). tPx=>4>y

which is a formula analogous with (B) of the LFC, but while (B) is invalid, 2FTI056 is clearly L-true. (The first clear formulation of 2FT1056 is due to Leibniz who wrote, "Things are identical (the same) if one can be substituted for the other and truth is preserved." This formulation is not exact since "things" and their names are not distinguished. One does not substitute "things" for each other in formulae. But, of course, the confusion is easily repaired. Frege saw the confusion and formulated the principle correctly in an axiom which is almost the same as 2FAS.) Note that a derivation like that given for 2F1053 is not available in LFLT' as formulated in Chapter 7. We have also in the present system: 290

HIGHER FUNCTIONAL SYSTEM AND CALCULUS OF CLASSES

2FTl057

X= y::> .( . (y 2. ~x= y::> y=X 3. ~y=x::> . ( x 4. f-x=y::> . ( x 5. f-x=y::> . (

Thus we have 2FT1058

x=y=. . ( . X = Y ::> : -.x=y::>:-.y=X

T95, +S,y=x/p,x=y/q 2FTl042, +Sr, y/Xl, X/Yl 2,3, -::> -.y=X 4,1,-::> (x::>:y=x(::» 2FTl055, +Sr,y/Xl,X/Yl - .y=x::>:-(x 6, T95, +S, (x/p, y=x/q, -::> 7, 5, - ::> -(x(::» (3 x 8, 2FT1024, -::>

3. y=X::> .X=y 4. - .X=y::>:- .y=X 5. 6. 7.

8. 9.

It should be noted that in steps 8 and 9 the functional correlate

of LFTl024 is employed; this formula is provable from 2FA9 and a generalized form of D4. The derivation D1059b establishing the other side of the equivalence is left as an exercise. The law 2FTl060

x=y&. - .y=z::>:- .X=Z

is now provable. From the antecedent of the theorem the formula (A)

( y&: (3x => ({>y &: (3({». ({>z & '" ({>y => : . (3({». ({>z & '" ({>x

(B)

Thus by modus ponens we have (C)

(3({». ({>z&"'({>x

which gives (D)

(3({»", . ({>z=>({>x

This result may be transformed into (E)

'" .X=Z

by 2FTl059. Thus 2FTl058 is provable. Likewise, of course, we have 2FTl061

x=y&. '" .X=Z=>:'" .y=z

Many other theorems, both with and without identity, are provable in the second order predicate calculus, but these are beyond the scope of the present work.

8.4 Abstraction and the Boolean Algebra In Chapter 7 the formula ({>x was read, "x is ({>", where x is an individual variable and ({> is a predicate variable. By this we do not mean that x = ({>, since, where the name "Socrates" is substituted for x and "wise" for ({>, we are not intending to identify Socrates with wisdom. In this case what is intended is to classify Socrates by saying that he is a member of the class of wise things, or, alternatively, has the property "wisdom". The fact that Socrates is wise does not exclude other beings from being wise, though if we intended to identify Socrates with wisdom, nothing else could possibly be wise since only Socrates is (identical with) Socrates. In general, the predicate expression designates, or names, or denotes a class. A class is here provisionally defined as any consistently specifiable collection or aggregate of entities where we do not assume that the collection must be space-temporally contiguous (or even spaciotemporal at all). Thus, New York is a member of or belongs to the class of cities, and so does ancient Babylon. Socrates is a member of the class of humans and so is the reader. Thus, "Socrates is wise", is a statement about two things, (i) Socrates, who is said to have the property of wisdom, or extensionally, to be a member of the class of wise things, and (ii) the

292

HIGHER FUNCTIONAL SYSTEM AND CALCULUS OF CLASSES

class of wise things which is said to have Socrates as one of its members. Now it is useful and clarifying to have a constant to express the relation of memberschip. The logical functor which is usually used is the epsilon, E, which is read, "is a member of the class." Thus "Socrates E wise," is read "Socrates is a member of the class of wise (things)". Likewise the matrix, xE is read "x is a member of the class named by ." It should be noted that whenever xE is assertable so is x, and whenever x is assertable, so is XE. Predicates are here taken as names of classes. Thus the meaning of a predicate, e.g., "wise", is a property. But the predicate "wise" names the class of wise things, as when we say, "Socrates is a member of the class of wise things." The same symbols will be used as class variables as were used for predicate variables. Thus we posit the following definition:

aEF=d! Fa (De! E) where F is a predicate variable or constant and a is an individual variable or constant. In an expression of the form aEF, a is said to be a member of or an element of F. When this definition is used in proofs its use is indexed by the sign 'Def. E'. Every theorem of LFLT' is a logical truth when all the expressions in it of the form Fa are changed into expressions of the form aEF. Thus, (A)

(x):. x=> 'l'x&: 'l'x=> ex=>:. x=> ex

is a theorem of LFLT', To (A) corresponds,

Though 'E' is not a symbol of LFLT', (B) will be regarded as a theorem of LFLT' when it has been extended so as to include this symbol. We will now introduce the notion of abstraction. Consider the wff of LFLT', (A)

Fx&Gx

(A) can, of course, be translated, using

(B)

E,

as

xEF&XEG

The statements formed out of (A) and (B) by substitution of an individual 293

SYSTEMS OF FORMAL LOGIC

constant for the individual variable x are true when the individual named by the individual constant is a member of both the classes F and G. But sometimes it is desirable to be able merely to name the property or designate the class named in (A) and (B) without reference to the fact (or non-fact) that x is a member or Socrates is a member or Thomas Blakeley is a member or the kitchen chair is a member. We do this by reference to the abstraction operator, "\". From any statement or matrix ofLFLT' (with e added) we can form a name which makes reference only to the predicate variables abstracting from the individual variables or constants having occurrence in it. Using the abstraction-operator we have: (C)

x\.xeF&xeG.

The expression (C) names a certain class, the class of those things which are both members of F and members of G. The abstraction-operator is used here to "abstract" from all succeeding occurrences of x following the "slash", \. It informs us, in effect, that we are to ignore all the following occurrences of x (to abstract from them) and to consider only the other functors and variables besides x. Thus, to represent it graphically, we have, (D)

... is a member of F and ... is a member of G

which is the name of the class of things which are members of both F and G. If we have (E)

x\.xeF:=JxeG

this is the name of the class of things which are members of G if they are members of F. If we have (F)

x\.xeFv xeG

we have the name of the class of things which are members of F or are members of G. The same can be done with any other functors or expressions of LFLT' supplemented bye. A name formulated by use of the abstraction-operator is called an

abstract. Now, given any individual constant, a, and given an abstract we have either

294

HIGHER FUNCTIONAL SYSTEM AND CALCULUS OF CLASSES

(G)

OE.X\ .... X ...

or (H)

"" .0E.X\ .... x ...

This follows by virtue of the law c]Jx=>:p=>(x)c]Jx

we obtain (B)

p=>c]Jx=> .p=>c]Jx.

And by application of the second rule of reduction to this formula we obtain

(C), the P-correlate of (A), is, of course, a tautology, a substitution on Tl. (iii) Application of the same process to LF AS

308

THE LOGICAL PARADOXES

(D)

(x)rpx~rpy

gives (E)

rpx~rpy

and (F)

p~p.

Thus the axioms of the system are either tautologies or have P-correlates which are tautologies. (iv) By an earlier result, the rules of the system are such that if the premises the rules have P-correlates which are tautologies, then the conclusion has a P-correlate which is a tautology. Since the axioms have the property of having tautologies as P-correlates and the rules preserve this property, it is clear that every theorem of LFLT' has a tautology as its P-correlate. Hence, for any theorem, T, of LF LT', its denial, ~ T, is such as to have a contradiction as P-correlate. Hence the Aristotle consistency of LFLT' follows, since by the foregoing result not both the wff A and its denial ~ A can be theorems. The absolute consistency of the system follows immediately since not every wff of LFLT' is a theorem of LFLT'. The Hilbert consistency of the systems is provable by selecting, say, the wff ~ :p~rpx~ :p~rpx as the test formula and showing that it is not a theorem.

9.5 The Decision Problem It might be hoped that the technique of reduction used to prove con-

sistency would afford a decision procedure for the system. This is not the case, however, because while every axiom and every theorem of LFLT' have P-correlates which are tautologies, it is not the case that every wff of LFLT' which has a formula as a P-correlate which is a tautology by application of the two reduction rules, QR2 and QR3, to it is a theorem of LFLT'. While it has been shown that if the wff A is a theorem of PLT' and if B is its F-correlate, i.e., if B is a substitution instance of A, then B is a theorem ofLFLT', it has not been shown, and cannot be shown, that if A is a theorem of PLT' and if A is the P-correlate of B, then B is a theorem of LFLT" If the latter could be shown then we would have a decision procedure for the system similar to that available for the Boolean 309

SYSTEMS OF FORMAL LOGIC

algebra. That this recourse is not open to us is shown by the following case. It can be proved that (A)

.XE Mortal is a universal class; only if all men are mortal; only if there are no members of the class of men who are not members of the class of mortal things. A class is a member of the class of classes E if and only if the class is non-empty, i.e., has members. Thus the class designated by the abstract

X\.XE Human &.XE Logician the class of humans who are logicians, is a member of the class of membered classes, E, since there are people who are logicians. But, apparently, the class denoted by the abstract

X\.XE round &XE square has no members: thus, it is false to say that the class denoted by this abstract is a member of the class of classes E. In an analogous way, with respect to A, it is false to say that the class denoted by the abstract

X\.XE Human &.XE Logician is a member of the class of classes A, since there are humans who are not logicians.) a, h, c, a', h', c', ... , an, ... (These are signs denoting "things" of any sort: prop'ositions, physical objects, relations, marks on paper, persons, etc.) The formation rules differ from those of LFi::T' in allowing unstratified expressions as wffs. Formation rule 2 of 7.23 is, accordingly, liberalized. 10.32 The present version of the Fitch system, differing in some details from Fitch's own formulation, contains as rules and definitions: (1) The rule of Abstractor-substitution, SA (as presented on page 295). (2) The following definitions:

DefE Dl. D2. D3. D4.

318

(see page 293) F=dfx\.XEF -F=dfx\,,-, .xEF FnG=dfx\.XEF&XEG FuG=dfx\.XEFvxEG

NON-STANDARD FUNCTIONAL SYSTEMS

D5. D6. D7. D8. D9. DlO.

=dfx\", .X=X V =dfx\.x=x FcG=df(x)[XEF=>XEG] #=dfx,y\"'.X=y (X)[: .y=Z::::> .X= y (Law of transposition for

FTl005

identity) 1

'" .x=y::::> "':y=z

hyp

2

y=z

hyp

3 4

x=yv '" .x=y x=y

+LEM= hyp

x=y

5

I-=:: .X= y

6

'" .X=y::::> "':y=z '" .y=z y=Z X=y x=y y=Z::::> .x=y 13 TH

7 8 9 10 11 12

324

4, reit hyp 1, reit 7,6, - ::::> 2, reit 8, 9, - '" 3,4-5,6-10, - v 2-11, +::::> 1-12. +::::>

NON-STANDARD FUNCTIONAL SYSTEMS

Fitch does not mention the fact in Symbolic Logic, but by virtue of the inclusion of his rule + LEM =, the system for the logic of identity possesses all the theorems of the classical propositional calculus - when (and only when) identity assertions are substituted for the propositional variables of, say, PLT" In other words, when we restrict the legitimate substitution instances on propositional variables to the formulae of the logic of identity, all the usual theorems of the propositional calculus - so restricted - are theorems of this system. The simplest way of showing that this is the case is to reduce this special system to a set of axioms resembling those of PLT', with its two rules and a restriction on the rule of substitution specifying that substitution be limited to the variables and constants of the logic of identity and to the propositional constants. The axioms of the special system, analogues of those of PLT', would be:

Al A2 A3

x=y~.y=z~x=y x=y~ .y=z~z=w~:x=y~y=z~ .x=y~z=w

"".x=y~"":y=z~:.y=z~x=y

Since it is provable that AI, A2 and A3 are theorems of the Fitch system and since modus ponens and substitution may be used as rules of his axiomatic, (though in Symbolic Logic, Chapter 1.4 he uses axiom schemata rather than employing an explicit rule of substitution), this system constitutes, when duly restricted, a part of his logic of identity. Thus, in this respect, the Fitch system is a more powerful instrument than the logics of intuitionism, since they do not possess this extension. It must be noted, however, that this extension, approaching S-logics, holds in Fitch's system only for a limited domain of logic, not for the entire system; otherwise the system would be an S-logic, which it is not, and would be contradictory, which it is not. 10.34 Some Laws of Class Memberships in LFF'p. Only a few more interesting laws using the usual class relation signs (n, U, - , c) will be mentioned here. (See next page.)

325

SYSTEMS OF FORMAL LOGIC

FTl006

'" .xEF=':XE-F

1 I--=.XEF

I XE.X\'" .xEF

2

hyp

1, SA

3 XE-F 4 "'.xEF:::J:XE-F 5 XE-F

2, D2 1-3, +:::J hyp

6 XE.X\'" .xEF 7 '" .xEF 8 xE-F:::J '" .xEF 9 '" .xEF:::J:XE-F&:. XE-F:::J '" .xEF

5, D2

10 TH

6,SA 5-7,

+:::J

4,8, +& 9,

+ =.

Likewise, using abstractor substitution, the definitions, and the previously presented rules it is easy to prove FTl007

XEF&XEG=. .xE.FnG

and FTI008

XEFv XEG=. .xE.FuG

These and similar results are left to the reader as exercises since they present no problems not previously confronted in earlier chapters. More interesting theses of the Fitch system are proved as FTI009

aE.x\x=X

and FTIOlO

X\X=XE.X\X=X

Note that FTlOI0 violates the rules of types and stratification. The Fitch system is not subject to any of the forms of these restrictions on the construction of wffs. Theorem FTl009 asserts that, for whatever be substituted for a, a is a member of the class of self-identical things, while FTIOlO, X\X=XE.X\X=X, asserts that the class of self-identical things is a member of the class of self-identical things. Fitch describes the class of self-identical things as a universal class, a class having all things (thus, of course, itself) as members. Such self-inclusive classes are rejected by type-theoretical and standard stratificational logics, but not by Fitch's. The proof of FTlO09 is as follows:

326

NON-STANDARD FUNCTIONAL SYSTEMS

11 a=a

hyp

21~·x\x=x

1, SA

(Step 2 follows from step 1 by abstractor substitution, since, as remarked on page 295 in the formula Wa (and thus aeF), we may substitute the property ... =a for Wand x\x=x for F; since a=a is a truth of logic we have that aex\x=x, "a is a member of the class of self-identical things", a universal class according to Fitch.) The easiest method of showing that FTIOlO holds is by simple substitution on FTl009, thus:

11 aex\x=x

2 x\x=xe.x\x=x

FTl009

1, S, x\x=x/a

a procedure perfectly legitimate in the type-theory-free Fitch system, but, of course, forbidden in systems like LFi.:'T" since in the latter system step 2 is not a wff. Fitch's own proof of this controversial result is slightly different: 1 x\x=x= .x\x=x 2 x\x=xe.x\x=x

+= 1, SA (In Symbolic Logic, "attribute introduction")

Note: Just as in the proof of FTl009, we have that the class named by x \ x = x is self-identical; thus, by abstractor substitution we have that this class is a member of the class of self-identical things, i.e., "selfidentity is self-identical"; or "the class of self-identical things is a member of itself." Likewise it is provable in the Fitch system that FTlOll

",:x\", .xexe:x\'" .xex

i.e., "it is not the case that what is not a member of itself is a member of the class of things not members of themselves", a formula also rejected as not wf by systems similar to LF~T" 10.35 The Laws of Class Membership and the Logical Paradoxes. It is clear that, unless due restrictions are made, the Fitch system will be subject to the Russell Paradox and similar logical antinomies described in Chapter 9. The Russell Paradox would be provable in the system in the following way, defining the Russell-class Z as before, namely, x\ '" . xex, we have as a theorem: 327

SYSTEMS OF FORMAL LOGIC

FTI012

ZEZV '" .ZEZ::):ZEZ&", .ZEZ

hyp

I ZEZ

I

2 ZE.X\'" .XEX 3 "'.ZEZ 4 ZEZ::)."'.ZEZ 5 "'.ZEZ 6 ZE.X\'" .XEX 7 ZEZ 8 '" .ZEZ::):ZEZ 9 ZEZ::) '" .ZEZ&:", .ZEZ::):ZEZ 10 I ZEZV '" .ZEZ

1, (Def. of Z) 2,As 1-3, + ::) hyp 1, As 2, Def. 5-7, + ::) 4,8, +& hyp

11

ZEZ

hyp

12 13 14 15

ZEZ::) '" .ZEZ&: "'.ZEZ::).ZEZ ZEZ::) '" .ZEZ "'.ZEZ ZEZ&", .ZEZ

9, reit 12, -& 13, 11, - ::) 11,14, +&

16

"'.ZEZ

hyp

ZEZ::) '" .ZEZ&:", .ZEZ::). ZEZ "'.ZEZ::).ZEZ ZEZ ZEZ&", .ZEZ ZEZ&", .ZEZ ZEZV '" .ZEZ::):ZEZ&", .ZEZ

17 18 19 20 21 22

9, reit 17, -& 18, 16, - ::) 19,16, +& 10, 11-15, 16-20, - v 10-21, +::)

The converse, of course, is also a theorem, since the Fitch system possesses the rule of negation elimination, - '" . Now the underlying motive for suspending LEM as a general law becomes apparent: ifLEM were a theorem of the Fitch system we would have (A)

ZEZV '" .ZEZ

as a theorem, since this system does not reject either

(B)

328

ZEZ

NON-STANDARD FUNCTIONAL SYSTEMS

of step 1 in the proof or

"'.ZEZ

(C)

of step 5 as do classical systems, on the ground that they are not wffs. But if the system allowed the proof of (A) as a theorem, the following would plainly result:

1 ZEZV '" .ZEZ:=J:ZEZ&", .ZEZ 2 ZEZV '" .ZEZ 3 ZEZ&", .ZEZ

FTl012 A 1,2, - :=J

IJthe above deduction were valid in the Fitch system, the system would, as above, be provably contradictory, since step 3 is an explicit contradiction. However, though step 1 is a theorem of the system, FTlOI2, step 2 is not generally provable. Hence, it cannot be introduced as the second step of a valid proof in this system. And, hence, the deduction is invalid. FTl012 is rendered "harmless" with respect to consistency by virtue of the absence of LEM as a general law of the system. The deduction also sins in other ways against Fitch's restrictions. But there are other antinomic results: one of them is the Curry Paradox, which could, even when LEM is suspended, be proved in the system LF~F unless further restrictions are placed on valid proofs in it. Following Fitch, but adapting his rules and definitions to the present articulation of his system, the Curry Paradox is derivable as follows, where Y is taken as an abbreviation of the abstract X\.XEX:=JP

The Curry Paradox YEY

2 YE.X\XEX:=JP 3 I;E Y:=JP 4

hyp

7 YEY

1, (Def. of Y) 2, As 3, 1, - :=J 1-4, +:=J 5, As 6, (Def. of Y)

8 P

5, 7, - :=J

5 YE Y:=JP 6 YE.X\XEX:=JP

329

SYSTEMS OF FORMAL LOGIC

It is notable that no negation functor of any sort appears in the Curry

Paradox. Hence, it cannot be avoided merely by the suspension of LEM or of the rule, + "'. Any system in which the Curry Paradox is provable is Post-inconsistent, (cf. Chapter 5); and since we can substitute, e.g., p&'" p, for the bare variable p, thus rendering it Aristotle-inconsistent, or can set up the proof in such a way that Y is defined as

the system's barriers against inconsistency are insufficient, if these consist only of the propositional logical suspensions of such laws as pv",p ",.p&"'p p-::::;q::::;. "'q"::::J "'p "'P"::::J "'q"::::J .q"::::Jp

and the like. Thus Fitch imposes a special restriction: this restriction is entirely systematic in character; unlike the original Russell-Whitehead Principia M athematica exposition of the theory of types, the prototype of the meta-logical solution, it is not presented as, in se, evident, but, rather, is justified because it, (a) prevents the proof of the known paradoxes, and (b) allows the proof of the absolute consistency of the entire, system. Thus, instead of appealing to its immediate or intuitive evidence, Fitch argues for it in terms of the logical and systematic consequences which this technical restriction makes possible. The special restriction is stated by Fitch as follows: A propositional item [element] p cannot appear in a proof [or derivation] in such a way as to be a resultant! of a subordinate proof [a main echelon or sub-echelon] that (i) has p as an hypothesis and (ii) has some proposition other than p as an item [element].2 1 Fitch's use of the term "resultant" differs somewhat from that introduced in Chapter 4. By the term resultant he specifies: (i) any element inferred from other elements in a proof by the rules of inference, that element then being said to be a resultant of all the elements needed for its inference: (ii) a sub-echelon is said to be a resultant of, (a) each of its own elements, (b) any elements reiterated into it from superior echelons, and (c) of each of the superior echelons themselves. Thus in the following proof 2 Op. cit., p. 109.

330

NON-STANDARD FUNCTIONAL SYSTEMS

Examination of all the proofs of PNDF and of LF~F shows that only one of them violates the special restriction. The proof of FTl 0 12 in particular does not violate it. However, examination of the Curry paradox - and the numerous analogous paradoxes which can be built up from it shows that it stands in violation of the restriction: The Curry Paradox has the element YeY

both as hypothesis (step 2) and as a resultant (step 7): step 5, Ye Y=>p

is a resultant of step 1, the hypothesis, and the echelon to which it belongs; likewise, step 7 YeY

is deduced from step 5, with the help of step 6 and the definition Y=df.x\.xex=>p

Thus step 7 is a resultant of an echelon having a formula of a form identical with it as hypothesis and which has formulae other than Ye Y

2 3

p:::>r

hyp

p

hyp q

p:::>r 4 p 5 r 6 q:::>r 7 8 p:::> .q:::>r 9 p:::>r=> .p=> .q=>r

hyp 1, reit 2, reit 4,5, -:::> 3--6, +:::> 2-7, +:::> 1-8, +=>

the second sub-echelon (with hypothesis q) is a resultant of both the main echelon and the first sub-echelon, the first sub-echelon, of the main echelon. Step 4 is a resultant of step 1 by the rule of reiteration, step 5 of step 2 by the same rule. Step 6 is the resultant of step 4 and 5 by - =>. Step 7 is the resultant, (this is the special use of the term employed in Chapter 4), of the entire echelon 3--6, step 8 of the echelon 2-7, step 9 of 1-8. The second sub-echelon is a resultant of each of its element, steps 3, 4, 5, 6, of steps 1 and 2, and of its superior echelons, with the sole exception of the theorem echelon.

331

SYSTEMS OF FORMAL LOGIC

as elements. Thus the Curry Paradox violates the special restriction and is, therefore, not a valid deduction in the system LF~F' The restrictions on the rule of negation introduction and the special restriction not only exclude the known paradoxes, but all antinomic results, as Fitch shows by means of a recursive proof of consistency for the system (cf. Symbolic Logic, Chapters 4.20 and 6.27), whose inclusion limitations of space forbid here. 10.36 Quantification in LF~F' The quantification rules are not primitive, but derived, in LF~F' In this the present system adheres to the original. Thus, there is a sense in which the system, taken in terms of its primitive basis, is a class-membership-based logic, (i.e., it is built up on the relation .... E ... as primitive) as opposed to LF~T' which is a property-relationbased system, (i.e., is built up on the basis of functions like CPx, rather than xEF). That this distinction, though formally significant for the structure of the system is not crucial with respect to content is shown by the mutual transformability of class and property statements indicated in Section 8.4f of Chapter 8. The significant differences with respect to content between LF~F and LF~' arise in the context of the differences between their restrictions on rules of formation and transformation, and hence between the number, form and character of their laws. It should not be assumed that every functional calculus built up on the basis of the technique of natural deduction and/or constructed as a class-membershipbased logic is a non-standard system. Methods for converting LF~F into a standard functional calculus are discussed at the end of the chapter. First, proofs are set out using the primitive functors A and E; then the derived quantification rules are deduced and a few laws proved; finally rules for negation with quantifiers allowing the proof of the functional analogues of the law of De Morgan are presented. FTl013

FEA~

.aEF

This law follows simply by resort to the rule of universality elimination, -A. (Note that the converse does not hold, since, from the fact that some entity, a, is a member ofthe class Fit does not follow that Fis a universal class.)

332

NON-STANDARD FUNCTIONAL SYSTEMS

FTlO14

FnGeA-::J .FeA

FnGeA 2 x FnGeA 3 xe.FnG 4 xe.y\yeF&yeG xeF&xeG 5 6 xeF 7 FeA 8 TH

hyp 1, reit 2, -A 3, D3 4,As 5, -& 2-6, +A 1-7, +-::J

The converse is not a theorem. To explain the proof: F nGeA is a universal formula, i.e., F n G is said to be a member of the class of universal classes, A. Thus it may be reiterated into the non-hypothetical subechelon 2-6 in accordance with the rule + A. Step 3 follows from 2 by universality elimination. In step 4, y is used to distinguish the abstract

y\yeF&yeG from the variable x which is assigned as a member of the class named by the abstract. The remainder of the proof follows clearly from the rules cited in its justification column. An important theorem, not proved in Symbolic Logic, is FTl015 YEA where V is regarded as an abbreviation for the class x\x=x, D6. 1 x a=a

+=

2

aex\x=x

3 4

aeV

As D6 +A

YEA

Likewise, we can prove: FTl016

1 2 3 4 5 6

FeA-::J .FuGeA

I FeA

x FeA xeF xeFv xeG xe.FuG FuGeA 7 TH

hyp 1, reit 2, -A 3, + v 4,D4 2-5, +A 1-6, +-::J 333

SYSTEMS OF FORMAL LOGIC

and FTI017 1 2 3 4

5 6 7 8 9 10

xe.Fn-F:::J:xeA

xe.Fn-F xe.y\yeF&.ye-F xeF&xe-F xe-F xe.y\- .yeF - .xeF xeF xe.y\- .y=y xeA TH

hyp 1, D3

2, As 3, -& 4,D2

5, As 3, -& 7,6, - 8,D5 1-9, +:::J

Some laws for the functor E are: FT1018

aeF:::J .FeE

This follows by the rule FR7, +E. Another law analogous with FTl014 is: FT1019

FnGeE:::J .GeE

1

FnGeE

hyp

2

x xeFnG

hyp

xe.y\yeF&yeG 3 4 xeF&xeG 5 xeG GeE 6 7 GeE 8 TH

I

2,D3 3, SA 4, -& 5, +E 1,2-6, -E 1-7, +:::J

Note here that the rule of existence elimination, permitting the deduction of step 7 from steps 1 and 2-6 is, like the rule of disjunction elimination, a rule which allows the conclusion of a sub-echelon to be shunted into the superior echelon to its immediate left. Thus the conclusion of such

334

NON-STANDARD FUNCTIONAL SYSTEMS

a sub-echelon is transformed into its resultant (in the sense of these terms as defined in Chapter 4). An interesting result provable in L~F is

FTl020

AeE

YEA 2 x\x=xE.A 3 AeE

TH1015

1, D6 2, +E

It is notable that while steps 1 and 2 abide by the stratification requirements of LFfT' since V and its equivalent, x\x=x, are classes and A is a class of classes, step 3 does not, since both A and E are on the same level, namely classes of classes. In general, LFfT' does not allow the unrestricted formulation of the classes A and E, (cf. Symbolic Logic, Appendix C). This theorem reads, in its main interpretation, "The class of universal classes is a member of the class of membered classes", or "Universality is non-empty". That the system allows for the formulation and proof of such results and at the same time is able to guard itself against the logistic and semantic antinomies is of no small significance, since it provides a consistant way of formulating certain forms of argument rejected by type-theoretical and standard stratificationallogics. The two standard quantifiers are available by means of the following deductions, following Fitch, where if>, P, e, denote arbitrary contexts. The Derived Rule of Universal Quantifier-Elimination (- UQ) 1 I (x)if>x

hyp

2 x\if>xEA 3 ae.x\if>x 4 if>a

D9 SA DefE

Thus we have (x)if>x::::::> . if>a

which we will justify by the citation "- UQ". 335

SYSTEMS OF FORMAL LOGIC

The Derived Rule of Universal Quantifier Introduction ( + UQ)

I

n x

(fJx n+i XE.X\(fJX n+i+l n+i+2 x\(fJxE.A n+i+3 (X). (fJx

n+i, SA n-n+i, +A n+i+2, D9

Thus any proof carried out in accordance with + A can be converted by the use of definitions into a proof terminating with a universally quantified statement and in general any formula of the form

(... )EA can be converted into a universally quantified formula of the form

...

( )(...)

where the first left and right parentheses embody a quantifier. We will regard the latter as an abbreviation for the former. The Derived Rule of Existential Quantifier Elimination ( - EQ)

I

n (3x)(fJx n+l x\(fJxE.E n+2 x XEX\ (fJx -

n+3

(fJx

n+i p n+i+ 1 I p

336

DIO hyp Def.E

n+ 1, n+2-n+i, -E

NON-STANDARD FUNCTIONAL SYSTEMS

Existential Quantifier Introduction ( + EQ)

1 (/Ja

hyp

2 aE.x\(/Jx 3 x\(/JxE.E 4 (3x)(/Jx

1, SA 2, +E

1

3, DlO

Thus by this derived rule we can always infer (3x)(/Jx from (/Ja: thus we have (/Ja:J (3x)(/Jx

(By the definitions of the former system, preserved in L~F> we are permitted to transform formulae of the form (/Jx

into XE(/J

and vice versa.) A few proofs illustrative of this quantification theory are given below. using the standard property and individual signs. FTlO21

(x). (/Jx:J Px&. PX:J eX:J: (x). (/Jx:J ex

1 I (x).(/Jx:J Px&. pX:Jex

hyp

2

1, reit 2, D9

3 4 5 6 7 8

x (x).tPx:J Px&. PX:J8x x\(/JX:J Px&. pX:JexE.A aE.x\(/Jx:J Px&. PX:J8x (/Ja:J Pa&. Pa:J ea (/Ja:J Pa Pa:J8a (/Ja

9

(/Ja:J Pa

10 11

Pa Pa:Jea ea (/Ja:Jea

12 13 14 (x)(/JX:J ex 15 TH

3, -A -UQ 3, -& 3, -& hyp 4, reit 9,10, -:J 7, reit

11, 10, -:J 8-12, +:J 2-13, +UQ 1-14, +:J 337

SYSTEMS OF FORMAL LOGIC

FTlO22

(X). tPX V 'I'X=>f~)X&.(y) 'I'y=> :(y)ey

(X). tPX V 'I'X=> eX&.(Y) 'I'y

hyp

2 3 4 5

(x).tPxv'I'x=>ex (y)'I'y x\tPxv'I'x=>exe.A y x\tPxv'I'x=>exe.A

1, -& 1, -& 2, D9 4, reit

6

ye.x\tPxv'I'x=>ex

tPy V 'I'y=> ey 7 8 (y)'I'y 'I'y 9 tPyv'I'y lO ey 11 12 ye.y\ey 13 ye.y\eye:A 14 (y)ey 15 TH

5, -A

6,SA 3, 8, 9, 7,

reit -UQ +V lO, - =>

SA 5-12, +A D9 1-14, + =>

In future proofs we will make a direct inference from steps like 11 to those like 14 without the intermediate steps. FTlO23 2

(3x).x=y&4>x=::4>y

(3x).x=y&4>x

hyp

x x=y&(/Jx

hyp

3 x=y (/Jx 4 (/Jy 5 6 I tPy 7 (3x).x=y&tPx=> :(/Jy 8 tPy

9 y=y lO y=y&tPy 11 (3x).x=y&tPx 12 tPy=> .(3x).x=y&tPx 13 (3x).x=y&(/Jx=>tPy&:tPy=> .(3x). x=y&(/Jx

14 TH

338

2, -& 2, -& 3, 4, - = (first form) 1,2-5, -EQ 2-5, + => hyp

+= 8,9, +& lO,+EQ 8-11, + => 7,12, +& 13, + ==

NON-STANDARD FUNCTIONAL SYSTEMS

Note that step 11 follows from step 10 by + EQ replacing y with x at its first and third occurrences. y is not here replaced at its second occurrence, since '= y' is the property assigned to y by the expression y = y. (x could be substituted for y, of course, at all places of y's occurrence in 10, but it need not be.) If f/J is a two place rather than a one place predicate we have: FT 1024

(3x)(y). f/Jxy~ : (y)(3x) . f/Jxy

1

(3x)(y). f/Jxy

hyp

2

y (3x)(y). f/Jxy

I, reit

3

x (y).f/Jxy

(y). f/Jxy~: f/Jxy 4 f/Jxy 5 f/Jxy ~ (3x). f/Jxy 6 (3x).f/Jxy 7 8 (3x).f/Jxy 9 (y)(3x). f/Jxy 10 TH

hyp -UQ 4, 3,-~ +EQ 6,5, - ~ 2,3-7, -EQ 2-8, +UQ 1-9, + ~

Fitch includes an illuminating discussion in terms of substitution instances. If f/Jxy is the proposition, "x is a cause of y", then the proposition (3x)(y)«Pxy would be the proposition, "There is an ex such that, for every wye, ex is a cause of wye", or in ordinary language, "There is something which is a cause of all things". The proposition (y) (3x) «Pxy would be, "For every wye, there is an ex such that ex is a cause of wye", or in ordinary language, "Each thing has a cause". Now we can see that if it is true that "There is something which is a cause of all things", then it must be true that "Each thing has a cause". But from the fact that each thing has a cause it does not follow that there is a cause of all things. Hence it is not possible, in general, to derive (3x)(y)f/Jxy from (y)(3X)«Pxy.l Now we present the negation-quantification rules of LF~F' lOp. cit., p. 155.

339

SYSTEMS OF FORMAL LOGIC

FR8

Negative Universal Quantification Introduction ( + "" UQ)

From (3x)",,x

and (x) . t1> x::> lJIx::> . '" lJIx::> '" t1> x

possessed by these systems but unprovable in LF

pF.

pF.

10.37 The "Standardization" of LF It is a relatively easy matter to make this system an S-logic for the lower functional calculus with identity. The procedure would be as follows: (i) Introduce stratification into the rules of formation, in a way analogous with those of LFi:T" (ii) Replace the propositional foundation of the system, PNDF, with PLT, or PLT', or some other S-propositionallogic. (iii) Remove the special restriction. (iv) Specialize the constants A and E in a way consonant with the limitations imposed by the stratification rules of LFi:'T" When these revisions are made, a number of theorems of LFpf' e.g., FTl 0 10, 10 II, 10 12, 10 15, etc., are no longer provable. But a large number of other theorems of the classical S-lower functional calculus, unavailable in LF~F' become available, e.g., the important and fertile law (x). '" Wx::> '" lJIx::> . lJIx::> t1>x

a theorem-by-substitution of LFi:'T" and the law of excluded middle, Tl 003 of LFi:'T" (x). Wx v '" Wx

343

BIBLIOGRAPHY

Books

AIusTOTLE, Metaphysics, 2 voIs. Text and translation by Hugh Tredennik, Cambridge, Mass., 1935. AIusTOTLE, Prior Analytics. Text and translation by Hugh Tredennik, Cambridge, Mass., 1955. BOCHENSKI, I. M., Ancient Formal Logic, Amsterdam, 1952. BOCHENSKI, I. M., A History of Formal Logic, South Bend, 1961. BoCHENSKI, I. M., A Precis of Mathematical Logic, trans. Otto Bird, Dordrecht, Holland, 1959. BooLE, George, An Investigation of the Laws of Thought, New York, 1951. BooLE, George, The Mathematical Analysis of Logic, Oxford, 1948. CARNAP, Rudolf, Introduction to Symbolic Logic and its Applications. Trans. William H. Meyer and John Wilkinson, New York, 1958. CHuRCH, Alonzo, Introduction to Mathematical Logic, vol. 1, Princeton, 1956. COPI, Irving M., Symbolic Logic, New York, 1954. FITCH, Fredric B., Symbolic Logic, New York, 1952. FITCH, Fredric B., A System of Combinatory Logic. Technical Report No.9, Office of Naval Research, Group Psychology Branch, New Haven, Nov., 1960. F'REoE, Gottlob, Begriffschrift, Halle, 1879. FREGE, Gottlob, The Foundations of Arithmetic. Trans. J. L. Austin, Oxford, 1950. FREGE, Gottlob, Grundgesetze der Arithmetik, Jena, vol. 1, 1893, vol. 2, 1903. HACKSTAFF, L. H., The Laws of Thought and their Function in Systems (Dissertation Yale), New Haven, 1958. HAcKSTAFF, L. H., Systems of Formal Logic (Mimeograph), Columbia, Mo., 1959. HERMEs, Hans and SCHOLZ, Heinrich, Mathematische Logik, Leipzig, 1952. HEYTING, Arend, Intuitionism, Amsterdam, 1956. HEYTING, Arend, Die formalen Regeln der intuitionistischen Logik, Sitzungsb. Preuss. Akad. Wiss., I. Phys-math. Kl., 1930. HILBERT, David and ACKERMANN, Wilhelm, Principles of Mathematical Logic (Translation by L. M. Hammond, G. G. Leckie, F. Steinhardt of Grundzilge der Theoretischen Logik, 1928), New York, 1950. HILBERT, David and BERNAYS, Paul, Grundlagen der Mathematik, 2 vols, Berlin, vol. 1. 1934; vol. 2, 1939. KANT, Immanuel, Critique of Pure Reason. Trans. N. K. Smith, London, 1953. KANT, Immanuel, Kritik der reinen Vernunft, Hamburg, 1952. KLEENE, Stephen, Introduction to Metamathematics, Princeton, 1952. 344

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MOSTOWSKI, Andrejej, Sentences Undecidable in Formalized Arithmetic, Amsterdam, 1952. POST, Emil L., The Two Valued Iterative Systems 0/ Mathematical Logic, 1941. (Cited in Church, op. cit.) QmNE, Willard Van Orman, Mathematical Logic, Cambridge, Mass., 1955. ROSSER, J. Barkley, Logic/or Mathematicians, New York, 1953. WmTEHEAD, Alfred North, and RUSSELL, Bertrand, Principia Mathematica, vol. I, Cambridge, 1935. WILDER, Raymond L., Introduction to the Foundations 0/ Mathematics, New York, 1952.

Articles BERNAYS, Paul, 'Habilitationsschrift', Math. Zeit., 25 (1926) 305-320. FITCH, Fredric B., 'A Basic Logic', J.S.L. 7 (1942). FITCH, Fredric B., 'A Demonstrably Consistent Mathematics', Ibid. 15, No. I (1950) 17-24; 16, No.2 (1951) 121-124. FITCH, Fredric B., 'A Further Consistent Extension of Basic Logic', Ibid. 14. No. 4 (1950) 208-218. FITCH, Fredric B., 'An Extension of Basic Logic', Ibid. 13 (1948) 95-101. GENTZEN, Gerhard, 'Untersuchung tiber Logische SchlieBen', Math. Zeit. 39 (1934-5) 176-210,405-431. HACKSTAFF, L. H. and BOCHENSKI, I. M., 'A Study in Many-Valued Logic', Studies in Soviet Thought, March 1962. JASKOWSKI, Stanislaw, 'On the Rules of Suppositions in Formal Logic', Studia Logica, No.1 (1934). JOHANNSON, Ingebrigt, Article in Compositio Mathematica 4 (1936) 119-136. KALMAR, Laszlo, 'Vber die Axiomatisierbarkeit des AussagenkalkUls', Acta Scientiarum Mathematicarum 7 (1934-35) 222-243. KOLMOGOROFF, A., 'Zur Deutung der intuitionistischen Logik,' Math. Zeit., 35 (1932) 58-65. KOLMOGOROFF, A., Article in Recueil Mathematique de la Societe Mathematique de Moscow 32 (1924--25) 646-667. LUKASIEWICZ, Jan and TARSKI, Alfred, 'Untersuchung tiber den Aussagenkalkul', Comptes rendues des seances de la Societe des Sciences et des Lettres de Varsovie, Classe III, 23 (1930) 30-50. POST, Emil L., 'Introduction to a General Theory of Elementary Propositions', Amer. Jour. Math. 43 (1921) 163-185. RUSSELL, Bertrand, 'Mathematical Logic as Based on the Theory of Types', Amer. Jour. Math. 30 (1908) 222-262. W AJSBERG, Mordechaj, 'Untersuchung tiber den AussagenkalkUl von A. Heyting', Wiadomosci matematyczne 46 (1938) 45-101.

345

INDEX

Note: For references to general discussions on the formation of systems the reader should consult the Table of Contents. Abbreviations 40, 47 Abbreviative techniques, derived rule of theorem introduction 64-66 for axioms 50 implicative series 67-68 in natural deduction 136-7 of proofs 40--41, 84, 115-6, 164-5, 176-181 points for brackets 1.66 use of theorems 1.532 Abstraction 8.4 Abstraction operator definition of, 294 Antecedent 1.324, 1.441 Argument (see also proof) bad (see Argument, invalid) conclusion of 1.01 formally invalid 1.041 formally valid 1.041 good (see Argument, valid) invalid 1.01, 1.02, 1.031, 1.032, 1.033, 1.037, 1.041, 1.531 logical definition of 1.01 logically invalid 1.037 logically valid 1.037 of a functor 1.213 premises 1.01 rigorous 1.12 sound 1.037 statements 1.01 valid 1.01, 1.02, 1.037, 1.041, 1.53, 1.531 Argument-form definition 1.031, 1.021, 1.03 invalid 1.031, 1.033, 1.05

346

truth-preserving 1.038 unreliable (see Argument-form, invalid) valid 1.031, 1.032, 1.05 Argument-rule (see Argument-form) Argumentation (see Argument) Aristotle 1.11, 1.211 on the Law of Contradiction 104-5 Aristotelian logic 8, 9, 64n theory of syllogism 272-7 Association to the left 41-2 Associative law for conjunction axiomatic proof of 75-6 Attribute introduction 327 Axiom, definition 31 Axiom set (see Axioms) Axiomatic proof 1.5, 1.53 construction techniques 2.611 elements in 1.54 Axiomatic systems definitions in 40 formation rules in 39 interpreted 1.7 primitive basis for 1.63, 1.67 uninterpreted 1.7 Axioms (Refer also to Table of Contents) independence of 45,3.9, 125-7 index numbers for 2.4 logical truth of 31 non-independence 45 sign of assertion 32 transformation-forms 80-1 use of 1.532, 34-5 Bernays, P. 48n

INDEX

Bochenski, I. M. 217n, 277n Boole, G. 9 Boolean algebra 8.41, 8.5 and propositional logic 301-3 Extended 8.41, 296, 298-303 Pure, and Pure with Inclusion 8.41, 296-298, 299-301 Bound variables 7,6, 241f, 256-7 Brackets 24 abbreviated by points 1.66 Brouwer, L. E. J. 222 Calculus of classes 291 in hierarchy of logic 45, 46 Carnap, R. 22, 253 Church, A. 44, 48n, 106, 193n, 194, 245n, 310, 311n, 312 Class-membership-based logic 332 Classes 9f membership 8.4f laws of 10.34, 10.35 self-membership 9.1, 9.2 Closed formula 183f, 244f Comey, D. D. 217n Completeness 1.11, 5, 9.6 absolute 200 Aristotle 200 Hilbert 200-1 "not complete" 7.1 Post 201 Comprehensiveness 1.11, 1.13, 5 Conclusion in systems of Natural Deduction 131, 147 of an argument 1.01 Conjunction and truth-values 76 truth-table for 1.322 truth-table using 1.441 Conjunction elimination 141 Conjunction introduction 141 Connective (see Functors) Consequent 19 of the implication 24 Consistency 44-5, 5, 9.4, 9.6 absolute 198, 308 Aristotle 198, 308 Hilbert 199, 308 in Boolean Algebra 302-3

Post 199 Constants (see Functors) Contingency 22-5, 27, 92,202,256,257 Contradiction 28 (see also Logicalfalsehood) determination of 4.9 in paradoxes 307 Converse law of transposition 97 Copi, I. M. 18,21 Curry paradox 10.35, 329-332 Decision procedure 4.9, 5.1,261-3,9.5, 303 Decision-rules 4.9 Deduction by substitution 54 conclusion of 131 primitive basis of 1.64 Deduction theorem 3.7, 118-21 for LFC 7.10, 282-3 logical operations in 114-5 proof by 110 Def. e 8.4 Definition of the existential quantifier 7.5 Definitions Dl, D2, D3 118 D4238 DI-D9, Boolean Algebra 296,301 DI-DlO, Fitch's LFC 318-9 De Morgan 9 (see also, Law of De Morgan) Denying the consequent 177 Derivation 35-6, 106-10, 113, 132, 4 Derived rules 1.65, 138, (see also, Abbreviative techniques and Rule) *2A5, *2A8 84 *2All 88-9 4*1 149 4*83172-3 4*84173-4 dm 176 dmi 176 QRI, QR!, 260 QR2, QR3, 262-3 Derived rule of substitution 2.61 Derived rule of theorem introduction 64-6 Derived rule of universal quantifier 347

SYSTEMS OF FORMAL LOGIC

distribution 267 Restricted rule of negation introduction 224-6 Substitution of equivalences (SE) 92 Disjunction exclusive, inclusive, material, weak 1.323 truth-table for 1.323 truth-table using 1.444 Disjunction elimination 142 Disjunction introduction 142 Distinct variables (see variables, distinct) Double negation elimination 4.625, 6.81 Double negation introduction 4.625, 6.81 Echelon 4.5, 132-5, 138-40 Element of a proof (see proof, element of) Elimination rules 54-5, 138-45 Equivalence and substitution 91-2 converse theorems 89 truth-table for 1.325 Equivalence elimination 4.624 Equivalence introduction 4.624 Equivalent steps 298 Equivalent systems 45, 126-7,169,313-5 Euclid 9 Euclid's axiom 288 Existential quantifier 7.5, 10.2 Existentially quantified formula 7.5 Extended functional calculus 1.8, 1.82 Extended predicate calculus (see Extended functional calculus) Extensional logic 1.32, 1.47 F t p234n F-correlate 7.9 F-truth 7.9 Falsehood 1.31 First syllogistic law, axiomatic proof of 61-4 First syllogistic law with conjunction, axiomatic proof of 78 Fitch, F. B. 95n, 96, 130n, 133 functional calculus of, 10.3 propositional calculus of 46, 6.1, 6.3, 6.9

348

system Q 317n Formal implication (see Implication, formal) Formal logic, objective of 1.05 Formation rules, in a meta-system 1.61 Formula, closed (see Closed formula) complex 1.42 construction of 1.213 contingent 1.41, 1.441 definition of 1.212 length of 24-5, 54-5 logical truths 1.441, 1.442 open (see Open formula) truth-table tautologies, 1.46 well formed (see Well formed formula) Free variable 7.6 Frege, G. 14, 290 propositional logic of 46, 3.9, 125-8 Functional calculus of the first order (see Lower functional calculus) Functional logic 1.13, 1.8 Functional variables 7.6, 246f Functors 1.213 (see also individual names of functors) compared with variables 1.213 extensional (see Functors, Truthfunctional) independence of 122-5 Intuitionistic interpretation of 6.82 main 24-25, 26 primitive 1.6311, 6.41, 7.21 strong 1.322 truth-functional 1.32, 20 verbal interpretations of 1.22 weak 1.323 Gentzen, G. 130n GOdel, K. 311 theorem 9.6 HFLT,389 Heyting, A. 95n, 96 lower functional calculus of 315 Hierarchy of logic 1.8-1.82 Hilbert, D. 9, 48n

INDEX

Hypotheses 131-7 determination of 57-9, 61-3, 4.55 two in one derivation 149-50 Hypothesis bar 133 Identity 8.2, 10.3 of classes 296, 297, 306n Immediate consequence 55 Implication, causal 21 common properties of various forms 1.324 formal 1.324 intuitionistic 1.15 material 1.15, 1.213, 1.22, 1.324 truth-table for 1.324 truth-tables using 25, 27, 28 strict 1.15, 1.324 strong 1.15 verbal interpretation of 1.15 Implication elimination 4.621 Implication introduction 4.621 Implicative series 67-8 "Imply", meaning of 1.324 Improper symbols (see functors, primitive) Inclusiveness of systems 1.72, 1.82 Inconsistency 10.35 Indefinite propositions (see Propositions, indefinite) Index column 1.441 Index lines 4.51 Indirect proof technique 171, 176-7 Individual constants 7.22c special 7.22d Individual variable 7.1, 7.22a, 8.4 relationship to propositional variables 7.9 Inductive logic 1.81 Interpretation of systems 6.82, 7.5, 8.4,8.41,9.3 Interpreted systems 1.7 Introduction rules 54-5, 138--45 Invalidity 7.81 Jaskowski, S. 130n Johansson, I. 95n, 96 Minimum calculus of 44n, 46, 207, 6.10

Justification column 1.55, 136, 140 Kalmar, L. 193n, 194 Kant, I. 95n KoImogoroff, A., calculus of, 207, 232-3 interpretation of I-logics, 222 Kiing, G. 217n LF+ 314 LFFF 47, 316--43 LFI314-5 LFIH 315 LFILNC 315 LFLT 47, 234n, 313 LFLT' 47, 234-83, ::84-301, 9.4-9.5, 313-6 LFLT' 284-9, 317, 318, 319, 320, 321, 323, 327, 332, 335, 343 LFLTF 313 LFMIN 47, 315 LFpM 313 L-contingency (see Lo ~ical contingency) L-falsehood (see Logi;al falsehood) L-truth (see Logical t"uth) Law compared with rule 72 Law for the definition of material implication 100-1 Law of assertion 72-: Law of assertion with conjunction 78 Law of cancellation 78 Law of commutation 71 Law of composition ~'8 Law of contradiction 104 as an axiom 6.10 axiomatic proof of 104 denial of 1.444 for Identity 324 in intuitionistic log,c 214 lower functional c dculus proof 265 truth-table for 1.444 Law of contraposition (see Law of transposition) Law of De Morgan 102-3, 144-5, 174-5,4.10 functional analogms of 332, 339--43 in intuitionistic logic 214, 6.81 in Minimum calcu'us 6.10

349

SYSTEMS OF FORMAL LOGIC

meta-sign for 176 Law of the denial of the antecedent 95n Law of double negation 95n, 97, 4.10 Law of excluded middle 95n, 104 as an axiom 6.82 and Fitch's system 6.9 double negation of 214 for identity 320, 325 indirect proof 171 Intuitionistic interpretation of 220-1 Lower functional calculus proof 265 suspension of 6.3 truth-table for 1.444 Law of existential generalization 7.10 Law of exportation 78 Law of identity 1.522, 56-60, 7.10 Law of importation 77-8 Law of interpolation 73 Law of negative proof 99 Law of the denial of the antecedent 97 Law of the factor 78 Law of the symmetry of conjunction 75 Law of transposition 3.1, 3.4, 6.8, 324 Law of quantifier distribution 7.8 Law of reductio ad absurdum 98-9, 6.8 Law of specialization 7.8 Leibniz, G. W. 8, 10,78,290 Lewis, C. L. 20 Logic of identity 325 Logic of probability 1.81 Logic, subject matter of 1.21 Logical contingency 1.4, 31, 33, 35, 92, 4.9 Logical falsehood 1.4, 1.444, 33, 35, 92, 4.9 Logical indeterminacy (see Logical contingency) Logical operations 3.71, 1I4-5 Logical rule, compared with logical law 72 Logical truth (see also Tautology) 1.4, 1.444,31,33, 35,92,4.9,293 Lower functional calculus 1.13, 1.8, 1.82, 7, 8, 9.4, 10 Lukasiewicz, J. 1I6

Material implication (see Implication, material) Medieval logicians 10 Meta-language (see Meta-system) Meta-system 1.61-1.67 Meta-variables 1.62 Metalinguistic signs 1.61 Method of reduction 298 Modus ponens (law) 21 Modus ponens (rule) 21, 1.5222, 34, 2.61,58 and law of assertion 72n in Lower functional calculus 7.7 incorrect application of 34 meta-symbols for 2.31, 6.52 modus tollendo ponens 184 Monadic predicate calculus (see Calculus of classes) Multiple quantification 244

Main connective (see Functors, main) Main echelon (see Echelon) Material equivalence (see Equivalence)

Object-language (see Object-system) Object-system 1.61-1.67 Open formula 183f, 245

350

Natural Deduction 6.9 and Minimum Calculus 232 Intuitionistic system of 6.81 Negated conjunction elimination 145 Negated conjunction introduction 145 Negated disjunction elimination 144 Negated disjunction introduction 145 Negated implication elimination 184 Negation 1.42 intuitionistic interpretation of 314 truth-table for 15-16 truth-table using 1.444 Negation elimination 144, 1901, 191,215 Negation introduction 144, 171, 190, 215 Negative existential quantification elimination 340 Negative existential quantification introduction 340 Negative universal quantification elimination 340 Negative universal quantification introduction 340 Non-equivalent systems 83, 45 Non-standard systems 6.2, 10.36

INDEX

Operand 242 Orders of logics 1.8 P + 1.6311, 1.6312, 1.66, 46, 2.1-2.64, 3.1,3.601, 122-3, 130, 131, 138, 141, 169, 193, 5.4, 207, 208, 211, 219, 223,229,230,263,313,314 PF 46, 6.3-6.33, 6.9-229, 232 PI 46, 6.3-6.8, 2.5, 6.82, 229-233, 314, 315 P(i -> 231-3 PIDM 47, 219 PIDN 47, 219 Pm 223, 232, 315 PILEM 47, 219 PILNC 47, 104, 232 PILT 47,219 PLT 46,3.1-3.71,3.81, 118, 121-5, 130, 131, 132, 138, 145, 169, 171, 190, 5.1-5.4, 208, 209, 21 On, 219, 223, 234,263,264,286,296,313,343 PLT' 46, 1.632, 115, 3.8-3.9, 190, 5.15.34, 207, 209, 7.1-7.4, 238, 258-64, 279-80, 282-3, 286, 287, 288, 296, 297,299,300,303,308,309,313,315, 325,343 PLTF 46, 125-8,206,209,219,313 PMIN 46, 6.10 P1MIN 229 p2 MIN 104,230-2, 315 PND 47, 4.1-4.10, 226 PND+ 169 (4.1-4.624, 4.7-4.74),190,215 PNDC 132 PNDF 47,6.9,317,331,343 PNDI 47, 6.81, 226, 232 PNDI,215n PNDMIN 47, 232 PND,Red. 130, 190-2,206 PPM 46, 128-9,206,207,209, 234n, 313 PPM-4129 P-correlate 7.9, 9.4 P-truth 7.9 Paradoxes (refer also to the Table of Contents) Curry, 10.35, 329-32 Liar, the, 307 logistic 9.3 meta-logical avoidance of 9.3 Russell 9.2, 10.35

semantic 6.3 Partial systems 4.74, 169, 6.2, 5.82 Particular affirmative proposition 272 Particular negative proposition 272 Peirce's Law 105, 179-81, 204-5 Positive logic 2.1 Post, E. L. 122n, 193 Praeclarum theorema 78 Predicate constants 7.22e, 8.1 Predicate variable 7.1, 7.22b, 8.1, 8.4 Primitive functors (see Functors, primitive) Primitive symbols 1.6311 Primitive variables 1.6311 Principal functor (see Functors, main) Principia Mathematica 40, 43, 45, 46, 82-4, 128, 174 Principle of addition 82 Principle of association 82 Principle of permutation 82 Principle of summation 82 Principle of tautology 82 Proof 30, 35, 132 direct and indirect, 171, 176-81 elements of, 1.521-1.522 structure, 64, 67-8, 69-71, 72, 4.5 variant, 1.06 variant techniques for, 176-81 Proof echelon (see Echelon) Proper symbols (see Symbols, proper) Property-relation-based logic 332 Propositional calculus 9, 45-6 interrelationships of systems 7.9 Propositional variables 1.211, 1.213, 7.9 compared with functors 1.213 in meta-systems (see meta-variables) Propositions 1.211 classification of 6.9 contradictories 7.10, 272-3 definite 6.9 inconsistent (see Propositions, logically false) indefinite 6.9 logically false 1.324 properties of 1.31 self-contradictory (see Propositions, logically false) self-referential 6.9

351

SYSTEMS OF FORMAL LOGIC

Q-open, Q-closed formulae 244-5 Quantification, range 10.31 Quantifier, range of 7.6 vacuous occurrence of 244

Reductio, Special law of (see Special law of reductio) Reductio ad absurdum (Law) (see Law of Reductio . ..) Reductio ad absurdum (rule) unrestricted (see Unrestricted rule of ... ) Reflexivity 69 Reiteration (see Rule of reiteration) Repetition (see Rule of repetition) Restricted calculus 7.22b Restricted predicate calculus (see Lower functional calculus) Restricted rule of negation introduction 6.9 Resultant 140, 147, 33On-31n Rigor 9 Rule (see also Logical rule and Derived rules) Rule for substitution on bound variables 254 Rule for substitution on individual variables 250 Rule for the analysis of the theorem 135 Rule for the definition of material implication 172-3 Rule of abstractor substitution 295-6, 304-5 Rule of existence elimination 321 Rule of existence introduction 322 Rule of existential quantifier elimination 336 Rule of existential quantifier introduction 337 Rule of formal substitution 253 Rule of formula substitution 259 Rule of identity elimination 319-20 Rule of identity introduction 319 Rule of inference, The (see Modus ponens) Rule of introduction (see Introduction rules) Rule of passage 238-9

352

Rule of reiteration 139 Rule of repetition 138-9 Rule of substitution (primitive) (see Substitution) • Rule 'of substitution for predicate variables 250 Rule of substitution for propositional variables 249-50 Rule of substitution on bound variables (see Rule for substitution on bound variables) Rule of substitution on individual variables (see Rule for substitution on individual variables) Rule of substitutivity of equivalence 92, 254 Rule of universality elimination 320 Rule of universality introduction 320-1 Rule of universal quantifier elimination 335 Rule of universal quantifier distribution 267 Rule of universal quantifier introduction 336 Rule of universalization 249 Rules in systems of Natural deduction 131-2 derived (see Derived Rules) for conjunction 141 of construction 4.6, 4.62 for disjunction 142 for equivalence 142-3 for implication 139--41 for negation 143-5 for the Law of De Morgan (see Law of De Morgan) of elimination (see Elimination rules) of formation (see Formation rules) of reduction 262-3 of transformation (see Transformation rules) Rules of transportation 138-9 Russell, B., (see also Principia Mathe-

matica) systems of 40, 128-9, 227 Russell Paradox 9.2, 10.35 S-systems (see Standard systems)

INDEX

Satz des Widerspruchs 95n Scholz, H. 104,223 Schroter, K. 104,223 Second law of quantifier distribution 265 Second order functional calculus 289292,9.6 Second syllogistic law 31, 65-7, 110-1 Second syllogistic law with conjunction 78 Sentential calculus (see Propositional calculus) Series for material implication (see Implicative series) Single negation elimination 144 Single negation introduction 144 Singular functional variables (see Predicate variable) Sound argument (see Argument, sound) Special law of reductio 95n, 98 Standard systems 207, 219, 313, 343 Statement matrix 239 Stoic-Megarian logic 9 Stoics 9 Strict implication (see Implication, strict) Strong implication (see Implication, strong) Substitution 12, 31-2, 50, 52, 235 and abstraction 295 and symmetry 2.61 as a method of proof 1.51 consistent 32 deduction by 54 equivalent formulae 91-2 examples of 2.61-2.611 in lower functional calculus 249-54 on variants 115-6 restrictions on 107, 249, 250, 253, 254, 255-7 simultaneous 51, 113-5 Substitution instances 1.03, 1.05, 235, 259 Fitch on 339 Substitution of equivalences 92 Substitution matrix 251 Symbols importance of 1.15, 1.211, 1.212 improper 1.213

primitive 1.6311 proper 1.213 Syntactical variables 1.62 Systems Equivalent 45 Natural deductive compared with axiomatic 131 non-equivalent 45 2FLT' 289-92 Tarski, A. 116n Tautology 22, 32, 33 (see also L-truth) TH (see Theorem) The rule of inference (see Modus ponens) Thesis (see Theorem) Theorem echelon (see Echelon) Theorem 1.53, 1.532, 1.54, 1.64, 2.51, 2.52, 2.53, 2.55, 131, 4.9 Theory of types 45, 227, 316-7 Third law of quantifier distribution 266 Transformation rules 34, 37, 39, 40, 42-3 (see also name of specific rule and refer to the Table of Contents) Transformation-forms (see Axioms, transformation-forms) Transitivity 69 Truth-preserving 7 Truth-tables 14-18, 19, 22-9, 116-7, 123~, 194, 303 Truth-values 1.31, 1.41, 4.9

Uninterpreted systems 1.7 Universal affirmative proposition 272 Universal class 10.31 Universal negative proposition 272 Universal quantifier 7.1 Universally quantified formula 239 Unrestricted rule of reductio ad absurdum 4.10 Valid (see Argument, valid; Argumentform; Valid formulae) Valid formulae 1.4 Variables (see also Propositional variable; Individual variable, etc.) distinct 26 primitive 39

353

SYSTEMS OF FORMAL LOGIC

Variant formulae 54, 106-7, 113-6, 286-7 Variant proofs 106

wi and wff (see Well formed formula) Well formed formula 38, 39-40, 42, 49, 94-5, 210, 237, 285, 307-8, 318

354

Whitehead, A. N. (see also Principia Mathematica) system of 40, 128-9 Wilder, R. 222n Zinov'ev, A. A. 217n