Logic, Everyday Discourse, and Metaphysics (UNIPA Springer Series) 303074597X, 9783030745974

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UNIPA Springer Series

Gianni Rigamonti

Logic, Everyday Discourse, and Metaphysics

UNIPA Springer Series Editor-in-Chief Eleonora Riva Sanseverino, Department of Engineering, University of Palermo, Palermo, Italy Series Editors Carlo Amenta, Department of Economics, Management and Statistics, University of Palermo, Palermo, Italy Marco Carapezza, Department of Human Sciences, University of Palermo, Palermo, Italy Marcello Chiodi, Department of Economics, Management and Statistics, University of Palermo, Palermo, Italy Andrea Laghi, Department of Surgical and Medical Sciences and Translational Medicine, Sapienza University of Rome, Rome, Italy Bruno Maresca, Department of Pharmaceutical Sciences, University of Salerno, Fisciano, Italy Giorgio Domenico Maria Micale, Department of Industrial and Digital Innovation, University of Palermo, Palermo, Italy Arabella Mocciaro Li Destri, Department of Economics, Management and Statistics, University of Palermo, Palermo, Italy Andreas Öchsner, Department of Engineering and Information Technology, Griffith University, Southport, QLD, Australia Mariacristina Piva, Department of Economic and Social Sciences, Catholic University of the Sacred Heart, Piacenza, Italy Antonio Russo, Department of Surgical, Oncological and Oral Sciences, University of Palermo, Palermo, Italy Norbert M. Seel, Department of Education, University of Freiburg, Freiburg im Breisgau, Germany

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Gianni Rigamonti

Logic, Everyday Discourse, and Metaphysics

Gianni Rigamonti Department of Philosophy University of Palermo Palermo, Italy

ISSN 2366-7516 ISSN 2366-7524 (electronic) UNIPA Springer Series ISBN 978-3-030-74597-4 ISBN 978-3-030-74598-1 (eBook) https://doi.org/10.1007/978-3-030-74598-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of RUTH MANOR (RURU) Tel Aviv, 1944–2005 A great woman

Acknowledgments

Many people have been helpful to me over the years, reading, commenting, and criticizing earlier drafts of this work, suggesting changes or further readings and encouraging me in various ways to go on. Without their contribution, either positive (in the form of appreciation and suggestion) or negative (in the form of criticism) this work—whatever its value—would be much poorer than it is. These people I have to thank include, to begin with, my anonymous referees, whoever they may be, and then Lucia Geraci, for letting me know Dharmakirti; Lorenzo Fossati, Gianluigi Oliveri, Sandro Mancini, Eleonora Riva Sanseverino and my former students Emanuela Campisi, Marta Clemente, Valentina Valenti and Roberta Zagarella for their constant encouragement; Silvano Tagliagambe, who was the first, years ago, to believe in this work and make the publication of an earlier, Italian version1 possible; Marco Carapezza, who fought so hard in my behalf; and, last but not least, Francesco (Ciccio) La Mantia, also a former student of mine and now a colleague, who encouraged me again and again to go on and was extremely helpful in making relevant literature available.

1 See

References, [36]. But this book is not a simple translation. It is much richer. vii

Introduction

I This work has changed a lot during its making. Of course, when you begin putting a book together, you must have an idea of the things you want to write, otherwise you will not get started at all; but once you have one, and are turning that idea in your head into actual signs on paper, the doing reacts on your initial mental plan and changes it into something different, new ideas crop up, while others that were looking important fade, and either dwindle to minor details, or disappear altogether. And I don’t know whether there are people for whom writing is simply turning ideas into signs on paper in a perfectly straightforward way, with no significant change, but I certainly am not one of them. This goes in a particularly strong way for Logic, metaphysics, and everyday discourse. At first, years ago, I was only interested in the fact that everyday discourse, unlike standard formal languages, also contains expressions obviously belonging neither to a proposition’s subject, nor to its predicate; but then, while reflecting on these expressions, many other ideas—about individuals and proper names, about dependent propositions, about non-declarative ones, about truth values and their not forming an algebra in everyday discourse, about Hume, Kant, and Buddhist philosophy—have occurred to me, and I have found them relevant and worth discussing. So this book has been written and re-written again and again, over a fairly long span of time, and I am afraid traces of these shifts in my viewpoint must have remained here and there. I hope, anyway, they will not disturb overall comprehension. Another important preliminary is that this is a very informal book about logic, but not a book about so-called informal logic. I appreciate, respect, and might even occasionally quote such authors as Pinto or Groarke, but the topics they mostly discuss—dialectics (in Plato’s sense), burden of proof, discussion strategies, good arguing practice, and the like—are far removed from my main focuses in this work. My main focuses are very different: they are the nature of giving names in everyday discourse and the behavior, also in everyday discourse, of truth values (very different from the one they have in formal systems). And, of course, the classics of formal

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Introduction

logic are relevant to these topics: but so are more traditionally philosophical classics that will be discussed, though very incompletely, in these pages.

II My initial focus, anyway, will be more modest: I will simply highlight, to start with, some limits of today’s elementary mathematical logic, and more exactly of its applicability to everyday discourse. With “today’s elementary mathematical logic” I mean something broader than so-called classical or ordinary two-valued logic, namely, the set of logics characterized by (1) the use both of extensional propositional connectives and quantifiers but of no other (e.g., modal or temporal) operators and (2) two classes of non-logical terms,2 predicates, needing saturation, and names,3 saturating them. Depending on the assumptions one makes, many logics can be built upon this foundation; today’s most popular (but by no means the only) ones are, of course, classical, intuitionistic, and linear logic. It is this nebula I call “today’s elementary mathematical logic”, but from now on I will often skip, for brevity, the qualifiers “elementary” and “mathematical”. And when I wonder whether, and how far, logic so understood is applicable to everyday discourse, I am wondering how far it gives rise to, respectively, insights and distortions about it.

III Now, modern logicians have both constructed important formal systems, such as elementary logic itself, in all its variants, or Dedekind-Peano-Russell’s arithmetics, or Zermelo-Fraenkel’s set theory, and studied the structure of everyday discourse.4 Evidence of this twofold development is impressive since Frege’s days, and logicians’ work has always been accompanied since then by the assumption, sometimes explicit sometimes implicit but clear, that the distinction of non-logical terms in two classes, names and predicates, allows logic to characterize the structure both of formal languages and everyday discourse in an essentially complete way. Now, one of my central points (and the one I started from, some six years ago) is that as far as everyday discourse is concerned this assumption is wrong, or, more exactly, that if weadmit just two classes of non-logical terms, names and predicates, we can get a satisfactory logical grammar of several formal languages, but not of everyday discourse. This 2 The

distinction between logical and non-logical terms is far from clear, but a discussion of this point can—and will—be omitted. It would not be relevant to the issues I am tackling in this book. 3 I am simply saying “names” for brevity. There will be time for finer distinctions. 4 I will say sometimes “discourse” and sometimes “language”, and use as qualifiers sometimes “everyday”, sometimes “ordinary” and sometimes “common”. All these expressions are meant as equivalent.

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point is not new, of course; but for all I know it has been treated by logicians, so far, in a way very different from the one I will choose here.5 This is not to say that Frege (or Russell) were mainly interested in a general logical grammar of everyday discourse; they were mainly interested in one for mathematics. This is well known. But in the making of that logical grammar some important byproducts having to do (sometimes also, sometimes just) with everyday discourse also came out, such as, e.g., Frege’s two capital distinctions between Sinn and Bedeutung and between Begriff and Gegenstand, plus the no less important one between identifying and predicative uses of the copula, plus his (unhappy, but factually existing) analysis of dependent propositions in the second half of Über Sinn und Bedeutung, plus his late (1918) paper about negation…; and Russell worked long and hard on the foundations of mathematics, but also developed a theory of definite descriptions certainly having to do with everyday discourse; and coming to later times, the discovery of the undecidable entirely belongs to mathematics, but still more recently the introduction of the (also all-important) distinction between rigid and non-rigid designators was pure philosophy of everyday discourse. So emphasizing, as some scholars do, that today’s elementary logic has been created, first and foremost by Frege, as a tool for a rigorous re-organization of mathematics is correct; but adding that, as a consequence, criticizing it for not giving certain other results is unfair seems wrong to me. It is true Frege himself wrote (BS, xi) that one cannot condemn (verurteilen) an instrument conceived for certain ends as being inadequate for certain other ones, but here I am not condemning anything—using that term to describe my attitude would be unfair. I am only emphasizing that over and above certain allimportant results of modern mathematical logic there are in everyday discourse, not in formal systems, some equally important facts having to do with meaning and truth (so that it is entirely reasonable to classify them as logical facts) but lying entirely out of the scope of mathematical logic; and this is not criticizing the latter, or its creators.

IV In fact, something has a big, big importance here: a duplicity the development of logic in the last 200 years testifies beyond any reasonable doubt. The thing is patently clear from more than one viewpoint, including the broadest possible one for us in the early twenty-first century. Let us make, namely, one big step back: if we wonder which were logic’s most original and creative periods in its almost twenty-four centuries long history, the obvious answer is there have been three: in Antiquity, in the Lower Middle Ages and (roughly) in the last two hundred years. But an important point, which ought perhaps to receive more attention than it usually does, is that in each 5 There have been, anyway, serious anayses of everyday discourse by formal logicians, and in my opinion the most significant one is Richard Montague’s. I will say something about his theories both in this Introduction and in § 7.3.

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of these periods different professional groups have been foremost in developing new logical ideas: philosophers in Antiquity, priests in the Middle Ages (for such were, with very few exceptions, teachers in Medieval universities), and mathematicians (Boole, Frege, Gödel, and others) in the contemporary age. Now, that a discipline’s development should be affected by the profession of those working in it is entirely normal; but something else is normal too, namely, that those also interested in that discipline but belonging to different professional groups should feel a need to bring into better focus aspects that already are central to their group’s attention. So far I have said nothing new, but the first of my central theses is coming just now, and I hope it is not so obvious: in fact if not in theory, most logicians have consistently started, since Frege’s times, from the assumption that there are no essential structural differences between everyday discourse and the formal languages of mathematics, or at least the most complex of these.6 Now, this assumption is wrong. The amusing thing is that deep within, I believe, most logicians do not really deny there are important differences, not reducible to a simplistic precise-imprecise polarity, between these two levels, but then these differences are not taken seriously, or at least not in logic classes for freshpersons. A practically impossible to know and steadily growing number of introductory logic handbooks are around in the world; some of them are excellent and some not so good, some are more technical than philosophical and some the other way round, and they also differ significantly in their choice of subjects to treat carefully or not so carefully or even to skip, but as far as I know they all have in common, except possibly for some inessential differences in notation, two notions, individual term and atomic, or elementary, proposition applying—so we are told—both to formal languages and common speech.7 An individual term (but “name” is quicker, and so will I say quite often) can be either (1) a constant (that if we want maximum generality we can write e.g., as an a plus a numerical index: a1, a2 ,…, an,…) or (2) a variable, in which case we can use an x instead of an a, writing x 1 ,x 2 ,…, xn,…, or (3) an n-argument function saturated by n individual already-saturated terms (the general pattern is usually given 6 See

for example this particularly strong claim by Richard Montague: “I reject the contention that an important theoretical difference exists between formal and natural languages” (“English as a formal language”, in Formal Philosophy, edited by R. H. Thomasson, Yale University Press, 1974, p. 188). 7 I believe this is the right place to say something about a work by W. C. Purdy that, given its title (“A Logic for Natural Language”, Notre Dame Journal of Formal Logic, Volume 32, Number 3, Summer 1991, 409–425), might be taken for a study having something in common with my book. This would be a mistake. Purdy’s is a very serious work, but (1) it has a massive formal apparatus, while the one I am going to use here will be as slim as possible; (2) the formalism Purdy uses to describe what he takes to be the deep logical structure of “natural language” (his term) is explicitly Boolean (see e.g., p. 410), while my view of “everyday discourse” (my term) is, also explicitly, nonBoolean (see above all §§ 5.1–2); (3) Purdy is hostile to the “popular view that spoken or written language is a ‘surface’ phenomenon, that its logical structure and meaning reside in underlying base language, and that complex transformations relate these two levels”, and in this I agree with him, but then he introduces, to describe the “natural logic” of “natural language”, a system L N “characterized by the absence of variables and individual constants” where “The role of bound variables is played by predicate functors called ‘selection operators’” (409–10): now, I must confess I find this mysterious and keep feeling far, far more at home with the terminology of good old Frege’s Begriffsschrift.

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as f(t 1 ,t 2 ,…, t n ), where t 1 ,t 2 ,…, t n are any individual already-saturated terms, either variable or constant, either atomic or not). This threefold division is meant as exhaustive and applying both to everyday discourse and various formal languages; the lettersymbols just introduced are simply a general framework, but the particular symbols actually used can vary ad libitum from language to language; in everyday discourse proper names in the usual grammatical sense of the term, such as “Ann”, are simple individual constants, pronouns, such as “he” or “she”, are simple individual variables, and if we want examples of functional individual terms we can take such expressions as “Ann’s father” or “her father”. Next come the notions of well-formed atomic8 proposition (in everyday discourse) and well-formed atomicformula (in formal languages); I am going to speak for simplicity just of propositions, but the basic structure today’s logic assumes is always the same. Now, the standard definition of an atomic (or elementary) proposition really conflates two distinct features: (i) containing just one n-place predicate, where n is any positive integer, plus exactly n names saturating it, and (ii) having no other proposition as a proper part, which entails that in a well-formed atomic proposition there is nothing but the predicate and the names saturating it. The general pattern is Pn (t 1 ,t 2 ,…, t n ,), where Pn is an n-adic predicate and t 1 ,t 2 ,…, t n are individual terms in the just defined sense. Nothing else is an atomic well-formed proposition, so no expression having any other parts over and above an n-adic predicate and n individual terms is one. Now, such a definition is correct for many and important formal languages, but not for everyday discourse, since in the latter (and in it alone) there are many perfectly grammatical expressions P satisfying condition (i) above but not condition (ii), for they have other components too over and above a predicate and the names saturating it, so that if we take these components away, what is left still is a well-formed proposition, and at the same time a proper part of P’s.

V Anyway, this point will only be discussed after other, equally important ones have been dealt with. In Chap. 1 I will briefly describe Aristotle’s syllogistics and show it is an absolutely rigorous formal logic using just one class of variables, instead of two as we do in modern logic; there will also be two sections, devoted to Lukasiewicz’s reconstruction of the whole theory in 1956, where I show how Lukassiewicz’s analysis, serious as it is, suffers from its author’s incapability of stepping out of his logic— based of course, in Frege’s style, on two essentially different classes of non-logical terms—and seeing things from Aristotle’s own perfectly rigorous viewpoint. It is well known an innovation can be, in any field, only imperfectly understood by scholars of a later age unless it is viewed in its own context, taking the historical

8 Or

elementary (and the same goes for formulae).

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background it emerges from into account. This also goes for today’s logic: to understand what it is one must consider the pre-existing received view. This is the aim of Chap. 1.

VI Chapter 2 will describe as simply as possible those basic technical aspects of contemporary logic my criticism will then be addressed to in later chapters. It has, so to say, a service function, although right at the end (footnote 7) a radical objection that will only be fully developed later, in Chap. 4, briefly emerges. Things change a lot, however, starting from Chap. 3. As I have already said, one of the two classes of non-logical words Frege distinguishes is that of individual terms, or proper names; and of course, on the other hand, in today’s logic handbooks such phrases as “individual constant” and “individual variable” occur everywhere. Well, if “individual” is used in such a systematic way as an adjective, then it really is difficult to exclude from logical vocabulary “individual” as a noun, so that the question “What is an individual?” becomes inescapable. Now, there are formal systems (such as e.g., Peano number theory) where a precise answer is possible, the set of individuals is neatly circumscribed and the concept of individual is well-defined, but in everyday discourse this concept is entirely undetermined and there simply is no well-defined set of individuals. The long, long Chap. 3 is devoted to this topic. But something stronger has to be added. It isn’t just that (whatever they may be) individuals, i.e., the names’ referents, do not form a set in the classical, Cantorian sense of this term, that of a well-defined manifold; the root problem is that the very process of naming goes hand in hand with another one, that of reification i.e., of circumscribing chunks of experience having some sort of inner unity and relative stability. Objects are nothing but such chunks, and a name is nothing but an expression referring to one of them9 ; but this means that new objects are constantly circumscribed, and consequently there is no set of objects.10

VII But there also is another problem. According to the standard notion of well-formed formula, all (and only) expressions consisting just of an n-place predicate (where n is any positive integer) plus any n individual terms are well-formed. Now, in everyday 9 Not

necessarily an actual one: “Hamlet” or “Raskolnikov” are also names.

10 This view of what objects (and consequently names) are has precedents, above all, in the first part

of Hume’s Essay on Human Nature and in the works of such Buddhist philosophers as Dighnaga (sixth-century A. D.) and Dharmakirti (seventh century), but I give to their ideas a non-idealistic turn. See. § 3.12.

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discourse this is wrong. There are plenty of examples of n-place predicates, n≥2, that are not saturated by any names whatever, but only by terms belonging to certain wellcircumscribed classes. This is both the opening theme of Chap. 4 (which goes then on with some reflections about Bolzano, Frege, and Wittgenstein) and the first part of an analysis of some classes of dependent propositions. This analysis is unsystematic but, as far as I know, new. The fifth chapter is an interlude devoted to some very important methodological questions that had emerged in the first four and could not be postponed any longer, while Chap. 6—well, I will not say it completes the analysis Chap. 4 had begun (that would require an amount of work I don’t believe I can do on my own), but it certainly extends it.

VIII After this long, long preparatory work, in Chap. 7 I tackle the last of my central subjects: namely, all standard introductions to logic, including mine,11 clearly say that well-formed atomic (or elementary) formulas are exactly those strings that contain an n-place predicate, n individual terms, and nothing else. The general pattern is, as we have already seen, Pn (t 1 ,t 2 ,…, t n ,); whatever does not fit this pattern is no wellformed elementary formula, so no string containing nothing else over and above the predicate, with its n-argument places, and the n individual terms saturating it, is one. Historically the notion of well-formed formula was born as a variant of that of grammatically correct proposition. In everyday discourse we can have grammatical utterances, such as “Mary packs her things”, and ungrammatical ones, such as “Mary packs his things”; every language has a grammar of its own and none accepts any string whatever of its basic signs, whether alphabet letters or phonemes, as grammatical. Now, in the notation of post-Fregean logic all and only strings of the form Pn (t 1 ,t 2 ,…, t n ) are accepted as elementary well-formed (i.e., grammatically correct) formulas (this is of course just a general framework, to be filled—given any language L where propositions, and so truth or falsity, are possible12 —using L’s alphabet). Suppose now we take Pn (t 1 ,t 2 ,…, t n ) as a general scheme, e.g., of the equalities and inequalities of elementary algebra (or also of some higher branch of mathematics, such as, say, calculus): things function neatly and all expressions getting used there respect this pattern. But as a general framework of elementary propositions in everyday discourse Pn (t 1 ,t 2 ,…, t n ) fails, for in it perfectly grammatical elementary propositions not satisfying this pattern are easy to find.13 More exactly, 11 Gianni

Rigamonti, Corso di logica, Bollati Boringhieri, Torino, 2006. languages, such as that of musical scores, are foreign to the notions of truth and falsity. I am not talking about these. 13 Of course, the first one to say not exactly this but something very similar to it and get rid of “Fregean good manners” was Wittgenstein in Philosophische Untersuchungen, a book I am indebted 12 Some

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mathematical sentences always are, so to say, respectful of Fregean good manners, but when we shift to propositions of everyday discourse we find that many of them respect ordinary grammar but not the foundations of Fregean logic, for they do not contain just the required predicate-and-names, but other things too. This is what I say, with many examples, in Chap. 7. What is the nature of these other things? They vary enormously, and I don’t feel like attempting an exhaustive classification, but a great dichotomy, aptly describable in traditional grammatical terms, is obvious from the very first: adverbs on one side, and all the rest on the other. However, supposing this dichotomy is granted, can such “other things” be classified using the notions of saturated and unsaturated term, or can they not? In Chap. 7 I show this can be done and the “other things”, that I call tertia (obviously tertium in the singular) for want of a better term, invariably are unsaturated terms with one argument place, but are nevertheless essentially different from predicates, for either predicates or whole propositions, not names, saturate them. This may well appear harmless, but as one digs deeper one very soon sees it is not, and traditional people, if they ever read me, will very probably attempt various rescue maneuvers to take the sting out of it. I do my best to anticipate and refute a couple of these possible maneuvers in the later part of Chap. 7. Lastly, in a very short Chap. 8 I synthesize the whole essay in eighteen theses (this is in practice a synopsis) and say a few words on the new research pathways these results can in my opinion open.

IX Just one last, very general point still remains to be added: there is a villain in this story, and that villain is Tarskian semantics.14 Leaving metaphors aside, the point here is that in all standard formal languages, given any expression P—be it a name, a predicate or a proposition—its reference has to be given with respect to a domain D (the “universe of discourse”), and (1) D is usually seen as something entirely given and unchanging, (2) a name’s referent is a well-specified, also unchanging element of D, and (3) a predicate’s referent is a changeless (of course!) manifold whose structure can have very different degrees of complexity, but whose ultimate constituents invariably are elements of D. Moreover, all elements of D, i.e., all individuals existing in the universe of discourse, and as a consequence D itself, are entirely given, once and for all: nothing can be added, nothing can be taken away, nothing can undergo the to more than I can say. Think of the language game where one word, e.g., “Slab”, can be a complete proposition: though this possibility is very different from the ones I am exploring here, I believe it has in common with them an underlying feeling that in everyday discourse there are many things making perfect sense that logic as it is today (but also, of course, as it was around 1950) could not possibly account for. 14 So called becuase it first reached its full-fledged form with Alfred Tarski’s Der Wahrheitsbegriff in den formalisierten Sprachen (1934).

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slightest change. In such a universe, the notion of truth is as precise and unvarying as possible, which of course is extremely advantageous in terms of simplicity, clearness, and rigor; but in real life this sort of precision and invariance apply next to never, so that unless we are extreme rigorists, unwilling to relax these strong requirements of Tarskian semantics, we will be naturally led to look for something milder and more realistic. I suppose it is clear I am for the second option, and regard Tarskian semantics as a “villain” as far as everyday discourse is concerned.

X One final word of warning. As I have already said at the beginning of this Introduction, mine is a very informal book about logic, but not a book about so-called informal logic. For, as the reader will find, my fundamental subject is—in a fully explicit way in § 3.9, but basically in the whole book—how we come to name things or, equivalently, what giving names is, and this seems to me as metaphysical a subject as possible. And now it only remains to wish my readers—if I ever have any—a pleasant reading. Palermo, Italy 2021

Gianni Rigamonti

Contents

Part I

Individuals and More

1 Good Old Aristotle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Foundations of Aristotelean Logic . . . . . . . . . . . . . . . . . . . . . . . 1.2 One Class of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Lukasiewicz’s Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Lukasiewicz’s Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 7 8 9

2 Some Minimal Technical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Frege’s Reform of Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A Last Important Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Modern Notion of Well-Formed Formula (or Proposition) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Essentials of Tarskian Semantics . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12

3 What Is an Individual? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 When We Know What We Are Talking About, and When We Do Not . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Wondering About the Meaning of This Word, “Individual” . . . . . . 3.3 Examples, Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 A Still Broader Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Short and Long Stories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Individuality and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Individuality and Substantiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Nelson Goodman’s Strange Predicates . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Non-individuals, at Last . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 An Open-Ended Universe? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Inventing New Individuals in an Open Universe . . . . . . . . . . . . . . . . 3.12 Metaphysical Consequencess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 16 19 19 20 22 25 28 29 30 32 35 38 42 45

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Part II

Contents

The First Great Gap

4 When Predicates Behave Badly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Those Embarrassing Propositional Attitudes . . . . . . . . . . . . . . . . . . . 4.2 Rescue Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 A Look at the Classics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Superman and Clark Kent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 I Know, You Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 57 57 63 65 68

5 An Interlude: Matters of Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The in-Out Making of the True and the False . . . . . . . . . . . . . . . . . . 5.2 What Is Correctness? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Said in Passing: Are They Really Extensional? . . . . . . . . . . . . . . . . 5.4 Of Grammatical Differences Among Spoken Languages . . . . . . . .

71 71 74 75 76

6 This, and This, and That . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Adversative Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Causal Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Two Sorts of Relative Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Final Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Wild Behavior of Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 82 84 89 91

Part III Not Just Names and Predicates 7 Of Many, Many Other Things . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.1 A Basic Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.2 Those Few, Very General Things That Can Be Said with Some Confidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.3 About (Above All) Montague . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.4 Those Surprising Nominal Tertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8 Sum Total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.1 Recapitulation and Further Perspectives . . . . . . . . . . . . . . . . . . . . . . 105 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Part I

Individuals and More

Chapter 1

Good Old Aristotle

Abstract The first two sections reconstruct Aristotle’s syllogistics, emphasizing its structural differences from modern logic. The third and the fourth discuss the most important modern interpreter of syllogistics, Lukasiewicz.

1.1 The Foundations of Aristotelean Logic The Prior Analytics have as their central subject “proof and demonstrative science” (Book I, Chap. 1, 24a 11–12),1 but before discussing this topic Aristotle needs some preliminary definitions, the most important ones being those of proposition and term. For him “a proposition is a discourse asserting or denying something about something” (24a 17–18). Every proposition talks then, to say it in a not so skeleton-thin way, about something and either asserts or denies something (else) of it. The thing it talks about is called by Aristotle hypokeimenon, the underlying; centuries later hypokeimenon was rendered in Latin as sub-jectum, an excellent translation; still later, subjectum became subject in English, sujet in French, Subjekt in German, soggetto in Italian, and so on, but its original meaning, “the underlying”, is no more visible in these modern terms. What is asserted or denied of the subject is called instead kategoroumenon, a term that would later be rendered as praedicatum in Latin. Today these two words, subject and predicate, become familiar to us, at latest, during lower secondary school, but they were invented by Aristotle. In his own days they were novel, at least in the technical sense they have in his works, and the novelty was no doubt an immense success. Subject and predicate are—we learn as we go on reading—the proposition’s terms.2 What does this mean? “I call terms those the proposition can be reduced to, that is, the predicate and what this is said about, adding or taking away being 1 Unless otherwise stated, all translations from works originally written in languages other than English are mine. 2 This is the standard, very well entrenched translation of the Greek horos, but in everyday discourse horos meant (as well as the Latin terminus) something like boundary.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Rigamonti, Logic, Everyday Discourse, and Metaphysics, UNIPA Springer Series, https://doi.org/10.1007/978-3-030-74598-1_1

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or non-being” (24b 17–18). “Adding being” and “taking non-being away” mean asserting the predicate of the subject, “adding non-being” and “taking being away” mean denying the former of the latter. We have here an almost imperceptible transition that probably escaped Aristotle himself, but is extremely important. Stating that a sentence3 is about something and says something about it is a truism; stating that something underlies it as its subjectmatter is not so trivial already; stating it can be reduced to subject and predicate, i.e., there is nothing else in it, is still less so. There is a hidden, subtle shift: Aristotle starts from the assertion that every proposition speaks about something and says something about it and concludes, in the end, that in every proposition there are two well-distinguished things, the “what it is about” and the “what is said about the latter”, and nothing else. I hope I have, at this point, made clear to the reader what is clear to me, namely, that the conclusion is stronger than the starting point, to which an equivalent of this “and nothing else” is subreptitiously added. So far, only the preliminaries of a study of “proof and demonstrative science” have been given, but now Aristotle enters medias res and explains that the standard form of a proof is syllogism, “a discourse whereby, putting certain things, something different from the premises necessarily follows from their being”. (24b 18–20).4 It is clear beyond any reasonable doubt, thanks to the context, that “putting certain things” means “given certain premises” and “from their being” means “if these are true”; so a syllogism has premises, and if these are true another proposition, different from them, is necessarily true. Aristotle introduces thus a general concept of logical consequence very similar to the one we normally use in logic today. Anyway, it is also clear from the context that there is a duplicity here. At first Aristotle introduces syllogism as generally speaking a relation between premises and conclusion such that if the former are true, the latter also necessarily is, but then, as he goes on, he only takes cases of a particular sort into account. More exactly, in all cases he discusses i. ii. iii. iv.

two premises and one conclusion are present; the premises contain three terms in all, so that one of these, the middle one, occurs in both; the conclusion contains both other terms occurring in the premises; the one occurring in it as subject is called minor, the other major; premises and conclusion are either affirmative universal (Every A is B) or negative universal (No A is B) or affirmative particular (Some A is B) or negative particular (Some A is no B).

Not all syllogisms have the same form; they are divided into three “figures”. In the first figure, the middle term is such “in position too” (25b 37), which means, 3I

will use proposition, statement and sentence indifferently.

4 Not all discourses are syllogisms, and in particular, given a couple of premises satisfying conditions

(i)–(iv), it is not always the case that something else necessarily follows from them. See infra, still in this section.

1.1 The Foundations of Aristotelean Logic

5

simplifying but not falsifying, that it is the subject in one premise and the predicate in the other; in the second it is “first in position” (26b 39), i.e., twice a predicate; in the third it is “last in position”, i.e., twice a subject. First figure syllogisms, such as “If every A is B and every B is C, then necessarily every A is C”, or “If every A is B and no B is C, then necessarily no A is C”, are “finished”, i.e. evident, and need no proof; all other ones (we can take as examples “If every A is B and no C is B, then necessarily no A is C” for the second figure and “If every B is A and every B is C, then necessarily some A is C” for the third) are “unfinished”, i.e., they stand in need of a proof and are indeed proved, always with rigorous techniques I briefly describe in the appended footnote.5 Not all couples of premises satisfying the conditions specified above give a conclusion: e.g., from “No A is B” and “Every B is C” nothing follows about the relation between A and C. When two premises give no conclusion, Aristotle proves it with a method based on contrary examples, i.e., giving two triplets of terms such that the premises are true for both, but with one the conclusion is universally negative and with the other universally affirmative. Let for example the first triplet be dog-catfeline and the second dog-cat-mammal: no dog is a cat, every cat is feline and no dog is feline, while no dog is a cat, every cat is a mammal and every dog is a mammal. This proves that nothing follows from “No A is B” and “Every B is C”. What sort of expressions are the terms that can occur as subjects or predicates in a syllogism? Aristotle never says it, but gives plenty of examples especially in Chaps. 4, 5, and 6 of Book I of Prior Analytics, where he defines and describes in full detail, respectively, the first, second, and third figure.6 In each of these he systematically, and leaving no gaps, discusses both the couples of premises giving a conclusion 5 Proving an unfinished syllogism means reducing it to a first figure (i.e., finished) one. This is done

either converting a premise, i.e., swapping its subject and predicate (which can be done completely with negative universal and affirmative particular premises, i.e., going from “No A is B” to “No B is A” and from “Some A is B” to “Some B is A”, and partially with affirmative universal ones, i.e., going from “Every A is B” to “Some B is A”), or “through the impossible”, i.e., denying the conclusion and showing that this, plus one of the premises, produces a first-figure syllogism concluding that the other premise is false. As an example of a proof by conversion, take the second figure syllogism “If every A is B and no C is B, then necessarily no A is C”. We convert the second premise getting “No B is C”, which together with “Every A is B” gives, through the first figure, “No A is C”. As an example of a proof “through the impossible” take the third figure syllogism “If every B is A and every B is C, then necessarily some A is C”. We deny the conclusion, getting “No A is C”, and this, together with “Every B is A” entails, through the first figure, “no B is C”, which contradicts the other premise, “Every B is C”. If you feel somewhat dizzy, don’t worry. It is normal. 6 For him syllogistic figures are just three while a later tradition, possibly beginning with Boethius (fifth-sixth-century A.D.), distinguishes four. Many scholars have variously tried, in different periods, to explain this “lack” of a fourth figure in Analytics, but I agree with Günther Patzig (Die aristoteliche Syllogistik, Vandenhoeck & Ruprecht, 1959) that according to the way the theory of syllogism is organized, the same syllogistic modes can be divided into three or four figures without any gaps, and the threefold nature of Aristotle’ classification does not entail any incompleteness. See also G. Rigamonti, L’origine del sillogismo in Aristotele, Manfredi, Palermo, 1980.

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and those giving none; for the second he uses—as I have already said—the method of contrary examples, so we can put together, thanks to his own examples, a short list of terms he obviously deems fit to be taken either as subjects or predicates in a proposition. They are: Animal Disposition Good Horse Ignorance Inanimate Man Raven Science Snow Stone Substance Swan Unity White Wild Wisdom.

(All these terms recur again and again or, as classical scholars say in such cases, passim, so that giving detailed references would be impractical). Two remarks must be made here. The first is that all seventeen terms just listed can occur both as subjects and predicates in Aristotle’s examples, and the same goes for his variables, i.e., A, B,  for the first figure, M, N,  for the second, and , P,  for the third. The second is not meant just for syllogistics, or even just for Aristotle, but has a wider scope: if one introduces a general concept, call it C, but then, when making examples, takes them all from a smaller class, call it C’, an ambiguity is created: is that person talking about the bigger class C or the smaller one C’? In Aristotle’s case, this is unclear: first he states, unobjectionably, that every proposition is about something and says something else about it, but then, when he has to make examples, it comes out that both the “something” a proposition is about and the (different) one it affirms or denies of the former almost invariably are, in fact, single words at the linguistic level and general concepts at that of meaning. This ambiguity is overcome neither in the Analytics nor elsewhere, and we also find it in later logicians up to the eighteenth century.

1.2 One Class of Variables

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1.2 One Class of Variables Let us focus now on the variables Aristotle uses to build his formal logic (historically, the first one deserving this label). It is clear from his examples that all these variables are mutually substitutable, and there is no trace in his writings of that distinction between two conceptually well separated classes of variables we could no more dispense with after Frege. Aristotle does all he has to do with variables belonging to one class, even where, respectively, predicative and individual terms would be needed for us today. There are several examples of this last point, but one will be enough. In Chap. 2 of Book α of Prior Analytics, to prove that negative universal propositions are convertible, i.e., that if no B is A, then no A is B, he uses ekthesis, a term translatable as “exemplification”. His reasoning is the following: “…if no B is A, no A will be B. For let  be an A that also is B: then it will not be true that no B is A, for  is a B” (25a 16–18). If it is true, says Aristotle, that an A, namely, , is a B, it also is true that a B (still ) is an A, but this is against the hypothesis. Today we would prove this, using natural deduction, in the following, very different way: let ∀x(Bx → ¬Ax)(i.e., “No B is A” re-written in today’s style) be given. Suppose now that ∃x(Ax & Bx) (“Some A is B”): from this we get, eliminating the quantifier ∃x, Ay & By (we can do it for in this context y is a new individual variable) and further, by conjunction elimination, both Ay and By. But from the hypothesis we get, eliminating ∀x, By → ¬Ay, and from this plus By we deduce, thanks to modus ponens, ¬Ay; now, we had inferred Ay already, so we have a contradiction. Therefore if ∀x(Bx → ¬Ax) holds, ∀x(Ax → ¬Bx) also does, i.e., going back to the ancient notation, if no B is A then no A is B. The modern proof essentially relies on rewrite rules for individual variables. But even not taking the obscurity of the notion of individual (that will be Chap. 3’s central theme) into account, generally speaking this is by no means necessary to prove what we want to prove here, for if we simply suppose there is a generic something being not just A, but B too, this “something”, call it , must also be not just B but A too, against the hypothesis (mind the swapping). Now, this reasoning goes through both if  is an individual, or if it is a subset of A (and B): so we have a correct proof of the convertibility of negative universal propositions even if we have no notion of an underlying Frege-type structure with two essentially different classes of terms, predicates, and names. I am not claiming, of course, that Aristotle’s tools were just as powerful as the ones we have today. With his tools, a very good logic of monadic predicates can be made; but not one of n-adic predicates, with n any positive integer—which entails, for one thing, that ancient logicians did not simply fail to discover the undecidable but could never have done it, with the tools they had. Anyway, within its limits Aristotle’s logic is self-contained and perfectly rigorous: the notions of proposition, subject, predicate, term, truth, and falsity are explicitly defined in it, it clearly says which syllogisms need a proof and which do not and the former are actually proved using well-specified techniques (conversion and reductio ad absurdum, or proof “through

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the impossible”). Moreover, whenever two premisses give no syllogism Aristotle proves it with his method of contrary examples. What more could one require? Syllogistics is a construction of a stunning precision and completeness, and I believe this is why it has been studied so long. Moreover, given its architectonic perfection it deserves being studied according to its own principles, without trying to reduce it to later (and certainly more advanced) systems.

1.3 Lukasiewicz’s Accomplishments Now, an incapability not of giving the post-Fregean paradigm up (that would be unreasonable), but of forbidding it to make Aristotle’s viewpoint incomprehensible is clearly visible in many twentieth-century logicians, including such a distinguished one as Jan Lukasiewicz (1878–1956), who became famous above all for being the first to introduce, back in 1918, “new” truth values different both from the true and the false. In Aristotle’s syllogistics from the standpoint of modern formal logic (North Holland 1956) Lukasiewicz works for some respects extremely well, but proves hopelessly incapable of putting his Fregean paradigm aside, and consequently of reconstructing Aristotle’s logic without distorsions. Yet he starts well, recovering the original formulation of syllogistics and sweeping modern interpreters’ misunderstandings away. The standard example of a syllogism, says Lukasiewicz, can and must not be the one usual in traditional handbooks, namely, “All men are mortal; Socrates is a man; therefore Socrates is mortal”. There are three reasons for this. The first is that when Aristotle makes examples, he does not use proper names but, as we have seen in Sect. 2, general terms,7 and such examples as “All men are mortal; all Greeks are men; therefore all Greeks are mortal” would be far more orthodox. The second reason is that Aristotle works, as a preference, not with words of common language but with symbols, so that writing, instead of the last example above, “All B’s are ; all A’s are B; therefore all A’s are ” would be far more “Aristotelean”. The third reason is that Aristotle does not write his syllogisms as simple sequences of propositions, with a “therefore” separating the conclusion from the premisses, but as implications with the consequent prefixed by an ananke (translatable as “necessarily”): “If all B’s are C and all A’s are B, then necessarily all A’s are C”. It is next to incredible, but in all comments of the Analytics (published roughly between 1800 and 1950) Lukasiewicz very often refers to one always finds an incapability of seeing, or perhaps of taking seriously, these technical aspects of syllogistics (that simply are syllogistics) and giving up the use of not just more recent, but irrelevant notions—or, said more bluntly, these authors understand nothing of the works 7 This

is not a 100% true, but comes very near to it. There are in Analytics a very few cases where a proper name is used as an example, but they really are sparse and unimportant.

1.3 Lukasiewicz’s Accomplishments

9

they believe they are explaining. Lukasiewic often quotes, e.g., one Prantl, working in Germany in the nineteenth century and quite authoritative in his days, who claims that the ananke (“necessarily”) invariably prefixing a syllogism’s conclusion expresses “the synthetic force of thinking” (die synthetische Kraft des Denkens). Prantl, in other words, superimposes on Aristotle Kantian categories having nothing to do with him, without even trying to reconstruct his theories faithfully!

1.4 Lukasiewicz’s Failures Lukasiewicz explains, on the contrary, all these things very faithfully and carefully, and this is a great merit; but he also suffers from his own blind spots, due to a rigid-to-paralysis faithfulness to the Fregean paradigm. The fundamental thesis of Aristotle’ syllogistics is that all and only the syllogistic modes8 Aristotle licenses as valid still remain so when re-written in standard modern notation; to give an example, in today’s logic “If every A is B and every B is C, then necessarily every A is C” becomes ∀x((Ax → Bx) & (Bx → Cx)) → ∀x(Ax → Cx), also a valid formula. And so on and so on, case after case. This is true, of course. Given an (always possible) adequate and faithful translation, the syllogisms that are valid for us today are exactly those Aristotle also licenses as valid, with the only exception (highlighted, of course, by Lukasiewicz) of those needing, from our viewpoint, the additional premiss that if every A is B (no A is B), then some A is B (some A is no B). For Aristotle this is trivially true, since he seems to have no notion of an empty class, while we do; but if we abstract from this, there is indeed a perfect agreement between his theory of syllogisms and the modern theory of classes. But if we did not go further than this (and Lukasiewicz does not), it would be as if we subjected the Aristotle-boy to an exam in modern logic, giving him, in the end, an a plus. Now, I believe the Aristotle-boy deserves better than that: what he deserves is not being examined by us about our theories, but being understood in his own terms, for what he says, and to understand him in his own terms we must first of all take seriously his working with one class of variables, and achieving in this way important results. If we stubbornly cling to the idea that he, too, sees behind predicates the individuals they hold of, as we do when we interpret predicates as sets in our standard semantics, in fact it is us who insist on seeing in him things that are not there, and on refusing to understand that important, technically valid results can be reached without seeing individuals behind universals. And to forestall a possible misunderstanding: I am not saying Aristotle ignores that an universal term, such as “horse”, can be truly said of many individuals. He knows it perfectly well. But I am saying that for him, as regards the construction of what he calls “demonstrative science” at the very beginning of Analytics, this is unimportant.

8A

syllogistic mode is a couple of syllogistic premises, i.e., premises satisfying conditions laid down in §1. A mode is valid if a conclusion necessarily follows from it, invalid if none does.

Chapter 2

Some Minimal Technical Foundations

Abstract The essentials of modern logic are introduced, with references to Bolzano, De Morgan, and Frege.

2.1 Frege’s Reform of Logic Except for a couple of brief remarks in § 2.4, this is a service chapter. It recalls some basic notions of philosophy of language without which contemporary logic would never have been born, and supplies the thinnest possible technical apparatus. But right at the end some weighty enough doubts are also voiced. In contemporary logic there is a complete separation between predicates and individual terms, or names; this separation was introduced and defended by Frege especially in Concept and Object (Über Begriff und Gegenstand, 1892). A long, long tradition, says Frege, claims that a term’s being a subject or a predicate is just a matter of position, but this is wrong. Subject and predicate have different natures: In brief, my position can be…stated thus (where I mean “subject and “predicate” in a linguistic sense): a concept is the meaning of a predicate, while an object is something that can never be the whole meaning of a predicate, but can be the whole meaning of a subject.1

If we put together Frege’s thesis that “an object is something that…can be the whole meaning of a subject” with his warning that “I mean ‘subject’ and ‘predicate’ in a linguistic sense”, we can safely conclude that for him a subject is an object’s name. Vice versa, an object never is “the whole meaning of a predicate”; therefore a predicate is not an object’s name. The function of naming and that of predicating, i.e., attributing properties, and consequently the terms performing, respectively, the

1 “Oggetto

e concetto” (orignally published, as “Über Begriff und Gegenstand”,), pp. 359–373 of Gottlob Frege, Aritmetica e logica, edited by Corrado Mangione, Boringhieri, Turin, 1967. The passage quoted is on p. 365. My italics and translation. All quotes from Frege have been checked on the German original, both for Über Begriff und Gegenstand and Über Sinn und Bedeutung, but I rely very much on this old, excellent anthology.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Rigamonti, Logic, Everyday Discourse, and Metaphysics, UNIPA Springer Series, https://doi.org/10.1007/978-3-030-74598-1_2

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former and the latter, must belong to two disjoint classes (of course these two functions, as we have seen, had already been distinguished by Aristotle; but not separated, not associated to grammatically distinct terms). Now, at a surface level everyday discourse is teeming with apparent counterexamples, but Frege disposes of them quite elegantly: …for example, in the proposition “All mammals have red blood” the predicative nature of the concept cannot be ignored, for it is permitted to use, instead of such a proposition, one of these other two: “Those beings that are mammals have red blood”, “If a being is a mammal, then it has red blood”.2

In both paraphrases invented by Frege, what was initially the subject has become a predicate (I am using both terms in their traditional grammatical sense). I love seeing in this passage the precise moment when modern logic was born, the exact conceptual turning point leading from “Every A is B” to “For every x, if x is A, then x is B” (and with a small further step, from “Some A is B” to “For some x, x is A and x is B”). So for Frege a proposition’s subject always is, contrary as appearences may be, an object’s—more exactly, a single object’s—name: a proper name. This is explicitly and very clearly said in Sense and Reference, another work of Frege’s annus mirabilis, 1892: …by…”name” I mean here any expression functioning as a proper name, i. e., whose meaning is a definite object (where “object” is meant in the widest sense). The indication of a single object can also consist of several words, or other signs. We will, for brevity’s sake, always call it “proper name”.3

If we put this quotation and the two former ones together we conclude that names are no concepts, i.e., no terms predicable of several things, or universals. Now, if they are no universals, then it seems reasonable and indeed unavoidable to conclude they are individual terms. But is this notion, individual, really clear? We will discuss this point in the next chapter.

2.2 A Last Important Preliminary However, before entering this theme of the meaning of “individual”, that will prove surprisingly thorny, we need another preparatory step. Until about 1850, when an Englishman, Augustus De Morgan, first proved that relations are not reducible to 2 Op.

cit., ibidem. Frege, “Senso e significato” (originally “Über Sinn und Bedeutung”, 1892; see references), in Logica e aritmetica cit., pp. 374–404. Quotation from p. 377, my italics.

3 G.

2.2 A Last Important Preliminary

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properties, all logicians believed that given any finite set of propositions there was a natural one-one correspondence between their respective subjects and predicates: given one subject one predicate, that of the same proposition, and vice versa. Not all of them inferred from this supposed correspondence the conviction, originating as we have seen with Aristotle, that subject and predicate were interchangeable, but before (roughly) 1850 they all admitted this one-one matching was there. As a consequence, the belief that relations were just more complicated properties—so that, for example, in “Matthew is richer than John” the property “richer than John” is predicated of Matthew—was generally accepted. Even such an extraordinarily talented thinker as Bernard Bolzano still maintained, in his 1837 Wissenschaftslehre, that in every proposition there are one predicate and one subject. Almost at the beginning of §127, Parts which the author takes all proposition to have, we read: In all propositions, first of all, there occurs the concept “to have” or, more precisely, the concept expressed by the word “has”. Along with this constituent every proposition contains two others, that “has” connects in the way indicated by the expression “A has b”. The first of these constituents, that which is expressed by A, occurs as if it represented the object the proposition concerns, and the other, b, as if it represented the attribute that the proposition ascribes to this object. Accordingly, I allow myself to call the part A, however it may be constituted, the basis or subject-representation, and b the attributive part or predicaterepresentation.4

One must explain that in his technical language Bolzano consistently uses has instead of is as a copula. He never explicitly says why, but he probably wanted to prevent all possible interpretations (frequent in his days among philosophers, and I am not implying they have disappeared today) of the copula as a sign for identity: “Giuseppe has intelligence” is farther from common use than “Giuseppe is intelligent”, but also more obviously non-identifying. It also has to be emphasized that Bolzano must have been well aware that subject and predicate had different natures, for he consistently used different variables for the first and the second, but still believed nevertheless that in every proposition there was not simply one predicate (something we still believe today), but one subject too (which keeps being true for us only in a strictly grammatical sense). It was De Morgan who, not long after Bolzano, destroyed this long-received view; but at this point I cannot help illustrating the reasoning by which he proved that relations cannot be reduced to monadic predicates. Except for a few inessential changes, it is the following: take the inference 2.2.1 If all tusks are parts of elephants and all elephants are animals, then all tusks are parts of animals. 2.2.1 is obviously correct, but such correctness is mysterious for us if we suppose that first the property “part of elephant” and then the property “part of animal” are 4 Bolzano,

The Theory of Science, translated and edited by Paul Rusnock and Rolf George, OUP, 2014, vol. II p. 5; but Rusnock’s and George’s translation constantly has idea instead of representation. I have made this substitution because I find representation more faithful to the original German term, Vorstellung.

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predicated of “tusk” (there is of course also another predication, ascribing “animal” to “elephant”, but that is no problem), for in this way we get five terms in all, “tusk”, “part of elephant”, “elephant”, “animal” and “part of animal”, and have the inference form 2.2.2 If all A’s are B and all C’s are D, then all A’s are E,5 obviously lacking logical validity. But we get out of trouble if we suppose that the binary relation “part of” is not just a component of such pretended monadic predicates as “part of elephant” and “part of animal”, but cannot be eliminated: for in this way we get the obviously valid schema 2.2.3 If all A’s are parts of B’s and all B’s are C’s, then all A’s are parts of C’s. So relations are not more complicated monadic predicates. The latter are predicated of one thing at a time, while binary relations are unreducibly predicated of two at a time, ternary relations of three, and so on; and if by subject of a proposition we mean, in Aristotle’s way, the thing it is about and not, in the grammarians’ way, just the term its verb refers to, then subjects, in a logical sense, can be more than one.6

2.3 The Modern Notion of Well-Formed Formula (or Proposition) Let us turn now to the effects of Frege’s radical separation of subject and predicate on the technical apparatus of contemporary elementary logic. In a standard presentation small letters x, y, z,…, possibly with numerical subscripts, are normally used as individual variables and a, b, c,…, also with numerical subscripts, as individual constants.7 These are simple individual terms, but we also need complex ones, such as e.g., “Ann’s father” or “her father” in everyday discourse or x + y in arithmetics. To get maximum generality we introduce, to begin with, a new symbol t (possibly with numerical subscripts: t 1 , t 2 , t 3 ,…) for a generic individual term, either variable or constant and either simple or complex. Now, a complex individual term is built, language by language, √by means of an n-adic function (“the father of”, “the prettiest one among…”, “log ”,…) and n individual terms, with n ≥ 1: the general pattern is f(t 1 ,…,t n ). An individual term is well-formed if and only if it is either simple (a 5 Perceptive

readers will notice that De Morgan, who lived before Frege, still works with just one class of variables. But the important thing here is, his reasoning is perfectly sound. 6 In “Tony is taller than John”, for example, the grammatical subject is one, Tony, but the logical subjects are two, Tony and John, because the proposition is about both (the relation “taller than” holds–one way—between both). 7 In everyday discourse pronouns are used as individual variables and proper names as individual constants.

2.3 The Modern Notion of Well-Formed Formula (or Proposition)

15

possibly indexed letter-symbol) or a function with n argument places saturated by exactly n individual terms.8 After the general concept of well-formed individual term, that of atomic (or elementary: I will use both adjectives indifferently) well-formed formula comes next: 2.3.1 X is an atomic (or elementary) well-formed formula if and only if it contains one n-place predicate, n well-formed individual terms, and nothing else. The last three words, “and nothing else”, are essential to exclude, for example, negative formulas from the class of atomic ones. If you have an n-place predicate, n individual terms and a “not”, that is a well-formed formula, but not an atomic one. Three remarks have to be made here. The first is that although according to today’s logic the above definition holds (though technical details have to be filled in on a case-by-case basis, and can vary a lot) both for everyday discourse and many formal languages—if you read such classics as Frege, Russell, or Quine, this is entirely obvious—everyday discourse is teeming with intuitively correct propositions that somehow strain the notion of proper name. Take, for example, 2.3.2 Sea water is not drinkable, or 2.3.3 My brother-in-law hates cheese and never eats it; now, the notion of individual, as we will see Chaps. 3 and 5, is extremely hard to circumscribe, but classifying such words as “sea water” or “cheese”, as they are used in these examples, as individual terms is intuitively awkward. The second remark is that this concept of well-formed atomic formula I have just introduced (not departing from standard treatments) really conflates two distinct properties, that of containing nothing but an n-adic predicate and n names, and that of containing no well-formed formula as a proper part. Now, in Chap. 7 we will see that as far as everyday discourse is concerned these two properties can depart, for there are well-formed propositions containing exactly one n-adic predicate, n names, and no propositional operator such as e.g., negation, or a modality, but where nevertheless some other well-formed proposition occurs as a proper part. The third remark is that in everyday discourse we do not always have (even with obviously well-formed expressions) just one possible way to distinguish the predicate from all the rest, but sometimes there can be more than one. For example, 2.3.4 Bob must talk to his wife has two possible analyses: 8 This

general definition of the concept of well-formed individual term involves a bootstrap, for it defines a concept using it in the definiens. But this bootstrap is no vicious circle, for each term is built in a finite number of steps, given in a well-defined order, from atomic i.e. simple ones.

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(a) (b)

2 Some Minimal Technical Foundations

the predicate is “must”, saturated by “Bob” on the left and by “talk to his wife”9 on the right; the predicate is “talk to”, still saturated by “Bob” on the left, but simply by “his wife” on the right, and “must” is just an inflection of this predicate.

A choice can only come from a theory of so-called servile verbs. Either these verbs, such as “must”, are predicates on their own, and in that case (a) is right, or such an expression as “must Q”, where Q is any verb, really is an inflection of Q, though not obtained in the ordinary way, i.e., by somehow suffixing Q, but through another word lacking a meaning of its own and only serving to specialize Q’s. Modern formal logic chooses the second option and so treats modalities as operators modifying a predicate and not as predicates on their own, but there seems to be no reason for seeing the first option as inconsistent.10

2.4 The Essentials of Tarskian Semantics So far, anyway, only syntax—i.e., the rules for writing grammatically correct formulas—has been given, but not semantics, i.e., those other, also indispensable rules that specify the meanings of a formal language’s expressions. Now, we shall not need all of today’s standartd semantics in a full-fledged, exhaustively detailed form, but some partial characterization is necessary, otherwise the very important philosophical conclusions that will be reached in §§ 3.10–12 would be incomprehensible. With his first, groundbreaking book, Begriffsschrift (1879), Frege created a language for logic11 ; now, a language’s expressions have meanings and, more specifically, for Frege the meaning of an individual term is, as we have already seen, an object. This point (i) already is quite clear in Frege’s writings, especially Über 12 Sinn und Bedeutung, and (ii) gets further developed in Wittgenstein’s Tractatus, but (iii) only reaches completeness and full rigor in 1934, with Tarski’s Der Wahrheitsbegriff in den formalisierten Sprachen (The concept of truth in formalized languages). For Tarski, a formula’s truth value in a formalizedl language L is determined by a 9 This

is no name; but as we go on, we will find that things are far more complicated than Frege believed, for there are in everyday discourse binary predicates (so-called propositional attitudes) that cannot be saturated by two names, but require for saturation a name and a proposition. An example can be 2.3.3 The captain understood a storm was approaching. I will say something more detailed about propositional attitudes in Chap. 5. 10 If we take option (a), then “must” in 2.3.4 is saturated not by two names, but by a name and a proposition; however, there certainly are (potentially) infinitely many cases, those of so called propositional attitudes, where exactly this happens (see Chap. 4). 11 To be punctilious, not the one–due to Giuseppe Peano—I have described in this chapter; but Peano simply modified technically (not conceptually), in the last decade of the nineteenth century, the Begriffsschrift’s notation, making it easier to use. 12 Ü is the German umlaut.

2.4 The essentials of Tarskian semantics

17

relation between L’s individual terms and certain elements of a domain D, on whose basis another, more general relation between (a) certain sets ultimately reducible to elements of D and (b) certain combinations of individual terms of L can be defined. Now an essential point of Tarskian semantics is that the domain D, or universe of discourse, must be well defined in Cantor’ sense, i.e., such that for any object x whatever it must be determined, with no ambiguity, whether x belongs to D or not. This is not seen as something that might change over time; in other words, according to Tarskian semantics (1) neither any existing individuals can either disappear or change, nor any new ones can ever come to being, and consequently (2) a formula’s truth value can never change. Now, we will see that in any ontology reasonably associable to everyday discourse it constantly happens that some existing individuals cease to be, new ones acquire existence, and all those having an empirical existence constantly keep changing as long as they are around, so that there can be no unchanging domain of individuals. This is a very important point, having momentous consequences; we will come back to it in a more detailed way toward the end of Chap. 3.

Chapter 3

What Is an Individual?

Abstract The first nine sections discuss at length the notion of individual, concluding it is not an ontological, but a linguistic one. The last three draw some very general conclusions that may well be called “metaphysical”.

3.1 When We Know What We Are Talking About, and When We Do Not Preliminaries are over at last, and we can start discussing the ideas this book is mainly devoted to. Now, there are cases where, working at a given theory, we know perfectly well what the individuals whose existence it admits or even mandates are. We can take as examples Peano-Russell number theory, or ZF (Zermelo-Fraenkel) set theory. In number theory natural numbers are individuals, all of them, and nothing else is one; so an individual term can only denote a natural number. In ZF sets are individuals, and nothing else is; so to the question “What is an individual?” one has to answer “Any set, and nothing else”. These are, in their own way, wonderfully clear situations, with a perfectly precise ontology associated with a well-defined domain1 of possible interpretations of individual terms. But if we look farther than these happy cases problems begin. This already happens, at metatheory level, with the general notion of formal theory. A formal theory is, of course, a (preferably finite) set of formulae in some standard logical notation, given as axioms, plus a finite set of rewrite rules allowing us to deduce consequences from axioms and then further consequences from the axioms plus the already deduced ones, to infinity. Now, what is such a structure a theory of? The 1I

follow Cantor in calling a domain M well-defined when and only when (i) for every x, it can be decided whether x is an element of M or not, and (ii) for any two elements x and y of M, it can be decided whether x = y or not. See about this Georg Cantor, Gesammelte Werke mathematischen und philosophischen Inhalts (Collected Works of a Mathematical and Philosophical Content), edited by Ernst Zermelo, Olms, Hildesheim, 1932, repr. 1962, p. 150. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Rigamonti, Logic, Everyday Discourse, and Metaphysics, UNIPA Springer Series, https://doi.org/10.1007/978-3-030-74598-1_3

19

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3 What Is an Individual?

orthodox answer is “This depends on the universe of discourse you choose”, and of course it is a correct one, since—with the exception of logical truths and contradictions—a formula’s truth value depends on the universe of discourse we refer it to. I will make just one example, stated in everyday discourse but easily translatable into some formal language: if someone asks “Is there a number smaller than all other ones?” we cannot simply answer “Yes” or “No”, but must inquire first what is the universe of discourse here, for there is indeed a smallest natural number, 0, but no smallest relative one. Specifying the universe of discourse, i.e., the set of possible interpretations of individual terms, is even more important in formal theories, where these terms have no meaning of their own but must be interpreted. Now, which universes of discourse are admissible? In principle, all but the total class.2 There are no constraints except this one, no general ontology of formal theories as such.

3.2 Wondering About the Meaning of This Word, “Individual” Let us go now to everyday discourse. I will use “individual” (as a noun, not as an adjective) and “object” as synonyms, and will not define them.3 The reason is not just that trying to do so would lead me, as centuries of philosophy show, into an incredibly entangled mess, but also that such an attempt would be useless from the viewpoint of the main conclusions I am interested in, and that are already clear enough to me. These conclusion are that (I) given any grammatically permissible property P, there are individuals that do not have P, but (II) it is not the case that for any grammatically permissible property P there are individuals having P.4 I will arrive at (I)–(II) through a long sequence of examples, beginning exactly now. “Individual” is currently used (as a noun) in various ways, some of them broader some narrower. In the most common and frequent of these uses it is a synonym of “person”, except that it often has a negative, sometimes disparaging hue: in pulp fiction, for example, we easily find such expressions as “an untrustworthy individual”, or “that individual disgusted me”. But (in biology publications, not in fiction) “individual” can also mean living organism at large, and we happen to read, for example, that an anthive can count up to one million individuals. That broadens the term’s scope a lot, for in this way both a giant redwood, growing tall up to 140 m and living up to 6,000 years, and a virus visible only with electronic microscopes and living half an hour before dividing or dying are individuals. 2 The

total class is the class of all things. It cannot be a set, and has features making it unusable not just in formal theories, but in all rigorous and precise ones. 3 As the reader goes on, anyway, it will become increasingly obvious to him/her that the view defended in this book is diametrally opposed to Wittgenstein’s claim in Tractatus, 2.02, that “An object is simple”. 4 At first sight points (I)–(II) might look trivial, but we will discover they are not.

3.2 Wondering About the Meaning of This Word, “Individual”

21

There is an obvious relation between this biological use of the term and its etimology: “individual” comes from “indivisible”. If we divide a stone into parts all of these are stones in their turn, but if we divide a living being either none or at most one of the parts we get is still a living being. To be very meticulous, this is not 100% true for there are animals, e.g., hydra, such that if you cut them to pieces each of these still is a living being, and of the same species as before: but these are atypical, marginal cases, and would tend moreover to push us toward a notion of “individual” narrower than “living being”, while as we go on we will be practically forced to accept a broader one. This broader notion will gradually emerge through many, many examples, all of them presupposing “proper name” (to be understood in Frege’s very wide sense, see §2.1) as a primitive notion. In connection with this, I also adopt, with a conservative move, Aristotle’s distinction in De Interpretatione between universal terms (such as man), that are meant of many objects, and singular ones or proper names, such as “Socrates”, that are meant of one; I say “are meant of”, and not “correspond to”, for there are both singular and universal terms—e.g., respectively, “the eighth king of Rome” or “barking cat”—to which no object corresponds. But in many cases, to a proper name (I leave universals aside, as not having to do with the present discussion) there actually corresponds exactly one object: this object is an individual. Ideally, an individual is a proper name’s meaning and a proper name is a term uniquely denoting an individual.5 Of course, the term “proper name” has a narrower meaning in grammar and a wider one in logic. For traditional grammarians, only terms from a special though open-ended list—such things as Achilles, Vanessa, Rome, Mississipi, War and Peace—are proper names, but for logicians also definite descriptions—such things as “Cinderella’s stepmother” or “the first king of Scotland”—are.6 But there is more to it, for many expressions are proper names in certain contexts and not in others (here, anyway, differences among languages can have an important role). Take for example 3.2.1 The dog was growling and showing his teeth: in that context “the dog” is a definite description, for it uniquely refers to one object. But in 3.2.2 The dog was tamed some ten thousand years ago its reference is altogether different.7 5 There

also are, of course, individuals with no name and names denoting no individual. I usually give no names to the gnats stinging me much too often in summer nights, yet they certainly are individuals; and if you ask where Lake Bologna is in Italy—well, there is none. 6 It is good to remind here that a proper name in the strictly traditional grammatical sense and a definite description might well refer to the same object: take for example “Europe’s longest river” and “Volga”. 7 This distinction is not so harmless as it might look. In 3.2.1, “the dog” certainly is a proper name, but (i) granting it also is one in 3.3.2 is, to put it mildly, rather generous, (ii) in any case, it has a different reference there: not a single dog, but a species.

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3 What Is an Individual?

3.3 Examples, Examples And now the examples’ flood comes. Let us begin with the most obvious things: 3.3.1 Rosie is thin. 3.3.2 Bob is an engineer. But it isn’t just humans that have a name: 3.3.3 Tige is a mastiff. 3.3.4 Dutch Belle is a champion, she gives forty liters of milk a day. And it isn’t just living beings: 3.3.5 Florida is a half-island. 3.3.6 There are many art masterpieces in Florence. 3.3.7 I am interested in visiting Patagonia. 3.3.8 The North Sea has an unpleasant climate. And about the North Sea: an individual (and it follows from the things said in §3.2 that the North Sea is one) does not necessarily have a sharply defined spatial boundary. Seas certainly do not, and the very idea of drawing on some map a precise line dividing any two of them is silly (even if nations and their governments often forget about this). But that does not go just for seas: all the world knows for example about Sahara, but where does Sahara exactly begin, or come to an end? The very question is stupid. This goes, by the way, not just for space boundaries, but also for the temporal ones of events or processes or periods. These, too, can be individuals in the sense I have tried to explain in §3.2; one can say, for example, 3.3.9 It was in Italy that Renaissance began (everybody writes “Renaissance” with a capital R), but if anybody asks when it exactly began, or finished, he (or she) will be invited not to say nonsense; yet grammatically such a proposition as 3.3.10 Renaissance was a corrupt age is just as singular as 3.3.11 Jill is very good at maths.

3.3 Examples, Examples

23

This has, of course, a hard-enough-to-swallow consequence, namely, that Renaissance is an individual just as much as Jill—something looking counterintuitive and unwelcome to commonsense, for commonsense easily induces us to reason that individuals are somehow “small”, so that a system comprising an immense number of them, e.g., of people, cannot be one. This problem has been discussed by many philosophers, belonging to very different schools and ages. Recalling all their discussions would be impossible, so— skipping many interesting things that have been said over the centuries—I will only contrast two possible criteria for individuality: (a) that individuals are denotations of proper names, and (b) that individuals are in some sense “small” things, as opposed to bigger ones (that we might call for example “systems”) having many, many individuals as members. According to the first view, all of a nation’s citizens are individuals but the nation also is; according to the second, the bigger system’s elements are individuals but the system itself is not. Neither option is unobjectionable. (a) has, as we have seen, some counterintuitive aspects; but (b) is disastrously imprecise, and quite often arbitrary in its conclusions. For example: are we humans individuals? I suppose nobody would deny that. And are nations individuals? No, of course, according to (b): single humans are individuals, nations are not. Anyway, each of us contains about a hundred million cells: are these individuals? Well, when we study cells, we are strongly tempted to conclude they are. But how is it that a big system of individuals is itself one at the cell-person level, and not at the person-nation one? There is a well-known sort of trouble waiting for us here: commonsense knows nothing about consequences—while criterion (a), unintuitive as it may look, does. So after being introduced early in this section, our general criterion of individuality—being the designatum of a proper name—can be reasonably seen as confirmed here. But it should begin to show already—and will show better and better as we go on—that it is a linguistic, not an ontological criterion. However, let us go back to our examples: we were considering some big ones, but none is bigger than the world, and grammatically “the world” is nothing but a definite description, so that two famous (and mutually incompatible) metaphysical assumptions, i.e., 3.3.12 The world exists from eternity and 3.3.13 The world has been created use a definite description, and are therefore assertions about an individual. But as individuals can be big, so they can be small. I might start e.g., from 3.3.14 Mary is ill and then add something more precise: 3.3.15 Mary’s heart is okay, but her liver is badly damaged.

24

3 What Is an Individual?

But things do not stop there. A histological exam is performed on a specimen of Mary’s liver, and someone comments 3.3.16 This cell is cancerous; and we can go still deeper down, treating mitocondria, chromosomes, DNA triplets, and so on as individuals. I grant that when one reaches the level of quantum physics things get much murkier, for ontology is really bizarre there; but even leaving quantum level aside, variations of individuals from small to big and from big to small can be enormous, both in space and time. But are at least all things, big and small, to which we give proper names bodies, or system of bodies? No. Here I will give an example associated to a personal recollection: 3.3.17 The rainbow visible in Milan at seven p.m. on September 22, 2015, was wonderful. I can claim this with certainty, for I saw that rainbow. It had been raining all day and all day had I gone around, feeling cold, because of various things I had to do; and then, at that very hour, I realized the sun was shining at last. Its light only reached the roofs of buildings, for it was sunset (at that date it would be at six p.m. with a solar hour, but in September we have a legal hour in Italy), so its last rays disappeared in a minute or so, and the rainbow with them; however, that rainbow was a real beauty. A rainbow is no body, but has nevertheless a physical existence; and that rainbow was individuated, as all rainbows are, by the place where and the time when it could be seen. By the way, good old Thomas Aquinas used to say that principium individuationis of natural things is given by place and time, and many have repeated this after him. But this thing Aquinas used to say is not always true. Take this example about something he could have no idea of: 3.3.18 Background cosmic radiation corresponds to a temperature of 2.7 K degrees. Background cosmic radiation was discovered in 1964. It is strikingly uniform and can be observed in all parts of the universe, with no exception, yet it has a proper name in Frege’s extended sense. I believe I need add nothing else. Again: all “individuals” I have discussed so far have spatial or temporal continuity in common, where the “or” of course is inclusive; indeed, when we periodize history, dividing it into ages that are as many chapters of a narrative, each of these ages has a unitary, unbroken course. As far as I know, no historian ever invented an age lasting, say, from the second to the sixth-century A.D. and then again from the eleventh to the sixteenth. And talk about a rainbow visible, say, in Paris, or in Rome, is OK, but talk about one visible half in Rome and half in Paris would not. Anyway, in fact spatiotemporal continuity is a feature of most individuals but not of all. Two examples will suffice to show this. For time, take the most famous bicycle race in the world, Tour de France. When they speak about it, newspapers of all countries, including those with languages different from French, call it Tour

3.3 Examples, Examples

25

de France or simply Tour, always with capital initials; so “Tour de France” is a proper name if there ever was one, and according to the criterion chosen in §3.2 and just confirmed in this section, denying its denotatum is an individual would be unreasonable. Now, this individual becomes physically real, year after year, in July, then loses this physical reality in less than a month and only recovers it the following year, but people use to call it simply “le Tour de France”, one individual thing. So much for temporal continuity. For spatial one, take this other example: mr. New Croesus is immensely rich, he owns lots of apartments in New York, Hong Kong, Capetown and London, building areas in Germany, Mexico and the States, farms in Brasil, Italy and France, forty cargoes always going back and forth through all three oceans, shares and cash in banks all over the world. Then he dies, and we read on the papers: 3.3.19 New Croesus’s property entirely goes to daughter. Son disowned. “New Croesus’s property”: this is a definite description, so a proper name, of a well-defined set of spatially scattered things. But we haven’t finished yet. People, animals, stones, rivers, cities, soap bubbles, waves, waterfalls, rainbows, storms, historical and geological ages and so on always have some form of physical existence; but there also are things having a name but not this sort of existence. Take the following example: 3.3.20 π is a transcendent number.8 π is, of course, the ratio between circumference and diameter in a Euclidean circle, i.e., that famous irrational number beginning with 3,14159…; but if it is a number, it can be no physical object. There also are, then, individuals not belonging to the physical universe. You do not need any religious faith to admit it. But at this point, finding non-trivial properties distinct from individuality itself and belonging to all individuals simply qua individuals begins looking like a desperate pursuit.

3.4 A Still Broader Domain We make a pause, after this very quick and incomplete choice of different sorts of individuals, with a small booty of possibly not entirely trivial conclusions. The first of these is that although language is undeniably capable of describing objects and telling stories, there is no structural analogy at all between it and the world, at least in the sense that grammatically similar terms may refer to things having no ontological similarity at all. Take for example 3.4.1 The soap bubble vanished and the kid cried: 8A

number is transcendent if and only it transcends ordinary algebra, i.e., if and only if it is no m-th root of a natural number n, for any m and n.

26

3 What Is an Individual?

this is a conjunction, and both conjuncts have the same grammatical form i, e. subject and one-place predicate, yet it is difficult to imagine a bigger ontological difference than that existing between kids and soap bubbles. Now, ontological differences among individuals have already been explored, though very quickly, in the previous section; but to complete the viewpoint I am trying to outline, something more must be said about “very big” things. However, I will introduce this theme starting from the small, not the big. October 12, 1492, May 8, 1945, January 31, 1940—and I could go on as long as I like—are precise dates, two of them famous and one not, but precise anyway. Dates indicate sometimes a day and sometimes a year, and those e.g., on tombstones can be of both sorts, but they all are proper names, either of one year or of one day, and can occur as subjects in propositions: 3.4.2 1917 saw two revolutions in Russia, first a bourgeois-democratic and then a communist one 3.4.3 February 21, 2004, is Marcello’s date of birth. Now, such dates aren’t simply well-circumscribed proper names, but once you accept the Gregorian calendar, as most countries do today, the existence of their denotata is obvious and undeniable (and for those who use other calendars other, equally well-defined dates correspond to the Gregorian ones). Moreover, no interpretation effort is required about them. But things change as soon as we shift from pure quantitative indications to qualitative individuation criteria. We use to talk about Middle and Modern Age, Cambrian and Cenozoic Era, Paleolithic and Neolithic Period, and this is not putting single events somewhere in a calendar, but introducing cuts in a potentially infinite course. In this connection, I will discuss the history of mankind far longer than the biological or geological one, but something will be said about the latter too. Now, human history always goes on, always will as long as humans exist, and (this does not apply just to humans) when a fact happens, others happened before, still others will happen later, and they never stop happening—starting again only after a while—for any time. On May 23, 1453, the Turks took Constantinople; on that day the last byzantine emperor, Constantin XII Paleologus, went down fighting. Constantinople had been for a thousand years a Greek and Christian town, and became a Turkish and Moslem one; its name also changed, now it’s Istanbul. For us today, when we study the history of this town, it is absolutely normal to divide it into a Greek and a Turkish period, but both before-and-after that May 23, 1453, there were in it men and women doing sex and having children, families growing or shrinking, people getting richer or poorer, behaving justly or unjustly, falling ill or recovering, being born, growing up, reproducing, getting old, dying; and there were sunny and rainy days, hot and cold days. And it is true the place was Greek-speaking, and became (not overnight, of course) Turkish-speaking; but people there never stopped saying, first in Greek then in Turkish, “I love you” and “I hate you”, while the Earth kept spinning, always at the same speed. After the Turkish conquest the world did

3.4 A Still Broader Domain

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not freeze, neither in Constantinople-Istanbul nor elsewhere, and neither for one year nor for one day nor for one minute. Then why do we divide history into periods and introduce splits in it, if the world keeps going on anyway, and never ceases doing so? To understand something about it. If history were for us just a sequence of years, all equal in their being years and simply piling up one upon another, as we do sometimes with coins, we would understand nothing about it. For our brains, beyond a limit coming soon enough a purely quantitative heaping up of monotonously homogeneous elements means nothing at all. We get lost in such repetitions, and to understand something about them—or rather, that there may be something for us to understand in them—we have to introduce turning points and crucial aspects. But we, and we alone, decree they are such. Not that we create them, of course: an honest historian invents nothing, but in the ocean of events she selects and emphasizes some, while silently discarding others as unimportant; and in another ocean, that of beginnings and ends succeeding each other day after day, she classifies some as “crucial”, “central” or “epoch-making”, while with others she does not even bother to say they count for nothing. If she did not do this, she could give no sense to the facts she studies, and if she writes about them, she could not make those who read her understand anything about them. She must introduce cuts in the unbroken flow of events and select from time to time a moment when a chapter having lasted for a certain number of pages comes to an end and another begins; she cannot help doing so, for we understand change only if we think it as a story, and a story is something with a beginning, a development, and an end. Of course, in history there can be moments that almost force us to see them as turning points: it really is difficult to write a sensible story of the city of Constantinople-Istanbul without emphasizing its fall (or conquest) in 1453, or one of Mexico without dwelling quite long on the fall of Tenochtitlan and the Aztec Empire in 1521. But emphases giving more importance to these events than to others are the historian’s; events do not emphasize themselves, the fall of a great empire is an event and a kiss between two lovers is an event. For us they obviously have very different weights, but these (possibly quite sensible and sound) valuations are ours. Now, if there is a subjective element in our giving weights to events, and we usually associate an epoch’s end and the beginning of another to some event we deem crucial, it is obvious we choose these beginnings and ends, and so they depend on our choices—to the point that we can choose them in different ways. The most glaring case is that of Middle Age: among professional historians themselves, some associate its beginning with the shift of the Roman Empire from Paganism to Christianity and so take Constantin’s Edict of Milan in 313 A.D. as a boundary event; for others the boundary is the deposing of Romulus Augustolus, the last western emperor, in 476 A.D., and for a very famous one, Henri Pirenne, the Middle Age really began with the advent of Islam and more exactly with Muhammad’s “great flight” from Mecca to Medina in 622 A.D. (corresponding, of course, to year 1 of the Moslem calendar). All three schools of thought can invoke very important reasons to support their choice, but in the meantime we keep, unperturbed, talking about the Middle

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Age, with capital initials of course, although it is not clear at all what this period we talk about exactly is. But these things do not happen just with human history. Paleontologists, for example, claim there was a Cenozoic Era beginning with the extinction of dinosaurs, and in the fossil record there is indeed an abrupt change, occurring some sixty-five million years ago in a geologically negligible time and with a dramatic difference between before and after, for then a dominant phylum, that of dinosaurs, disappeared rapidly and completely. But the fact remains that the extinction of dinosaurs has been unanimously chosen by paleontologists, who are less litigious than political or cultural historians, as a First Class Turning Point in the history of planet Earth. And to confirm this change from Mesozoic to Cenozoic Age was a choice, I remind readers that dinosaurs’ is not the only mass extinction we know of, but simply the only one to which such a shift from a whole geological Era to another has been associated. Dinosaurs were extinguished, I repeat, some 65 million years ago, but today we know there was about half a billion years before, in the Paleozoic Era, an even bigger mass extinction that caused about 96% of existing species to perish, yet nobody has ever identified that catastrophe with the end of a geological Era and the beginning of another.

3.5 Short and Long Stories So we divide durations into parts or, if you prefer, stories into chapters, treat these sections or cuts as individuals, giving them names, and create in this way Eocene or Paleolithic or Classical Antiquity or the Golden Age of Latin literature or El Siglo de Oro of the Spanish one, where “create”—let this be said explicitly to avoid misunderstandings—means that as long as we do history, and not fiction, we group together and at the same time divide, select and at the same time omit real events as we decide, with the only constraint that no temporal gaps must be allowed. The durations we work upon are cumulative processes going on—as I have already said— without interruptions of their own, but the grouping together and subdividing is ours, and when seen from a distance the whole looks governed just by a before-or-after relation, understood as a mere being out of each other of events following each other in time. Of course, we look for causal links in this flow, we need that just as we need a partition into chapters to make sense of the whole, but this only happens on the basis of previously given before-and-after relations. However, in this monotonous before-and-after we also find episodes (typically brief ones) with a different and richer structure. Take for example the Great Depression, in itself not a brief affair at all, since it began in 1929 and lasted well into the Nineteen-Thirties. I am not going, for brevity and simplicity, to discuss its very important consequences in the rest of the world, and will only say something about its beginnings in the United States. It is well-known that the Great Depression was triggered by the so-called Black Wednesday panic in Wall Street, and that consisted of a very big number of individual sales following each other at a breathtaking speed

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and systematically, compulsively, catastrophically leading to a plummeting of prices. But such individual transactions were not, properly speaking, the simple elements of this sudden crisis starting the Great Depression, simply because no simple elements existed. For it is true Black Wednesday was in a sense the summation of many many sales in a short short time, but each of these, except possibly quite a few at the very beginning, was in turn the result of panic. Each individual sale was caused by panic, and panic was caused by a big number of individual sales. Typically, the way so-called complex phenomena function is this: the behavior of the whole is due to that of a huge number of components, but these can only do what they do within the whole. An army wins or loses a battle if and only if its individual soldiers’ behaviors sum up in a certain way, but a soldier fights the way he does thanks to the overall situation around him; and at another level, a macroscopic organism functions thanks to what its single cells do, but each cell can only live and function inside the organism. We have, to sum up, found “individuals” (so treats them everyday discourse) having time as their basic dimension, but also discovered that such individuals can be of two very different sorts: mere divisions of the flow of time we must introduce to make sense of it, and processes (usually of brief duration) caused by the action of many components whose workings are, in turn, only possible within a given overall situation—such tings as Middle Age or Eocene on one side, and a battle, a stock exchange panic or a football game on the other.

3.6 Individuality and Uniqueness This view of what individuals are in everyday discourse is certainly different from most traditional ones, where two conditions, indivisibility and an exact extent in space and time, usually are taken for granted. But it is not entirely unprecedented, for one could (maybe with a pinch of goodwill) see something similar to it in Quine’s claim that “being is being a value of a variable”; and in any case, Quine’s position seems to be compatible with this extremely liberal treatment of the notion of individual I have tried to sketch, though it does not seem to imply it (and something similar could be said about the later Wittgenstein). However, it is obvious at this point that there is an important difference between the view I am defending here and a more traditional one, seeing every individual as something unique. Popular as it is, the latter view is not easy to clarify. I will try to do so recalling, in a simplified way, a theme developed in Chap. 2 of Peter Strawson’s Individuals, “Sounds”. After discussing the concept of physical individual in Chap. 1, “Bodies”, Strawson imagines a world entirely made of sounds. In such a universe the normal individuating tool of our bodily world, space-time position, fails, for its spatial component is no more present. We live in a world where, given two qualitatively and quantitatively indistinguishable bodies, with the same length and breadth and width and shape and color and smell, etc., these must, if they really are two,

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occupy instant by instant different positions in space, and only in different times can they be in the same place. Bodies’ principium individuationis is position in space and time. Here of course Strawson echoes a traditional idea, going back, as we have seen in §3.3, at least to Aquinas. But in a world entirely made of sounds space does not exist. It is true that for us every sound has a direction (though not so precise as that associated with vision), for it is in front of us or behind us, to our right or to our left, above or below us; anyway, it has such a direction, says Strawson, owing to cenesthesy, i.e., to one’s general perception of one’s body, of its moving or not, of what its various muscles are doing. We cannot perceive this direction with hearing alone, so in an universe entirely made of sounds it is not there; space is not there. Here in fact Strawson makes a mistake, for we have two ears, with two drums and labyrinths, each functioning on its own. Coordinating the stimuli from both systems is up to auditive brain cortex, just as coordinating those coming from both eyes is up to visual cortex, and it is such coordination that gives sounds their direction, so the latter is also due to hearing as such and not just to cenestesy. But I choose to sweep such an objection aside and concentrate instead on this extremely interesting (although Strawson does not discuss it in a flawless way) idea of a spaceless universe of sounds. In a universe of bodies two distinct things cannot be in the same place at the same time, so it is space-time position that distinguishes two otherwise identical objects. But in Strawson’s all-sound universe there is no space, so how can one distinguish two qualitatively identical occurrences of the same sound—e.g., the same melody? Consider such a melody. It has precise notes, each with a well-defined pitch and intensity (ranging from pianissimo to fortissimo) and duration, and in the case of polyphony it also has well-defined harmonies. Let us call M this melody. In the world of sounds nothing forbids M to occur over and over again: how can we tell whether any two of these occurrences really are the same individual or not? Here Strawson invents a very remarkable bravura exercise: he imagines that over and above ordinary sounds, that always have a beginning and an end, there is a master sound never ceasing or pausing, but whose pitch can change. Now, if they are accompanied by different pitches of the master sound two of M’s occurrences are distinct individuals, while they are the same one occurring twice if such pitch is both times the same. This is extremely ingenious, and shows how a principium individuationis can also be invented, if one has fantasy enough, when space-temporal position is out of question.

3.7 Individuality and Substantiality But we also have another important problem to take into account: is an individual always a substance? Thus stated, the question has all the vagueness of a tradition centering around the word “substance” for over two thousand years, but also taking all possible liberties with it, and stretching and twisting it in countless ways. However,

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since Aristotle’s times many have tried to make this notion more precise saying that if something is a substance, then its being does not consist in inhering in something else as an attribute; and if we accept this, as it is usually done, then the correct answer to the question opening this section is “No”. The most interesting example I know of an individual being no substance is that, already discussed in §3.2, of a rainbow, obviously in a certain place at a certain time. From a naïve viewpoint a rainbow is as much of a “thing” as a tree or a stone, though the impossibility of finding a place where it ends is perplexing anyway, even at this unsophisticated level. But today a not exactly modern but highly plausible theory (we owe it to Witelo, a Saxon thirteenth-century monk) explains that rainbows are caused by a vast number of very small spherical water drops reflecting and refracting sunlight. These drops alone are substances; a rainbow is just something they do in the presence of sunlight. The existence of different historical layers in our knowledge has to do with this, of course. Humans believe certain things at a naïve, more ancient level and certain others at a more sophisticated and recent one. But for all the differences among distinct layers of our knowledge, the examples of individuals being no substances abound. I will make a few more. Take, to begin with, waterfalls; some, e.g., Niagara Falls, are famous and visited by many tourists. Now, waterfalls are things that can have a proper name, and we know what follows from this; but a waterfall, as the word says, is nothing but a movement of something else, water. Another example of something being nothing but a movement of something else, water again, are waves. Think of such a sentence as 3.7.1 The wave was enormous, it scared me, or imagine to see a real or a painted wave (I have Hokusai’s masterpiece in mind). So the elements of a class being no class of substances can also be individuals. Another example of a non-substantial individual is 3.7.2 That scirocco wind knocked me out with its burning heat. In Italy we call scirocco a very hot wind coming straight from Sahara. Now, 3.7.2 is about that scirocco wind, that individual case of a certain sort of air movement. My last example will be about hurricanes. A hurricane is a violent movement of a portion of the atmosphere, it inheres in that portion of the atmosphere, so it is no substance; but these non-substantial things are given names. Americans have begun, giving them feminine names in odd and masculine ones in even years, then others too have followed suit. Since accurate meteo records are taken, the most terrible of these non-substantial things was Katrina, that half-destroyed New Orleans and killed some 1800 people in 2005. At this point, the lesson is obvious: an individual is anything we give a name to, i.e., anything we treat as such. Trying to define the notion of individual ontologically is ill advised. It is much wiser to treat it as a linguistic concept. Of course, this conclusion is not entirely new: we can find it (or something very like it) both in Quine and the later Wittgenstein. But maybe there is something new in the way it is reached here.

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3.8 Nelson Goodman’s Strange Predicates However, I must warn readers against a possible confusion: saying that anything we treat as such is an individual is not equivalent to saying we can treat anything whatever as an individual. I will try to explain this point via a rather long detour, whose initial step will be a rehearsal of a famous paradox on predicates (not on individuals and proper names) by Nelson Goodman.9 Using the adjectives “blue” and “green” we can, Goodman says, define these other two color predicates: 3.8.1 bleen = seen for the first time within day X and blue, or after day X and green. 3.8.2 grue = seen for the first time within day X and green, or after day X and blue. As Goodman knows perfectly well, the “naïve” reaction to these two concepts of bleen and grue is that they are put together artificially, combining two already given and simpler notions, blue and green. But, he says, simplicity (associated, here, to naturalness) and complexity (associated to artificiality) have nothing to do with this, for there is in fact a complete symmetry. We can, namely, define “green” and “blue” in terms of “bleen” and “grue” in the following way: 3.8.3 green = seen for the first time within day X and grue, or after day X and bleen. 3.8.4 blue = seen for the first time within day X and bleen, or after day X and grue. (The reader should check personally whether both definition couples function equally well and with a really complete symmetry). According to Goodman this similarity shows there is, in principle, no reason to conclude “blue” and “green” must be preferred, logically, to “grue” and “bleen”. It is true we have in fact a preference and that preference is the same for all, but this is a merely historical fact and has nothing to do with a putative conceptual superiority of “blue” and “green” over “grue” and “bleen”. No such superiority exists, but it simply happens that the couple “blue-green” is well entrenched in use while “grue-bleen” is not, and this is just a matter of habit; not something conceptual, but something merely factual. Goodman was a logical empiricist, and it is natural for me to compare twentiethcentury logical empiricists to ancient sophists (this is not disparaging: I very highly admire ancient sophists, and consider not simply delightful, but profound such arguments as the heap, the cart or the person with horns).10 Logical empiricists too (not 9 This paradox has been discussed a lot of times by a lot of people, often in a way far more nuanced

and complex than mine in this section. But I am not aware of any counterexample similar to the one I am discussing here. 10 The heap: You can never put a heap of corn seeds together, for one seed does not make a heap, and if n seeds do not, neither do n + 1. The cart: What you say gets out of your mouth. You said “Cart”, so a cart got out of your mouth. The person with horns: If you haven’t lost something, you have it. You haven’t lost horns, so you have them.

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just Goodman, but also e.g., Hempel) have invented paradoxes, proving in this way most useful to philosophy, for a paradox forces us to reflect and so leads us—if all goes well—to understand something more deeply. So I take, metaphorically, my hat off before this amusing invention of Goodman’s: but once this homage he fully deserves has been paid, we must understand where it is that his reasoning fails (for it obviously does), or his paradox will have been useless. It is the moment, then, to invent a small story. Phil is very fond of the Alps, and has his holydays there every summer. One year he visits the West Valley, a place with a beautiful lake surrounded by an equally beautiful meadow. Phil, who uses “grue” and “bleen” for colors, sees the West Lake is bleen and the meadow grue (day X is still to come; it will be December 31st of that year). Next year Phil takes his summer holidays in the East Valley, bordering with the West one. There, too, he finds a lake, very like the West one except that it is grue and the surrounding meadow is bleen. A mountain divides the valleys, and Phil climbs on top of it. From its summit he can see both lakes (and both meadows) together: they are just as big and so are the meadows, have the same shape and so have the meadows, but their colors are different. Or not? Anyway, he takes pictures of both, the grue lake with the bleen meadow and the bleen lake with the grue meadow, and when back at home looks at them, but can’t tell which is which, and wonders: “How is this possible, if their colors are different?” But there is a sequel to the story. Nora is Phil’s girlfriend. They are madly in love and do everything together, but Nora uses traditional color words, “blue” and “green”. She, too, can’t tell which is the picture of the West Lake and which that of the East Lake, but does not find it so strange, for both lakes are blue and both meadows green. The conclusion to be drawn from this story ought to be clear: although we can only answer the questions (a) “What color is this?”, and (b) “What’s the name of this color?” with one word, and that word is the same in both cases, (a) and (b) are not equivalent and a thought experiment sweeping their supposed equivalence away can be invented, pace positivists such as Goodman, so hopelessly incapable of accepting a world of meanings not reducible to our flatus vocis. But now, after predicates, let us repeat the same trick with names (Goodman never did it of course; he was too much of a nominalist—I mean, he respected proper names too much—for that). I will start from Jack and Jill, who have nothing unusual about them, and then invent two new “individuals”, Jick and Jall, in the following way: 3.8.5 Jick is Jack’s right-hand half plus Jill’s left-hand half. 3.8.6 Jall is Jack’s left-hand half plus Jill’s right-hand half. What I have in mind is no prodigious surgery of an imaginary, faraway future, one capable of cutting and reassembling living bodies in today impossible ways, and even of putting together two new ones, each consisting of half a man and half a woman; I need not imagine anything like that to invent Jick and Jall. Grouping together in thought things already existing physically in a new (not entrenched, Goodman would

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say) way is enough. So I am not disturbed at all, if a male body people call “Jack” and a female one people call “Jill” keep going around, and no half-male half-female one does; I simply individuate things in the world in a not entrenched way. Why should this not be legitimate? There is no spatial continuity between Jick’s right half and his (or her?) left one, but we already know (see §3.2) that spatially discontinuous individuals can be accepted as existing. We have, moreover, a perfect Goodmanian symmetry between Jack and Jill on one side and Jick and Jall on the other, for just as we have defined the latter by means of the former, so we can do the reverse: 3.8.7 Jack is Jick’s right-hand half plus Jall’s left-hand half. 3.8.8 Jill is Jick’s left-hand half plus Jall’s right-hand half. This certainly sounds bizarre, but it is here we unavoidably arrive at if we put no constraint whatever on the notion of individual, and none seemed to exist so far: neither continuity in space or time, nor being a substance, nor even having some sort of being, since we can talk of such things as the smallest positive rational number or the eighth king of Rome with a perfect respect for grammar. There seems to be no property P distinct from being an individual but that every individual must possess to be one; on its own, being an individual seems to be not a (non-linguistic) property, but simply the fact of being denoted by a proper name, and so the meaning of the subject of a singular proposition. But maybe we have gone too far. Let us reflect: the statement (that I keep considering correct). 3.8.9 There is no neither-strictly-contradictory-nor-strictly-tautological property distinct from being the meaning of a proper name that every individual must possess simply as such. does not entail this other one, 3.8.10 Given any neither-strictly-contradictory-nor-strictly-tautological combination of properties with the grammatical form of a proper name, its meaning is an individual.11 In fact, if we formalize 3.8.9 and 3.8.10, which can only be done in a second-order language, we get, respectively, 3.8.11 ∀P∃ × (¬Px) and 3.8.12 ∀P∃ × (Px), that are certainly not equivalent.12 11 I

say “meaning” and not “denotatum” to circumvent Meinongian troubles. course, I am supposing P is as required in clauses 3.8.9 and 3.8.10.

12 Of

3.9 Non-Individuals, at Last

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3.9 Non-individuals, at Last But once we have taken stock of all that has been said so far, can we tell whether there are or not expressions with the grammatical form of a proper name (in the wider sense), but that cannot be reasonably considered as names of anything? I have a caveat here, namely, that I am venturing on a ground far more tricky and poor in certitudes than most traditional themes of logic, but once this caveat is explicitly stated, my answer is “Yes, there are some”. The reasons of this “Yes” do not amount to a rigorous proof but look nevertheless plausible, if only because a “No” would force us to accept really strange things. But here we have reached the level of basic metaphysical options, and some of the things I am going to say now will certainly not persuade those with basic metaphysical options different from mine. However, maybe some others can be accepted by everybody, or at least by many. I begin with two questions. The first is: is the existence of a name a necessary condition for the existence of the thing it names? It is not. In Antarctica the Russians have discovered a very big lake, Lake Vostok,13 hidden under a layer of ice roughly 3,500 m thick. Now, lake Vostok obviously existed well before humans discovered and named it. The second question is: is the existence of a name a sufficient condition for the existence of the thing it names? Neither. In the second half of the nineteenth-century astronomers looked quite a long time for a planet nearer to the Sun than Mercury. They even gave it a name, Vulcan; finding Vulcan would have explained some anomalies in Mercury’s motion the age’s physics could not account for. But Vulcan has never been found, and today the conviction prevails that it does not exist (by the way, what was formerly seen as an anomaly in Mercury’s motion has been explained and, so to say, “normalized” by Einsteinian general relativity). Once we have these answers, to solve somehow the problem whether all neitherstrictly-contradictory-nor-strictly-tautological expressions that, given their grammatical form and position, could be proper names really are such, I find it useful, as a first step, to call “world” the whole of our experiences. (Of course the term “world”, being polysemic in the highest degree, also has other meanings, and the strongest one is notoriously that of sum total of things present in time and space; but at the moment I am not interested in discussing this totality.) We introduce divisions and classifications into the world, thus understood; we have to, or we would understand nothing about it, would not even know which things to look for and which to avoid, and so would not survive long, either as single humans or as humankind. Now, for other animals this re-ordering of experience, indispensable to survive, grow up and reproduce, may well be an individual undertaking, but for homo sapiens it is not. For homo sapiens it is a shared one. It is not simply that we live in societies, but also that a human society could not last long if each of its members had ideas 13 It

has a surface of around 15,000 km2 . Vostok means East in Russian, but I don’t know why they have called it like that.

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entirely of one’s own, independent of those of others and utterly unrelated to them, about the things one must avoid and those one must look for—and before that, about the things that are there, for to be good or bad a thing must first of all be there. And in our species, the tool for building an idea of the world (i.e., of which things are there and which are not) is language. But a word of warning is in order here: I am not saying that the world in the strong sense, i.e., as cosmos, is of our making, or that it does not exist beyond its social production as the overall system of experience. This nonsense has been and still is around, but at the price of a total contempt for consistency and intelligent reasoning; and the dominant idea of the world as cosmos has always been, since the most ancient times we know of, an idea of something far bigger than human experience. But the fact remains that we, not as single persons but as a society, divide things (understood as objects we think of) into real and unreal; and of course, most things begin and finish on their own, without waiting for our descriptions, but in the meantime we, to understand our environment and survive in it, must form ideas about them. And these are social ideas. Now, having an idea of the world does not mean having first of all a cosmology. Of course, every culture has one and none can afford to have none, but every cosmology presupposes a system of shared ideas about which things are there and which are not, and there can be no such idea without subjects and predicates, proper names and class names, and a vocabulary to denote things and attribute properties to them, plus a grammar to construct phrases linking the first to the second. Here very different solutions are notoriously possible, but there is no unlimited liberty, for a necessary constraint holds anyway: language must be an efficient tool for the organization and survival of society. We can easily imagine (happily, de facto impossible) situations where it would not be one: for example, if a nation living in a very cold climate had no words to say “ice” and “snow”, or one surrounded by vast swamps had none to say “gnat”, or one of seamen and fishermen had none to say “wave”. And I don’t see how a culture capable of building shelters of any sort—houses, tents, igloos or whatever—could lack words naming them; or—last example—how people cooking at least some of the food they eat, as all peoples do (except—so they say—traditional Inuits, if there still are any) could lack words to distinguish cooked from uncooked food. It seems reasonable, then, to conclude that in creating, as it certainly does, its own vocabulary a culture does not enjoy an unlimited liberty, but is strongly conditioned by its environment. I can recall some very well-known examples concerning predicates, but we will see that there can be a counterpart for them among names too: (1)

Arabs have different words to denote camels, respectively, of one, two, three, and four or more years, and for each age one word for the male and one for the female—so eight words in all; a Westerner feels okay for being simply able to say “camel” (with local variants in orthography and pronunciation, of course). But the Arabs’ ability to draw finer distinctions Westerners ignore obviously has to do with the great importance camels had in their traditional economy (though not, or not so much, in today’s).

3.9 Non-Individuals, at Last

(2)

(3) (4)

37

Fireland is very windy, and I have read somewhere that its natives, Fuegians, who today exist no more, having been exterminated partly by us Europeans and partly by measles (that we brought to them), had some thirty different words to say “wind”. Inuits have different words for fresh-water and salt-water ice. Peoples living in lands where the sea never freezes do not make this distinction. Still Inuits have fourteen different words to say “snow”, while usually peoples living farther south merely distinguish snow from hail; and in the extreme South of Europe, where snow is rare, Sicilian peasants (not educated Sicilians) call “snow” hail too. But I have come to know something even more interesting: once, while I was lecturing, a Kikuyu student from Kenya (a very good-looking young man who was a Catholic priest, and had come to Italy to study theology) told me that in his native language the same word meant “snow” and “God”. Kikuyus live at the feet of Kilimangiaro, and it never snows in their homeland; but when they look up, to the almost six thousand meters of the mountain’s peak, they see this very white, very bright thing, high up in the sky. No wonder it is God for them.

It is not strange for your language to be adequate to your world. If it were not, that would literally be a tragedy. I will invent a few (happily imaginary) examples of inadequateness: “A perfectly white, soft, cold thing was falling from the sky”. Isn’t it better to say snow? “Today I will try and catch some of those oblong things moving in water”. Isn’t it better to say fishes? “if you go down the whole slope, at the bottom you will find the moving water. It always goes the same way, is much longer than broad and one can see neither whence it comes nor where it goes” Isn’t it better to say river? “Although it was daytime we were in the shade, for the sun was covered by a partly white, partly grey thing moving unceasingly”. Isn’t it better to say cloud? “Those two parts of mine that when I close them I see nothing more are aching”. Isn’t it better to say eyes? And now, after predicates, proper names: “Do you know the story of that wooden lad whose nose would grow longer every time he lied?” Better say Pinocchio. “This year I want to go to that place near London where they have a famous tennis tournament”. Better say Wimbledon. “The man who created relativity theory…” Better say Einstein. “The dancer the French executed in 1917 because she was a German spy…”. Better say Mata Hari. But we also need a few examples of the opposite sort, i.e., not of how we would fare if we lacked certain terms, but of terms that don’t get used because they are of no use and would only cause confusion. Predicates come first:

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“Why doesn’t the Central Statistical Institute give data about the average hage (height plus age) and the average wincome (weight plus income) of citizens?” “This is crazy! I submitted a research project about the sexual behavior of alliphin, the cross-breed of alligator and dolphin, and it didn’t get funded!” And now proper names. Imagine this mad dialogue: “Yesterday Jack and Jill went to the movies together” “Just they two?” “No of course. They were four” “And who were the other two?” “Jick and Jall, obviously. But they paid just two tickets, since the four of them only took two seats”. We use both class terms and proper names (always in Frege’s broader sense) because they allow us to squeeze vast amounts of information into reasonably brief expressions. We must give proper names to single people, animals, towns, and so on simply to have ourselves understood in a reasonably short time. Just try and speak about the Russian Revolution without ever saying “Bolsheviks”, “Lenin”, “Stalin”, “Trotzky”, etc., or about Italian opera without ever saying “Rossini”, “Donizetti”, “Verdi”, “Puccini”, “Traviata”, etc.! And concerning specifically individuals, de facto we can only speak about one if a reasonably brief term denoting it (or him, or her) is available to us—but we only use a term if it is of some use to us. This, and this alone, is the standard by which to decide both what “individual” means and which individuals are there at the level of everyday discourse: something is an individual if and only if its (proper) name gets used, and a name gets used if and only if using it is to the advantage of our idea of the world, or at least of a part of it.14 This is, in a way, a very Wittgensteinian conclusion (in the later Wittgenstein’s spirit, of course)15 : being an individual is having a proper name, and being a proper name is being used as one in a reasonably effective and socially acceptable way. The difference between Jack and Jill on one side, and Jick and Jall on the other, boils down to this. But some very important, very general conclusions still have to be drawn from the things that have just been said. I will do it in the next three sections.

3.10 An Open-Ended Universe? Both names and predicates emerge as components of our construction of a worldview—in us humans a social, not an individual pursuit. Now, this social pursuit is open-ended: it never comes to a halt, never stops introducing new names and predicates and deleting old ones. Spoken languages evolve, and it isn’t just grammar 14 Of course, this does not mean that all our ideas about which things exist in the world and which do

not are right. But that does not refute the theory I am putting forward here about which expressions can be called proper names, or names of individuals, in everyday discourse. 15 See §4.2 for something more detailed about Wittgenstein.

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that does—it is vocabulary too; and concerning more specifically names, I believe treating their domain as a set in the classical, Cantorian sense of this word, i.e., as an entirely given, unchanging manifold, is a mistake. In daily used, real-life language games new names are constantly introduced, and not simply because new humans (as well as other living beings) constantly keep being born, but for countless other reasons too. To make just one very brief example: in many countries there have been, or still are, political parties called Christian Democracy, but for all I know none of these existed in 1900 A.D. Now, the theory we find in standard introductions to formal logic portrays something very different, for there we always read, with unessential notational variations, that propositions’ meaning and truth value have to be defined with respect to an entirely given, closed domain—the set of individuals, also called “universe of discourse”. Take the simplest possible case, that of an atomic first-order formula Px i with a one-place predicate P: in standard treatments, to find its truth value the first step is choosing a closed, entirely given domain D. Then, to see whether Px i is true in D we must (1) associate to xi exactly one element, say d i , of D; (2) associate to the predicate P a subset, say DP , of D; (3) check whether d i is in Dp . If it is, then Px i is true in D; otherwise it is not. With more complex formulas the procedure is more complicated technically, i.e., longer to state and more tiring to follow, but the basic ideas are the same: individual terms must be associated to elements of an entirely given, changeless domain, predicates to sets built out of this domain, and if a formula has in the domain a counterpart with exactly the same structure, though typically with different basic constituents, then it is true there; otherwise it is false. This closed, unevolving semantics is standard in today’s formal logic—I mean, not just in handbooks but in research papers too; moreover, it can have quite extreme developments. To make just one by now classical example: in his Meaning and Necessity (1956) Rudolf Carnap introduces and massively uses the notion of state description. A state description is a set containing, for every atomic formula P, either P or its negation ¬P; in a standard formalized language, this means that a state description is an assignment of truth values to all atomic formulas, and as a consequence to all formulas, atomic or not. But to achieve this result, the domain of individuals must be entirely given and unchanging. Carnap had respectable motivations we need not go into here, but the theory resulting from his notion of state description is incompatible with the one I am putting forward in this work, for in §3.9 it has been shown—I hope—very clearly that an essential requisite of “the world”, as I characterize it there, is that there be in it an open-ended, indefinitely renewable domain of individuals. However, Carnap’s theory has also been attacked by at least one author working in formal logic, Jaakko Hintikka. In his Logic, Language Games, and Information (OUP, 1973) Hintikka challenges the notion of state description; his motive for doing so is that involving, as they do, all individual variables and constants, state descriptions make one of deductive logic’s most important moves, elimination of the existential quantifier, useless.16 Therefore, Hintikka substitutes model sets to state descriptions. 16 See

infra, still in this section.

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A model set is a set of statements such that for every conjunction P&Q, if P&Q is in it both conjuncts P and Q also are, and for every disjunction PvQ, if PvQ is in it, then at least one of its disjuncts, P and Q, also is. According to Hintikka, the main defect of Carnap’s theory in Naming and Necessity is that a knowledge of all properties of all individuals is invoked there as an ideal benchmark by reference to which actual knowledge has to be assessed. More exactly, since a state description is a set containing as an element either P or ¬P for every atomic formula P, and uniquely determining, as a consequence, the truth value of any formula, atomic or not, it also is a maximal consistent set, i.e., one getting inconsistent any time we add to it any new literal17 ; and it is, further, a complete description of its universe of discourse, for in (classical) elementary formal systems once the truth values of atomic formulas are given, those of complex formulas also are as a consequence. Carnap’s ideal, then, is that of a knowledge that is completely determined, once and for all, if a certain basic layer is. This layer may well be a set of infinitely many formulas, but it is anyway exactly what we need to know all that can be known.18 Now it isn’t just, says Hintikka, that in non-trivial cases this benchmark, being infinitely complex, is de facto unattainable; there is something deeper, namely, that logic loses much of its power if it has to work in a domain where no new individuals can be introduced—as it is the case with state descriptions, since for all individuals x and all properties P a state description determines whether x has P or not. Here Hintikka has in mind an inference rule he—quite rightly in my opinion— makes very much of, and often refers to in all of Logic, Language Games, and Information: the so-called “E-elimination” (where E is the existential quantifier), according to which from an existentially quantified formula ExPx we can deduce Py provided y is a “new” variable, not used in previous deductive steps. Examples abound, both in ancient and modern times, and it amuses me reconstructing at this point the oldest and most famous of all, the proof by Hippasus of Metapontus, around 480 B.C., that there can be no rational square root of 2. I am giving it in a slightly modernized language, but the reasoning is exactly Hippasus’s: √ There can be no √two integers such that their quotient is 2. For suppose two integers a, b such that a/b= 2 exist; then b should be both odd and even, which is impossible.

PROOF To begin with we can suppose that a and b are mutually prime: for if they are not, we can divide both for all their √ common divisors. This given, we prove first that b is odd in the following way: if a/b= 2, then a2 /b2 = 2, and a2 = 2b2 , so a2 is even. But only even numbers have even squares, therefore a is even too; and b must be odd, for they are mutually prime. But now we prove b is even in the following way: if a is even, then there is a c such that a is 2c, so a2 is 4c2 and b2 is 2c2 , i.e. an even number, and as a consequence b also is. So we have a contradiction.

This theorem contains two steps going from a claim that there generically are things of a certain sort to something stronger, “Let so-and-so be such a thing (or such things)”, where “so and so” must be one or more new names, different from all those 17 A

literal is either an atomic formula, or the negation of one. view obviously has an ancestor in Wittgenstein’s Tractatus. See infra, §4.3.

18 This

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given to previously known things, because of that thing we call “so-and-so” we only know it is of a certain sort, and nothing else. Today we call this move E-elimination, or elimination of the existential quantifier,19 for in the context of modern formal logic it becomes a shift from an existentially quantified formula to one where the quantifier has disappeared and a bound variable has been substituted by a previously unused free one; but basically this move has always been there in deductive thinking, that without it would be crippled and unable to prove many of its results. Now, call it ekthesis as Aristotle would do, or E-elimination as we do today, the fact remains that in logic, once we know suppose that certain P’s exist, we can make a new assumption, “let so-and-so be such a P”, provided “so-and-so” is a new name, not associated to any previously known object. This is because of that new thing we are introducing we only know it is a P, and nothing else; but if we gave it the name of anything we already know we would say something not following from our premises about it, which good reasoning forbids. Aristotle knew it, and would use this move any time he needed it; modern logicians know it, and use—with a different notation—this move any time they need it. It is an extremely important move, without which a lot of theorems, in many branches of mathematics, could never be proved. Hintikka’s view, in its innovating aspects, is basically an illustration of the importance of this move, and it certainly represents a big progress in our view of logical inferences, for a consequence of his analysis is that in non-trivial cases proofs always involve the introduction of previously unknown individuals, a point that cannot be found either in the earlier Wittgenstein, or in Carnap, or in many other authors. Anyway, Hintikka’s innovation is merely logical, not ontological; in other words, it is perfectly compatible with a closed ontology such as that, e.g., of number theory or set theory,20 for its gist is that in infinite domains we can always introduce new names to complete our proofs, and should better do it any time we can, since without this move our conclusions would remain much poorer than they could otherwise be, but not that in proving theorems in formal systems we must admit the possibility that new individuals come to being. Nor could he have claimed this: for such a possibility does not make sense in disciplines dealing—as all branches of mathematics, including mathematical logic, do—with timeless entities. All this can be summarized in two very concise points: Hintikka advocates the introduction of previously unknown individuals in proofs as an essential tool for logic, but that does not mean he also advocates that of open-ended universes of discourse where new individuals can come to being. Anyway, he has good reasons not to do so: for in such a universe the standard, Tarskian semantics of elementary logic would be lost.

19 For 20 Cfr.

another, slightly more formal proof using it go back to Chap. 1, §1.2, p. 14. this book, §2.3, right at the beginning.

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3.11 Inventing New Individuals in an Open Universe In §3.10 I have made a detour through formal theories, but basically this book is a study of everyday discourse, and now I am back to my central subject. In this section my main question will be: “What’s the difference, if any, between the possibility of inventing new individuals in formal systems, as described e.g., by Hintikka, and that of doing so in everyday discourse?” My answer will be that in everyday discourse this invention of new individuals has a far wider scope, for it is not constrained by an already given, unchanging universe of discourse. But to argue this point I need first a long step back, that will bring me to some very general, very foundational things. To begin with, I am a realist. I believe many, many propositions (though not all) are true (or false) independently of anybody’s knowing they are.21 Proposition P is true if and only if it is the case that P, not if and only if somebody knows that P. Said in another way: when is P true? When it is the case that P. And when does somebody know that P? When (i) that person believes that P, and (ii) it is the case that P, where (i) and (ii) are mutually independent. Of course, there is nothing new in this position of mine. So, given this definition, if there is knowledge there also is something else in virtue of which it is knowledge. But between knowledge, understood as a state of a thinking being, and this “something else” we might also call the known thing (here I am going to repeat something I have already said, but I do it because I believe it is useful), there must be a relation that can be called in various ways—e.g., centuries ago it was often named concordantia, similitudo and the like, while more recently such terms as agreement, similarity, etc. have been and still are used. In most of these terms, ancient and modern, the underlying idea is that the knowing (which, remember, according to a very old and well-entrenched definition is true belief) and the known thing are somehow alike. Now, this idea is also present, and in a particularly strong version, in the semantics of formal logic. Given the simplest possible case, that of an atomic first-order formula Pxi with a one-place predicate, take a domain D. According to orthodox treatments, to see whether Pxi is true in D we must (1) associate to x i exactly one element, say d i , of D; (2) associate to the predicate P a subset, say DP , of D; (3) check whether d i is in Dp . If it is, then Pxi is true in D; otherwise it is false in it.22 With more complex formulas the procedure is more complicated technically, i.e., longer to state and more tiring to follow, but the basic ideas are the same: individual terms must be associated to elements of a domain, predicates to more or less complex sets built out of this domain, and if a formula has in the domain a counterpart with exactly the same structure, though with different basic constituents, then it is true; otherwise it is false. More specifically, if an n-place atomic formula Px 1 , …, x n is true, then the meaning of its “subject” x 1 , …, x n is an ordered n-tuple of elements of the domain 21 This also is Bernard Bolzano’s view. See especially his Fundamentallehre (§§ 1–45 of Wissenschaftslehre). 22 Of course, to be exact this is not unqualified truth, but truth under an interpretation. But when it is not strictly required, complete rigor is intolerably tedious.

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D, so that, if an atomic formula is true in D, then in a sense its structure is exactly replicated in D. According to this view, then, knowledge and the known have (typically) disjoint basic constituents but (always) the same structure, and this is expressed saying they are isomorphic. Such is, for example, Wittgenstein’s position in the Tractatus: and this is, of course, an extremely strong variant of the centuries-old claim about a concordantia, or conformitas, or similitudo, between knowledge and the known. But for a non-naïve realist this claim is wrong, for from the assumption that proposition “P” is true if and only if, quite independently of anybody’s uttering or thinking it, it is the case that P, it does not follow, in everyday discourse, that between a true proposition and the state of things whose occurring it truly asserts there must be an isomorphism, or even some weaker sort of structural similarity. I will argue this point in two steps; the first one will be about representations23 and the second about propositions. (i) It was Bernard Bolzano who in his Wissenschaftslehre (1837) discovered an essential point from which all that has just been said follows. In §63 of that work, whose title is “Whether the parts of a representation are representations of the attributes of the object of that representation”, we read: It has often been said that a representation of an object…has a certain agreement with it. The obscurity of this expression caused some philosophers to think that this agreement between a representation and its object is some kind of similarity in the composition of the two. Thus they assumed that the constituents of a representation must be representations of the parts of its object…But this is not always the case…Consider the following representations: “a land without mountains”, “a book without engravings”, etc. In these cases the constituent representations “mountains” and “engravings” obviously do not indicate parts that the object has, but those that it lacks. The same can be shown even more conclusively with representations such as “the eye of the man”, “the gable of the house” etc. Who could deny that in the first of them the representation “man” and in the second the representation “house” occur as constituents?24

Two remarks are important at this point. (I) Here Bolzano is not discussing the meaning (or truth conditions) of propositions; nevertheless, he is doing semantics, for he is discussing certain terms and their meanings—and showing there are cases where no similarity at all exists between the former and the latter. This happens either where (a) “not P”, or something equivalent, occurs, for “P” is present there as a representation while no P is present in the represented object (take such examples as “a shirt without buttons” o “a shoe without laces”)25 or (b) a representation reverses a 23 I

borrow this notion of representation from Bolzano. For an illustration, see §4.2. Bolzano, The Theory of Science, edited by Paul Rusnock and Rolf George, OUP 2014, vol. I p. 194 (first published in German as Wissenschaftslehre in 1837). Anyway, my quotation is not entirely faithful, for wherever it has “representation” the editors’ translation of Bolzano has “idea”. I have made this substitution (and not just here but also in §4.2, where there are other quotations from The Theory of Science, or Wissenschaftslehre) because in my opinion representation is a more faithful translation than idea of the original German term, Vorstellung. Something more about Bolzano will be said in §4.2. 25 “Without” is an obviously negative term; “an x without y’s” is interchangeable with “an x that has no y’s”. 24 B.

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part-whole relation existing in its object (“the team’s best player”, “the monument’s pedestal”). (ii) He is discussing the (pretended) similarity between knowing and the known analytically, i.e., considering the linguistic expression of the first and not, as it was standard practice in his days, its “emerging to consciousness”—a hopelessly imprecise notion; but many acknowledge it is imprecise today, while in 1837, when Bolzano’s Wissenschaftslehre was first published, as far as I know, apart from that book’s author next to nobody did. So these few lines by Bolzano already are a semantical revolution on their own, for they show beyond all reasonable doubt that any general similarity, of whatever sort, between a representation and the object it represents is out of question. (ii) Now, this can easily be extended—though Bolzano did not do it—to whole propositions. Let us imagine this scenario. I am in the railway station, and have just checked that 3.11.1 The 6.35 p.m. train from Rome has arrived. How can I say so? Well, I have gone to Platform 6, for I knew the 6.35 p.m. from Rome was due there, and—there it is; and I know that train I am seeing is the 6.35 p.m. from Rome, for this is written, very big and well visible, at the platform’s near end. But suppose now I see something different, making me conclude that 3.11.2 The 6.35 p.m. train from Rome has not arrived. What can I have seen, to conclude that? Platform 6 again, with the same, well visible advice, but this time it is empty. Suppose, further, I go to the station and to platform 6 at 6.35 p.m. (for whatever reason) in two distinct days, and in one of these 3.11.1 is true, in the other 3.11.2 is. That truth, and that falsity, depend on how things are, not on any belief or uttering of mine; and of course I can claim this if and only if I have a realist theory of truth and falsehood—but if I follow that theory, I must conclude I can speak truly about something (in this case, a train) when it is not there. So the states of things justifying my uttering, respectively, 3.11.1 and 3.11.2 differ because something—a train—is present in the first and absent in the second, while the true propositions describing them do so for the opposite reason: one word, not, is there in 3.11.2 but not in 3.11.1.26 Where is then isomorphism, or even some weaker sort of similarity? Recapitulating, (1) there is no general similarity between a representation and the object it represents (this we owe to Bolzano), and (2) the same can be said of a true proposition and the state of things it truly describes. This goes above all for negative propositions, and the reason is that negation is an operation of the mind. It is not in things, though many philosophers have argued it was.27 26 Not all languages have a separate word for negation. In Japanese, for example, negation is an inflection that gets added to the verb. But this is no essential difference. 27 A funny thing is that one of them was Hegel, who from this viewpoint may well be labeled as a naïve realist.

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So, recapitulating, (a) we need a worldview, (b) we must be able to express this worldview verbally, and (c) our utterances about the world are true if and only if they describe the world or, more often, a (usually much smaller) part of it as it is, but this does not entail that our true propositions are isomorphic to the states of things they truly describe. They very often are not. And this decoupling between truth (or faithfulness of description) and isomorphism was discovered almost two hundred years ago by a gentleman, Bernard Bolzano, who under many respects was well ahead of most twentieth-century logicians, but in his lifetime was less successful than he deserved. Anyway, at the level of propositions this decoupling can also be considered in the light of Wittgenstein’s distinction, in Philosophische Untersuchungen, between “seeing” and “seeing that”. An image is formed on the retina, and then becomes vision thanks to a processing by the visual cortex: that’s seeing. But it can also happen that through vision someone understands something going beyond it thanks to other factors present in one’s mind, while someone else in whose mind there are no such factors doesn’t understand this “something beyond”. Suppose, for example, two people are looking at old photos, and suddenly one of them exclaims “Oh, that’s mother, as a girl!”. Both see the picture; but just one sees that it is a picture of somebody he knows well, his mother, taken long, long ago. Now, 3.11.1 and 3.11.2 are instances of seeing that; and even granting something that might well be too generous, namely, that there is some sort of similarity between the simple seeing and the seen thing, there certainly is none between P and seeing that P. And now the last step, to complete the description of how new individuals are continuously invented and spoken about in everyday discourse. This invention is so common, that trivial examples are easy to find. Suppose, then, that while walking—it does not matter when or where—a lady sees a wallet on the ground: if she is honest, she will bring it either to the police or to its owner (whose name and address can easily be discovered examining the wallet’s contents), if she is not she will keep it to herself, but in either case she mentally goes through the invention of two new individuals, the wallet (that she sees and touches, but understanding it is a wallet goes beyond this) and its owner (whose existence she infers). We couldn’t survive long without such inventions, and I am not even sure there are persons incapable of them. But if they exist, they are mentally impaired in an extremely serious way.

3.12 Metaphysical Consequencess (A) So we construct concepts and judgments out of experience, but these concepts and judgments are in no sense copies of the objects or states of things they represent. If I know what a cow is, this does not mean there is somewhere inside me anything

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resembling a cow—a venerable mistake we find in Egyptian papyruses of the Third Millennium B.C. as well as in contemporary authors, but a mistake anyway.28 However, the opposite view, namely, that no similarity between our thoughts and the objects these are thoughts of exists, has long been associated with scepticism; and the same can be said of an extreme variant of this view, namely, that, given a representation R, there is no object OR distinct from R, but such that R is its representation. The first to argue at length in favor of this view in modern times was David Hume (who accepted and even claimed the label of sceptical philosopher). In his Treatise of Human Nature (1739), e.g., we read: …since nothing is ever present to the mind but perceptions, and all ideas are deriv’d from something antecedently present to the mind; it follows, that ‘tis impossible for us so much as to conceive or form an idea of any thing specifically different from ideas and impressions. Let us fix our attention out of ourselves as much as possible: Let us chace our imagination to the heavens, or to the utmost limits of the universe; we never really advance a step beyond ourselves, nor can conceive any kind of existence, but those perceptions, which have appear’d in that narrow compass.29

Moreover, They are the successive perceptions only, that constitute the mind; nor have we the most distant notion of the place, where these scenes are represented, or of the materials, of which it is composed,30

which means—Hume concludes—that we can never perceive, and therefore know, anything but perceptions. Now, in fact there is a serious mistake here, for Hume makes no distinction between two very different claims: that (a), whatever X may be, one can think of X only if one has an idea (a representation, Bolzano would say) of X, so that (trivially), one can only think of something one is thinking of, and (b) that one can only think of one’s own thoughts. (a) is correct, but tautological, (b) is wrong: one can only think one’s own thoughts, not of one’s own thoughts. If a young man is thinking of his girlfriend, he is having thoughts about her, but not thinking of those thoughts: he is thinking of a girl, not of a thought. And a thought is never its own object: if it were, this would be a vicious circle. Another, very important point is that if we analyse ideas, i.e., mental representations, that can be either sensations (impressions, says Hume) or the results of our mental processing of sensations, according to Hume significant results come out. One of these is about identity: 28 The most extreme case I know of in modern times is that of poor Friedrich Engels (1820–1895), who in his Antidühring (1878) claimed that concepts, when correct, must be “mirror images” (Spiegelbilder) of outside objects. Engels was a pleasant writer and—judging from his works—a very nice man, but this Spiegelbilder idea is comically wrong. 29 David Hume, A Treatise of Human Nature, edited by L. A. Selby-Bigge, Clarendon Press, Oxford, 1987, 67–68. Italics mine. 30 Op. cit., 253.

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We have a distinct idea of an object, that remains invariable and uninterrupted thro’ a suppos’d variation of time; and this idea we call that of identity… We have also a distinct idea of several objects existing in succession…and this to an accurate view affords a perfect notion of diversity, as if there were no manner of relation among the objects. But tho’ these two ideas of identity, and a succession of related objects be in themselves perfectly distinct…yet ‘tis certain, that in our common way of thinking they are generally confounded with each other. That action of the imagination, by which we consider the uninterrupted and invariable object, and that by which we reflect on the succession of related objects, are almost the same to the feeling…This resemblance is the cause of the confusion and mistake, that makes us substitute the notion of identity, instead of that of related objects.31

So identity understood as uninterrupted existence through time, Hume concludes, is not in things; it is something we ascribe to things, but goes well beyond them— or, which is the same, beyond sense perception. There has been, then, a Western philosopher—and one of the greatest in the last five hundred years for his very rigor, his capability to go all the way to consequences, however upsetting they may be—who believed that ascribing reality to anything not reducible to a bundle of perceptions, and so to anything having a continued existence of its own over and abover any such bundle, was incorrect. (B) But the same view is present, though with very different motivations, in a tradition far removed from Western thought, that of Buddhist philosophy. For Buddhism life is dukkha, suffering; this can’t be avoided, and does not finish with death, for once the present incarnation is over a new one will take its place, and the mind will be caught again in samsara, the pointless, potentially infinite cycle of birth-death-rebirth, with all its unavoidable evil. But there is a way out: you can stop reincarnating and access nirvana, an eternal, blissful, purely mental state, if you reach enlightening, i.e., knowledge—very difficult but not impossible to conquer— of what all things ultimately are: and ultimately they are void.32 Take something entirely at random and call it X: now X, whatever it may be—a human being, an ant, a mountain, a snowflake—never exists on its own. It depends on “causes and conditions”, but these depend in turn on further causes and conditions, and so on to infinity—where “to infinity” implies that there must be no God, for if there were one it would be (as it is the case for many Western thinkers) the first cause giving being and meaning to all the rest—while there must be no ultimate meaning of things, for if there were one (distinct from void-ness), enlightening and ascent to Nirvana would be impossible. This I have described, not to be too long, in a very simplified form may well be called the Buddhist faith; but there have also been Buddhist philosophers, defending that faith (as Christian, Jewish or Moslem philosophers have done with their ones) with rational or would be rational arguments. I will briefly discuss one of these thinkers, Dharmakirti, who lived in India in the seventh-century A.D. But Dharmakirti 31 Op.

cit., 253–254. My italics. is not mere, ordinary intellectual knowledge; it is not simple assent to some verbal statement, such as “All.things are ultimately void of real existence”. It is a state where your mind sees this not through discursive reasoning but immediately, and with an absolute certainty and clearness. Moreover, once this state is achieved it cannot be lost, but lasts unaltered for ever and ever.

32 Enlightening

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is very difficult, very technical, and quoting him would be impractical, for a text of two lines might well require two full pages of explanation; so I will simply give a personal reconstruction. I know this is risky, and am well aware that the expertise of Buddhist specialists is lightyears above mine: but I hope there may be some not entirely wrong ideas in my reconstruction. At its most basic level all of our knowledge, says Dharmakirti, is instant knowledge, for it all derives from sensations, and there are sixty-four distinct ones in an eyeblink (according to all commentators, here “sixty-four” means “indefinitely many”). All ideas of anything having duration are secondary and derivative: when certain instantaneous sensations form a sequence whose elements are very like each other, and a greater and greater likeness corresponds to shorter and shorter time gaps, we impute the whole sequence to something having duration, e.g., a dog met during a walk (but of course “walk” too is a construction of ours going well beyond sensation, just like “dog”). Now, though both motivations and developments are different, there is in such a vision of human knowledge something important in common with Western empiricism: the basis of all knowledge, and at the same time its most genuine component, free from any secondary, subreptitiously added superstructures, is sensation. In itself, sensation has no duration at all (sixty-four distinct perceptions in an eyeblink), so it cannot be altered—consciousness has literally no time to alter it; but all the rest of our knowledge is cheating, for it contains superstructures foreign to primary sensation, and among these superstructures there are objects, understood as things with an independent existence (for example, this chair I am sitting on—or myself). Affinities with Western empiricism in its most rigorous form are obvious: there is an original, genuine, unaltered layer of knowledge and there are superstructures built upon this foundation but subreptitiously adding to it things that are not in it, objects: rabbits, leaves, fingers, books—generally speaking, items with a duration, that are not to be found in the primary, crucial layer whence all “things” come. But the sooner we understand that those we call “things” are not in that layer, the better. So Buddhist and Western empiricist philosophy, although their basic motivations are very different, have in common an exclusion from genuine knowledge of anything going beyond immediate, unprocessed perception. (C) Now, this exclusion, whatever its context may be, has consequences incompatible with commonsense. Buddhist philosophers knew it perfectly well, and developed a very sophisticated theory about the difference between ordinary and ultimate knowledge to deal with this problem—but Hume also knew it well, and toward the end of the first part of his Treatise of Human Nature he avows that They are the successive perceptions only, that constitute the mind,33 and consequently we must conclude that That action of the imagination, by which we consider the uninterrupted and invariable object, and that by which we reflect on the succession of related objects, are almost the same to the

33 A

Treatise of Human Nature cit., p. 253.

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feeling…this resemblance is the cause of the confusion and mistake, and makes us substitute the notion of identity, instead of that of related onjects.34

The lesson one can derive from this is that the structures of, respectively, raw, unprocessed sensation and knowledge proper are, at the level both of commonsense and top, highly sophisticated science, so different that the former can never grant any validity to the latter, unless…but before explaining what this “unless” amounts to, I will further illustrate the difference between these two layers with a fictitious dialogue between Adam and Eve—or rather two characters resembling them, but both with a (factually impossible) strictly empiricist worldview: EVE ADAM

EVE ADAM EVE

ADAM EVE ADAM

You are good at making love. So are you. But you know, in the dark time before this one I have made love with a girl exactly like you—same voice, same face, same boobs, same way of kissing and panting and moving—indistinguishable, two water drops. Gee, me too have been35 with one exactly like you in the last dark time! Was he good? Fantastic, just as good as you! Aw, I’m a lucky woman! Every time it gets dark a man, and they all are exactly like each other, and so good at making love! But, you know, it’s the same for me! Every time it gets dark, a girl, the new one always exactly like the last one, and they all are superb lovers! What a coincidence! How would you explain it? Aw, no headaches, please! Let’s just make love, and stop at that!

But let’s forget now about Buddhist philosophy and concentrate just on David Hume and the consequences of his radical scepticism. Kant was grateful to Hume for waking him up from his “dogmatical slumber”, that is, for showing him that many things he had long taken for granted—basically, the continued existence of objects we only perceive discontinuously, at longer or shorter time intervals—were not in experience itself. For example, many people would describe an important part of their lives as leaving home and reaching workplace in the morning and then leaving workplace and getting back home in the evening, where “home” and “workplace” are always the same, day after day; now, this sameness is something we add to perception, but nothing in perception itself. If we consider just perception and memory, Adam only perceives, and then remembers, that every night he makes love with a very pleasant woman, and all those women are extremely like each other; and Eve only perceives, and then remembers, that every night she makes love with a very pleasant man, and those men are extremely like each other. Now, identification, i.e., telling oneself “She (he) is always the same one” goes beyond perception and memory; nevertheless, we constantly identify distinct perceptions, ascribing thus to experience things that are not in it. And Hume had intelligence enough to discover this and courage enough (for a lot was required) to publish his discovery. 34 Op. 35 The

cit., 253–54. poor girl is not highly educated.

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But there also was somebody—Immanuel Kant—who realized Hume was right in claiming that we put into so-called factual knowledge things that are not in sense perception, but also understood, as nobody had done before him, that if we re-organize sense data, putting into this re-organization things that are not in them, then—if we do this in an appropriate way36 —our knowledge is exactly as it ought to be. Owing to his unfortunate tendency to obscure writing, he did not manage to express this basically correct insight in a flawless way: e.g., what does that famous claim that “the Intellect is Nature’s lawgiver” mean? This is unclear; anyway, it certainly can be—and has often been—understood as meaning that “the Intellect”, whatever it may be, really gives nature its laws. Now, I have nothing in common with this view; my point here is simply that in constructing knowledge out of sense perception we use (as Hume and Buddhist philosophers say) tools that are not in perception itself, but this (as they don’t say) does not imply human knowledge is essentially cheating. Two points (neither of which is paradoxical or unexpected; but neither, also, can be found in Kant) are very important for a correct understanding of this view: (a) that knowledge interacts with practice, and (b) that human knowledge, though having, thanks to language, important aspects that are exclusively of its own, also has much in common with animal knowledge. Concerning (a), Hume is right in saying that ascribing identity to the objects of distinct perceptions is going beyond those perceptions: but if we never did it, we could have no normal lives. Everyone with a family loses sight of one’s family members in the morning, going to work, and then sees them again, and recognizes them, in the evening. Moreover, any time you go to work, or back home, you recognize places; if you didn’t, this would mean you have some very serious brain trouble (there are cases of such troubles, usually in old age, and doctors unhesitatingly classify them as mental disorders). So being able to recognize objects is necessary for life; and this does not go just for us humans, but for animals too. Bees can go back to their bee-hive, ants to their anthive, birds to their nest, just as we can go back home. And as we can recognize certain specific humans we know well, so do animals, or at least those high enough on the ladder of evolution, with some ones of their own species. Some years ago, high up on a mountain (the height was roughly 1500 m) I met a big flock of sheep; they all were either adult females or small ones, about two months old (adult males are very aggressive, very difficult to control, and are kept apart, except in reproduction times). The small ones were allowed to stray, while their mothers fed on grass; but late in the afternoon both had to come together again, to give or suck milk, and then for night rest, and it was evident no mother would ever milk a lamb different from her own, no lamb would ever accept milk, except from one’s mother. So they were looking for each other, and finding each other, through voice: to my human ear, all the mothers’ deep voices were indistinguishable, and so were all the lambs’ ones, whose pitch was much higher, but both obviously had something very fine-tuned in their auditive cortex, for they all managed to find each other through their mutual calls. It was a beautiful show. 36 Very

difficult to describe, of course.

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Recognition is an essential tool for life. A lot of animals, including many that don’t even have a centralized brain, being invertebrates, are capable of it: and so are we. Moreover, this I am calling recognition hasn’t to do just with individuals, but with classes too (and in that case it may well be called classification). Bees recognize flowers, and never mistake leaves for flowers, or vice versa; if they did, their species would probably disappear. So recognition and classification already existed and were necessary on this planet long before language, and humans; now, I can’t imagine any other planet where some not-entirely-primitive-life exists, but recognition and classification don’t; so I conclude that Hume or Dharmakirti are right in claiming, or implying, that both processes go beyond sensation, but their other claim (that they are delusions) is wrong, for it takes knowledge for what it is not—mere contemplation, comparable to watching a show without being a character in it. Each of us is one, not merely watching, but acting, in the show of life; and he, or she, would be a seriously impaired person, doomed to quick disaster, without a capability of recognizing or classifying. To finish, I further illustrate this point with one last example. Finding an English translation has been difficult, but somehow I have managed. Long ago, there was a little boy approaching his second birthday. So he was, as all kids of that age, somewhere in the middle between speech and non-speech: he already knew a lot of words (and their number was growing daily), and was even beginning to construct some—well, not exactly phrases, but certainly word-sequences. Friends would ask his parents: “Does he say mama?” “He does”. “And papa?” “He does”. “And water?” “No”. “Funny. They usually say water, at his age”. Well, he was an intelligent little boy (and is an intelligent adult man now), but still couldn’t say “water”. Also, among his favorite words there was one, “tumber”, his parents couldn’t understand. He used it quite often. To make it short: he had a personal tumbler, of plastics so it wouldn’t be shattered to pieces by a fall, and with handles so grabbing it would not be too difficult for his small hands, and his parents very seldom left it empty, so that whenever he was thirsty, he would go to it and drink (and in those few cases he did find it empty, he was looking quite disappointed). That little boy had constructed the wrong object, a thirst-quenching solid individual, instead of a thirst-quenching liquid substance. Of course, his mistake was quickly corrected: but I believe we should watch very carefully how children learn to speak, in their second and third year. We would understand a lot more about us adults, and about how we get to know things.

Part II

The First Great Gap

Chapter 4

When Predicates Behave Badly

Abstract The main topics of this chapter are two: a discussion (at times very critical) of certain themes in Bolzano’s, Frege’s, and Wittgenstein’s work, and an analysis of some propositional attitudes. A new polarity correct–incorrect, distinct both from true–false, and from grammatical–ungrammatical, also begins to emerge.

4.1 Those Embarrassing Propositional Attitudes So in §3.9 an answer to the question “What is an individual?” has been given, after all: the notion of individual is not ontological but linguistic, the denotatum of any proper name is an individual, and nothing else is. But what is a proper name? The received view is that all and only constants saturating an argument place in a predicate are proper names, and nothing else is. Now, this view may be okay in many formal languages and theories, but runs into serious trouble with everyday discourse, where counterexamples abound to a basic tenet of contemporary logic, namely, that all and only individual terms, either constant or variable, can saturate predicates—except that unfortunately, in everyday discourse this is wrong. Let us see why. There are binary predicates—so-called propositional attitudes1 —that do not invariably give well-formed formulas whenever both their argument places are saturated by any individual terms, for one of these must indeed be an individual term, either constant or variable, but the other has to be a proposition. I am giving, as I so often do, a list of examples: 4.1.1 I concluded Magdalen had left. 4.1.2 Max feels sure one cannot trust Fiat-Chrysler shares. 4.1.3 I suppose Congress will pass this law in the end. 1 Analyses of propositional attitudes far more detailed and sophisticated than mine have been given,

but I am not interested in competing with them. My only aim is emphasizing certain aspects of the behavior of some at least of these attitudes from the viewpoint of (1) truth values and (2) conditions of well-formedness. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Rigamonti, Logic, Everyday Discourse, and Metaphysics, UNIPA Springer Series, https://doi.org/10.1007/978-3-030-74598-1_4

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4.1.4 My coach has said I am playing from the first minute next Sunday. 4.1.5 The witness maintains she is not the defendant’s lover. 4.1.6 He finally realized his wife had left him. “Concluding”, “being sure”, “supposing”, “saying”, “maintaining”, “realizing” are all two-place predicates giving rise to grammatical, i.e., well-formed, propositions if and only if their argument places are saturated by, respectively, an ordinary individual term and a proposition. Now, 4.1.1–4.1.6 are well-formed propositions, but (taking, as we always ought to do, actual use as a benchmark) none is among 4.1.7 I concluded Magdalen 4.1.8 I suppose Congress 4.1.9 Max is sure Fiat-Chrysler shares 4.1.10 My coach has said next Sunday 4.1.11 The witness maintains the defendant 4.1.12 He finally realized his wife. Take e.g., “Max is sure Fiat-Chrysler shares”: “being sure” is a two-place predicate, so it needs—says Fregean orthodoxy—two individual terms, “Max” is one— and here things are okay—“Fiat-Chrysler shares” is also one and…what’s gone wrong this time? So, as we can see, it simply is not true that whenever an n-place predicate is accompanied by n individual terms we have a well-formed formula; or at least, not in everyday discourse. Now, the rules for constructing proper names, i.e., expressions whose denotata are individuals, as they are given (in an entirely standard way) in §2.3 of Chap. 2, are the following: first there are simple proper names. They can be either constants or variables. In formal languages they are written as indexed letters, usually different for constants and variables. In everyday discourse variable simple proper names are pronouns, while constant ones are, in standard cases, exactly those words traditional grammar calls so, i.e., a far smaller domain than the one logic grants to the term; then there are complex proper names, that are put together saturating an n-place function with n already constructed proper names. The process can be repeated as many times as we like; it is recursive. Now, the denotatum of any expression built in this way invariably is one of those things I have discussed so long in Chap. III, individuals, and at this point we have a question to ask: given that a propositional attitude must be saturated (also) by a proposition, can a proposition be seen as a proper name of anything? Certainly not in the sense of §2.3, for a proposition is (in standard cases) true or false while a proper name in the sense described in §2.3 never is; but we would better refer to the classics here for, e.g., Bolzano, Frege and Wittgenstein discuss either this topic, or related ones. Anyway, first I will dispose (or try at least to do so) of a possible objection.

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4.2 Rescue Maneuvers The extreme plasticity of everyday discourse allows (maybe not in all cases, but certainly in many) rescue maneuvers whose aim is replacing objective or subjective propositions2 with syntagms being no propositions at all. Here, too, I try to make what I mean clearer through some examples. The sentence with an objective or subjective dependent clause is on the left of the arrow, the one replacing it with a non-propositional phrase on its right: 4.2.1 X believes (does not believe) God exists → X believes (does not believe) in God’s existence. 4.2.2 Marx believed capitalism would unavoidably collapse → Marx believed in the unavoidable collapse of capitalism 4.2.3 Euclid proves there are infinitely many prime numbers → Euclid proves the infinity of prime numbers. But other maneuvers seeming to take the sting away from propositional attitudes are also possible, e.g., 4.2.4 My uncle feels the government is not saying the truth → In my uncle’s opinion, the government is not saying the truth. In 4.2.4, right of the arrow “the government is not saying the truth” saturates no more a two-place predicate together with “my uncle”, but becomes an independent proposition the propositional operator “in my uncle’s opinion” acts upon.3 Anyway, these maneuvers aiming at rescuing the full validity in everyday discourse too of standard definitions (such as that in §2.2) of “well-formed formula” in modern logic have to face an important objection: they can at most present us with two (both grammatically correct) versions of the same statement one of which is compatible with that definition and so with today’s logical orthodoxy, but from the viewpoint of this orthodoxy the scandal remains alive even if we simply admit it is possible for some predicates to be saturated, fully respecting grammar, by propositions.

4.3 A Look at the Classics Basically, the problem here is that, with the only exception of conditionals, in today’s logic there is in fact no space for those propositions traditional grammars call dependent or subordinate. Conjunction and disjunction do not subordinate, but coordinate; 2 But

also other sorts of dependent propositions I will discuss later. are many things to say about the expressions one can find in a proposition over and above the predicate and the names saturating it. I will discuss this topic in Part III.

3 There

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negation, quantifiers and, going beyond elementary logic, modal and temporal operators simply modify one proposition; the only exception is the antecedent of implication, and that really isn’t much, compared with the vast variety of subordination forms existing in everyday discourse. Yet some at least of the classics were far more liberal in this, for they did not ignore subordinate propositions at all, and some of the things said by Bolzano, Frege, and Wittgenstein about them are worth recalling here. Bolzano’s position cannot be understood correctly without considering preliminarily what a representation (Vorstellung) is for him: …the combination of words “Caius has wisdom” expresses a complete proposition. The word “Caius” itself expresses something that can be part of a proposition, as we have seen, although it does not by itself form a proposition. This something I call a representation.4

A representation, then, always is a proper part (in the standard sense of smaller than the whole) of a proposition. Nevertheless, some representations can contain in turn another proposition as a proper part: …we should never think of representations in themselves as being propositions in themselves, but only as being actual or possible parts of such propositions. This does not mean that a representation cannot contain a whole proposition or even several of them as parts, for even complete propositions can be combined with certain other representations in such a way that the whole which is thus formed does not state anything unless further parts are added. Hence, such a whole cannot be called a proposition, but merely a representation. Thus, e.g., the words “God is almighty” express a whole proposition which recurs in the following combination of words, “knowledge of the truth that God is almighty”; but this new combination of words no longer expresses a complete proposition. However, a new proposition can be generated through a further addition, e.g., when we say”Knowledge of the truth that God is almighty can give us much consolation”. Consequently, what is expressed by the words “knowledge of the truth that God is almighty” alone, is a mere representation, although one that has a complete proposition as a component.5

In these passages (both belonging to Elementarlehre, the second of the four parts of the Theory of Science or Wissenschaftslehre), to explain his concept of representation Bolzano refers back to that of proposition in itself he had introduced in the first part of his work, Fundamentallehre or Theory of Fundamentals: I wish to show as clearly as possible what I mean by a proposition in itself. In order to accomplish this, I want to define first what I mean by a spoken proposition or a proposition which is expressed in words. With this name I wish to designate any utterance, if through it anything is stated or claimed; that is to say, whenever it has to be one of the two, either true or false…Given that it is understood what I mean by a spoken proposition, I should like to note that there also are propositions which are not presented in words but which somebody merely thinks, and these I call thought propositions. Obviously, in the expression “spoken proposition” I differentiate the proposition itself from its articulation. In the same way I differentiate a proposition from the thought of it in the expression “thought proposition”. I call 4 B. Bolzano, The Theory of Science, ed. cit., vol. I, §48, p. 158. Here the same caveat as in §3.9 is necessary: I am quoting from Rusnock’s and George’s translation, but substituting idea with representation. 5 Op. cit., vol. I, §49, p. 161.

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a proposition in itself that which one must necessarily associate with the word “proposition” if he wants to follow me in the above distinction.6

If we put these quotes together, two extremely important things come out. The first is that according to Bolzano there is no strictly one-way part-whole relation between representations and propositions: a representation always occurs, in fact, as a proper part of some proposition, but this does not make it impossible for a proposition to be a proper part of a representation. The second, that for all I know is stated here for the first time in the history of logic,7 is that a proposition that is true or false when uttered independently can also occur in contexts where its truth value either disappears or becomes irrelevant. If we put together the just quoted definition of proposition in itself in §19 of the Wissenschaftslehre and the (also just quoted) things Bolzano says in §49, this will be quite clear. Frege, too, says stimulating things about subordinate propositions in the second, usually undervalued part of Über Sinn und Bedeutung; but his remarks, interesting as they are, contain some mistakes. To begin with, he claims that a period presents itself, from a logical viewpoint, as a proposition and more exactly as the main proposition, which implies that the main proposition of a declarative period must be true or false; but this is not always correct. Take for example such a declarative period as 4.3.1 Looking at the horizon, the captain concluded a storm was approaching: now, 4.3.1 is true or false as a whole, but its main clause, “the captain concluded”, is neither, for it is too incomplete to have a truth value. What has a truth value, and the same one as the whole period, is “the captain concluded a storm was approaching”, but according to standard grammar textbooks this is not the main proposition alone. Anyway, his considerations bring Frege to conclude—making, as we will see, a mistake—that within the period they belong to subordinate propositions are neither true nor false. Now, many of these propositions would be declarative, if stated independently; and if a declarative proposition is characterized first of all—as a long, long tradition maintains—by the possession of a truth value, it follows, says Frege, that in a context where it is not independent, an expression that on its own would be a declarative proposition is no longer one but can be interpreted as a name—more exactly, as the proper name of that thought (Gedanke) to denote which it has been included in the period. Thus, in example 4.1.1 “a storm was approaching” would be, reasoning à la Frege, the name of a thought of the captain’s, and so something very different from the independent (and true or false) proposition “A storm is approaching”.8 Here Frege says something extremely interesting and something quite objectionable at the same time. We have already seen that the context can modify the logical nature of some simple and brief syntagms, such as those of the form determinate article + noun (“The oak is a big tree”—“The oak was hit by the thunder”), but here 6 Op.

cit., vol. I, §19, p. 58. Italics in the original.

7 I must add this clause “for all I know” because I don’t feel like excluding Bolzano had predecessors.

The history of logic is an immense domain I have only partially visited. 8 But things are more complicated. See §4.2.

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Frege claims that even a whole proposition is no more one in certain contexts. Is it really so? Can we say so, e.g., about such claims as “All knowledge originates with experience” and “Locke maintains that all knowledge originates with experience”? Things are really entangled here, and—to begin with—we can only disentangle them if we reflect on Frege’s theory of meaning as stated in the initial sections of Über Sinn und Bedeutung, so I must recall some very well-known things. I apologize for this, but cannot help it. Frege’s distinction between the sense of a name and its reference justifies what no philosopher had been able to justify for over two thousand years, identity judgments; for an identity judgment is either of the form a = a or of the form a = b, and in the first case it is void, in the second—that was the blunder nobody had ever been able to correct—it is false. For how can a = b be true, if a is different from b? Frege’s solution is well-known. Every name is a name of something; the thing it is a name of is its reference. But the way an object is given (die Art des Gegebenseins) by a name can vary, and the sense of a name, not to be confused with its reference, is exactly the way it gives its object. Take for example these two definite descriptions, “Italy’s capital town” and “Italy’s biggest town”: they are different, but both refer to the same object, Rome, and “Italy’s capital town is its biggest town” is a true and non-trivial identity judgment. Understanding the distinction between the reference of a proper name and its sense was, of course, a great achievement. But Frege also extended this distinction from names to whole propositions, and there things do not work so well. The reference of independent declarative propositions is discussed in §5 of Über Sinn und Bedeutung, a section beginning with these words: So far we have only discussed the sense and reference of those signs (words, expressions) we have called proper names. But now we face this new problem: what are the sense and reference of a whoie declarative proposition?9

Now, Frege seems to take for granted that his sense-reference dichotomy, so precious in connection with proper names, can usefully be extended to whole propositions, and in practice does not discuss at all whether this can be usefully done or not. In fact, just one page later we do find some sort of justification for this, but it is a very poor one, that simply begs the question: If we are concerned with the reference of some part of a proposition, this means we generally recognize and require one for the whole proposition too.10

Now, according to Frege the reference of a (declarative independent) proposition is its truth value, while its sense is the thought, or Gedanke, it expresses. He concludes then that

9 Frege,

Aritmetica e logica, edited by Corrado Mangione, Boringhieri, Turin, 1967, p. 383. See aslso footnote 17 on p. 17. 10 Op. cit., p. 384.

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Every declarative proposition (where, as we have seen, the important thing is the reference of its words) must be considered then as a proper name; and its reference—given that there is one—must be either the True or the False. Whoever states judgments, whoever holds something is true, even the sceptic himself, must—perhaps silently—acknowledge these two objects.11

This is, notoriously, one of Frege’s most controversial theses, and one I challenge too, though not for the reason positivists do. Positivists have a ban against abstract objects, while I have none, and it is no scandal to me if someone claims there are two things called, respectively, the True and the False. But the idea that both have infinitely many names, and—which is even stronger—every time we utter an independent declarative proposition we name one of them sounds bizarre, and dangerously near an identification of stating with naming. I don’t believe such confusions ought to be accepted in logic. There also is another difficulty. With names in the ordinary sense of the word, if one names something, one means it. One says for example “Madagascar”, and means Madagascar. This also goes for “the True” and “the False”: who says “the False”, means the False, and who says “the True” means the True. But if I, being (let us suppose) very ignorant, honestly claim “Madagascar is near the North Pole”, I mean it, for I believe Madagascar is near the North Pole; and if we follow Frege we must also admit I name the False—but do not mean it. I say something false, but what I name—not what I say—is not simply something false, it is the False—except that I, believing that Madagascar is near the North Pole, do not mean it. Naming something without meaning it—and not just in one, very special case but in a whole, very big class of cases, that of false utterances! I am afraid this really is too strange to be plausible. Now, this idea that truth value is the reference of independent declarative propositions is openly reasserted as a matter of course in Frege’s discussion of dependent ones. So we read, e.g., …Here the problem immediately arises, whether the reference of secondary propositions also is, as that of primary ones, a truth value.12

Now, there are cases where The secondary proposition’s reference cannot be a truth value, but must be a thought…this happens after such verbs as “to say”, “to hear”, “to be persuaded”, “to conclude”, and similar ones.13

But in fact things are not so simple, for there are cases where a secondary proposition’s truth value simply is not there, and others where it is there but has no relevance for the correctness of the main clause, but there also are ones where the secondary proposition must be true for the whole period to be stated correctly, cases where it must be false, cases where it can be either way and cases where the idea of giving it 11 Op.

cit., p. 385. My italics. cit., p. 387. 13 Op. cit., p. 388. 12 Op.

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a truth value does not even make sense. Things are enormously complicated here, as we shall see in the course of the very incomplete, but nevertheless quite long, study of subordinate propositions that is going to occupy most of this chapter. Lastly, I believe this also is the right place to say, as briefly as possible, something general about Wittgenstein; but first a preliminary remark is needed. There has been, there is, and there will probably be for a long time to come a sometimes heated discussion about the relation between the “earlier Wittgenstein”, i.e., the author of the Tractatus, and the “later” one, the author, above all, of Philosophische Untersuchungen. A majority of scholars believe there is a radical break—not to say a polar opposition—between them, but there are also some who see in Wittgenstein’s work a basic continuity, from beginning to end. I am somewhere in the middle: I believe the relation between the “earlier” and the “later” Wittgenstein is very complex, very dialectical, and has aspects both of continuity and (sometimes radical) change. More precisely, Wittgenstein always is faithful to himself, from first to last, in having language as his only real focus; he begins and finishes his career as a philosopher of language. But in the Tractatus he says, e.g., that “Something can happen or not happen, and all the rest stay equal”. (1.21) “An object is simple”. (2.02) “States of things are independent of each other”. (2.061) “A mark of an elementary proposition is that no elementary proposition can be in contradiction with it”. (4.211) “Even if the world is infinitely complex, so that every fact consists of infinitely many states of things and every state of things consists of infinitely many objects, there must be anyway objects and states of things”. (4.2211) “A proposition is a truth function of elementary propositions”. (5) “All propositions are the result of truth operations with elementary propositions”. (5.3)

The overall view emerging from these Tractatus theses is very clear: Wittgenstein takes the basic methodological choice of Fregean logic—namely, (1) conceiving every elementary proposition as a statement of an n-ary relation among n individuals and (2) combining or modifying elementary propositions just by means of extensional connectives and quantifiers—and adds to this the idea that as in the language of formal logic there are atomic individual terms, the letter-symbols, so there are in the world atomic, simple objects (2.02, 4.2211). Moreover, in the Tractatus this is not just a methodological choice about how to do logic, but a worldview (1.21, 2.02, 2.061). Now, the view of world and language I am putting forward in this book is polarly opposed to this one; but—this is important—not to the central ideas of Philosophische Untersuchungen. It will be enough to recall two themes of that work. (i) At its very beginning Wittgenstein quotes a passage from Book I of St. Augustin’s Confessions where the idea of a basically uniform one-one correspondence between words and objects is put forward, and criticizes it, showing with his famous example of “five red apples” that even if we grant there is a correspondence between every single word and something in the world (which is granting a lot), this correspondence can vary so much that the idea that all terms have in common some thing that can always be called their

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“reference” looks perfectly empty; and this—Wittgenstein adds—does not refute just St. Augustin but many others too, including Frege and “the author of the Tractatus”. (ii) Any interaction among humans including verbal (but usually also non-verbal) exchanges and working, normally, to the satisfaction of those involved in it is a legitimate language game. No conformity to other uniform a priori criteria is required for an (also) verbal interaction among humans to be one. (i)–(ii) are lightyears away from Tractatus 5, “A proposition is a truth function of elementary propositions”. In Philosophische Untersuchungen, every use of language making intersubjective agreement and cooperation possible is a legitimate language game; now this is an extremely liberal view, where there certainly is room for such ideas as those I am developing in this book.

4.4 Superman and Clark Kent As far as classics are concerned, what has been said in §4.3 can be enough for this book’s purposes, but before that discussion of dependent propositions I have promised in §4.1 can take off there is a difficulty to dispose of: “referential opacity”, to use Quine’s term, of proper names in contexts depending on a propositional attitude— in other words, the fact that in such contexts ascribing a well-defined reference to a name may be surprisingly difficult. Since Quine invented, back in 1960, his famous example about the Spartans, the Lacedaemonians and Jones, who knew that the former were warlike but not that the latter also were though they were exactly the same people, this has been endlessly discussed and rediscussed, and so many examples in this vein have been added to the original one, that commenting them all would be too long; but I believe that choosing one and analyzing it carefully can be enough, in order to achieve a reasonably good understanding of the problems behind referential opacity and their possible solution. The one I select is Kent Bach’s, and reads as follows: 4.4.1 Lois Lane believes that Clark Kent is a wimp. 4.4.2 Lois Lane believes that Superman is a wimp. According to commonsense, 4.4.1 is true and 4.4.2 false. Lois and Clark are colleagues in the same workplace, they meet there everyday, and he is extremely shy; while Superman is her hero, an invincible man, afraid of nothing. But the problem, of course, is that Clark and Superman are the same person, though Lois would never believe it. Now, how can she believe that the same person in and is not a wimp? According to Bach, answering “Because she does not know they are the same person” is simplistic and inadequate, because it ignores three principles that can never be given up: direct reference, compositionality, and semantic innocence. Direct reference is the principle that if a name occurs in a proposition, it contributes to the meaning of that proposition and its contribution is its referent; compositionality, the principle that the meaning of a composite expression is derived from the parts

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that make it up; semantic innocence, the assumption that embedding a term in a “that” clause, i.e. in one depending on a propositional attitude, must not change its “semantic value”. Accepting direct reference would be going well beyond Frege, but I will not elaborate on this because discussing compositionality and semantic innocence will be enough for my purposes. Now, compositionality is not there in everyday discourse, where sentences are not built assembling pieces that are already there on their own and do not become anything different when we put them together. Formal languages are like that, they join pieces according to certain rules and the joined pieces always remain what they were; but everyday discourse does not function that way, it is no Lego-game, it does not simply put some pre-existing things together, leaving them as they were, but the context reacts upon them, making their meanings different in different occurrences. Take these examples: 4.4.3 Music is all my life. 4.4.4 My favourite music is Verdi’s Requiem, or 4.4.5 He’s got a keen, brilliant mind. 4.4.6 If you smoke in my presence, I don’t mind. 4.4.7 Mind the step! The same word, music, means something extremely general, a whole art form, in 4.4.3, and something far more specific, a single musical work, in 4.4.4; and mind is a noun in 4.4.5 and two different verbs in 4.4.6 and 4.4.7 (I don’t mind = I am not disturbed, mind the… = pay attention to…). By the way, dictionary authors know this perfectly well: think of how many different meanings can be listed in an entry, e.g., of the Oxford Dictionary! So looking for compositionality in everyday discourse is looking for something that is not there. As for semantic innocence, what is “semantic value”, to begin with? Is it sense, or reference, or some third thing? Or sometimes sense, sometimes reference? An example Bach gives seems to make things clearer: 4.4.8 Lois Lane believes that Clark Kent is a wimp, but he is not. That “he” in 4.4.8 is an anaphora, referring to the “Clark Kent” a few words before. The following seems to me a reasonable interpretation of 4.4.8: (i) (ii) (iii) (iv)

There is a man; The name of this man is Clark Kent; Lois Lane believes this man is a wimp; but he is not.

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But this reasonable interpretation is lost if we are unwilling to give semantic innocence up, for it essentially relies on a distinction between what somebody really is and what someone else takes him to be. And something similar can be said of that old Quinean “enigma” about Jones, the Spartans and the Lacedaemonians. Namely, making things as simple as possible, (v) (vi) (vii)

The word “Spartans” has a meaning to Jones; The word “Lacedaemonians” is a blank to Jones; The Spartans were the Lacedaemonians.

(v) and (vi) are about certain words and what they mean to Jones, (vii) is about the referents of those words. With this distinction, such pretended “riddles” are easily solved; without it they are not, and cannot be. But why not give up composionality and semantic innocence, if they cause such trouble?

4.5 I Know, You Know Now that preliminaries, at last, are over, the first propositional attitude I discuss is knowing, and the first thing I say about it is that it can introduce both objective and interrogative dependent propositions, as we can see from this example: 4.5.1 I know that Israelis have been hunting Nazi criminals hiding in South America, but do not know whether they have succeeded in finding them all. Here the first know is of the know that sort (it introduces an objective proposition), while the second is of the know whether sort, it introduces an (indirect) interrogative one14 ; if we want to use labels, we can speak of an assertoric and an interrogative use. I will discuss first know that and then know whether. (1)

“Knowing that P” is only correct when P is true. This also holds when “know” is denied. Consider, to see this, the following examples:

4.5.3 Now Caesar knew Pompey had become an enemy for him. 4.5.4 Charles knows France is a monarchy. 4.5.5 Bob does not know the Nile flows into the Red Sea. 4.5.6 Ann does not know there are eight universities in Milan. 14 I

call all occurrences of “know” followed by an indirect interrogative proposition “occurrences of the know whether sort” for brevity, but as a matter of fact interrogative propositions can also be introduced by words different from “whether”, as it twice happens, e.g., in 4.5.2 I know who shot me, but do not know why he did it.

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I will start from the bottom. 4.5.6 is correct, for it is true that Milan has eight universities; 4.5.5 is not, for it is false that the Nile flows into the Red Sea, and just as you cannot know a false thing, so you cannot ignore it. 4.5.4 is also incorrect for the same sort of reason, since France is a republic, not a monarchy, while 4.5.3 is correct, for Pompey really became, after being his main ally, an enemy for Caesar. I have been speaking in all four cases of the dependent proposition’s truth or falsity, but also of the whole period as being correct or incorrect, and now of course I must justify this new correct-incorrect opposition, different from the far better known and far more often discussed one true-false. To begin with, if we work seriously and do not forget about the discipline logic always imposes on any language, we must conclude that properly speaking 4.5.4 and 4.5.5, which have been labeled as incorrect, are not false (or true, of course). A true or false proposition is first of all an expression where every component— every representation, Bolzano would say—is used according to its meaning, and with “know” this happens neither in 4.5.4 nor in 4.5.5, two statements we cannot call false because it would be unreasonable to abandon the principle that if P is false then Not-P is true, while the negations of 4.5.4 and 4.5.5, i.e., 4.5.7 Charles does not know France is a monarchy and 4.5.8 Bob knows the Nile flows into the Red Sea,15 are just as incorrect as the propositions they deny. (2)

The negative form of knowing that has a remarkable property: in the present tense its first person, both singular and plural, necessarily produces a contradiction. Take

4.5.9 I don’t know Florence is in Italy or 4.5.10 We don’t know Siberia is extremely cold: they are both equivalent, for different P’s, to “P, but I (we) don’t know that P”, and so inconsistent; and the same sort of inconsistency is also present when P is false, as we can see from 4.5.11 We don’t know ants produce honey,16 which is just as absurd as 4.5.9 and 4.5.10, for it also implies both that a proposition P is true and that the speaker is unable to assert it. But the contradiction disappears when “not knowing that” has a second or third person subject: 15 I am discussing 4.5.5 and 4.5.8 as if “Not-not-P” were strictly equivalent to “P”. This should not be understood as a choice in favor of classical vs. intuitionistic logic, for I believe no such choice is relevant here. I am simply taking a shortcut. 16 Of course, with believe things would be entirely different. But here we are discussing know.

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4.5.12 You (he/she/they) do not (does not) know Catholics have lost elections does not involve any inconsistency. But the impermissibility of “I (we) don’t know that P” only concerns the present tense, not the past. Both 4.5.13 I was six years old, and did not know it was not Jesus who brought gifts to kids at Christmas17 and 4.5.14 We did not know the wine had turned sour are correct sentences. With future tense the only example occurring to me is sad, but perfectly adequate: 4.5.15 If I get Alzheimer, in the last times I will no more know you are my children. Another important point is that “Knowing that Not-P” is stronger than “not knowing that P”. So 4.5.16 Marcello knows the grapes are not ripe entails 4.5.17 Marcello does not know the grapes are ripe but not vice versa: for if somebody does not know whether P or Not-P, then that person knows neither that P, nor that Not-P, and there is no contradiction in this. (3)

Let us take “knowing whether”. I will use, to be brief and simple, this label for all cases where “knowing” introduces an (indirect) interrogative clause, but in fact the uses of language I put together under it are very different. The birthplace, so to say, of these differences are direct interrogative propositions, but the distinctions originating there carry over to indirect ones. The things I will say about this point are well-known, but worth recalling.

We can partition interrogative propositions between two classes, those requiring a yes-or-no answer, such as 4.5.18 Have you forgot to switch the light off?, and those neither requiring nor admitting this sort of answer, such as 4.5.19 Who is Arthur’s girlfriend?

17 In

Catholics countries kids believe it is Jesus, not Santa Claus.

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Now, a yes-or-no question can be turned into an independent declarative sentence, and vice versa. There are languages, such as Italian, where a change, respectively, in intonation in the oral form and in the final punctuation in the written one are enough, and others, such as English, where these changes are also required but grammar, too, is different; anyway, in all languages the correct answer to the question “P?” is “Yes” if and only if the statement “P” is true, and “No” if it is false. Answers to questions not of the yes-or-no sort are no statements, but here too intonation (and in some languages also grammar) is different in questions and answers.18 After making this distinction, I will say a few things first about the uses of “knowing whether” where the indirect question is just a different form of a direct one requiring a yes-or-no answer. “x knows whether P” implies nothing about P’s truth value. When x knows whether P, x knows either that P or that not-P, but on its own “x knows whether P” does not specify which is the right option. The basic difference between “knowing that” and “knowing whether” is that the first is a bridge term, comparable to a sheet with a description of a subjective state on one side and one of a well-specified fact on the other, while the second also has to do both with subjectivity and objectivity, but the latter is undetermined (and as a consequence in a statement of the “I don’t know whether P” sort there is no inconsistency). However, except for this important difference, if a statement with “knowing that” is correct the one we get from it substituting “that” with “whether” also is. Some doubt is possible, anyway, about the first person affirmative use of “knowing whether”. For what reason—we might wonder—should one state “I know whether P”? When the speaker knows whether P, he/she knows either that P, or that not-P— but why give, then, an incomplete information, where an important part is missing? However, if we reflect we see there are situations where this behavior is perfectly logical (I am not saying nice): for example, when we explain to somebody we have a certain information but are unwilling to share it. One might say, e.g., 4.5.20 I know whether this loan has been granted, but am not going to tell you.

4.6 Beliefs Let us consider belief now. The first important difference between believing and knowing is that only “believing that” exists, while “believing whether” does not. The second is that “believe” is not a bridge term, but only expresses a subjective state. It is true this mental state has a factual state I will call, as usual, P as an object, but while a correct use of “x knows that P” involves a commitment to P’s truth both when the speaker and the knower are the same person and when they are not, “x 18 I cannot believe there are languages without a special intonation for questions, but I have something

to ask linguists: is interrogative intonation the same in all?

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believes that P” involves such commitment only when they are. To be schematic: “I know that P, but it isn’t true that P” and “So-and-so knows that P, but it isn’t true that P” are both inconsistent; “I believe that P, but it isn’t true that P” also is; “So-and-so believes that P, but it isn’t true that P” is not. And there also is another important difference between the negative first-person uses, respectively, of “know” and “believe”, for “I don’t believe that P” is not inconsistent at all. There are, then, propositional attitudes Q such that “X Q’s that P” is only stated correctly if P is true, and others such that both its truth value and its logical correctness19 are independent from that of the Q-ed proposition. “Believe” is of the second sort; to take all possible combinations, 4.6.1 Ptolemy believed the Sun went round the Earth 4.6.2 Ptolemy believed the Earth went round the Sun 4.6.3 Copernicus believed the Sun went round the Earth 4.6.4 Copernicus believed the Earth went round the Sun (1) are, respectively, true, false, false, and true and (2) in all of them “believe” is used in a way consistent with its meaning. There are, then, propositional attitudes Q such that “x Q’s that P” is only stated correctly if P is true, and others such that its correctness is independent of P’s truth value: but are there any such that “x Q’s that P” is only correct if P is false? The answer is yes. Take for example 4.6.5 I cheated myself that Henry was honest. or 4.6.6 I had a wrong conviction I had failed that exam: “Henry was honest” and “I had failed that exam” must be false for, respectively, 4.6.5 and 4.6.6 to be conceptually correct. So if we take the whole class of expressions of the “Q-ing that P” form, there are in it all possible combinations between the correct use of the main clause and the truth value of the dependent one. This does not mean that “I believe that P” is always correct, whatever P may be; it certainly is not when P is obviously false, where this adverb, “obviously”, is essential: for when P is something false, but whose falsity is difficult to discover, then “I/we believe that P” is OK. However, even this limitation imposed on “I/we believe” disappears with “You/she/he/they believe (believes) that P”, something correct20 whatever P may be.

19 I

discuss this notion of correctness in §5.2. forget this is not equivalent to “something true”.

20 Never

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The situation is exactly specular with “not believing”: “You/she/he/they don’t (doesn’t) believe that P” is conceptually correct whatever P may be, “I/we don’t believe that P” is so only when P is not obviously true.21 Lastly, the relation between “not believing that P” and “believing that Not-P” is also interesting. The second is stronger, for a person who believes that Not-P certainly does not believe that P, while if that person simply does not believe that P, she may well not believe that Not-P either (in that case, she suspends her judgment, as the Stoics would say), and there is no inconsistency in this. In other words: believing neither that P, nor that Not-P, is different from believing that neither P, nor Not-P. The second belief is inconsistent, the first is not.

21 The boundary between obviously and not obviously true (or false) statements is blurred, and so is, as a consequence, the distinction I am drawing here; but this cannot be avoided, for it depends on the way things really are.

Chapter 5

An Interlude: Matters of Method

Abstract The discussion of everyday discourse developed so far makes two entirely new problems inescapable. The first is that in everyday discourse there is no algebra, either Boolean or non-Boolean, of truth values. The second is that a new polarity (conceptually) correct–incorrect, distinct both from true–false, and from grammatical–ungrammatical, becomes indispensable.

5.1 The in-Out Making of the True and the False The fragment of casuistics I have developed in Chap. 4 has already touched some important methodological issues concerning specifically everyday discourse. But taking such issues into account without explicitly naming them has become impossible by now, so in Chap. 5 I momentarily interrupt, to discuss these issues, the casuistics I have begun in Chap. 4 and will resume in Chap. 6. A new basic difference, and possibly the most important of all, between formal systems and everyday discourse emerges from the examples I have given so far, namely one about the central point of all logics, formal and not: the production of the True and the False. In classical formal systems there are formulas originally having a truth value and others having one just derivatively; the originally true or false formulas are the elementary ones, the others do have a truth value, but that of a non-elementary P is determined by those of its elementary subformulas, i.e. of those among P’s proper parts that are elementary formulas. So the truth value of any formula P is either originally given, or determined by those of other formulas, and an infinite regress is impossible because every formula has a finite length, and therefore a finite complexity,1 so that in a finite number of steps one gets down to its elementary subformulas.2 But in everyday discourse things are not always so simple, 1 In formal propositional logic, the complexity of formulas has degrees. if P is atomic, its complexity

degree is 0; if P is not-Q, then it is that of Q, plus one; if P is “Q and R”, or “Q or R”, or “If Q, then R”, then it is that of Q, plus that of R, plus one. 2 Strictly speaking, this only goes for propositional logic and not for first (or a fortiori second) order one. But in these, too, the truth value of a formula depends on those of a certain set of elementary ones, and the only difference is that typically these are infinitely many. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Rigamonti, Logic, Everyday Discourse, and Metaphysics, UNIPA Springer Series, https://doi.org/10.1007/978-3-030-74598-1_5

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for there, too, P’s truth value can depend on other expressions, but these can also be either (i) no subformulas of P’s or, more generally, (ii) no propositions with a truth value of their own, or even (iii) no propositions at all. Take for example this very short dialogue: 5.1.1 “Yesterday the oxygen blonde of the flat below was nervous”—“But she’s not oxygen!” Normally, the second speaker3 firmly believes he or she is saying something true. This true thing is a negative proposition, and its utterance is triggered not by another proposition but by a simple qualifier, “oxygen”; without that, it would neither be uttered nor have reason to be—so a term doing, here, the work of a simple adjective causes the uttering of a declarative, i.e. true or false, proposition. This is possible thanks to context, of course; but if we took away from context that simple qualifier, “oxygen”, the answer, which is a complete proposition with a truth value of its own, would be entirely out of place and make (pragmatically) no sense at all. There is of course an orthodox solution to this difficulty: if we regularize “Yesterday the oxygen blonde of the flat below was nervous” using the well entrenched standards of ordinary logic, 5.1.1 becomes 5.1.2 x is a woman and x is an oxygen blonde and x lives in the flat below and yesterday x was nervous. The question, anyway, is not whether this “regularization” can be done or not: it obviously can. But does it explain more than it conceals or conceal, vice versa, more than it explains? 5.1.2 is a conjunction with (if the second speaker’s statement in 5.1.1 is correct) a false conjunct so it is false, period. But let us go back to 5.1.1: it begins with a good old ascription of a property to an individual. The term for this property, “nervous”, is commonly used and easy to understand for both characters in the dialogue, and in that context its carrier is well specified. The speakers could be e.g., man and wife, and whoever they may be, between them the expression “the blonde of the flat below” obviously functions as a definite description, uniquely denoting somebody both are acquainted with. “Oxygen” is, so to say, a surplus term, not affecting reference. So the truth of “Yesterday the oxygen blonde of the flat below was nervous” is perfectly compatible with that of “But she’s not oxygen!”,4 and the latter is an objection against a single word. We do object to single words, without questioning the overall truth value of their context, but ordinary two-valued logic does not explain this. The central point here is that in everyday discourse the functioning of the True and the False is not an algebra; not a closed game with white or black chips, invariably having just white or black chips both as inputs and outputs. It is such a game, of course, in ordinary propositional logic: if P is true—white chip—not-P is false— black chip—and vice versa; if both P and Q are false—two black chips—“P or Q” 3 And

the first one too, of course. But here, for reasons that will become very quickly clear, I am especially interested in the second. 4 Things would be different, of course, if the answer were “She was not nervous!”.

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also is—another black chip -, while it is true—a white chip—in all other cases; and so on. But in everyday discourse, if an imperative proposition obviously having no truth value, such as 5.1.3 Call Janet’s husband is used correctly, then 5.1.4 Janet is married must be true, so something true has come out of something neither true nor false. This is no closed game with white or black chips, that is, no algebra—Boolean or otherwise—of the True and the False. And, vice versa, the truth of such a proposition as 5.1.5 Rome had seven kings in all makes it very difficult to ascribe truth values to next-to-all propositions about the Eighth King of Rome.5 In everyday discourse there is, more generally, something really peculiar. It is familiar to us, we move in it as easily as a fish in water, we use it with a perfect ease, and it very seldom gives us those feelings of obscurity and difficulty formal languages can on the contrary cause, sometimes up to outright incomprehensibility. But being able to use it is one thing and understanding how it works is another, and we are much better in the first than in the second. Thus it very often happens that a correct use of expressions having no truth value leads us to conclude that certain propositions are true, but there seems to be no general theory of this phenomenon though it certainly exists, as I have tried to show with my examples. Here the main difficulties are two: we need a notion of logical correctness wider than that of truth, but so far we have no tolerably precise definition for it; and we also are unable—if we except a few interesting but fragmentary remarks—to analyze the effects of context in a precise and comprehensive way.

5 There

have been and—I am afraid—will be lengthy discussions about propositions with nondenoting individual terms. What is the truth value of 5.1.6 P(the Eighth King of Rome), or more generally of P(t), when t is non-denoting? According to Bolzano and Russell all these P(t)’s are false, with the only exception of “There is no t” (and variants, of course). Reviving this discussion would be entering a maze, and I prefer to avoid it. But I have instead a remark of a different sort to offer: there are statements containing non-denoting terms that are subject nevertheless to the normal discipline of truth values: we know perfectly well, for example, that 5.1.7 Hamlet and Ophelia were married is false, and its negation true.

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5.2 What Is Correctness? Let us start from the first difficulty. In this work I call more or less systematically “correct” or “incorrect” various expressions of everyday discourse. Now, what is this correctness I so often talk about? Neither truth, for expressions having no truth value can also be correct, nor mere respect of grammar, for grammatically flawless but incorrect expressions abound; take, for example, 5.2.1 Napoleon was disappointed to discover at Marengo he had won after all. In 1800 the battle of Marengo against the Austrians had begun very badly for Napoleon, and his army was retreating when, late in the afternoon, a cavalry charge led by General Desaix reversed the situation, and in the end the French won. By the way, Desaix was kind enough to die during this charge, leaving the whole glory of victory to an undeserving Bonaparte. Now, of course the latter must have been surprised and possibly even amazed by this reversal of the situation, but not disappointed, for disappointment is an unpleasant surprise. Two more examples will, I hope, make the point even clearer: 5.2.2 The general encouraged his soldiers to flee. 5.2.3 John was delighted to hear Rose did not love him back. A general can encourage soldiers to attack, to advance, to resist, even to retreat orderly, but not to flee. When an army is in flight, which is something very different from retreat, it has ceased to behave as an army, and no general can encourage his soldiers to do so. Maybe he might urge them to, if he thinks the whole war is hopelessly lost and further resistance is pointless: but encourage, a synonym of giving courage! And with 5.2.2 the “incorrectness” is even clearer: when one is in love, learning not to be loved back is a tragedy, not something one can be delighted about. In 5.2.1–5.2.3, then, some words are not used according to their meanings. Of course, from a traditional viewpoint one might object that 5.2.1–5.2.3 are simply false—and false they are, but not simply so. Of course, Napoleon was not disappointed by his Marengo victory, no general ever encouraged his soldiers to flee, and no person in love was ever delighted, on discovering not to be loved back. But with 5.2.1–5.2.3 and the like, one need not inquire whether things really went that way or not, for supposing they did makes no sense. This is something one should better hammer in again and again, deeper and deeper, so I hope this further example will be even clearer than the former ones: 5.2.4 When he at last understood his wife was not faithless at all, Othello was disappointed. How can something unexpected but good be disappointing? This makes no sense; it is an incorrect way of speaking. Again: take

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5.2.5 She was confident her lover was fed up with her. “Being confident” about something makes sense if and only if that something is good; 5.2.6 She was confident her lover was faithful makes sense, 5.2.5 does not. I believe, then, we need a correct–incorrect polarity, distinct both from true-false and grammatical-ungrammatical. I can offer no precise definition for it, and only feel like asserting that every true or false proposition is correct too and every correct one is grammatical, but that is all. Thus this category, “correctness”, has blurred boundaries; but so do the more traditional ones “true” and “grammatical”, for we know, thanks to Saussure and others but above all to history, that grammars of spoken languages continuously—though slowly—evolve, and thanks to Tarski that there can be no really general and precise definition of truth. This notion of correctness I am introducing is, then, partially indeterminate; but so are other fundamental and more traditional concepts. Anyway, two points are in my opinion beyond question: correctness has to do with the conceptual inner harmony of a statement, so it is a logical notion—and it only emerges from a study of everyday discourse, not of formal languages. So this new polarity, though having blurred boundaries, is inescapably there.

5.3 Said in Passing: Are They Really Extensional? The domain I have begun to explore does not include at the moment one-place propositional operators (anyway, I will discuss them fairly long in Part Three), but only two-place ones (such as propositional attitudes), and not all of them, since I can see no reason to go back systematically to conjunction, disjunction, and implication, that have already been studied in depth long ago. I do have, nevertheless, one small brief thing to say about the three traditional two-place connectives. In theory, if we connect any two well-formed formulas by means of one of them, we always get another well-formed formula; but here the basic duplicity of the notion of well-formed formula becomes important. I like comparing this notion to a sheet with “grammatical” on one side and “making sense” on the other. In fact, most logicians are not very interested in the notion of grammatical correctness per se; to them, it only is important as a precondition for the existence of a meaning, but in fact it is just a necessary condition for that, not a sufficient one. Finding examples of this non-sufficiency is not difficult. Take such a conjunction as 5.3.1 Pigs grunt and Joyce is a difficult writer: it certainly is grammatical, but what about its making sense? Its purported sense is, to put it softly and sweetly, exceedingly well hidden, and keeps hiding also when we substitute an “or” for that “and”.

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Now, if sense becomes a problem so does truth value,6 and I am quite unsure that and, or and if…then can be classified as strictly extensional connectives, if extensional two-place connectives are those that, when binding any two propositions, invariably produce a new one being (i) significant and (ii) with a truth value determined by those of its arguments. And I am also afraid the only really extensional propositional connective is negation: for I do believe that denying the true we invariably get the false and vice versa, but negation does not connect two propositions: it only modifies one.

5.4 Of Grammatical Differences Among Spoken Languages But a further caveat about the long casuistic I have started in the Chap. 4 and will resume in the next one cannot be postponed any longer. During this casuistic I have made many examples of expressions of everyday discourse. Obviously, I have given (and will give) such examples in the language I am using to write this essay, English; but every language has a grammar of its own, partially different from those of all other ones, and I have discussed so far and will go on discussing grammatical matters, so that much of what I have said (and will say) has been (and will be) culture-bound. This is a problem, of course; if all the things I say were as culture-free as equations or musical scores, that all those able to read them read in the same way all over the world, things would be much easier for me. But real facts do not go that way; and differences among languages do not simply exist, but are important and interesting. I will give just two examples of this, namely two very short words, the French si and the German sondern. The French do not always use the same word for affirmative answers, but they say oui when the question they are answering contains no negation and si when it does: 5.4.1 As-tu vu Jeanne?—Oui. (Have you seen Jane?—I have) 5.4.2 Tu n’as pas vu Jeanne?—Si (Haven’t you seen Jane?—I have) English speakers answer both questions in the same way (and so do Italian, Spanish, German speakers); French speakers do not. Using oui where si is required, and vice versa, would be a grammatical mistake. Their si is in fact a double negation, a not-not, not to be confused with a simple yes (oui). As French speakers give different affirmative answers to questions English (and Italian, German, Spanish…) speakers always answer in the same way when their answers are affirmative, so German has two different words where English always has but (and Italian ma, and French mais): given an opposition, if its first part is 6 Michael

Dummett discusses this point at length in his Logical basis of metaphysics (Harvard University Press, 1991).

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affirmative the second is introduced by aber; if the first is negative, then sondern is required, not aber. So we have 5.4.3 Er arbeitet, studiert aber abends. He works, but in the evenings he studies. 5.4.4 Sie ist keine Tänzerin, sondern eine Sängerin. She is not a dancer, but a singer. Anyway, these grammatical (and at the same time logical) differences among languages should be neither concealed or forgotten, nor magnified. In principle, the meaning of a sentence in a language can always be rendered in another one, as I have just done above, translating “Sie ist keine Tänzerin, sondern eine Sängerin” as “She is not a dancer but a singer”. Some authors, such as Sapir, Whorf, or Quine, do not accept this last point and claim there are cases of radical untranslatability, but this claim has a fatal weakness: one cannot sensibly assert in language L that “Proposition P* of language L* is untranslatable into L” without understanding P*, and—with the possible exception of mystical experiences, supposing they do exist—the very idea of understanding something but being unable to describe it is inconsistent. What is true, is that a word by word translation between any two different languages is very often impossible; but a good translation is never word by word.

Chapter 6

This, and This, and That

Abstract More dependent clauses are discussed, and more surprising results emerge. The most important of these are that distinct clauses belonging to the same period can sometimes have an overlap, and that there are clauses that, though being neither interrogative nor imperative, are such that wondering about their truth value does not make sense.

This chapter will review—once more, without unrealistic attempts at completeness—some classes of dependent propositions different from objective or subjective ones, i.e., from those depending on propositional attitudes. A very varied and often surprising overall view will emerge.

6.1 Adversative Constructions I will discuss adversative constructions first; I say “constructions”, a term more general than “propositions”, for gathering grammatically very different cases under the label “adversative” not only makes sense, but is highly instructive.1 Such constructions can in fact connect both simple adjectives or nouns, as in 6.1.1 an elderly but very good-looking lady… and whole propositions, as in 6.1.2 She is old, but still looks very good or in 6.1.3 Although she is small, Valerie plays volleyball. When the antithesis is between propositions, the form it is expressed in may be both subordinative, as in 6.1.3, or coordinative, as in 6.1.2 or in 1 Webster’s

New World Dictionary has this entry for adversative: “Expressing opposition or antithesis, as the words but, yet, however”. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Rigamonti, Logic, Everyday Discourse, and Metaphysics, UNIPA Springer Series, https://doi.org/10.1007/978-3-030-74598-1_6

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6.1.4 Valerie is small, but plays volley ball, and the ordering of the opposition can be reversed in both cases: 6.1.5 Valerie plays volley ball although she is small, 6.1.6 Valerie plays volley ball, but is small. Moreover, in subordinative forms it does not really matter which is the main proposition and which the dependent one, so that there is no important difference between, say, 6.1.3 and 6.1.7 Although she plays volley ball, Valerie is small. Now, all these symmetries are there because adversative constructions are inherently symmetrical. They simply say there is an opposition between two elements, stopping at that; and of course in an utterance one of these elements must come first and the other second, but this is only due to the fact that most languages and linguistic productions, including everyday discourse, are linearly ordered,2 and to no other reason. This symmetry reminds one of conjunction, and there also is another similarity. Suppose Valerie is tall: then all of 6.1.3–6.1.7 are false, and the same goes if Valerie does not play volley ball. Now, since there seems to be nothing unusual about this example, it is reasonable to conclude that an adversative period is only true if both members of the opposition3 are. This is why adversation looks very similar to conjunction; and in fact such an important author as Quine in his 1950 Methods of Logic claimed that “and” and “but” are essentially the same thing, the difference between them being merely “rhetorical” or “emotional”, not logical. But this is wrong. The “only if” we have just found is not an “if and only if”, as we can see from this new example: 6.1.8 Although she is very tall, Claudia plays volley ball. Suppose Claudia really is tall, and really plays volley ball: then 6.1.8 certainly is not false, yet we feel something is wrong with it, while we have no such feeling with 6.1.9 Claudia is very tall and plays volley ball. The point is, no opposition exists between being tall and playing volley ball. It is the other way round: height is a big advantage in volley ball and being short a big disadvantage, so most players, both males and females, are tall and very few are small. Therefore 6.1.3–6.1.7 are correct adversations because there is an opposition in all of them, while in 6.1.8 there is none, and so the use of “although” is incorrect there. As for 6.1.9—well, pace Quine, conjunction has nothing to do with opposition, 2 The

first to notice this was Ferdinand de Saussure in his Cours de linguistique générale, Geneva 1913. 3 When this is one between propositions, of course.

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and this is a conceptual (but we might as well say logical) difference, not a rhetorical or emotional one. Now, what is the nature of this opposition or antithesis underlying adversative constructions? At the beginning of my reflections about it, I was inclined to view it as probabilistic—to think, in other words, that in acceptable, well constructed periods of the “P but Q” sort the probability of P given Q must be inferior to the prior probability of P and, symmetrically, the probability of Q given P must be inferior to the prior probability of Q. But there are plenty of absolutely normal, standard cases of adversation where there seems to be no probabilistic relation of this sort: we can take such examples as. 6.1.10 She has a lovely face, but her legs are short. 6.1.11 He stutters, but can play chess in a fantastic way. 6.1.12 He has fine manners, but is a clumsy dancer. For all I know, there is no negative (or positive) statistical correlation between, respectively, having a lovely face and short legs, stuttering and being a good chess player, or having fine manners and dancing badly, yet the “buts” in 6.1.10–6.1.12 sound perfectly correct. We have no especially low posterior probability here, but an entirely different sort of opposition, one between something good and something bad. It is good having a lovely face, bad (in the majority’s opinion) having short legs; bad stuttering, good playing something (e.g. chess) well; good having fine manners, bad dancing clumsily. I am not saying there never are negative statistical correlations in adversative constructions; they can be there, we have already met some in previous examples and it amuses me adding this new one: 6.1.13 Although he weighed over a hundred kilos, Muhammad Ali, formerly Cassius Clay, was extremely agile. In fact such an agileness as that of Clay-Ali, that someone called “a hundred kilos’ dragon fly”, is rare among heavyweights and far more common among smaller fighters, from fly- to lightweights; but negative statistical correlation is just one form of opposition. There are many others. How many? I don’t know, and don’t even believe one can be precise about this. I am claiming, in fact, that adversative constructions are founded on a generic, inherently imprecise notion of opposition. But everyday discourse is itself imprecise; it is wonderful in its plasticity and expressive power, but at the same time very distant— not always of course, but more often so than not—from the beautiful simplicity and exactness of mathematical structures. In fact, an “opposition or antithesis” vaguer than the one needed to construct an acceptable adversation is difficult to imagine. The presence of an affirmative and a negative polarity is invariably sufficient for that. Take , e.g.,

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6.1.14 Claude plays violin, but cello: it cannot be accepted. However, 6.1.15 Claude does not play violin, but cello can, and the same goes for 6.1.16 Claude plays violin, but not cello, while 6.1.17 Claude plays not violin, but not cello is nonsensical again. But there is another wrinkle to add here. To change nonsensical things of 6.1.14 type into acceptable ones, it suffices adding an “also” after the “but”. I will show this with a new example. Let us begin with 6.1.18 Ann has studied Latin and Greek: this is said correctly, while 6.1.19 Ann has studied Latin but Greek is not. However, 6.1.20 Ann has studied Latin, but also Greek is correct again.

6.2 Causal Propositions There is an important analogy between adversative and causal propositions: the latter, that in the simplest case (the only one I am going to discuss) consist of a main and just one dependent clause, are also only correct if these are both true, but that is not enough, for a certain relation between the contents of both is also needed. Anyway, this relation is far more elusive and vague even than that quite imprecise “opposition”, required for adversations to be correct, I have discussed in the previous section. I will proceed by way of examples, as usual. Among all conjunctions4 that can be used to introduce a causal dependent proposition I will only use, for brevity and simplicity, because. To begin with, for a causal explanation to be correct the main proposition must be true. Whatever we substitute for the dots, 4 Not

in the logical but in the grammatical (and far wider) sense of the word.

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6.2.1 Alpine groundhogs can be seen all year long because… will be incorrect because alpine groundhogs can only be seen (of course, in the high mountain valleys where they live) in summer. So from now on I will only use as main clause “Alpine groundhogs can only be seen in summer”. Let us try then with 6.2.2 Alpine groundhogs can only be seen in summer because they are rodents. Now, it is true they are rodents but this cannot be a correct causal explanation, for other rodents, such as mice, squirrels, and beavers, are visible all year long while still others, moles, can be seen next to never. Let us try then with 6.2.3 Alpine groundhogs can only be seen in summer because in winter they migrate. This, too, cannot be accepted, for the purported causal clause is false. They do not migrate at all. The correct explanation, of course, is 6.2.4 Alpine groundhogs can only be seen in summer because in winter they hibernate. To sum up, a causal explanation goes from something true to something true. Only something that really happens can have a cause or be a cause; moreover, if P causes Q, then P and Q must be different.5 However, this does not mean that any true proposition causally explains any other true proposition.6 The thing ought to be obvious, but since sense of humour is never wrong, I will illustrate it with a funny example. Let us suppose mr X is broke, and constipated too; this does not mean we can accept 6.2.5 Mr. X is broke because he is constipated. If that were a good causal explanation, then anyone with serious financial problems could improve one’s economic situation simply taking a laxative: and were it so easy! What is necessary, then, over and above the truth both of the main and the dependent clause, for a putative causal explanation to be OK? Though we usually are able, in individual cases, to assess with certainty whether a purported causal explanation is good or not, a clear and generally shared answer to this question has never been found, except for one obvious but trivial point: that a candidate-to-be-a-cause must be somehow relevant to the fact it ought to explain. Now this is correct but next to perfectly void. Being able to explain in a clear, exact, and obviously right way what 5 If

we also admit causation de re, and not just de dicto, then it must be said that according to Christian theology there is exactly one exception to this, for God is his own cause (causa sui), and nothing else is. But this need not concern us here. 6 This would imply, by the way, that causal explanations are symmetrical, i.e., something certainly wrong. It is smoking that causes lung cancer, not lung cancer that causes smoking.

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“relevant” precisely means would be wonderful, but the difficulties opposing this undertaking are enormous. Reminding readers of just one will be enough here: this fuzzy, foggy concept of causal relevance is strongly culture-bound. In some cultures, for example, thinking that an illness has been caused by witchcraft is plain commonsense, in others it is nonsense. But consensus about acceptable causal explanations is often missing even within a single culture; in these early years of the twenty-first century, for example, in Italy there are many who find it plausible to explain recovery from a serious illness thanks to a miracle by a Padre Pio from Pietrelcina (Pietrelcina is a village in southern Italy) who is today an official saint of the Catholic Church, but many others, including me, see Padre Pio’s cult as mere idolatry and superstition, and find it disgusting. I will close this section with a few words about consecutive, i.e., reverse causal, periods. I call them reverse causal because, if we except a swapping of the roles between cause and effect (i.e., the fact that here it is the main clause that states the cause, and the dependent one the effect) in consecutive periods things go exactly as in causal ones. If we reflect about this example, 6.2.6 In the nineteenth century the introduction of railways changed the pattern of commercial exchanges so that many small local transport dealers were ruined,7 we will see that the things said above about causal propositions can be exactly repeated here, except that now the cause is stated by the main proposition, and the effect by the dependent one.

6.3 Two Sorts of Relative Propositions All dependent propositions we have discussed so far, except of course interrogative ones, have truth values of their own; but from now on things are going to change. Let us consider at this point relative propositions, so called because they are introduced by a relative pronoun (it is true that in some languages, such as English, this pronoun can often be omitted, but standard grammar textbooks maintain—correctly in my opinion—that in such cases it is implicit). Now, relative propositions have very different logical natures when they, respectively, (i) enrich or even complete, making it definite, a description, or (ii) say something new about some already identified person or thing. I can illustrate the difference with these two examples: 6.3.1 The tall woman we have just met is my landlady. 6.3.2 Mary Jane, whom we have just met, is my landlady.

7 This

really happened in many places. Small cart-and-horses transport workers could not compete wity the new thing, and lots of them went bankrupt.

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In 6.3.2 the relative proposition has an independent and definite truth value of its own, in 6.3.1 it has none, and as a consequence, we can imagine an objection making sense against 6.3.2, but not against 6.3.1: 6.3.3 But she was not Mary Jane, she was Emily! I will call incidental relative clauses like 6.3.2. These clauses are either true or false, just as independent declarative propositions are. We have e.g., a false one with 6.3.4 Madagascar, which is the biggest island in the world, has a tropical climate (the biggest island in the world is not Madagascar, it is Greenland), and a true one with 6.3.5 Shakespeare, who was from Stratford-on-Avon, moved to London. Now, in 6.3.1 the relative clause functions as an adjective, and so has a completely different nature. Both 6.3.1 and 6.3.2 say a certain person “is my landlady”, but in 6.3.2 that person is already well identified independently of this qualification, while in 6.3.1 she is identified not by a single proper name but by three properties (being a woman, being tall, having just been met by speaker and hearer) that together characterize her uniquely, and one of these is expressed by a relative clause. That clause, then, is as good as an adjective, and it does not make sense wondering whether an adjective is, per se, true or false. I will call adjectival relative clauses of this sort. So something very important, that we will meet again, first emerges with 6.3.1: while some dependent propositions do possess a truth value of their own, for others it does not even make sense asking whether they are true or false. For all I know, this was not anticipated by any of the classics, but there would certainly be room for it in a view à la (later) Wittgenstein (while there would be none, e.g., in one à la Quine, characterized as it is by a cult for elementary formal logic’s “elegant standard grammar”). With adjectival relative clauses the situation is, indeed, far more complex than with incidental ones. Consider such cases as 6.3.6 There was a girl with curly hair by me—There was a girl who had curly hair by me or 6.3.7 A man with a blister on his forehead came in—A man who had a blister on his forehead came in. These seem to be situations where the presence or absence of an extra clause does not make much difference. One is even tempted to call these extra clauses apparent propositions. There is anyway a maneuver for rescuing the propositional nature of such clauses: substituting the relative pronoun with a conjunction (in the grammatical sense of the word, so not necessarily “and”) and explicitly naming the object it refers to:

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6.3.8 There was a girl with curly hair by me—There was a girl by me, and this girl had curly hair, 6.3.9 A man with a blister on his forehead came in—A man came in, and this man had a blister on his forehead. But these transformations work both ways, and I can see no reason to prefer one variant over the other in everyday discourse, i.e., in our actual standard language games, to use a Wittgensteinian term; so I will not put adjectival relative clauses aside (and we will soon see they are very instructive). However, we do something extremely useful if we conceive adjectival relative clauses as non-propositions, at least from the viewpoint of their relations with truth values (in the orthodox sense of the term, that of modern bivalent logic). We do not find anything like that in such clauses, but just in their counterparts, the second halves of 6.3.8–6.3.9 (“this girl had curly hair”, “this man had a blister on his forehead”), while “who had curly hair” and “who had a blister on his forehead” are no independent statements, and asking whether they are true or false does not make sense. “…who had curly hair”: who is “who”? Of course, the context explains it, but in the clause the grammatical subject is just “who”, a word specifying, on its own, exactly nothing. But the most interesting point is not that an adjectival clause has a double nature, being both an attribute and a proposition. There is something still deeper I will try to illustrate using as a starting point a short, short story, very popular in England and very amusing, for those at least who do not believe devils really exist. There is a crowded, noisy pub where everybody is having fun. Then a devil suddenly appears and says: 6.3.10 I will take away the one who remains in here last. Just imagine what happens then. There is by the way an English idiom connected with this story, “the devil takes the hindmost”, meaning that in certain situations the worst off are completely lost and can find no salvation at all. But now let us do some logical analysis with 6.3.10. Normally take has three argument places, somebody takes something somewhere (or to someone else), but in the take away variant argument places are just two, somebody takes something (or someone) away. Now, who will the devil take away? “The one who remains in here last”, he says. The whole relative clause, plus “the one”, name the object. But this clause has a predicate, “remains”: which is its subject? Of course “who”. So the whole relative clause is the object of the predicate of the main one, and its subject is part of this object. This means that two distinct predicates, “take” and “remain”, have overlapping saturations. Here of course there is ample room for normalizing or would-be normalizing maneuvers; we could, for example, emphasize that “the one” is accusative and belongs to the main clause, while “who” is nominative and belongs to the dependent one, so there really is no overlap; but this is not to the point and the overlap is there anyway, for the object of “take away” is not just “the one”, but “the one” plus the

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whole relative clause. Moreover, culture can make a significant difference here: the devil in 6.3.10 is English, he speaks English. What would he say in another language? I can only put together examples in a few European languages here, so I must apologize both with non-Europeans and with Europeans using languages different from English, French, German, Italian…and Latin. However, even within these narrow bounds some interesting variations can be found. To begin with, a French devil would say 6.3.11 Je m’emporterai celui qui reste ici le dernier. So French devils, too (and of course French speakers at large), introduce the object of the main clause and the subject of the subordinate one with different words (celui, qui); and German devils do the same, for one of them would say 6.3.12 Ich werde den, der hier als letzter bleibt, wegnehmen (den is accusative and der nominative, of course). But with an Italian devil things would change, for he would say 6.3.13 Mi porterò via chi resta qui dentro per ultimo, “I-will-take-away-who-remains-in-here-last”, where chi (who) is doing double duty, as object of one verb and subject of another. Italian grammar allows it, and so does e.g., classical Greek grammar, while in English, German, or French this cannot be done. But we have not finished yet. Imagine a big, important ancient Roman devil, acting, more or less, around the end of the Empire; he speaks Latin of course, and an elegant, Ciceronian one, for he is highly educated (if you are very Protestant, you can even imagine he is a close friend of the Pope’s). So he would say, more or less, 6.3.14 Postremum hic manentem mecum auferam. He says the same thing as his colleagues, yet there is no relative clause here, and no relative pronoun; we find in their place a so-called present participle in the accusative, manentem. But the same trick is also possible in English: 6.3.15 I will take away the last one remaining in here. At the moment I will stop here with examples, for so many problems are coming out that one must halt and tidy things up, before going over to further complications. To begin with, 6.3.14 and 6.3.15 bring a new fact to light, namely that the boundary between verbs and non-verbs, and so between propositions and non-propositions, is blurred. This is especially obvious in such a language as Latin, so rich in verbal adjectives—words with a double nature, that can be seen both as predicates, i.e. cores of a separate proposition, and (but this is a theme that will only be developed in Part III) as mere modifications of an already given one. Which of these two things is that manentem?

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This is an idle question, of course. Manentem in 6.3.14, as well as remaining in 6.3.15, can be seen both ways; but the important thing is, they can also be seen as instances of something far deeper and more general, namely that in a complex period, with more than one clause, there may well be no sharp demarcation between two different clauses. I am not denying there are cases (such as, e.g., propositional attitudes) where the demarcation is sharp: but with adjectival relative clauses (and I am not saying just with them) it is not. This is something completely different from anything else we have met so far, and also, to my knowledge, from all views ever defended about connections among propositions. It is well known since Aristotle (see Chap. 1) that there can be propositional chains, and propositional logic, past and present, could well be seen as the science of these chains; but even when one goes further than extensional connectives, as I am trying to do here, the propositions language combines are normally seen as something similar to pearls in a necklace, i.e., things that for all their belonging to a chain keep having no common parts, no overlap. Now, this is correct as far as objective, subjective, adversative, causal, consecutive, and of course main clauses are concerned, but not with adjectival relatives. Suppose A says 6.3.16 I share all opinions B has defended. Share and defend are two transitive verbs. Both need an object. Now, what does A share? All opinions B has defended. And what has B defended? Opinions. But the thing I want to emphasize here has to do with grammar, not with content; it is not the fact that two people share certain opinions, but the way this is said, namely by means of two transitive verbs whose objects overlap. 6.3.16 is very different from anything like 6.3.17 B has defended certain opinions and A shares those opinions, where each verb has an object of its own, distinct from that of the other, for all their equiformity (opinions—opinions: two distinct occurrences); in 6.3.16 “opinions” has one occurrence, and this means the object of “has defended” is a proper part of that of “share”. Now, two transitive verbs can have the same object, but normally the term saturating both has two occurrences, one for each verb, and when there is just one this can be understood as an abbreviation due to aesthetic and practical reasons, not to reasons of content. Thus, we normally say 6.3.18 I have read and appreciated this book, not 6.3.19 I have read this book and I have appreciated this book, but we know that in principle each verb is saturated on its own and each conjunct can be asserted separately. Now, with adjectival relative clauses things are completely different, and I do not see, at this point, how one can escape the conclusion that in

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some cases two propositions are distinct, for each has a predicate entirely of its own, but nevertheless overlap. The essential novelty connected with adjectival relative clauses is this, but a brief further comment on the putative truth value of such clauses as “B has defended” in 6.3.16 is worth adding: there is no such value, and looking for one does not make sense. For a proposition P to have a truth value, all of its component parts must have a meaning of their own, not involving other components, external to P, of the utterance this is a part of, and with adjectival clauses things are not like that. There is one last point to add. An adjectival clause has no truth value, but if it is used correctly some other proposition must be true. In 6.3.16, “B has defended” has no truth value, but if the speaker has talked sense, then “B has defended some opinions” must be true. There is nothing strange in this: as I have already said in Chap. 5, the game of True and False is a game with white and black chips, but not one with nothing but white and black chips. There can be an algebra of the True and the False in formal languages, but there is none in everyday discourse.

6.4 Final Propositions The last sort of dependent propositions I will consider in this chapter are final ones; they make a very difficult case. Let us start from this example: 6.4.1 The coach made that substitution to reinforce the defense. This final clause is certainly equivalent to a causal one with “want”: 6.4.2 The coach made that substitution because he wanted to reinforce the defense. But is there behind this equivalence a general pattern going from “So-and-so does P to get Q” to “So-and-so does P because he/she wants Q”? Now, with such examples as 6.4.1–6.4.2 this idea certainly works; but not with such ones as 6.4.3 Bees gather pollen from flowers to produce honey, 6.4.4 Spiders spin their web to catch preys, or 6.4.5 Imperial penguins reach the interior of Antarctica to reproduce. Properly speaking, will is something conscious and clear, but with certain animal behaviors, both asserting they are governed by will and denying there is an end driving them seem quite implausible; and this does not concern just non-human animals for, e.g., our mighty human sexual desires often drive our actions without waiting for (and sometimes against) our will. It really gets difficult, at this point, to escape the conclusion that an unreducible notion of end-in-view is at work here; but it must be clear I am thinking of a linguistic,

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not an ontological notion. I am not claiming that ends not reducible to matters of fact are at work in the world; I leave that point unjudged. But I do claim that in everyday discourse there are secondary propositions unreducibly ascribing ends to behaviors, where “unreducibly” means that these propositions cannot be restated faithfully, i.e., without any essential loss of meaning, as descriptions of states of things. Saying this is not taking sides in the centuries-old quarrel between finalists and mechanists, neither is it claiming that those animal or human behaviors that seem to be voluntary really are governed by physical processes, or that they are not; here I am not interested in this sort of debate. But it is claiming there are in everyday discourse statements about ends-in-view, as things essentially different from matters of fact, and these statements are nothing but the so-called final propositions. I know of no other way of accommodating under this common label, as it has been normally done for millennia without any serious inconvenience, such examples as, respectively, 6.4.1 and 6.4.3–6.4.5. Let us go back to these examples for a moment. The coach made a substitution to reinforce the defense: but was it reinforced? We don’t know; I mean, not from 6.4.2 alone. Bees gather pollen from flowers to produce honey: but does a certain one of them, randomly chosen, actually produce it? If all goes well, yes, but generally speaking, we cannot say. Spiders spin their web to catch insects: but does a certain one of them, randomly chosen, succeed in this? Well, I suppose some do but some do not, and starve to death. But let us take another (quite different) example, this time from ancient Roman times: 6.4.6 The consul sacrificed victims to the gods that they might give him victory. Well, did they give him victory—or, said in a more modern and sceptical way, did he win that battle he was going to fight? The answer is always the same, monotonous one: we do not know—I mean, not from this final proposition alone; and we do not simply because it is final, and final propositions do not state any matter of fact, so that asking whether any one of them is true or false does not make sense. A final proposition says a certain action has a certain end,8 and this is not saying things are going a certain way. But there is a further (and very important) wrinkle to add: saying a final proposition has no truth value of its own is not saying it causes no new truth values to be there. It does, but indirectly; we can see this in two stages. Stage one: has “The consul sacrificed victims to the gods” a truth value? It does. Now, if and only if it is true we can go over to stage 2: has “The consul sacrificed victims to the gods, that they might give him victory” a truth value? It does: and it is true if and only if what he wanted was victory, while it is false if he wanted anything else, e.g., rain if his legions were in the desert, sun if they were in the far North, health if there was plague among legionaires, etc. To sum up: let us take as general pattern of final propositions 8I

say end, and not specifically end-in-view, for it really is difficult to think of certain ends as endsin-view. Entering an egg-cell is surely an end for a sperm, but I believe inferring that sperms have ends-in-view, and so views, is arbitrary and unscientific.

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6.4.7 x does P, that it may be the case that Q. What is necessary and sufficient, for 6.4.7 to be true? There are three conditions. First: it must be true that x does P. Second: it must be true he does it for some end. Third: it must be true that such end is Q. If x does not do P, or does it but for no end at all (there might be, for example, someone else forcing him to do it), or for an end different from Q, then 6.4.7 is false. So a final proposition has no truth value of its own, but can be a necessary condition of the truth (or falsity) of some more complex statement it belongs to.

6.5 The Wild Behavior of Negation My review of dependent clauses stops here. It has reached—I hope—some interesting results, but is very incomplete; however, a complete one would cost me an awful amount of work, and require an awful amount of patience from the reader, so I believe stopping now is wiser. But I feel, nevertheless, I absolutely must say something about a point, not limited to dependent propositions but with an even wider scope, that so far has been neglected in this work: the incredible complexities of negation’s use. These complexities also depend, of course, on grammar, so much of what I am going to say might well be culture-bound; however, I can’t believe there are spoken languages where no confusions and ambiguities are possible, when negation is concerned. In fact, such ambiguities can be there even in formal languages. This was discovered by Andrej Kolmogorov, then 22 years old, in 1925 (“O principe tertium non datur”, Matematiceskij Sbornik, 1925, 646–667; of course, in the nineteenthirties Kolmogorov would become famous for a very successful axiomatization of probability theory). Take the formula ¬Px, says young Kolmogorov: it may be interpreted either as ¬(Px), or as (¬P)x. In the first case, its meaning is that the proposition Px is false, in the second, that x has the property ¬P. The two statements, says Kolmogorov, are different, and tertium non datur might well apply to the first, but not to the second. Now, from the viewpoint of Fregean orthodoxy only ¬(Px) is well-formed and (¬P)x is not, for there ¬ is saturated not by a whole proposition but just by a predicate, something the rules introduced in Chap. 2 forbid; but the fact remains, that if we slightly relax these rules, an ambiguity about negation’s scope emerges in a formal system. However, in everyday discourse these ambiguities are far more important. To see this, take the following four examples, two of them causal and two final. The causal ones are: 6.5.1 Whales do not breathe under water because they have lungs. 6.5.2 Brutus did not kill Caesar because someone else had ordered him to. And the final ones are:

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6.5.3 She did not take that job in a foreign country to stay near her very old, ill parents. 6.5.4 We have not won elections to govern like our rivals. These four periods have the same formal structure, main clause, containing a “not”, plus a dependent clause, containing none; but every competent speaker understands perfectly well that in 6.5.1 and 6.5.3 the main clause is denied, for it is true that whales do not breathe under water (they must come to the surface to do it), and given the context it also is obviously true that the lady (whoever she is) 6.5.3 is about really gave up that possible job. But in 6.5.2 and 6.5.4 the “not”, though it is exactly in the same position, denies the whole period, instead of the main clause alone: Brutus (6.5.2) did kill Caesar but, for all we know, he decided to do so on his own, and (6.5.4) when a party wins elections—well, officially they never want to govern in the same style as their rivals. So in such (non infrequent) cases we can decide what the “not” really denies only thanks to an utterance’s content: with certain words around it the “not” goes, so to say, somewhere, with certain other words it goes somewhere else. Ah, Wittgenstein! Ah (again) Tractatus 4.002, “The unspoken agreements for the understanding of common language are enormously complicated”!

Part III

Not Just Names and Predicates

Chapter 7

Of Many, Many Other Things

Abstract Propositions of everyday discourse can have, unlike those of formal languages, other components over and above the predicate and the terms needed to saturate it. These components are of two very different sorts: adverbs and further individual terms. Only adverbs have already been discussed by other writers in this connection; but they are not the whole story.

7.1 A Basic Difference While in Part II I have discussed those chains of propositions a long-lasting tradition calls periods, in Part III I will return to individual, independent propositions. The standard I invoke to decide whether an expression is an individual proposition or something more complex is the traditional one, uniqueness of the predicate. This said, let us take, to begin with, 7.1.1 Tony is drinking beer. 7.1.1 has a two-place predicate (all transitive verbs have two argument places) and two individual terms, Tony and beer, saturating it.1 So far, so good. This is exactly the sort of structure elementary logic envisages since Frege’s times. But now we add something else, writing 7.1.2 Tony is merrily drinking beer. In everyday discourse this is an obviously well-formed proposition, but what is that “merrily”? It is neither a predicate nor anything saturating a predicate, and we add it to an already complete, well-formed proposition with no gaps for us to fill, so what is it? I mean—how can we classify it logically? (Grammatically, of course, it is an adverb). 1 In fact, classifying “beer” as an individual term sounds artificial (see about this also example 7.1.2);

but this is not relevant to the present discussion, so I will forget about it.

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Before answering this question, it is better to go on with examples. Indeed we can, in principle, keep adding new things as long as we like: 7.1.3 Tony is merrily drinking beer with Bob and Alice. 7.1.4 Tony is merrily drinking beer with Bob and Alice late in the evening. 7.1.5 Tony is merrily drinking beer with Bob and Alice late in the evening at the Flying Dutchman, and so on. Another important difference between everyday discourse and the formal languages of logic and mathematics is coming to light here. In the latter a formula, once its predicate is saturated, is not simply complete, but cannot be expanded any further. Take for example 7.1.6 tg(x) = sin(x)/cos(x), that can be paraphrased thus in everyday discourse: “The tangent of x is equal to the sine of x over the cosine of x”. On the left of = we write a one-place function, on its right a quotient, i.e., a two-place function, with two one-place functions, respectively, as dividend and divisor. This makes in all three argument places to be saturated; we saturate them with as many occurrences of an x, and we have done all there was to do. We neither must nor can go any further. The formula now is like a puzzle with all its pieces in place, and trying to add new pieces to a puzzle after it has been completed makes no sense. Anyway, with 7.1.1, “Tony is drinking beer”, although here too there are no gaps to fill, we are not obliged to add anything else, but are nevertheless allowed to do so.2

7.2 Those Few, Very General Things That Can Be Said with Some Confidence There are, then, things that everyday discourse has while many formal languages, including those of first- and second-order logic, have them not, and that are neither predicates nor terms saturating predicates but some third sort of thing, so I will call them tertia (tertium in the singular; it is Latin for “third”). I know the word is extremely generic, but tertia are so widely different from each other that the term’s very vagueness is among the reasons leading me to choose it. Tertia can, indeed, vary enormously also from a strictly grammatical viewpoint. Take 7.1.2–7.1.5: we have there 2 An

important point is that here the difference between everyday discourse and formal languages is not in the subject-matter, but specifically (and only) in the language. Take the paraphrase of 7.1.6 in everyday discourse, “The tangent of x is equal to the sine of x over the cosine of x”: we can add to it, e.g., an “always”, and write “The tangent of x is always equal to the sine of x over the cosine of x”. That “always” might well be classified as useless and silly, but it is not ungrammatical.

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merrily, an adverb; with Bob and Alice, an expression containing a preposition, with, two proper names, Bob and Alice, and a conjunction joining them, and;

an “adverbial expression”, late in the evening, consisting of an adverb, late, and further qualified by an expression consisting of a preposition, an article and a noun, in the evening;

and another proper name, (at the) Flying Dutchman. And all this in one simple phrase. Now, one thing is obvious: tertia are of two sorts, nominal and adverbial. For example, merrily is an adverb; it specializes a predicate, is drinking, giving it, so to say, a finer-grained meaning. The same goes for late in the evening: it says when an action is done, not how it is done, but also acts on the verb, while with Bob and Alice and at the Flying Dutchman are (inflected) proper names. It is obvious enough that some tertia (typically, adverbial ones) act specifically on the predicate and some on the whole sentence. If you want more examples, you can take, respectively. 7.2.1 Yesterday, at the Scala, she sang marvelously and 7.2.2 He finished his homework with a big effort. There are, then, adverbial tertia and nominal tertia. They are very different. Nevertheless, they all have something in common, namely (even when they are proper names) an argument place that has to be saturated either by a predicate or by a whole proposition. Please forgive my repetitions, but…merrily: what gets done merrily? Tony is merrily drinking beer. With Bob and Alice: what gets done with Bob and Alice? Tony is drinking beer with Bob and Alice. Late in the evening: what gets done late in the evening? Tony is drinking beer late in the evening. At the Flying Dutchman: what gets done at the Flying Dutchman? Of course, Tony is drinking beer there. So in everyday discourse, and in it alone, (1) proper names do not necessarily saturate argument places in a predicate. They may well be added to a sentence where this has already been done. (2) A predicate may be further qualified by an adverb. Now, an important question is: can we indefinitely keep adding new tertia to a well-formed sentence in such a way that, step after step, the result always remains well-formed? Here a distinction between matters of principle and concrete possibilities has to be made. Of course, beyond a certain complexity expressions become de facto unmanageable. Human brains cannot master sentences with too many words, and the comprehensibility of expressions is not independent from length, even if the

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boundary between those utterances that are brief enough to be mastered by a normal human being and those that are not varies a lot from speaker to speaker and is blurred, moreover, even from a strictly individual viewpoint. But the limitations of our brains should not be confused with the laws languages are subject to in themselves, independently of their users’ capabilities; and those intrinsic laws of everyday discourse that any serious reflection shows us to exist hold independently of our nervous system’s limitations. So it is reasonable to conclude that in principle new tertia can always be added to a sentence in a way respecting both grammar and sense.3 Now, as regards this possibility of indefinitely adding to a sentence newer and newer tertia in such a way that it never stops making sense, a comparison with negation may be useful: if we squeeze too many negations into one period, de facto we get lost—is that period affirmative or negative, as a whole? In such cases one must count, and if the number of negations is even then the period is affirmative, otherwise it is negative.4 Now, one does not feel forced to count negations, unless one has lost control of discourse: should we stipulate then, to avoid this, that in a single period the number of negations cannot exceed a certain threshold? Certainly not. For this would imply there are well-formed periods H such that ¬H is not well-formed, and this would violate a basic principle of any logical grammar worth this name, namely the recursiveness of logical terms; but such a disastrous conclusion cannot be avoided once we forget about the (all-important!) difference between the intrinsic properties of languages and the limitations of our nervous system, and consequently of our capability of handling language, formal or not. So I conclude that in principle in everyday discourse one can always add a new negation to a well-formed sentence in such a way that the result is well-formed too; and we can also conclude, similarly, that in principle (though

3 Anyway,

respect for grammar and for sense are two different things, and the first may well be there without the second. There are a lot of examples, and one I find particularly amusing is by Burchiello, an Italian fifteenth century poet who wrote a sonnet beginning like this, Nominativi fritti e mappamondi e l’Arca di Noé fra due colonne cantavan tutti Kyrieleisonne per l’influenza de’ taglier mal tondi…, that is Fried nominatives and world maps and Noah’s Ark between two columns were all singing Kyrie eleison under the influence of badly rounded trays…,

and going on in the same vein for ten more verses. Here grammar is OK, but these funny, funny verses are perfect nonsense. 4 Some Medieval philosophers used to criticize their age’s “dialecticians” using exactly this argument.

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certainly not in practice) one can indefinitely keep adding to a sentence newer and newer tertia.5

7.3 About (Above All) Montague Now, adverbial tertia (not nominal ones, or not so carefully) have already been analysed, for example, by Donald Davidson and Richard Montague. Davidson calls them predicates about predicates, or second-order predicates, and these terms aptly describe what adverbs do, i.e., changing a predicate that already is there into something with a more specialized meaning. He made, then, an important contribution. Montague, however, went well beyond this result, and deserves a longer analysis. As I have already said in § 7.1, in this chapter I forget about complex periods and only discuss simple, one-predicate propositions, with no distinction between main and subordinate clauses. Now, Montague wrote at least two essays, “English as a formal language” and “Universal grammar”,6 where he analyses the English variant of this fragment of everyday discourse, completely neglecting subordinate clauses; but he also seems to imply—rightly in my opinion—that what he says there about English can be adapted, with minor adjustments, to any other spoken language. So far, I share his general attitude; but I can’t agree with what he says at the very beginning of his “Universal Grammar”, namely “There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians”.7 To begin with, this is sheer dogmatism, for here a claim only gets asserted, but not argued at all. Moreover, that claim is simply wrong: for, as we have seen in Chap. 5, and especially in §§ 5.1–2, if we take (any) spoken language in all of its uses, going beyond declarative propositions, then the True and the False still have a very important role in it, but are governed by no algebra at all, either Boolean or non-Boolean, for they can both come out of and produce certain features of nondeclarative (and therefore neither true nor false) propositions.8 Now, this point seems to be entirely foreign to Montague: he is extremely good at creating richer and richer formalisms, but also looks completely unaware of the possibility of developing a fruitful informal sort of logical research and of the important discoveries this affords. 5 It

might be objected that the analogy with negation is flawed, for negation can (in principle) be indefinitely repeated while when we add to a sentence a new tertium, it normally is different from all previous ones, and in a dictionary there only is a finite number of entries, so that after a finite (though very big) number of steps we would get stuck, having exhausted all possibilitie; but such argument would be conclusive only if tertia could never exceed a certain number of words, and this limitation looks implausible. One-word tertia are a finite number (otherwise, dictionaries would be infinitely long); but there also are two-words, three-words,…, n-words,… ones. 6 Respectively, pp. 188–221 and 222–246 of R. Montague, Formal Philosophy, edited by Richmond H. Tomasson, Yale University Press, 1974. 7 Formal Philosophy, p. 222. 8 Go back to those sections for examples.

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Not that he doesn’t perceive at all the limits of his research style: sometimes he does, and for example at the beginning of “Universal Grammar” calls “cryptic and unsatisfactory” the theory he is going to lay down in that essay9 ; but if he is unsatisfied with some formalism, the remedy for him is developing a new, more sophisticated one, where every single step of the old, unsatisfactory system can be substituted by a (possibly ramified) sequence of steps, so that if at a certain stage formalization cannot do all we would like it to do, problems can be solved pushing formal complexity further, for example developing a semantic theory where certain variables range not over single formulas but over possible worlds, i.e., infinite sets of formulas.10 Montague was not the first to introduce possible worlds. There had been Henkin, with his maximal consistent sets; Carnap, with his state descriptions, and of course Kripke, who was led to develop his possible-world semantics as a way out of the impossibility of treating modalities by means of truth-functions. No bivalent truth function (but also no m-valued one, for any m ≥ 2) describes the behavior of such terms as “possible” or “necessary”? Well, let us take as primary units possible 9 “Universal Grammar”, in Formal Philosophy cit., 224. He was also planning to write a new, richer,

and more comprehensible treatment of his “universal grammar”, but his sudden, untimely, and tragical death made this impossible. 10 This is a point deserving a long footnote, for it is too important to be neglected, but discussing it in the main text would divert attention from the central subject of this section, i.e., Montague’s vision of everyday discourse. Montague uses a lot possible worlds as basic units of his semantics; now, the “possible worlds” of formal logic are, at root, nothing but maximal consistent sets of first-order formulas, i.e., more exactly, sets containing for every first order atomic formula H either H or ¬H, so that adding any new atomic formula to one of them would produce a contradiction; but this means these “basic units” are infinite sets. Of course, Montague was not the first to use possible worlds as a tool to expand semantics. The first one had been Kripke, who developed his possible-worlds semantics as a way out of the impossibility of treating modalities by means of truth functions. No bivalent truth function (but also no m-valued one, for any m ≥ 2) can describe the behavior of such terms as “possible”, or “necessary”? Well, let us take as primary units possible worlds, i.e., not single truth-value assignments to single atomic propositions, but maximal (and therefore infinite) consistent sets of such assignments, and then, with suitable combinations (I am skipping technical passages), an ultimately extensional semantics for modalities can come out. This attitude seems to have won out, but I am not among its admirers. In fact, semantical ideas of contemporary mainstream logicians often remind me of epicycles in Tolemaic astronomy. According to Tolomeus, planets (i.e., for him, the Moon, Mercury, Venus, the Sun, Mars, Jove, and Saturn) moved circularly in an epicycle around a moving point, the deferent, moving in its turn circularly around a motionless Earth; all these movements were uniform. Tolomeus, an excellent mathematician, succeeded in calculating both for deferents and epicycles relative diameters (periods had been known for a long time in his days, thanks above all to Babylonian astronomers) giving predictions that agreed reasonably well with observation, as it was possible with the naked eye (he lived in the second century A. D.). But then, in a long, long course of centuries, some discrepancies with his predictions would be discovered, again and again, and every time astronomers would react introducing second- or third-order epicycles, i.e., supposing that a planet was not moving directly along an epicycle, but along a smaller circle, say A, and it was A’s center that moved along the epicycle (and the latter’s center did so, in turn, along a deferent—which was moving, at last, around the Earth). Then Copernicus and Kepler came, with a new, simpler hypothesis, and astronomers stopped worrying about epicycles. And now take this very streamlined reconstruction of how formal logic developed in the twentieth century: at first, many believed (see e.g., Tractatus, 5) that all propositions, however complex, could

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worlds, i.e., not single truth-value assignments to single propositions, but maximal (and therefore infinite) consistent sets of such assignments, and then, with suitable combinations (I am skipping technical passages), an adequate, ultimately extensional semantics for modalities can come out. Now, I have serious objections I try to explain in footnote 7.10 against this general attitude, and so against Montague’s method too; but this does not mean I don’t see anything worth appreciating in his ideas. I do, and find many of his results, including some about everyday discourse, very advanced, interesting, and stimulating. The best example (indeed, a paradigm case) is for me this passage: Traditional grammar groups together modifiers of formulas, of verbs of various numbers of places, of adjectives, and of adverbs, and calls them all adverbs. It will be important, however, to distinguish these various sorts of modifiers. Accordingly, we shall speak of adformulas and adverbs, the latter in the narrow sense of modifiers of verbs, and subdivided into adone-verbs and ad-two-verbs according as the verbs modified are of one or two places;11 and we could, in an extension of the present fragment, speak of adadjectives, adad-oneverbs, adad-two-verbs, adad-adjectives, adadad-one-verbs, and adadad-two-verbs as well. (The word very~ would for instance belong to all of these categories).12

Here Montague is distinguishing adverbs modifying, respectively, intransitive verbs (ad-one-verbs), transitive ones (ad-two-verbs), adjectives (adadjectives), adone-verbs i.e., adverbs modifying in turn intransitive verbs (adad-one-verbs), and so on; in other words, he is creating a recursive hierarchy, for any time we have an ad-so and so, we can construct an adad-so and so—in principle, to infinity. Now, in a way all those I call “adverbial tertia” can be found in this list by Montague, since my “tertia” are either adverbs or noun phrases (adverbial or nominal tertia). Nevertheless, I do not think I am simply repeating something that has already been said long ago, for there are two important differences. The first is readability: Montague often is—his own term—“cryptic”, I try not to be so, and believe readability is important. The second is that I also discuss nominal tertia.

be understood as truth functions of atomic ones. But soon it came out that with first-order logic, and more specifically if quantification over n-adic predicates, with n ≥ 1, was allowed, then truth tables were no more sufficient to calculate their truth values; and it was also found, more importantly, that there were consistent statement logics, e.g., the intuitionistic one or Lewis’s S1-S5 systems, such that some of their formulas could in no way be seen as truth functions of any number of truth values. Well, I see some analogy between that old proliferation of epicycles, and possible world semantics. Of course, this is a minority viewpoint; but it is mine, anyway. And I also have another, even more fundamental objection. What such authors as Kripke or Montague do is, basically, building more and more complex structures by combining again and again certain basic cells of Tarskian semantics with each other. Now, in Tarskian semantics all predicates are subsets of a certain entirely given domain, the set of individuals; but sets are (at least in orthodox set theory) entirely given, unchanging entities and so are, as a consequence, their elements, while for me (go back to §§ 3.10-11) there is no such domain of individuals but we construct individual concepts using experience as a raw material, and never stop constructing new ones. 11 That is, respectively, intransitive or transitive. 12 From “English as a formal language”, in Richard Montague, Formal Philosophy cit., pp. 190–191.

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7.4 Those Surprising Nominal Tertia In fact, even Montague left nominal tertia untouched, and we don’t even know whether he was aware of them; but speculating about the reasons of this blind spot would be idle, so I choose not to do it and concentrate instead upon nominal tertia in an entirely independent way. I believe this choice can deliver some possibly embarrassing, but non-trivial results. Now, a funny thing is that so-called old-style “logical analysis”, the one in some countries (I’m afraid not in all) kids in lower secondary school must do, willy-nilly, looking for the (grammatical) subject, the predicate, the object (if there is one), and all so-called “indirect complements” (again, if there are any) of sentences, is very efficient in detecting those things I am calling nominal tertia. I will give examples, as usual. 7.3.1 They made love under a tree. They is subject, made, a transitive verb, is a two-place predicate, love is object. The rest is a complement of place. 7.3.2 He won elections eighteen years ago. The same analysis as in 7.3.1 for the first three words, but here the other three make a complement of time. 7.3.3 He’s come back for simple home-sickness. He, subject; has come back, intransitive verb; and four more words, making a complement of cause. 7.3.4 She’s been recovered for a check-up. She, subject; has been recovered, one-argument predicate, fully saturated by “she”; for a check up, “complementum finis”, or complement of end-in-view. 7.3.5 He sings Verdi like Placido Domingo. He, subject; sings, transitive verb; Verdi, object; like Placido Domingo, complementum modi, translatable as complement of manner (it has to do with the manner something is done). 7.3.6 According to Homer, Achilles killed Hector with his spear, not with his sword. Achilles killed Hector is a complete proposition on its own; with his spear and (not) with his sword are complementa instrumenti, a term translatable possibly as complement of means; according to Homer could be substituted, without any significant change in meaning, by Homer says that…. As usual, in principle examples could go on and on, but, as usual again, insisting too long would be pointless.

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Now, these examples are very different from each other. In some of them the “complement” can be substituted by a dependent proposition (for simple home-sickness— simply because he was home-sick, According to Homer—Homer says that), but in others (under a tree, eighteen years ago, like Placido Domingo, with his spear) this can’t be done—or at least, not in such a straightforward way. The most sensible working hypotheses for a further analysis of these expressions (for which the traditional term “complements” is, in my opinion, perfectly adequate) seem, then, to be the following: (a) (b) (c) (d)

complements can normally be added to a well-formed proposition of everyday discourse leaving it well-formed; some complements cannot be substituted by subordinate clauses, but some others can, so that the boundary between complements and subordinate clauses is blurred.

By the way, traditional grammar textbooks clearly say all this, while I can find no trace of points (b)–(d) in Montague. This seems to me an important drawback.

Chapter 8

Sum Total

Abstract This is a synoptic table in 18 points plus two overarching conclusions: that in everyday discourse there is no algebra of truth values, and that the domains of predicates and proper names are open-ended.

8.1 Recapitulation and Further Perspectives Summing up is the only thing that still has to be done at this point, and I do it first recalling very briefly, in the form of a synoptic table, the main results of this work and then offering an also brief general comment. Some of the results I am going to list have been known for a very long time, but maybe some others are newer. A—THESES ABOUT PROPER NAMES A1—No expression is a proper name, or no proper name, in itself. A2—Whatever is a proper name, is one in a given context. The same goes for whatever is no proper name. A3—Every proper name corresponds, in principle,1 to exactly one object. I am saying “object” and not “entity”, or “thing”, for there are names corresponding to no entity (“the eighth king of Rome”, “the smallest circle”). A4—The same expression can be a proper name of different objects in different contexts (“the winner of the last Tour de France”). A5—An expression can be currently used, both as a proper name or in some other way, if and only if this is compatible with a reasonably efficient communication. B—THESES ABOUT THE STRUCTURE OF PERIODS B1—The main clause of a period can only be true or false if it is declarative. B2—Some dependent clauses have a truth value of their own, others do not. B3—Every grammatical expression can be used in a conceptually correct or incorrect way. Such correctness and incorrectness depend on context. 1 Everybody

knows there are, de facto, indefinitely many Johns and indefinitely many Janes—and, moreover, old ones keep dying and new ones keep being born. However, on this point I am quite traditional: this is a mere accident, and has nothing to do with the logical nature of “John” and “Jane”.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Rigamonti, Logic, Everyday Discourse, and Metaphysics, UNIPA Springer Series, https://doi.org/10.1007/978-3-030-74598-1_8

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B4—When a dependent clause is the object of a propositional attitude, it has a truth value of its own. This truth value is also present when the main clause is not declarative. B6—Other dependent clauses, different from propositional attitudes, also have a truth value of their own… B7—…while others have none. B8—Some dependent clauses have a non-empty overlap with the main one. C—THESES ABOUT TERTIA C1—In a proposition of everyday discourse there can be expressions, tertia, belonging neither to the predicate nor to the terms saturating it. C2—Tertia can be of two sorts: (i) adverbs, that in standard cases neither are nor saturate predicates, and (ii) other sorts of expressions, divided in turn in two: (iia) nouns and adjectives, that in a different context can function as predicates, and (iib) proper names, that in a different context can saturate predicates. C3—Given a well-formed proposition of everyday discourse, one can always add to it a tertium in such a way that the result is in turn a well-formed proposition. C4—When in a well-formed proposition there is a tertium, if we take it away what remains still is a well-formed proposition. C5—Some well-formed elementary propositions of everyday discourse have no tertia. I call such propositions minimal. But there also are two overarching, extremely important theses extending to all topics discussed in this work: In everyday discourse, the true and the false do not form a closed algebra but can be both inputs and outputs of neither-true-nor-false expressions. In everyday discourse the domains of predicates and proper names are open-ended. I have tried to illustrate all these points through a rich choice of examples, and don’t believe they would have all been comprehensible without those examples, but am also aware that this work has, as it were, a ratio with the whole of everyday discourse comparable to that of our human diggings with the whole of this planet. Earth has a ray exceeding 6000 km, while we humans have dug a comparatively very small portion of its surface down to a maximum depth of about 6000 meters. Now, as mining industry has only scratched—happily in my opinion—very lightly our globe, so this research of mine has only explored (but here I am not saying “happily”) a ridiculously tiny fraction of that enormous reality, everyday discourse. I hope this tiny fraction does have some interest, but what remains to explore is far, far bigger. This is good news, anyway, for it means that for all those interested in this sort of research a lot of work remains—and will long remain—to do.

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