Conventionalism in logic: A study in the linguistic foundation of logical reasoning 9783111349497, 9783110995411

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JANUA

LINGUARUM

STUDIA M E M O R I A E N I C O L A I VAN W I J K DEDICATA edenda curat C. H. VAN S C H O O N E V E L D Indiana University

Series Maior,

46

CONVENTIONALISM IN LOGIC A STUDY IN THE LINGUISTIC FOUNDATION OF LOGICAL REASONING

by

CARLO BORROMEO GIANNONI RICE UNIVERSITY

1971

MOUTON THE H A G U E • PARIS

© Copyright 1971 in The Netherlands. Mouton & Co. N.V., Publishers, The Hague. No part of this book may be translated or reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publishers.

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 70-154525

Printed in The Netherlands by Mouton & Co., Printers, The Hague.

To Janet

ACKNOWLEDGMENTS

It is my privilege to acknowledge the debt which I owe to Professor Nuel Belnap, Jr. Without his many helpful suggestions and criticisms, this book would have been woefully inadequate. I am especially thankful for the generosity with which he gave of his time and for his constant encouragement. I must, of course, remain solely responsible for any errors which still remain. I also owe a debt to Professor Bruce Aune, who originally stimulated my interest in the problem of logical conventionalism; and to Professor Wilfrid Sellars, who has affected my philosophical thinking profoundly. I also wish to thank my wife, Janet, not only for her patience and understanding, without which I could not have completed this undertaking, but also for her many stylistic suggestions. A portion of Section IV.A.4 first appeared in my essay: "A Defence of Logical Conventionalism," Ratio XI (1969), 89-101. I wish to thank the editor and publisher for their kind permissions. The Appendix was first published as "A Note on an Impossibility Proof of the Conventionalist Thesis," in Philosophical Studies XXI (1970), 61-64. Finally, I wish to express my gratitude to Rice University for a generous grant that helped to make the publication of this work possible.

TABLE OF CONTENTS

Acknowledgments I. Introduction and Historical Background A. Introduction 1. Character, Content, and Function 2. Preliminary Terminological and Philosophical Remarks . a. Propositions b. 'If . . . then . . Implication, and Entailment . . B. Historical Background 1. The Nature of Logical Truth and Necessary Connection . 2. The Interpretation of Logical Signs 3. The Semantics of Sentences and General Terms . . . . 4. Logical Rules 5. Summary

. . .

7 11 11 11 12 12 13 14 15 23 27 30 35

II. Rules, Conventions, and Entailment A. Introduction B. The Concept of Rule C. Linguistic Rules 1. Linguistic Rules as Conventions 2. Logical Rules of Inference as Linguistic Rules . . . . 3. The Logical Rules and Entailment Statements . . . . D. Quine and the Tortoise E. Implication and Inference F. Related Concepts: Syntactical Rule, Semantical Rule, Metalanguage, and Artificial Language

37 37 40 44 44 48 52 58 62 64

III. Conventionalism, Semantics, and Ontology A. Necessity, Analyticity, and the A Priori B. Exposition of the Conventionalist Thesis C. Criticisms and Replies D. Further Criticisms Based on Semantical and Ontological Theories 1. Truth by Definition

69 69 75 79 87 87

10

TABLE OF CONTENTS

2. Semantical Rules 3. The Modes of Meaning 4. The 'Meaning' Critique of Conventionalism 5. Necessity and Propositions 6. Linguistic Rules and Validity E. Conventionalism in Geometry and Physics

90 96 99 101 106 108

IV. The Spirit of Conventionalism A. Propositions, Meaning, and Truth 1. Quotation Marks and the Translation Argument . . . . 2. Logical Constants and the Meaning of Sentences . . . . 3. Validity 4. Logical Truth and Logical Necessity B. The Conventionalist Thesis Reviewed 1. Final Statement of the Conventionalist Thesis in Logic . . 2. Logical Constants vs. Descriptive Predicates . . . . C. The Conventionalist Thesis Extended 1. Analytic Sentences 2. Synthetic A Priori Sentences 3. Conventionalism in Geometry 4. Conventionalism in Physics 5. Conventionalism in Physics and Logic Combined . . .

112 112 112 117 120 125 129 129 133 136 136 139 141 143 146

Appendix: An Impossibility Proof of the Conventionalist Thesis .

147

.

References Cited

150

Index

153

I INTRODUCTION AND HISTORICAL BACKGROUND

A.

INTRODUCTION

1. Character, Content, and Function F. B. Fitch has made the following comment about the conventionalist thesis in logic: "Some thinkers have advocated the view that all facts are contingent and that what appear to be non-contingent, logically necessary truths, are merely arbitrary conventions about the use of symbols, or are somehow merely the outcome of conventions. . . . This view has never been developed very satisfactorily, or in sufficient detail, in my opinion." 1 This is our opinion as well, and the motivation for this book. We hope to develop this thesis both satisfactorily and with sufficient detail. Specifically, we intend to discuss the following three closely related theses: (A) The thesis that logically necessary truths (logical truisms) are necessary by convention. This is the thesis which is usually termed the 'conventionalist thesis in logic'. (B) The thesis that logically valid arguments are valid by convention. This thesis is seldom discussed in the literature, although, as we shall see, it really comes closer to the heart of the problem than does the first. ( Q The thesis that the logical rules of inference are valid by convention; that is, that they are in some sense linguistic rules. We begin by noting that although the three theses stated above are very similar, they differ in referring to statements of an object language, arguments in an object language, and rules of inference which are formulated in a metalanguage, respectively. It is important for a clarification of the conventionalist thesis to keep these three groups distinct while at the same time being fully aware of the relationship between them. We will want to see, for example, to what extent logicians historically were able to make these distinctions and how they saw the relationships between these groups. In regard to the logical rules of inference, which play such a crucial role in the conventionalist thesis, we will want to know if these rules are language dependent 1

Frederic B. Fitch, Symbolic Logic: An Introduction (New York, 1952), p. 7.

12

INTRODUCTION AND HISTORICAL BACKGROUND

or language independent. Can the logical rules of inference be conceived of as linguistic rules? What is a linguistic rule, and in what sense can a linguistic rule be said to be a convention? Inasmuch as the logical rules govern correct reasoning, we will also want to know how reasoning proceeds as a mental activity. Because the logical truths and rules of inference are intimately bound up with the logical signs of a language (the so-called syncategoremata), we will want to ask the following questions about these signs: Do these signs exist only in language, or do they also exist in thought? If they exist in thought, do they then also exist (or subsist) in the world? Are these signs meaningful alone, or only in conjunction with other signs? If the former, in what way do they have meaning? If the latter, how is it possible for them to be meaningless alone, and yet to somehow mysteriously get meaning when conjoined with other symbols? The Scholastic logicians, among others, addressed themselves to some of these questions, and we will want to see what they had to say in order to contrast it with the view developed later in the book. There are two other related topics which we will consider, (a) Since the logical truisms are necessary, we will want to ascertain what it is for a statement to be necessary; what is the nature of logical necessity? Is the term 'necessary' a term of the object language or metalanguage? Is it sentences that are necessary, or is it propositions or facts or states of affairs that are necessary? (b) Since valid inference is usually characterized as leading from true premises to true conclusions only, we will attend to the question of what it is for a sentence or a proposition to be true. We will be especially concerned with this problem with reference to sentences which involve logical constants. In particular, we will ask the following question: "Are sentences involving logical constants, i.e., molecular sentences, true in the same way that sentences which do not involve logical constants, i.e., atomic sentences, are true?" 2. Preliminary Terminological and Philosophical Remarks a. PROPOSITIONS. There is both a terminological and a philosophical problem connected with the term 'proposition', for it is used in at least three distinct senses: (a) to refer to sentences from the purely formal point of view, that is, as certain grammatically well formed strings of expressions; (b) to refer to sentences as pieces of language but only in so far as they have meaning; and (c) to refer to the meaning of sentences. We will use the word 'proposition' in the third sense, assuming INITIALLY that sentences have meaning in the sense that meaning is a relation between a sentence and a proposition. In order to REFER to the meaning of a sentence - to a proposition - we will use a noun clause. For example, the phrase 'that John is tall' will refer to the proposition which is the meaning of the sentence 'John is tall.' We must emphasize that we make this terminological assumption about the word 'proposition' (and the

INTRODUCTION AND HISTORICAL BACKGROUND

13

word 'meaning') only initially, in order to adapt our vocabulary to that of our principal antagonists. In this sense of the term, it may turn out that there are no propositions. While the terminological problem can be solved by fiat, the philosophical problem is more difficult, for it concerns the nature of propositions. In so far as a proposition is taken to be the meaning of a sentence, and therefore different from the sentence, there are two main views: (a) a proposition is an abstract entity akin to Plato's Forms; it is an extramental, nonmaterial entity which subsists rather than exists; and (b) a proposition is a mental entity; it is the thought or judgment which is expressed by means of a sentence. According to this latter view, there is a parallelism between thought and language. One can either think about the world or talk about it. The difference between the two, however, is that sentences obtain significance by being related to thoughts, while propositions (thoughts) are significant by their very nature. We will initially use the word 'proposition' ambiguously with respect to these positions. There is also a third position, taken by Carnap and Pap, according to which while it is true that propositions exist, that sentences have them as their meaning, and that noun-clauses refer to them, it is not true that they are Platonistic abstract entities or thoughts. That is, while Carnap and Pap accept the conceptual framework of propositions, they reject the traditional ontological characterizations. While we will discuss this position on pages 105-106, for the present we will assume that the use of noun ('that') clauses commits one to the existence of either abstract entities or thoughts as propositions, although little will depend on the particular way in which or even whether they are ontologically characterized. b. 'IF . . . THEN . . . ' , IMPLICATION, AND ENTAILMENT. Again we have both a terminological and a philosophical problem. The terminological problem arises from the fact that each of these terms has multiple uses. For example, while the blanks in an 'if . . . then . . .' sentence are to be filled in with sentences, the resulting sentence can be used to mean a number of different things. In particular, an apparently object language sentence, such as 'If A then B', can be used to mean 'If 'A' is true, then 'B' is true', or " A ' implies 'B", which are metalinguistic statements. Similarly, 'A implies B' can be used to mean either " A ' implies 'B" or simply 'If A then B'. Moreover, the ambiguous use of these sentences would suggest either that they are synonymous or that they have a significant commonality of core meaning. Quine has attempted to distinguish between conditional statements and implications on the grounds that 'if . . . then . . .' is a sentential connective while 'implies' is a relation word, and he concludes from this that there is no danger in using the truth-functional ' r>', for in using it we are symbolizing 'if . . . then . . .', not 'implies'. We are not, therefore, making implication truth-functional. But is 'implies' not used sometimes as a sentential connective, particularly by logicians, as well as at other times as a relation word? If so, Quine's distinction collapses; however, even

14

INTRODUCTION AND HISTORICAL BACKGROUND

if Quine were right that 'implies' is only used correctly as a relation word, does it follow then that the sentential connective 'if . . . then . . . ' is truth functional? The fact that 'if . . . then . . s t a t e m e n t s and implication statements are used virtually interchangeably would argue against that. Regardless of what the correct grammatical form for these sentences is, the logical analysis of them should show that they are in some significant way synonymous. While I disagree with Quine with regard to the logical significance of distinguishing between 'if . . . then . . . ' and 'implies', I agree that there is a semantical difference. More precisely, a semantical difference can be drawn. 'Implies' and 'if . . . then . . . ' can be used to make three semantically distinct statements. Carnap has made this distinction by distinguishing between object language statements, semiotical metalinguistic statements, and nonsemiotical metalinguistic statements2 the latter two being statements about sentences and propositions, respectively. In order to standardize our vocabulary, we will choose one locution for each of these semantically distinguishable statements. Specifically, we will use the following terminological convention: when a statement is used to say something about propositions, we will use either the locution 'that A entails that B' or 'that A implies that B'; when a statement is used to say something about sentences, we will use the locution " A ' entails ' B " or " A ' implies 'B"; and when a statement is used to say something about the world, we will use the locution 'if A, then B'. We will also follow the convention of using capital letters as abbreviations for particular sentences and small letters as sentential variables, unless otherwise noted. The object language-metalanguage distinction can also be extended to categorical statements, such as, 'All men are mortal', corresponding to which we have the semiotical and nonsemiotical metalinguistic statements: "man' entails 'mortal" or " x is a man' entails 'x is mortal" and 'humanity entails mortality', respectively.

B.

HISTORICAL BACKGROUND

Before we begin our discussion of the conventionalist thesis in logic, it will be useful for us to examine a few of the crucial concepts from a historical viewpoint. In particular, we will discuss these four notions: (a) the nature of logical truth and necessary connection; (b) the interpretation of logical signs; (c) the semantics of sentences and general terms; and (d) logical rules. We will limit our discussion to the following logicians and philosophers who have said something relevant about one or more of these points: Plato, Aristotle, the Stoics, the Scholastics, Hobbes, Antoine Arnauld, John Stuart Mill, Frege, and Russell. This is not intended as an exhaustive study of the philosophy of logic of these men, but rather as background with which the later defense of the conventionalist thesis and accompanying philosophy of logic can be compared. 2

Rudolf Carnap, Introduction to Semantics (Cambridge, 1942), Section 17.

INTRODUCTION AND HISTORICAL BACKGROUND

15

1. The Nature of Logical Truth and Necessary Connection In the following passage from Plato, we find the foundation for a theory of necessary connection which to this day stands as one of the major opponents of conventionalism: " A n d the man who can do that discerns clearly one Form everywhere extended throughout many, where each one lies apart, and many Forms, different from one another, embraced from without by one Form; and again one Form connected in a unity through many wholes, and many Forms entirely marked off apart." 3 A single Form may be related to many other Forms, as the Form Bachelorhood is related to the Forms Unmarriedness, Adulthood, and Humanity, although these Forms are not so related to each other. Moreover, the truth about these relations is eternal and necessary, for the Forms are unchanging. The Philosopher, by means of the Dialectic, is able to know about these relations between Forms, and, thereby, to know the necessary truths. The Forms are unchanging, extramental, extraempirical, extralinguistic entities, and necessary connections are the relations between them. These relations are DISCOVERED by means of the Dialectic or by intuition; they are not created by man. The two essential elements of any nonconventionalist approach are here: (a) the necessary connections are independent of man; and (b) they are discovered by man. The different nonconventionalists differ only on the nature of the relata, denying either that the relata are extramental or extraempirical or both. The relevance of Platonism to the conventionalist thesis IN LOGIC, however, is not completely clear. Plato did not deal with logically necessary connections, with the connections necessary to make such logical truisms as the Law of the Excluded Middle true, and it is not at all clear how one would go about being a Platonist in logic. Platonism as we receive it from Plato stands as a nonconventionalism in regard to a limited class of necessary statements, viz., those necessary statements which are of the form 'All A are B'. Aristotle's theory regarding this same kind of statement is, of course, similar to Plato's. The difference between their viewpoints arises over the nature of the relata. For Aristotle the relata are not extraempirical; the existence of the universals is dependent on the particulars in which they are 'imbedded'. The relation between universals becomes in Aristotle's theory a relation of essential predication. A certain kind of object is necessarily also another kind of object because it is of the essence of this kind of object. A l l men are necessarily mortal because it is of the essence of man to be mortal. Necessary connection is a relation between the essential forms of an object. Perhaps the following quotation from Aristotle will make clear what he has in mind: " N o w attributes attaching essentially to their subjects attach necessarily to them; for essential attributes are either elements in the essential nature of their subjects or contain their subjects as elements in their 3

Plato, Sophist 253D

16

INTRODUCTION AND HISTORICAL BACKGROUND

own essential nature. . . . All attributes must inhere essentially or else be accidental, and accidental attributes are not necessary to their subject." 4 We have in both Plato and Aristotle a theory of REAL definition, which give the ESSENTIAL characteristics of an object. It is important to note, however, that Aristotle does not confine necessary statements to real definitions. He makes it a point in Book II, Chapter 3, of the Posterior Analytics to prove that definitions cannot be demonstrated, while only necessary truths are the conclusions of demonstrations. In the following passage he makes the point that demonstrated truths are informative while definitions are not: "Never yet by defining anything - essential attribute or accident - did we get knowledge of it." 5 Aristotle's theory of necessary connection - of essential predication - is wider than his theory of real definition. An attribute can be of the essence of an object, even if it is not a defining characteristic of the object. It is not clear that this is the case in Plato's theory, and we seem, therefore, to have in Aristotle a foundation for synthetic necessary truth which is not found in his predecessor. Our concern, however, is specifically with logical truth, and once again Aristotle has something to say, although it is not easy to discern the import of his arguments. He sets himself the task in the following way: "It is clear, also, that the philosopher, who examines the most general features of primary being, must investigate also the principles of reasoning. . . . So that he who gets the best grasp of beings as beings must be able to discuss the basic principles of all being. . . . And the surest principle of all is that about which it is impossible to be mistaken." 6 This principle is the Principle of Non-Contradiction. One would gather from this passage that Aristotle considered the principles of reasoning to be also basic PRINCIPLES OF BEING; i.e., metaphysical principles. Aristotle begins his discussion of the Law of Non-Contradiction by noting that while it cannot actually be demonstrated, a 'negative' demonstration is possible. In order to give a negative demonstration, the opponent of the law must admit to saying something significant (meaningful). If the person will make such an admission, he is committed to holding the Law of Non-Contradiction. Aristotle argues that if one denies this law, one has as a result that everything is true (and false). The man who denies the law says nothing definite "for he says neither 'yes' nor 'no', but 'yes and no'; and again he denies both of these and says 'neither yes nor no' ".7 Nothing significant is said. As long as the argument remains on this level, however, there is no reason to believe that these principles of reasoning are anything more than principles of significant speech. Aristotle does bring a metaphysical element into his argument when he asserts that a denial of the Law of Non-Contradiction entails a denial of 4

s 6 7

Aristotle, An. Post. I.6.74b5-13. Ibid., II.3.90M4-15. Aristotle, Metaphysica IV.3.1005b7-12. Aristotle, Metaphysica IV.4.1008a31-33.

INTRODUCTION AND HISTORICAL BACKGROUND

17

substance and essence - all attributes become accidental. For if some attributes were essential, and if a person were essentially a man but only accidentally not a man, then the law would hold for the essential attributes of an object. But a person cannot be essentially a man and also not a man, for the assumption is that a term such as "being a man" means something different than "not being a man". Therefore, Aristotle does seem to think that the substance-accident distinction, a metaphysical distinction, depends on the Law of Non-Contradiction. This argument does not seem to be particularly relevant, however, since denying the Law of NonContradiction wipes out not only this distinction but any other as well. Not only is the substance-accident distinction eliminated, but all significant discussion as well. There is a striking similarity between Aristotle's argument and a much more recent one. I quote here from Strawson: "The general and standard purpose of making statements is to communicate information, to state facts. The purpose is frustrated when something false is said. It is also frustrated, though in a quite different way, when a man contradicts himself." 8 When a man contradicts himself, we say that he has said nothing. Strawson, a recent conventionalist, uses the same argument to uphold the Law of Non-Contradiction as is used by Aristotle. The Stoics play an interesting role in our study of the nature of logical truth, for they held a theory of propositions ilekta) which, despite their materialistic prejudice, was quite Platonistic. Propositions are that which are signified by sentences, and the lekta in general are the significates of signs. Moreover, in the first and foremost sense of the word 'true', it is propositions that are true for the Stoics, and this specifically includes molecular propositions, as well as atomic propositions. While the semantical and ontological aspects of their theory will be discussed more fully in sections 2 and 3, we would like to point out here that this theory prevents any conventionalistic interpretation of logical truth. If it is nonmaterial propositions, atomic and molecular, which are properly speaking true, then they cannot be true by convention, for convention cannot in any way alter the character or create the character of nonmaterial entities. If a nonmaterial proposition is true, it is by virtue of its nature, and not by virtue of convention. While there was some disagreement among the Stoics regarding the theory of propositions, there was far more disagreement in regard to the nature of NECESSARY truth. In particular, they debated whether or not necessarily true propositions are necessarily true in a fundamentally different way than contingently true propositions. This disagreement over the nature of necessary truth also infects their notion of logical validity, for, as Benson Mates points out, "probably the Stoics agreed that an argument is valid if and only if the corresponding conditional is necessarily true." 9 Philo held a position closely resembling a modern view. '"The necessary' is defined as 'that which, being true, is in its very nature not susceptible of false8 9

Peter Strawson, Introduction to Logical Theory (London, 1952), p. 22. Benson Mates, Stoic Logic (Berkeley, 1961), p. 76.

18

INTRODUCTION AND HISTORICAL BACKGROUND

hood'." 10 On the other hand, Diodorus defined necessity as "that which, being true, will not be false." 11 A necessary statement for Diodorus is one which is always true, or is true for all times. Necessity becomes identified with universality. Again, anticipating our discussion later, we might note here that Diodorus' definition of necessity bears a close relationship to Quine's definition of logical truth. Both are based on the universal truth of a statement. The difference between the two is that Quine holds that a logical truth is universally true for all values of the descriptive constants, while Diodorus holds that a necessary statement is universally true for all time, where the variable in this case is time. In either case necessity is universal truth, rather than a special kind of truth. A necessary truth is not necessary because it CANNOT be otherwise, but rather because it is IN FACT not otherwise for a certain range of values. The theory of William of Ockham forms an interesting transition between Stoic Platonism and the theory which we will expound later in the book. He characterizes logic in this way: "Logic is distinguished from the real sciences in the following manner. The real sciences are about mental contents which stand for things; for even though they are mental contents, they stand for things. Logic, on the other hand, is about mental contents that stand for mental contents." 12 This is especially interesting when it is combined with Ockham's conceptualism, for the mental contents, the universals, exist only in the human mind. Logic is a study of mental contents using second-order mental contents in the process, but it is a study of mental contents not qua psychic states, but rather qua modes of signification, that is, in so far as the mental contents have an extramental reference or signification. The logical relations are no longer man-independent relations which are discovered. The relata of the relations are mental contents, while the relations themselves are "fabricated by the mind, not outside of itself as artificial things are fabricated, but within itself".13 As he says, logic treats of "those things which cannot exist without the reason".14 Thus we have the first major step toward conventionalism, for it is of the essence of the conventionalist treatment of logic that the logical relations, such as entailment, are created by convention - they are fabrications. This step toward conventionalism is bolstered by Ockham's insistence that the logical constants are syncategorematic, that is, they do not have signification by themselves. "So likewise a syncategorematic term does not signify anything, properly speaking; but when added to another term, it makes it signify something or makes it stand for something or things in a definite manner, or has some function with regard to a categorematic term." 15 This is then combined with his theory of formal consequence 10 11 12 18 14 15

A s quoted in Mates, ibid., p. 40. A s quoted in Mates, ibid., p. 37. William of Ockham, Philosophical Writings, trans. Ph. Boehner (London, 1957), p. 12. A s quoted in Ernest A. Moody, The Logic of William of Ockham (New York, 1935), p. 34. A s quoted in Moody, ibid. William of Ockham, Philosophical Writings, p. 51.

INTRODUCTION AND HISTORICAL BACKGROUND

19

as a relation which holds by virtue of the formal structure of the propositions, and not by virtue of the categorematic terms involved.16 Since the syncategorematic terms exist as mental contents,17 but do not signify anything, and since the formal consequences are true by virtue of the syncategorematic terms involved, the relation of formal consequence is purely a mental relation. In Ockham's terminology, the term 'consequence' is a term of second intention for it stands for terms - mental contents - rather than things. Ockham, however, would have had to have taken a stand on the question of why the truth of logical consequences depends only on the syncategorematic terms for us actually to attribute a conventionalism to him. What is the relationship between the mental contents signified by the logical words and true formal consequences? What are true formal consequences true statements about? We can get more or less definite ideas about his approach to categorematic terms. While Ockham denies that there are any universals independent of the mind (and language), the universals in the mind are not free creations of the human mind; they have an objective basis. For example, he says that "we can change the designation of the spoken or written term at will, but the designation of the conceptual term is not to be changed at anybody's will." 18 There is nothing conventional about categorematic mental contents; they are determined by reality and are the same in everyone. The categorematic mental contents are NATURAL signs of what they signify for they signify what they signify by their very similitude with their objects. It is for this reason that they cannot be treated conventionally. Syncategorematic terms, however, do not, properly speaking, signify nor can they be called natural signs. Can one say that by their very nature they change the signification of categorematic terms as they do? These are problems of detail to which Ockham does not address himself. Nevertheless, one can already see how both a conventionalist and a nonconventionalist theory could be developed within Ockham's framework. The direction in which one develops the theory depends on the treament given to the syncategorematic terms. Are they nonconventional signs or are they created by the human mind? Moody discusses two ways that Ockham distinguishes of determining the valid modes of arguing. One of them is of particular interest inasmuch as it is relevant to our discussion of Aristotle as well as to our later discussions. According to Moody, Ockham says that the valid forms of the syllogism can be distinguished from the invalid forms "inductively, since no instance of the valid forms can be found in which a false conclusion follows from true premises".19 This method depends on "judgments concerning the truth or falsity of actual propositions which, though generally conceded to be true, are not known to be true in the strict and 18

See Philotheus Boehner, O.F.M., Medieval Logic: An Outline 1250 to c. 1400 (Chicago, 1952), pp. 57-58. 17 See William of Ockham, Philosophical Writings, p. 51. 18 Ibid., p. 48. 18 Moody, Logic of William of Ockham, p. 213.

of Its Development

from

20

INTRODUCTION AND HISTORICAL BACKGROUND

demonstrative sense". 20 A s I understand the argument, in order to use the inductive procedure, we must be able to judge that the conclusions of a valid inference are never false when the premises are true. But to judge that the conclusion is true or false in the "strict and demonstrative sense", we must deduce it from premises which are better known. Therefore, judgment that the proposition is true or false depends on a prior knowledge that a certain inference is valid. It follows then that validity is prior to truth for we determine truth by making valid inferences from premises known to be true. This is contrary to the approach common today, since we would say that validity depends on and is determinable by the truth or falsity of the premises and conclusion of an argument. That is, we can determine that an argument is invalid by noting that it leads from true premises to a false conclusion where the falsity of the conclusion is determinable independently of the validity of the inference. But, if validity is prior to truth and if the determination of truth depends on the validity of the inference (as it seems to be in the case of both Ockham and Aristotle), this has important consequences. We see a sense in which the logical rules of inference could be seen to be valid by convention. If we say not merely that the DETERMINATION of the truth of the conclusion of an argument, but the very truth of the conclusion depends on the validity of the inference, then there is a sense in which the logical rules of inference can be said to create the truth of the conclusion. Whichever argument form we choose to be valid, the conclusions will be true (if the premises are true) by virtue of the argument form being valid. If in fact the truth of the conclusion is created by the choice of valid argument form, there seems to be no reason for choosing one argument form as valid over another. The transition from Aristotelian nonconventionalism to conventionalism turns on transferring an inability to determine truth into a lack of truth. In line with the Logical Empiricist verifiability criterion of meaningfulness, one might argue in favor of this transfer by saying that a proposition which cannot be determined to be true or false is neither true nor false. We have in Hobbes' Fourth Objection to the Meditations of Descartes the first statement of the conventionalist thesis in anything like the form it takes today. "But what shall we now say, if reasoning chance to be nothing more than the uniting and stringing together of names or designations by the word is? It will be a consequence of this that reason gives us no conclusion about the nature of things, but only about the terms that designate them." 2 1 Reasoning is not from one set of facts to another but from one string of names (sentences) to another. Reasoning is aboul words, not about things. A s Hobbes' statement stands as a model from which further conventionalist theses have been developed, so Descartes' reply is a model for arguments against the thesis. Descartes argues as follows: " W h o doubts whether a Frenchman and a 20

Ibid., pp. 213-214.

In René Descartes, Objections Urged by Certain Men of Learning against the Meditations: with the Author's Replies, Third Set of Objections, Objection I V . 21

Preceding

INTRODUCTION AND HISTORICAL BACKGROUND

21

German are able to reason in exactly the same way about the same things, though they yet conceive the words in an entirely diverse way?" 22 This is an early statement of the present-day Translation Argument. How can reasoning merely be about words, since the words of a Frenchman and a German are different and yet their reasoning is the same? This is an important difficulty for the conventionalist thesis, which usually depends on discussing sentences rather than the thoughts signified by them, and it is a problem which we will attempt to answer later. While a great deal of attention has been paid to John Stuart Mill's theory of the nature of mathematical truth, if only to refute it, less attention has been paid to his theory of deductive inference, which is equally interesting. Since Mill's theory about mathematical truths is that they are generalizations from experience, empirical statements about the world, one would expect him to have the same type of theory about the truths of logic, but this is not the case. Mill considers any proper inference to be a progress from the known to the unknown, a view which conforms with the logical views of Aristotle noted above. The question then arises as to whether or not deductive inference is a progress from the known to the unknown; if not, it is a mere petitio principii. But Mill notes that it is "universally allowed that a syllogism is vicious if there be anything more in the conclusion than was assumed in the premises".23 If Mill's statement is correct, why is the syllogism thought of as a means for gaining knowledge? One might argue, for example, that by means of a syllogism I can come to know that I am mortal, even though I am not yet dead. I cannot come to know this by observation, at least not yet. But I can come to know it by means of the syllogism, for I can deduce that I am mortal from the premises that all men are mortal and that I am a man. Mill argues that I am only as certain of the conclusion that I am mortal as I am of the premise that all men are mortal, and to be absolutely certain of this, I already must be certain that I, among other people, am mortal. Therefore, before I can know that the premise is true, I must already know that the conclusion is true. Mill does admit that there is a way out of this difficulty. One could conceive of the premise that all men are mortal not as asserting that every man is also mortal, but as asserting a relation between two universals. That is, if the statement 'All men are mortal' is conceived of as meaning 'the universal HUMANITY entails the universal MORTALITY' or simply 'humanity entails mortality', then the inference from the statement that I am a man to the statement that I am mortal indicates that there is an intercommunity between the universals and the individuals, the critical inference being from the statement 'humanity entails mortality', a second-order proposition about universals, to the statement 'every man is mortal', a first-order statement about men. This would be an inference from the known to the unknown, from the a priori to the empirical. Mill, however, denies the theory of universals 22 23

Ibid., Reply to Objection IV. John Stuart Mill, A System of Logic, 8th ed. (London, 1911), p. 120.

22

INTRODUCTION AND HISTORICAL BACKGROUND

involved in this justification of syllogistic inference, and whether or not we accept the theory of universals, we can still go along with Mill's reasoning as long as we confine ourselves to inferences which are solely from first-order statements to firstorder statements. If we thus confine ourselves, the rest of Mill's argument is of special interest. All inference for Mill is from a particular which is known to a particular which is not known; all inference is inductive. How are we, then, to account for the inference from generals to particulars; is this not a deductive inference? For Mill, the real inference involved is the inference from a particular to a general proposition, and it is enlightening to see how he considers the general proposition. As he says, "Generalization is not a process of mere naming; it is also a process of inference. From instances which we have observed, we feel warranted in concluding that what we have found true in those instances holds in all similar ones, past, present, and future." 24 Inference proper for Mill is a progress from what we know to other similar cases that we do not know about; it is a prediction about them. He describes the logical role of the general proposition in the following passage: "We then, by that valuable contrivance of language which enables us to speak of many as if they were one, record all that we have observed together with all that we infer from our observations in one concise expression, and have thus only one proposition, instead of an endless number, to remember or to communicate."25 General propositions are a contrivance of language; we, as users of the language, have contrived these general propositions to record together in a single proposition and endless number of propositions both which are known and which are inferred - predicted. The movement from the general proposition to the particular is merely a process of "deciphering our own notes".20 General propositions are linguistic contrivances which are deciphered via the logical rules of inference. This is a particular case of a general thesis which we will defend later in the book. We will assert that all statements involving logical words, such as 'all', are linguistic contrivances, and the logical rules of inference are rules for deciphering them. The inferences to new facts which are contained in molecular propositions are drawn out of them by the logical rules of inference. Frege's views on the nature of logical truth are, for the most part, in accord with those of the Stoics. For example, he says: "What is true is something objective and independently of the judging subject." 27 The objective is that which exists independent of man; in the case of sentences and judgments, the objective is the proposition. As has been noted by Mates,28 there is a considerable parallel between the semantics of the Stoics and Frege, and as a result, their theories of the nature of logical truth also coincide. It is interesting to note that Frege explicitly disavows 24

Ibid., p. 122. Ibid. 28 Ibid. 27 Gottlob Frege, The Basic Laws of Arithmetic, 1964), p. 15. 28 Mates, Stoic Logic, Chap. 2. 25

trans, and ed. Montgomery Furth (Berkeley,

INTRODUCTION AND HISTORICAL BACKGROUND

23

the kind of argument used by Aristotle and Strawson noted above. In regard to the argument that we must accept the laws of logic, "unless we wish to reduce our thought to confusion and finally renounce all judgment whatever",29 Frege says: "What is given is not a reason for something's being true, but for our taking it to be true."30 Frege was quite aware that such arguments do not tell us WHY a logically true statement is true, and it is for this reason that the same argument could be used by a conventionalist and nonconventionalist alike. It is difficult to get either a well defined theory of logical truth or a well developed theory of logical truth from Bertrand Russell. He does, however, make comments in various books which have a bearing on the question. In The Problems of Philosophy he makes a characteristically Aristotelian statement about logical truisms. He says: "The belief in the law of contradiction is a belief about things, not only about thoughts. . . . It is a belief that if a tree is a beech, it cannot at the same time be not a beech." 31 He becomes more Platonistic when he goes on to say the following: "The fact seems to be that all our a priori knowledge is concerned with entities which do not, properly speaking, exist, either in the mental or in the psysical world."32 Later he makes the following comment in the Introduction to the Second Edition of the Principles of Mathematics about the nature of logic: "Logic becomes much more linguistic than I believed it to be at the time when I wrote the 'Principles.'"33 At this point he seems to be inclining toward a conventionalistic theory, with its emphasis upon the linguistic character of logical truths; however, on the next page, in regard to the decision as to whether or not to accept the axiom of infinity and the multiplicative axiom, he says: "I cannot myself regard such a matter as one to be decided by our arbitrary choice. It seems to me that these axioms either do, or do not, have the characteristic of formal truth which characterizes logic."34 He goes on to admit, however, that there is a problem in defining what one means by saying that a statement is true by virtue of its form. In any case, it is apparent that he has not really made up his mind as to which path to follow. 2.

The Interpretation

of Logical Signs

In both this section and the next we will be concerned with the semantics and ontology of the various logicians, confining ourselves in this section to their views on the logical constants. These are particularly relevant in any discussion of the nature of logical truth, since logical truths always contain logical constants, and the logical rules of inference, with minor exceptions, always involve either a 29 30 31 32 33 34

Frege, Basic Laws, p. 15. Ibid. Bertrand Russell, The Problems of Philosophy (New York, 1959), p. 89. Ibid., pp. 89-90. Bertrand Russell, The Principles of Mathematics, 2nd ed. (New York, 1937), p. xi. Ibid., p. xii.

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INTRODUCTION AND HISTORICAL BACKGROUND

premise(s) or conclusion, or both, that contain a logieal constant. In the next section we will be concerned with their semantics of sentences and general terms. The first group of logicians to analyze the logical words in any detail was the Stoics. They apparently had extended debates regarding 'if . . . then . . .', and they were able to differentiate two basic types of 'or', the exclusive and the inclusive. They also make a distinction between 'since', 'because' and 'if . . . then . . .'. For our purposes there are two points to make about the Stoic discussions of the logical words. First of all, propositions whose sentences involve logical words were regarded as being composed of the propositions which are its parts by the logical connectives. For example, Sextus Empiricus says, "Molecular propositions are such as are, as it were, 'double', and are composed from two occurrences of the same proposition or from different propositions, and are composed by means of a connective or connectives."35 They view the molecular PROPOSITIONS as being complex entities with the logical connectives as part of the compound. This is indicated further by the fact that a molecular proposition can be composed from two occurrences of the same proposition. But it is not clear to me how a single proposition can 'occur' twice. One might also ask if the SAME proposition can occur both by itself and in a compound with another proposition, or is it different occurrences which occur in the different propositions? But then the first question reappears: How can a proposition have different 'occurrences'? I think this is the kind of problem of detail which a Platonistic approach to logical truth confronts and until this is solved, the Platonistic approach cannot even be evaluated. Another characteristic of the Stoic approach to the logical connectives is their 'formalistic' approach; that is, they hold that a molecular proposition is always expressed by a molecular sentence. A logical junction in the proposition is always expressed by a logical connective in the sentence. "A molecular proposition is always marked by the occurrence of a connective."36 The Scholastic discussion of the logical words centers primarily around words, such as 'all', 'some', and 'none'. It is these terms specifically which are known as the syncategorematic terms, for they are treated as 'co-predicates', which does not seem to be possible for the propositional connectives.37 They make the same distinction between the vocal, written, and mental term for these terms as for the categorematic terms. The syncategorematic terms are, therefore, not in any sense specifically linguistic entities. Neither are they in any sense extramental. The syncategorematic terms do not signify any object or objects. More specifically, the syncategorematic terms 'function' in a proposition. They do not have meaning alone, but rather they serve to fix the meaning of the proposition as a whole. At this point two general tacks are taken by the Scholastics; one is the formal 35

A s quoted in Mates, Stoic Logic, p. 96. Mates, Stoic Logic, p. 32. 37 Precisely this point is made by Philotheus Boehner, "Ockham's Theory of Supposition and the Notion of Truth", Franciscan Studies VI (1946), reprinted in his Collected Articles on Ockham, ed. Eligius B. Buytaert (St. Bonaventure, N.Y., 1958), pp. 266-67. 3

«

INTRODUCTION AND HISTORICAL BACKGROUND

25

approach and the other involves their theory of supposition.38 According to the formal approach, a distinction is made between the form and matter of a proposition. The matter of a proposition is given by the categorematic terms; these determine what the proposition is about. The syncategorematic terms determine the form of the proposition; they delimit the kind of proposition which is being asserted, that is, the quantity and quality of the proposition - whether it be universal or particular, affirmative or negative. These are the formal characteristics of a proposition. A valid inference, then, is an inference which is valid by virtue of the forms of the propositions involved, and independent of their matter. The emphasis of the second approach to the logical words is on what is asserted by a proposition, rather than on the kind of proposition it is. While the matter of a proposition tells us what the proposition is about, it does not tell us what the proposition says about it. The syncategorematic terms give us the assertion or denial part of a proposition and also tell us the extent of the assertion or denial. For example, according to Ockham, a proposition like "No men are mortal" is about men and mortals. The 'no' indicates that there is a disparity between what are men and what are mortals. It also indicates the extent of this disparity; it is not merely that someone who is a man is not a mortal or that some who are men are not mortals, but that no one who is a man is mortal. It is only with the syncategorematic terms, such as 'no' and 'are', that the subject and predicate of a proposition are formed into an assertion (or denial). The subject and predicate alone do not say anything; they are merely names. While Frege does not seem to talk about the logical constants per se, he does talk about functions, and, in particular about truth functions. The expression of a truth function can be considered to be the logical word together with appropriate blanks to indicate where the arguments go. There is, therefore, a close relationship between the expression of a truth function and some logical constants. The truth function approach is somewhat limiting, however, because there are molecular statements whose truth is not a function of the truth of the atomic statements of which it is composed. But what is a truth function for Frege? He says it is incorrect to identify functions and their expressions, "because a function is here represented as an expression, as a concatenation of signs, not as what is designated thereby".39 But he also notes that we must be careful in calling the function the denotation of the expression, for the function is usually written, for example, 'x2', and we must not think of the 'x' in 'x2' as itself denoting a number, 'x' merely marks a gap in the expression, and the corresponding function is said to be 'unsaturated'. The essence of a function manifests itself in a connection it establishes.40 From these remarks we can draw one positive and one negative conclusion about Frege's views on functions. We can be certain that Frege wants to distinguish 38 38 40

See Boehner, Medieval Logic, p. 20. Frege, Basic Laws, p. 33. Ibid.

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INTRODUCTION AND HISTORICAL BACKGROUND

between the expression of a function and the function itself, between the logical constant and what the logical constant stands for. On the other hand, we cannot be certain that Frege would want to admit that his functions are abstract entities. For instance, one could interpret Frege to be asserting that a functional expression is descriptive of a relationship that exists between the arguments of the function and its values. Just as the use of descriptive predicates does not commit us to the existence of universals, although we can use them to make an assertion about a state of affairs, so the use of functional expressions may not commit us to the existence of functions as abstract entities, although it may be the case that they are used to make an assertion about numbers or propositions. The question is whether or not the connection established between extralinguistic entities by a function is to be considered as an entity over and above the entities connected by the connection. Frege does not seem to be explicit on this point. As we noted in our discussion above, Bertrand Russell does not have a consistent view of the nature of logical truth; this is likewise true of his views on the logical constants. He says, for example: "Not even the most ardent Platonist would suppose that the perfect 'or' is laid up in heaven, or the 'or's' here on earth are imperfect copies of the celestial archetype. . . . Logical constants, therefore, if we are able to say anything definite about them, must be treated as part of the language, not as part of what the language speaks about." 41 These are characteristics of the conventionalistic-linguistic approach to logic. On the other hand, Russell is willing to assert that there are GENERAL FACTS, such as "All men are mortal." If general facts do exist, 'all' must in some sense be part of the extralinguistic fact, and he does take the general fact to be extralinguistic. Russell makes this claim, because the world cannot be described completely in terms of particular facts alone. There is a difference between the conjunction of all facts, such as "Socrates is a man and mortal", and the general fact that all men are mortal, for the general fact also contains the fact that Socrates, Plato, and so forth, are all the people in the world, and this is not contained in any of the particular facts. It is interesting to consider how Mill would approach this question of general facts. He would deny the existence of general facts and argue as follows: a general statement is merely a shorthand notation for a large number of inferences from particulars to particulars. It does not assert a fact at all, and, in particular, it does not assert that Socrates, Plato, and so forth, are all the people in the world. Moreover, it does not assert either that Socrates is a man or that Socrates is mortal. Rather, it allows us to infer that Socrates is mortal if we know that Socrates is a man. While Mill would agree with Russell that a general statement is not equivalent to a conjunction of particular statements, he would not agree that it is a conjunction of particular statements PLUS another statement. In a sense, he would say that a general statement is not a statement at all, but a contrivance of language. We have, then, historically three major views regarding the nature of the logical 41

Russell, Principles,

pp. ix-xi.

INTRODUCTION AND HISTORICAL BACKGROUND

27

signs: (1) the Stoic-Platonistic view that they exist as part of molecular propositions where the latter are regarded as abstract entities; (2) the Scholastic view that they are syncategorematic co-predicates; and (3) Mill's view that they are linguistic contrivances. It is this latter view which we will develop in detail. 3. The Semantics of Sentences and General Terms Most of Aristotle's semantic theory is presented in the first few sections of his work De Interpretatione. Written words are the signs of spoken words, and spoken words are the signs of affections or impressions of the soul. The signification of spoken and written words is a matter of convention. The words vary from one race to another, but "the mental affections themselves . . . are the same for the whole of mankind".42 He does not say that mental affections themselves are signs, but he does say that they are "representations or likenesses, images, copies" 43 of objects. Presumably, by object Aristotle means the secondary substance, the genus and species of a particular object, rather than the object itself in its entirety. It is interesting to note that Ockham agrees enirely with this semantic theory insofar as it is stated in De Interpretatione. He would even agree that the mental affection is a representation or likeness of the object. For this reason, P. Boehner characterizes Ockham as a 'realistic conceptualist'.44 It is only with Aristotle's specifically ontological theory of secondary substances that Ockham is in disagreement. Aristotle makes a distinction between simple and composite propositions which bears a resemblance to the current distinction between atomic and molecular propositions. A simple proposition is one which affirms OR denies some one thing of another, while a composite proposition is composed of simple propositions.45 Although Aristotle differs from the current theory in treating negative statements as simple, a more fundamental difference lies in the fact that he considers statements, such as "every man is white" and "some man is white", to be simple statements. In both of these statements white(ness) is affirmed of man (humanity). The subject is one thing, but universal, rather than individual, and one thing is affirmed of it. The two propositions stated above differ in that the first is universal IN FORM while the latter is particular in form, but they are both simple propositions. One can see how the movement from a realistic approach to universals to a nominalistic one has brought with it a corresponding change in the logical treatment of universal and particular statements, for with the elimination of universals they could no longer be interpreted as referring to universals. These statements and in particular the words 'every' and 'some' had to be reinterpreted. The latter words 42

Aristotle, De Int. I.17a7-9. Ibid., I.16a9. 44 See his "The Realistic Conceptualism of William Ockham", Traditio IV (1946), 307-35, reprinted in his Collected Articles on Ockham. 45 Aristotle, De Int. V.17a27. 4S

28

INTRODUCTION AND HISTORICAL BACKGROUND

came to function more like the words 'and' and 'or' rather than syncategorematic terms (co-predicates). The logical connectives of modern logic are not, strictly speaking, syncategorematic. This movement away from treating terms as syncategorematic has been aided by the sharp distinction introduced between object language statements and metalanguage statements, especially where the metalanguage is taken in a broad sense to include both a semiotical and a nonsemiotical part. 'All men are mortal' is an object language statement, while the statement 'humanity is affirmed of mortality' is a nonsemiotical statement of the metalanguage. This distinction coincides with that between what a statement means and a theory regarding the conditions under which a statement is true. 'All men are mortal' does not MEAN that mortality is affirmed of humanity, but it may be that the statement is true BECAUSE humanity bears a certain relation of entailment to mortality. Accordingly, the statement 'All men are mortal' is NOT a simple statement; it is rather the semantically different statement 'Humanity entails mortality' which is the simple statement. While nominalism historically has led to this kind of interpretation of general statements, the latter thesis is independent of the former. We can return to a conceptualistic or realistic treatment of universals, but this time with the awareness that we are not giving an analysis of language, but an ontological theory about the truth conditions for general statements. The ANALYSIS of language per se is independent of one's ontological theory. One's ontological theory enters when one proceeds to give a PHILOSOPHY of language; or, if you will, when one gives an interpretation of language. One might, therefore, accept Russell's analysis of general statements as formal implications as correct and accept Aristotle's theory as the correct philosophy of language. The Stoic semantic theory is very Fregean, as has been illustrated in detail by Mates in his book, Stoic Logic, referred to above. The main point of both the Stoic and Frege's semantical theory in regard to the conventionalist thesis is the belief that the sense of statements is an extramental lekton or proposition. As we have already noted above, this is the cornerstone for most recent attempts at refuting the conventionalist thesis in logic. Both the Stoics and Frege were careful to distinguish between the proposition which is extramental, public, and objective from the subjective thought which the Stoics called the presentation. While Frege uses the word 'Gedanke', it is clear that he means something independent of the mind. The Stoic's concept of truth is also in accord with some contemporary thought, for they hold that truth is IN the lekton, maintaining this belief against the Epicureans, who held that truth was a property of sounds.46 The latter theory, of course, is similar to the semantical theory of truth. These two positions are both opposed to the Aristotelian position (as well as the Scholastic position, which in general parallels the Aristotelian position in all semantical matters), which holds that truth 46

See Mates, Stoic Logic, pp. 25-26.

INTRODUCTION AND HISTORICAL BACKGROUND

29

is a property of the thought (judgment). Truth and falsity involve a synthesis of 47 CONCEPTS for the Aristotelian. We also find in the Port Royal Logic the distinction between a sentence and the proposition it expresses. A proposition, Arnauld says, is a judgment we make about things. The proposition is true when the judgment is correct. Truth, therefore, is a property of propositions for Arnauld, rather than of sentences. Since a judgment for him is a uniting or separating of two ideas, truth is a property of the thought as in the Aristotelian scheme. An interesting aspect of the Port Royal Logic semantic scheme is its distinction between the extension and comprehension of an idea. The comprehension of a universal idea, Arnauld says, is "the constituent parts which make up the idea, none of which can be removed without destroying the idea".48 He gives the example of the idea of a triangle which is made up of the ideas of three sides, and so forth. At least some ideas are composite ideas, being composed of simpler ideas. These composite ideas seem to be nothing more than several simpler ideas joined together under a common name and treated as a single idea. If the relation between the comprehension or sense meaning of an idea and the idea itself is one of parts to a whole, then this has an interesting consequence for our discussion of necessary truth. For example, in discussing universal affirmative propositions, Arnauld says that for such a proposition to be true, "the entire comprehension of the idea expressed by the predicate must be contained in the comprehension of the idea expressed by the subject".49 The consequence of taking the comprehension of an idea in the sense discussed above is to reduce all universal affirmatives to the status of trivialities. To say that the comprehension of one idea is contained in the comprehension of another is to say nothing more than that the first idea is literally a part of the latter. This is no more exciting than saying that the phrase 'unmarried' is part of the phrase 'unmarried, adult, male'. If the idea of bachelor is literally a collection of simpler ideas, such as the idea of unmarried, adult, and male, the proposition that all bachelors are unmarried is trivial in that it simply asserts that among the collection of ideas that make up the idea of bachelor is included the idea of unmarried. To avoid this consequence, Arnauld would have to distinguish between an idea and its constituent parts in such a way that the relation between them is not simply the part-whole relation. Arnauld distinguished between the extension and comprehension of IDEAS, while still retaining the Aristotelian and Scholastic thesis that words signify ideas. While John Stuart Mill adopts this distinction, he applies it to words rather than ideas. Mill's thesis is that words directly have an extension and connotation, rather than ideas. As Mill says: "Names are names of things, not of our ideas." 50 The connotation of a word for Mill is neither other words nor ideas, but properties (attributes) 47

Aristotle, De Anima III.8.432a9. Antoine Arnauld, The Art of Thinking, trans. James Dickoff and Patricia James (Indianapolis, 1964), p. 51. 48 Ibid., p. 169. 50 Mill, Logic, p. vii. 48

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INTRODUCTION AND HISTORICAL BACKGROUND

of things. Mill is careful to distinguish between words like 'men' which denote men and only indirectly signify or connote certain properties and words like 'humanity' which name the properties themselves. Sentences such as "All men are mortal" are about men, not about humanity; they are first-order, object language sentences about particular things, rather than second-order sentences about universals as the Aristotelian tradition considers them. While Mill was probably led to this conclusion by his nominalism, it is important to see that this result is independent of any ontological bias. The point is simply that to speak about the universal, humanity, we should have as our subject a word which names the universal, rather than one which merely connotes or signifies it. 4. Logical Rules Several distinctions, useful to us throughout the book, will now be made. We distinguish between valid arguments, entailment statements, conditional statements, and rules of inference. We take a valid argument to be a collection of statements such that one of them is considered to be the conclusion and the others the premises and such that the conclusion statement can be validly inferred from the premise statements. A valid argument, therefore, is a collection of first-order, object language statements such that one of the statements bears a certain relation to the other statements. The relation which holds between the premises and the conclusion statements when and only when an argument is valid we take to be the entailment relation. One can affirm this relation by making an entailment statement, which is a secondorder, metalanguage statement ABOUT the statements (or propositions) which occur in the argument. The statements (or propositions) which are mentioned in the entailment statement are used in the argument. One might say that the relation between the valid argument and the entailment statement is such that the former is valid BECAUSE the latter is true; that is, the entailment relation justifies the inference. An example of a valid argument is: "All men are mortal. All mortals are animate. Therefore, all men are animate." The corresponding entailment statement is the following: " 'All men are mortal and all mortals are animate' entails 'All men are animate'." Some philosophers would prefer to write this as: "The proposition that all men are mortal and all mortals are animate entails the proposition that all men are animate." An alternate statement of the entailment would be: "'All men are mortal' and 'All mortals are animate' together entail 'All men are animate'." A conditional statement, in contrast to an entailment statement, is a first-order statement; for example: "If all men are mortal and all mortals are animate then all men are animate." 'If . . . then . . .' is a sentential connective of the object language, while 'entails' is a relation word of the metalanguage. The conditional statement corresponding to a valid inference is, of course, necessarily true; it is a logical truism. The fact that it is necessarily true must be differentiated from the

INTRODUCTION AND HISTORICAL BACKGROUND

31

statement which asserts that it is necessarily true. One might say that an argument is valid BECAUSE the corresponding conditional statement is necessarily true, but this is equivalent to our other formulation, which says that an inference is valid because the corresponding entailment statement is true. Having characterized arguments, entailment statements, and conditional statements, we must now distinguish those expressions and collections of expressions which are like these except for containing variables in place of descriptive constants or particular statements. An argument-like collection of expressions which contains variables we will call an argument (or inference) schemata. An expression which is like an entailment statement or conditional statement except for containing variables we will call entailment statement and conditional statement forms, respectively. Arguments will be distinguished from argument schemata by being called arguments proper, and similarly for entailment and conditional statements proper. A logical rule of inference as formulated is on the same level as an entailment statement; the difference between the two is a matter of debate which we will consider in detail in the next part. They are related, in particular, to entailment statement forms rather than entailment statements proper, for the logical rules are generally taken to contain variables rather than descriptive constants. For the time being we can distinguish the entailments and the rules by characterizing the rules as 'prescriptive' and the entailments as 'descriptive'. If this is tenable, an issue of central importance to the conventionalist thesis arises as to the relation between them. We must also concern ourselves with the nature of the logical rules per se as distinct from formulations of them. In discussing the views of logicians in regard to logical rules, we want to ask two questions: (a) To what extent were they able to make the distinctions indicated above? (b) If they clearly distinguished between a rule and an inference, what view did they take of the former? Lukasiewicz and Bochenski specifically note that Aristotle's syllogism is a conditional statement form.51 A typical example is: "If A is predicated of all B and B is predicated of all C, then A is predicated of all C." 52 Lukasiewicz even seems to imply that Aristotle consciously wrote the syllogism as a conditional statement rather than as an entailment statement, rule of inference, or even inference schemata. Likewise, Bochenski says: "The Aristotelian syllogism is not a rule but a proposition."53 On the other hand, Austin believes that Aristotle may simply have been unaware of the distinction between a rule and a conditional statement, and that he did not consciously write his syllogism as one rather than the other.54 I am inclined to agree with Austin on this matter, for while Aristotle does use 51

See Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Logic, 2nd ed. (Oxford, 1957), pp. 1-6; and Innocentius Bochenski, History of Formal Logic (Notre Dame, Ind., 1961), p. 69. 52 See Aristotle, An. Prior I.4.25b40-26al. 53 Bochenski, History, p. 69. 54 J. L. Austin, "Review of Aristotle's Syllogistic", Mind LXI (1952), 397-98.

32

INTRODUCTION AND HISTORICAL BACKGROUND

the syllogistic form noted above, he also uses another form, where the consequence is written: "A must be predicated of all C." 5 5 How are we to interpret the 'must be'? Under one interpretation, the syllogism can be considered as equivalent to an entailment statement form, and therefore, closely related to a rule of inference. To obtain this interpretation we would consider the 'must be' as a modal operator which operates over the entire conditional. That is, putting 'must be' in the consequent can be taken as equivalent to prefixing ' 'it is necessary that" to the conditional statement without 'must be'. One can then interpret such a statement as talking about inferences. On the other hand, Lukasiewicz interprets the 'must be' as equivalent to universally quantifying the variables in the conditional statement; that is, as equivalent to prefixing "(A) (B) (C)" to the conditional statement without 'must be'. If this is the way that Aristotle intended it, he was treating logical necessity in much the same way as Quine, Martin, and others presently treat logical necessity. Logical NECESSITY is universal TRUTH. A logically necessary statement differs from empirically true statements in that the former is true for all substitutions of the constants, while the latter are true only for some. The fact that Aristotle's 'must be' in the syllogism can be equally well interpreted in either of these ways indicates that Aristotle himself was unaware of the distinction. Moreover, it is only when we want to give a theory about logic that it makes a difference whether we treat the 'must be' as a modal operator or as a string of quantifiers; the syllogism as a rule or conditional statement. Even Bertrand Russell 2200 years later was not fully aware of the distinction between rules and conditional statements, and while this has resulted in a number of philosophically puzzling statements, his logic was probably affected very little by it. It must be noted that in the Prior Analytics Aristotle does make some clearly metalogical remarks about the syllogism; he gives what are clearly rules for determining whether or not a syllogism is valid.56 There is a definite implication also that these rules specify necessary conditions for a valid syllogism. In addition, he attempts to show why they are necessary, but his explanation involves either examples or logical argument, rather than philosophical justification. Aristotle does not give a philosophy of logic per se. Since Aristotle was aware, at least at one point, of the difference between rules and statements, one might ask why he was not able to distinguish between rules of inference and conditional statements in regard to the syllogism. Why did Aristotle have rules stating necessary conditions for valid syllogisms and not treat the syllogisms themselves as rules stating sufficient conditions for a valid argument? Possibly Aristotle is actually considering his syllogisms to be a justification of something else, viz., of scientific syllogisms involving concrete terms, and, therefore, as a kind of second-order statement. Stoic logic centers around inference schemata; for example, "If the first, then 55 58

Aristotle, An. Prior I.4.25b37. See, for example, Aristotle, An. Prior I.24.41b5-6.

INTRODUCTION AND HISTORICAL BACKGROUND

33

the second. Not the second. Therefore, not the first." The Stoics had five of these basic inference schemata, which were "called 'undemonstrated' because they had no need of demonstration, 'since their validity is immediately c l e a r ' T h e Stoics, however, recognized that this did not exhaust all the possible valid arguments, even of propositional logic. They therefore had rules by which an argument could be reduced to one of these five basic forms. These rules differ from the inference schemata in two respects: (a) they are explicitly second-order statements, in contrast to the inference schemata which are merely collections of first-order statement forms; and (b) they indicate how to derive an inference from a given inference rather than how to derive a statement from given statements. That is, these rules are nonelementary, in contrast to the rules corresponding to the undemonstrated arguments which are elementary rules.58 These rules can be used either for proving the validity of nonsimple arguments proper or for proving the validity of nonsimple inference schemata. Only three of their four metarules are extant, and only two of these three seem to be distinct rules. One of the rules corresponds to a definition of DERIVATION in the ordinary logical systems where the elementary statements are object language statements; viz., if a statement A is derivable immediately from the premises (by an undemonstrated or derived rule) and if a statement B is derivable from the premises together with the statement A (by an undemonstrated or derived rule), then the statement B is derivable from the premises alone without the statement A. In the logical systems which have as their elementary statements entailment statements, this rule is known as 'cut' (Gentzen) or transitivity. While this rule does not have to be taken as undemonstrated, it does seem to be essential for any logical system which attempts to capture in any way the logical structure which we ordinarily use and which is imbedded in natural language. It is interesting to see, therefore, that the Stoics realized the importance of this rule. The other extant distinct rule which the Stoics used is the following: "If from two propositions a third is deduced, then either of the two together with the denial of the conclusion yields the denial of the other." 59 Whether this rule is actually independent of the other rule depends on whether or not the Stoics had a rule of conditionalization along with the other rule. Mates notes that there are certain Stoic passages which are virtually statements of the principle of Conditionalization, but at the same time there is no evidence that it occurs among the Stoic's four metarules. With a rule of conditionalization, this second metarule can be derived from the other metarule together with the 'undemonstrated' inference schemata Modus Tollens in the following way:

57

Mates, Stoic Logic, See the discussion of Mathematical Logic 58 Mates, Stoic Logic,

58

p. 67. of elementary and nonelementary rules in Haskell Curry, (New York, 1963), p. 68. p. 77.

Foundations

34

INTRODUCTION AND HISTORICAL BACKGROUND

(a) Hypothesis:

A B C

(b) By Conditionalization from (a):

A If B, then C

(c) Modus Tollens:

If B, then C notC not B

(d) By metarule (1) from (b) and (c):

A notC .'. n o t B

While the Stoics came much closer than Aristotle to making the relevant distinctions, they said nothing more about the nature of the validity of the undemonstrated inference schemata other than that they are 'immediately clear'. Since for Hobbes reasoning is a purely linguistic matter, it would seem that he would say that the logical rules of inference were linguistic rules. Likewise for Mill, DEDUCTIVE reasoning seems to be a purely linguistic matter, for the conclusions we deduce are from statements which are merely linguistic contrivances, i.e., the general statements. Reasoning from general statements to particular statements, Mill says, is a matter of "deciphering our own notes". 60 It would seem then that Mill would also say that the logical rules of inference were linguistic rules, although it must be mentioned that neither he nor Hobbes discuss rules per se. There is no question that Frege understands the difference between an inference, a conditional statement, and a rule of inference. For Frege a rule of inference is a metalinguistic proposition. Moreover, he is aware of the prescriptive or quasiprescriptive character of the logical rules. For example, he writes his first method of inference in Grundgesetze der Arethmetik as follows: "From the propositions ' | - | - P ' and ' — | p ' we may infer ' — | A'-" 61 Frege's use of quotation marks within the statement of the rule is significant, for he says that he is going to use quotation marks when he is talking about the sign within the quotation marks as distinct from the denotation of the sign.82 It is clear, then, that for Frege a rule of inference is about statements rather than about what they denote; a rule of inference is metalinguistic. This careful use of quotation marks is unfortunately something that we do not find in Principia Mathematica, and in this sense it is a retrogression from Frege's Grundgesetze. Moreover, Frege's use of the locution "from . . . we may infer . . . " makes it evident that he means something different than "if . . . then . . . " . Frege is fully aware that a rule of inference is something outside of an inference which justifies one in making the inference. 60 61 82

See above, p. 22. Frege, Basic Laws, p. 57. Ibid., p. 32.

INTRODUCTION AND HISTORICAL BACKGROUND

35

When we come to the question as to how exactly a rule of inference justifies one in making an inference, Frege is not as helpful, for he has an ulterior motive in explicitly distinguishing the rules of inference, rather than simply following them intuitively. He is trying to prove that arithmetic is reducible to logic. To do this, Frege says that one must demonstrate the laws of arithmetic using only the rules of logic. But unless the rules of inference used are made explicit, one does not know whether or not a demonstration uses only the rules of logic. By making explicit the rules that are to be used in all demonstrations and making clear that only these rules may be used, one can be certain that the only rules that are used are rules of logic. Frege notes that the validity of these rules must be 'self-evident', but goes no further in giving a philosophical justification of these rules. While Russell and Whitehead had the same ulterior motive as Frege in setting up their logical system, it is unfortunate that they were not as aware of the nature of a rule of inference as Frege was. They state what is essentially the same as Frege's first method of inference as follows: "*9.12 What is implied by a true premise is true. Pp." 63 While Frege distinguishes between the methods of inference (rules) and the basic laws of logic (axioms), Russell and Whitehead call both the rules and the axioms 'primitive propositions'. They are partially aware that *9.12 is different from the conditional statement, "if p is true, then if p implies q, q is true", for they say that the latter does not allow us to assert 'q' simply, while *9.12 does. They also say that *9.12 cannot be expressed symbolically, for any symbolic representation of *9.12 would treat 'p' as a hypothesis, while in *9.12 it is not hypothesized that 'p' is true, but rather it is assumed that 'p' is asserted to be true. They also say that *9.12 is used to DEDUCE a proposition from a proposition. But their groping to distinguish *9.12 as a rule from the other primitive propositions which are axioms is vitiated by the fact that they read the ' =»' of their logical system as 'implies', which is the same terminology as they use in stating what are their rules of inference. It is also vitiated by the fact that they do not distinguish between use and mention, a lesson which they should have learned from Frege. One point to note is that Russell and Whitehead's rules are stated as propositions, as descriptions, rather than in the quasi-prescriptive terminology which Frege uses. This procedure is, of course, justifiable, as long as it is realized that the rules are second-order propositions about first-order propositions or statements. But in lieu of any specific terminological decision on their part, it is not clear that they even fully realized the difference in character between the rules and the axioms. 5.

Summary

In the succeeding parts we will attempt to develop the thesis which is merely hinted at by Hobbes and Mill, but with full awareness of the opposition, as presented in the theories of Plato, the Stoics, Aristotle, and others. Our theory will be at 63

Alfred North Whitehead and Bertrand Russell, Principia bridge, 1925), I, p. 132.

Mathematica,

2nd ed. (Cam-

36

INTRODUCTION AND HISTORICAL BACKGROUND

variance, however, with most of the historical positions described in the preceding pages. We will aim to construct a comprehensive and coherent theory of the nature of logical truth, logical validity, and logical necessity. A semantic theory will be given for sentences as well as for the logical constants, a theory which is amenable to treating the logical rules as linguistic rules; that is, to treating deductive reasoning as essentially linguistic. In particular, we will defend the conventionalist thesis in logic by attempting to show that necessary connection has its source in rules, and that these have their source in linguistic regularities. Our immediate attention in the next part will be directed toward proving this latter point.

II

RULES, CONVENTIONS, AND ENTAILMENT

The concept of a linguistic rule is central in most discussions of conventionalism. It is widely held that analytic truths are true by virtue of linguistic rules or that they function as surrogates for linguistic rules. While we will discuss these various theses in Chapter III, here we intend to analyze and clarify the general notion of a linguistic rule, considering a variety of different concepts which are either related to the notion of linguistic rule or can actually be taken to be kinds of linguistic rules. For example, we will discuss the concepts of syntactical rule, semantical rule, rule of inference, entailment, linguistic regularity, custom and convention. We will then be in a better position to see what exactly the relation is between truths of logic and linguistic rules, and, in particular, to see whether or not some form of the conventionalist thesis in logic is tenable.

A.

INTRODUCTION

Before we begin our detailed discussion of these various concepts it will be well for us to discuss what logic is the study of and to relate, in a very general way, the notions of entailment and rule of inference to each other and to the study of logic. Even at this very general level we find some disagreement. We learn from Lewis Carroll 1 that to justify an inference from premises to a conclusion, we need something outside of the argument, as well as the premises. As he has so aptly illustrated, we cannot justify an inference simply by adding more premises. The conclusion is justified by the premises together with some principle or rule of inference which is outside of the argument. While one school holds that logic is the study of valid inferences, others hold that logic is the study of these very principles or rules of inference. The former school holds that we use these rules or principles in deciding whether or not an inference is valid; the latter school holds that these principles or rules are the very object of our study. Similarly, the first school, including people such as Quine,2 would use the rules of inference in 1

Lewis Carroll, "What the Tortoise Said to Achilles", Mind n.s. IV (1895), 278-80, reprinted in Readings on Logic, ed. I. Copi and J. Gould (New York, 1964), pp. 122-24. See his discussion of ' ' and 'implies' in Mathematical Logic, 2nd ed. (Cambridge, Mass., 1940), Section 5.

1

38

RULES, CONVENTIONS, AND ENTAILMENT

deciding whether or not a certain statement is logically true or a logical truism, their aim being to discern which statements are logically true, while the second school would hold with Kneale 3 that the rules or principles of inference are themselves the truths of logic. If we say that logical truisms, such as 'A or not A', are statements of the first order, then the rules or principles which justify first-order inferences will themselves be of the second order. These second-order statements can be written as rules, such as "From 'if p, then q' and 'p' to infer 'q'", or they can be written as statements of entailment (or deducibility), such as "The propositions that if p then q and that p together entail the proposition that q", or "The sentences 'if p, then q' and 'p' together entail the sentence 'q'." They can even be written: "If 'if p, then q' and 'p' are true, then 'q' is true." If we wish to study all of the rules or principles of inference, we must have recourse to third-order rules or principles, because there is an infinity of second-order statements. If we wish to study only first-order logical truisms and inferences, it is not necessary for us to revert to this third level. Given a finite number of rules together with a definition of valid argument or proof based on these rules, we can prove all the valid arguments and logical truisms of the first order. It is only when we wish to study the principles of inference for their own sake that we must make reference to third-order principles of inference. Thus far it would seem as though the differences between the two schools are of a minor technical nature, rather than a topic of philosophical interest. The theorems of the two kinds of logic will, of course, be different, and there will be more logical truisms than entailments, the entailment statements being essentially isomorphic to a proper subset of the material implications of the first-order logic. The differences, however, become striking when one includes in the class of entailment statements not only the rules of inference of logic, but also rules of inference of physics, that is, physical entailments, for when we change our second-order statements by adding the physical entailments, our third-order principles of inference are principles concerning the relation of entailment between entailment propositions themselves, whether they be logical or physical entailments. This scheme lends itself very readily to a nonconventional view of the truths of logic. It presupposes that logically necessary and physically necessary connections are of one kind, and logic itself is the study of necessary connection in general. (It has been noted, for example, by Pap,4 in a totally different context, that the rules of inference for laws of nature are the same as the rules of inference for the laws of logic.) Prima facie it would seem that any physically necessary connections must be independent of language, and therefore if logically necessary connections are of 3

See his discussion of modal logic in William and Martha Kneale, Development of Logic (Oxford, 1962), pp. 548-68, esp. pp. 557-58; see also his "Truths of Logic", Proc. Aris. Soc. n.s. XLVI (1945-46), 207-34, esp. pp. 209-10. Cfr. Curry, Foundations, p. 185, for a similar point of view. 4 Arthur Pap, An Introduction to the Philosophy of Science (Glencoe, 111., 1962), pp. 304-05.

RULES, CONVENTIONS, AND ENTAILMENT

39

the same kind, they too must be independent of language. Logic then becomes the study of this language independent necessary connection. While an advantage of this school is that it emphasizes the close relation between rules of inference and entailment statements, it produces a difficulty in that it does not enable us to distinguish between them. A rule is ordinarily conceived to be a prescription, while a statement is ordinarily conceived to be a description and can therefore be either true or false. If what justifies an inference is an entailment relation, we must first ask the question: Does the entailment relation hold between sentences or propositions? If the entailment relation holds between propositions, the rules qua prescriptions have no function. The only plausible function of a logical rule qua prescription is to permit an inference; i.e., to make the inference valid (permissible). But if the premises as a matter of fact entail the conclusion of an argument, then no prescription can change this fact or make it so. What is a matter of fact relation between extralinguistic PROPOSITIONS cannot be changed one way or another by a prescription concerning the use of SENTENCES. On the other hand, if the entailment relation holds between sentences or between propositions in some language-dependent sense of 'proposition', then we can go on to ask what the relation is between rules of inference and entailment statements. For example, the entailment might hold because of the rule, or, to put it another way, an entailment statement may be just a surrogate for a rule. There are other possible relations here, but before we can decide upon the correct relation, we first have to decide if a rule of inference is a prescription, and if it is, how it differs from other kinds of prescriptions. We will also want to know if a rule of inference bears any relation to a linguistic regularity, and if a linguistic regularity bears any relation to an entailment, the latter two being matters of fact. As we will be doing metaphysics to a certain extent, it will be well to make clear what our view of metaphysics is. We hold that the statements of metaphysics, while meaningful, are undecidable. Our criterion for an adequate metaphysics is twofold: it must hang together satisfactorily and it must be able to answer the crucial questions in its area. In so far as we will be giving a metaphysical analysis of logic, our analysis must give us an adequate explanation of logical necessity and valid inference, and, at the same time, answer the difficulties which have been posed by previous philosophers as well as any difficulties which we discover. On the other hand, we do not propose that our analysis is the only adequate one. The advantage of our system will be increase of simplicity in certain respects, gained, however, at the expense of a loss of simplicity in other respects. In particular we will construct a system which presupposes a minimum number of metaphysical entities and which can deal with the notion of necessity from a purely linguistic viewpoint. At the same time, individual theses of our system will not be as natural or as intuitive as in alternative systems. If we were to compare our system with a system in another part of philosophy, it would be with Kantianism. Both systems admit the existence of necessity, yet neither system places it where it most obviously seems

40

RULES, CONVENTIONS, AND ENTAILMENT

to be. Again we must emphasize that while we feel our system has certain advantages, we do not claim to have discovered the 'true' metaphysical basis of logical necessity. For example, when we ask whether or not there is an entailment relation between propositions, what we will really be asking is whether we can adequately deal with the nature of logic without presupposing such a relation. We believe that the question of whether or not there actually is such an entailment relation is an undecidable question. One answers the question not by "Yes" or "No" together with arguments, but rather by building a system which either includes or does not include such a relation.

B.

THE CONCEPT OF RULE

We will begin by discussing the concept of rule in general, after which we will tie together the notions of rule and regularity via the intermediate concept of convention. We will show that some rules and some regularities are essentially kinds of conventions. The force of this analysis will then be brought home by analyzing the concept of "following a rule", where we will see that whether or not someone correctly follows a rule is itself a matter of convention. With this analysis in mind we will go on to apply it in criticizing an objection made by Quine against conventionalism in logic. We will begin our discussion of rules by examining a paper by Max Black called "The Analysis of Rules".5 This paper is especially useful for our purposes because it is written with the same intent as we have; that is, with the intent of relating the general concept of rule to the particular concept of linguistic rule and ultimately of basing a conventionalist theory of necessity on the notion of linguistic rule. Black distinguishes four main senses of 'rule': (a) the 'regulation' sense; (b) the 'instruction' or 'direction' sense; (c) the 'precept' or 'maxim' sense; and (d) the 'principle' or 'general truth' sense. An example of a regulation is "No Smoking" or "One must drive on the right side of the road except when passing." An example of an instruction is "Do not plant tomatoes until after the last frost." We have two types of maxims, prudential rules and moral rules. An example of the former is "Eat in moderation"; of the latter, "Do not murder." An example of a principle is "Heavy objects fall when dropped." The particular characteristics of a rule in the regulation sense are the following: (a) we can talk about a regulation being put into effect (i) by a particular authority 6 and (ii) at a particular time; and (b) we can talk about disobeying, breaking, enforcing, changing, revoking and reinstating regulations. While Black admits that characteristic (a) is not applicable to linguistic rules and that characteristic (b) is s

In Max Black, Models and Metaphors (Ithaca, N. Y., 1962), pp. 95-139, printed originally as "Notes on the Meaning of 'Rule'", Theoria XXIV (1958), 107-36 and 139-61. 6 We take 'authority' in a broad sense to include amy person or group which has -the power to put a rule, regulation, or law into effect.

RULES, CONVENTIONS, AND ENTAILMENT

41

at best only partially applicable, he seems nevertheless to want to hold that if there are linguistic rules they must be regulations. He defends this not by appealing to the characteristics of regulations which he has distinguished, but by appealing to our pre-analytic notion of a regulation as something that regulates. Black thus considers linguistic rules to be regulations because he thinks linguistic rules are prescriptions for the use of language. The other three types of rules which Black distinguishes are all characterized partially by the fact that characteristic (a) is not applicable to them. Neither instructions nor principles nor precepts have histories nor are they put into effect by authorities. Since characteristic (a) is not applicable to linguistic rules either, one would think that they should also be classified under one of these three headings. There is a very close connection between instructions or directions and principles or general truths. Principles are clearly not rules in any prescriptive sense, for they are merely true or false statements about the world asserting some actual or alleged unformity in the world. The sense of 'rules' in which principles are rules is exemplified in the phrase "as a rule". When we say that something happens "as a rule", we are saying that something is generally the case; we are making a statement about the world. This is certainly one sense in which we use the word 'rule', but it seems to bear little relation to our prescriptive uses of the word. As previously noted, an example of an instruction or direction is "Do not plant tomatoes until after the last frost." While this seems to be a prescription, it clearly differs from a regulation not only in the fact that it does not have an author or a history, but also in the fact that it is a prescription relative to a particular end or purpose. The rule is to be followed IF one w^nts to achieve a certain result. Whether or not one is bound by the rule depends on the prior decision as to whether or not one wants to achieve the particular goal. Likewise, certain precepts or maxims, namely, the prudential rules, are also rules relative to a certain goal, although the goal in these cases is much more general. For example, a rule the purpose of which is to maximize happiness would be considered to be a maxim, while a rule the purpose of which is to maximize the growth of tomatoes would merely be considered an instruction. We must ask in regard to both instructions and prudential rules whether they are really prescriptions or merely surrogates for descriptions. Corresponding to both instructions and prudential rules, there are statements of fact which are, in effect, general truths or principles. For example, corresponding to the instruction, "Do not plant tomatoes until after the last frost", there is the statement of fact, "Tomatoes do not grow very well (if at all) when planted before the last frost", or "Tomatoes grow only if they are planted after the last frost." While there clearly is a correspondence between certain kinds of rules and general truths, the problem is with the nature of this correspondence. Is an instruction like a regulation or is it more like a general truth? In particular, is a direction prescriptive in the same way that a regulation or moral rule is prescriptive?

42

RULES, CONVENTIONS, AND ENTAILMENT

The rules which are most clearly prescriptive in the full sense are moral rules. While characteristic (a) of regulations is clearly not applicable to moral rules and characteristic (b) is applicable at best only in part, moral rules do participate in the most characteristic component of regulations; they are regulatory. Moral rules, as well as regulations, control human behavior. Also, if we look at the notion of moral rule pre-analytically, we would characterize such rules as nonvoluntary; that is, human beings are bound by moral rules regardless of any choice on their part. One can break a moral rule, but one cannot perform the same actions under the same circumstances without breaking the rule. That is, if an act of killing is morally wrong, it is morally wrong even if I decide not to adhere to the rules of morality. There are, of course, exceptions to moral rules. For example, a soldier is an exception to the rule that killing is wrong. But a soldier does not reject the rule. That soldiers are exceptions to the rule is built into the rule itself. On the other hand, I can plant tomatoes before the last frost without breaking the rule about planting tomatoes, as long as my intention in planting the tomatoes is not to grow them. For example, I might be interested in seeing what happens when one plants tomatoes before the last frost. The goal or purpose of the rule is to help one grow tomatoes, but this goal itself is not prescribed. I can, therefore, voluntarily reject the rule if I do not wish to achieve the goal. For that reason I call such a rule voluntary. But by no change in intention or purpose on my part can I change an immoral act into a moral or amoral act. In moral rules we have the clearest instance of a prescription, a control of my actions over which I have no contravening power. Regulations have this prescriptive character almost to the extent of moral rules. I can avoid being bound by regulations only by drastic acts on my part. For example, I can avoid the laws of one state only by living in another state; I can avoid being bound by "No Smoking" regulations only by keeping out of areas in which the regulation is in effect. I cannot by a simple change of intention avoid being bound by a regulation. The difference between regulations and moral rules can be summed up as follows: regulations are put into effect by a particular authority at a particular time, and they remain in effect for a particular area and for a particular time. The latter characteristics, while applicable to some moral rules - "One ought not to use a great deal of water in the desert or in New York City during the summer" - are not generally applicable to moral rules and the former characteristics are not applicable at all. On the other hand, both regulations and moral rules have the characteristic of being prescriptive and nonvoluntary; that is, they are a control over one's actions which one cannot avoid simply by a change in intention. The goal or purpose of regulations and moral rules is itself a prescribed goal, a goal which one cannot voluntarily choose not to achieve. It is for this reason that I call moral rules and regulations nonvoluntary. Both moral rules and regulations differ sharply from instructions and prudential rules in that there are no statements of fact corresponding to the former types of rules as there are in the case of the latter. (We must note, however, that we are

RULES, CONVENTIONS, AND ENTAILMENT

43

using our pre-analytical notions of moral rule and regulations. If some social contract or utilitarian theory is correct, we could find corresponding statements of fact. But on a pre-analytic level it would not seem as though there are these corresponding statements of fact.) It is, I think, this fact which makes moral rules and regulations prescriptive and nonvoluntary in a sense in which instructions and prudential rules are not. One can go from an 'is' statement to an 'ought' statement, if the 'ought' statement is an instruction or prudential rule. For example, it is perfectly valid to infer from the fact that tomatoes grow only if they are planted after the last frost, the rule that if one wants to grow tomatoes, one ought to plant them after the last frost. On the other hand, one cannot go from 'is' statements to moral rules or regulations without adding a great deal of analysis and theory which could be questioned. To bring out more clearly the kind of 'ought' that is involved in an instruction let us compare the rule regarding the planting of tomatoes with the statement, "If one wants to grow tomatoes, one will plant them after the last frost." This kind of formulation is sometimes used in place of the 'ought' formulation. In particular, it is used to state a rule when one wants to emphasize the fact corresponding to the rule. For example, when one says to a child, "If you want to get toys for Christmas, you will be a good boy", one is emphasizing to the child the fact that if he is not good, he will not get toys for Christmas. When one does not want to emphasize the fact or when the fact corresponding to the rule is only a probability statement, one often uses the 'ought' formulation. The 'ought' formulation and the 'is' formulation differ pragmatically, but in terms of content they are the same. The content of both formulations is the content of the 'is' formulation. In certain contexts the pragmatics of the word 'ought' is desired. In these cases one is using the statement as a 'linguistic prod' in Black's terminology, as an incentive to action. The reason, then, that one can infer an instruction from a statement of fact is that they are identical in content, and inference presumably involves only the content of statements. Moral rules have a prescriptive character which instructions lack precisely because the former are not derivable from 'is' statements. They do not differ merely in pragmatical function from statements of fact. And we can see that this is the case because of the nonvoluntary character of moral rules and regulations. If we were to seek a statement of fact corresponding to a moral rule, such as "Do not murder", we might find a statement, such as "This world is a happier place to live in only if murders are not committed." If we were to derive an 'ought' statement from this statement of fact as we did in the case of the rule regarding tomato planting, it would be the statement, "If one WANTS this world to be a happier place to live in, then one ought not to commit murder." But our moral rule is nonvoluntary, for we are bound by the moral rule whether or not we W A N T this world to be a happier place to live in. In other words, even if moral rules have goals, they are themselves prescribed goals. Therefore, it is because a moral rule is non-

44

RULES, CONVENTIONS, AND ENTAILMENT

voluntary that it is not derivable from an 'is' statement, and thus takes on its specifically prescriptive character. Hence we essentially have only two basic types of rules, prescriptive and nonprescriptive. Among the prescriptive rules we can distinguish those rules which have an author and a history, i.e., regulations, from those which do not have an author or history, i.e., the moral rules, both being nonvoluntary, while among the nonprescriptive rules, we can distinguish those which are relative to a very general end, such as happiness, i.e., prudential rules, and those which are relative to more specific ends, such as growing tomatoes, i.e., directions. The content of these voluntary prescriptive rules is a statement of fact or general truth, the rule formulation differing from the statement of fact in its pragmatical function only.

C. LINGUISTIC RULES

1. Linguistic Rules as Conventions Prima facie, it would seem that if linguistic rules were to fit anywhere in this scheme they would be like moral rules, since they are regulatory, and do not have an author or history. But are linguistic rules nonvoluntary? Is it possible, for example, to make an invalid inference without BREAKING a rule of logic? One might be inclined at first to say "No." If we look at language from a broad perspective, however, we see that it is used for many purposes, only one of which is to discover the truth. For example, if a novelist were to consciously have one of his characters reason illogically, would we say that the author has broken the rules of logic on the grounds that he wrote an invalid argument? Should we criticize the author for being illogical? Obviously not. It seems to me that a case can be made for saying that one has not BROKEN the rules of logic unless one is trying to argue logically or is at least engaged in an activity in which he implicitly accepts the logical rules. The novelist here was not trying to argue logically, but in fact he intentionally wrote a fallacious argument. In writing fiction one does not commit oneself to following the ordinary rules of logic. Similarly, in writing fiction one does not have to commit oneself to the ordinary rules of grammar. Faulkner and e e cummings have not broken the rules of grammar, because by their very style they have indicated that they have not committed themselves to the rules of grammar. I do not think that it would be correct to say that Faulkner and e e cummings are 'exceptions' to the rules of grammar. It is rather that they have chosen to be ungrammatical because this suits their purposes. The fact that something is not in accordance with the rules of grammar or the rules of logic does not in itself indicate that it OUGHT to be in accordance with these rules. Whether it ought to be in accordance with the rules depends at least in part on the intention of the author.

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We have not as yet established that there are linguistic rules. We have merely indicated that if there are linguistic rules they would seem to have some of the characteristics of moral rules, although they are not nonvoluntary, and consequently not as binding as moral rules. Possibly linguistic rules are a third type of rule, neither prescriptive nor nonprescriptive, but capable of being identified with some known characteristic of language. In a paper by George J. Bowdery entitled "Conventions and Norms" 7 we find an analysis of this third type of rule. Bowdery distinguishes three commonly used senses of the word 'convention': "1) a gathering of people; 2) a generalized mode of behavior either a) unformulated or b) formulated as a rule or prescription; and 3) the choosing of a mode of behavior or rule". These are related to each other as authority, action, and result. Interestingly enough, though, when a group of people get together in convention (1) to make a convention (3) the result is not ordinarily called a convention. The result might be called an agreement or policy, but not a convention. Therefore, the first and third senses of 'convention' are related not to the second sense of 'convention' but to another concept entirely. It is sometimes objected to conventionalism in logic that if the truths of logic were true by convention, there must have been a convention (1) or (3) at which the truths of logic were decided. But clearly when we talk about the logical truths of natural language being true by convention, we do not have in mind either the first or third sense of 'convention'. The second sense of 'convention' expresses more exactly what we have in mind, at least when we are talking about natural languages.8 Here by a convention we mean a generalized mode of human behavior which might or might not be formulated by a rule or prescription, where by 'generalized' we mean that the mode of behavior is followed regularly under similar circumstances. In the first case, where the mode of behavior is not formulated by a rule, we generally call it a custom. For example, it is a custom or convention of our society that only the bride wears white at the wedding. The important characteristic of a regular mode of behavior is that even though it is not formulated, it may nevertheless have an imperative function. The very fact that a mode of behavior is regular, that most individuals do in fact engage in this mode of behavior, constrains each individual to follow this mode of behavior. According to Bowdery, the mechanism whereby regular modes of behavior take on an imperative function is a disputed question in sociology, but it is accepted as a fact by sociologists that they do. Whether the imperative function comes from a desire to imitate and a fear of penalty or whether it comes directly in reaction to the regularity has not been determined by them. We can use the locution 'unformulated rule' as an abbreviation for 'unformulated regular mode of behavior with an imperative function'. The mode of behavior is in 7

Philosophy of Science VIII (1941), 493-505. The next few paragraphs are particularly important for we shall use the terms 'convention' and 'conventional' throughout the rest of the book in the sense specified herein. 8

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this instance itself functioning as a rule or prescription. Again we have a close connection between 'is' and 'ought' statements but in this case the connection seems to be causal rather than logical. The fact of the regular mode of behavior causes the prescription that one ought to conform to the mode of behavior. In contradistinction to instructions, which we noted were actually statements of fact with an added pragmatic function, we must in this case say that the rule itself does not have a content. I think we must also say that the ought statements related to conventions are pure prescriptions (as are moral rules), but in this case caused by a prior regular mode of behavior. The prescription is in force because and only because of the prior regularity in behavior. Moral rules, on the contrary, are causally independent of regularities of behavior. Before we specify more precisely the kind of unformulated rule that an "unformulated regular mode of behavior with an imperative function" is, let us look at the regular modes of behavior that are formulated by a rule. A convention in this sense is formulated by a rule only after the unformulated convention has already taken on an imperative function. That is, prior to the formulation of the rule there is both a regularity and a prescription to conform to the regularity. A formulated convention is merely a formulation of these two facts. That it is a formulation of both the regularity and the prescription can be seen from the fact that the mere statement of the regularity is sufficient to act as a prescription if pronounced at certain times. To use Bowdery's example, if one were to assert the statement "People eat pie with a fork at formal dinners" just as someone were picking up a spoon to eat his pie at a formal dinner, the statement would act as a prescription. It would have the same effect as the rule formulation, "One ought to eat pie with a fork at a formal dinner", for the regularity and the prescription are inextricably joined. Stating either the rule or the regularity is sufficient to state the other. If a linguistic rule is a convention in this second sense, we can also explain the fact noted by Black that a rule cannot be identified with any of its formulations. The rule is the prescriptive function of the regularity and exists prior to its formulation. The rule might be formulated as an 'is' statement or an 'ought' statement and it might be formulated in any language. But the rule itself, that is, the regular mode of behavior with an imperative function, precedes its formulation. Now that we have seen that there is another type of rule besides the moral ruleregulation type and the instruction-prudential rule type, how should we describe this type of rule in terms of our dichotomy? First of all, our rule is prescriptive, and, therefore, it belongs to the moral rule-regulation type. However, our rules are neither nonvoluntary nor independent of 'is' statements. That conventions are not nonvoluntary is evidenced by two facts. We note, first of all, that many times a person can completely disregard conventions without being censured. We would not, for example, say that e e cummings broke the rules of grammar or that he ought to have used better grammar. One can intentionally reject the rules of grammar, and thereby avoid being bound, but one cannot avoid being bound by moral

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rules simply by rejecting them. A second fact which indicates that conventions are not nonvoluntary is that a convention can be changed by the conscious activity of a group of individuals. On the other hand, a moral rule may change with time, but not as a result of the desire of individuals to change it. No matter how many people want cheating on their income tax to be moral, it is still immoral. In the case of instructions, we noted that on the basis of a uniformity of nature, we formulate statements of fact which describe this uniformity and also rules which both describe the uniformity and act as a linguistic prod. In the case of conventions, we noted that there is a regularity of human behavior and that in some cases this regularity also functions as an imperative. Because of this causal connection between the regularity and the prescription, a statement of the regularity can itself act as a rule, and, likewise, the rule can describe the uniformity. Again the 'ought' is not independent of the 'is' but in this case the connection is causal; the descriptive element is distinct from the prescriptive element but causally related to it. In the case of instructions, there is no prescriptive element outside of the rule formulation itself, and the prescriptive element of the rule formulation is seen to be a superimposition over the description. If we continue to look at moral rules on the pre-analytic level we see that there is neither a causal nor a logical relation between 'is' statements and moral rules. In any case the prescriptive element is not caused by a regularity in human behavior. Moreover, the prescriptive element is not a mere pragmatic addition to a descriptive statement, for moral rules are nonvoluntary, while if there are any descriptive statements corresponding to moral rules, they all have a condition of the form: "If one wants the world to be a better place to live in, then ." Consequently we have a form of rule which is both voluntary and prescriptive and which we can call a 'conventional rule'. As in the case of moral rules, and in contradistinction to regulations and instructions, they can be unformulated as well as formulated. The question now is whether this kind of rule can be identified with known characteristics of natural languages. That there are regularities in our use of language is undeniable. We regularly begin a sentence with a capital letter and end it with a period, question mark, or exclamation point. That this regularity is not merely accidental but self-perpetuating is evidenced by the fact that when we do not conform to the rule we are told that we should have. Therefore, we can say that the regularity has a prescriptive force which encourages conformity to the regularity. That this prescriptive force is voluntary can be seen from the fact that our linguistic conventions do change, sometimes merely as a result of a change in human behavior, sometimes as a result of conscious action on the part of certain humans to change the convention. When our mode of linguistic behavior changes sufficiently so that we can say that there is a new mode of linguistic behavior, the new mode of behavior becomes the prescribed mode. For example, as more and more people generally use the word 'ain't', the prescription against the use of the word dissolves. This is the special characteristic of conventional rules; they change

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when modes of human behavior change. And each time they encourage the retention of this new mode of behavior. One might say that there is a social law of inertia, and as a particular aspect of this law we have the linguistic law of inertia. Language tends to continue as in the past, and when modified, tends to continue in the new direction. This is due to the fact that the prescription attaches itself to the currently observed regularity. Therefore, in natural languages we have conventional rules, that is, voluntary prescriptive rules. 2. Logical Rules of Inference as Linguistic Rules That these rules apply to the grammatical aspect of language is easy to see. That the logical rules of a language are also conventional rules is what we hope eventually to show. According to this view, logical rules of inference would be regular modes of linguistic behavior which have a prescriptive function. Consequences of this view are that the rules of inference can be empirically discovered, that they exist unformulated before they are formulated, that one can make a mistake in formulating a logical rule, and that they are not unalterable. While the nonconventionalist would deny that the logical rules can be discovered EMPIRICALLY, he would agree with the second and third consequences. The last consequence is controversial among exponents of both schools, and we will look at it in detail in Chapter III. By a study of several related concepts, we hope to show now that the analysis of logical rules of inference as conventional rules is a complete analysis of the logical rules, that is, that it is not necessary to postulate anything beyond linguistic regularity to account for the existence of logical rules. Let us ask the following questions about rules of inference: "How does one go about deciding whether an alleged instance of correctly following a rule is a correct instance?" "Is it necessary to know the rule in order to correctly follow it?" "What is the relationship between the rule and the rule formulation?" We can take it as brute fact that we must learn how to infer validly; that is, that it is not an innate idea. But in what does this learning consist? One might think that learning how to follow a rule first involves learning the rule, and then learning how to follow it correctly. But is learning the rule an essential part of learning how to follow the rule?" Here we have in mind either of three senses of 'rule': (a) the rule formulation itself; (b) the logical rule as being a nonvoluntary prescriptive rule, the kind of rule that is independent of language and independent of human regularity, as are the moral rules; or (c) an entailment relation between propositions. But would knowing the logical rules in any of these senses help us to infer validly? It would not because in order to follow the rule in making valid inferences, we would still have to know that the particular inference falls under the rule, and • The following discussion is patterned after Wilfrid Sellars, "Some Reflections on Language Games", reprinted in his Science, Perception, and Reality (New York, 1963), pp. 321-58.

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thus we must make a transition from the rule to the particular inference. If this transition is a logical transition, where the rule is a premise and the particular inference is a conclusion, then we must in the process use logical rules of inference. Therefore, if learning how to follow a rule presupposes first knowing or being aware of the rule itself, we must have already known at least one of the rules of logic. This approach to how we learn to infer validly will never do for it presupposes that we already know how to infer validly. Surely learning how to be logical cannot presuppose our already being logical. On the other hand, if this transition from rule to particular inference is other than a logical transition, we are still left in the dark as to how we learn to infer validly. However the proponent might describe this transition, he must allow for the possibility of mistakes which people make when learning how to infer. Let us assume that learning how to follow the rules of logic cannot necessarily involve knowing the rule; that is, it is necessary and sufficient for us to see particular instances of valid reasoning for us to LEARN HOW to infer validly.10 Therefore, knowing HOW to infer is logically prior to knowing THAT the logical rules are such and such. On the one hand, as a result of seeing valid inferences we might acquire an insight into arguments so that we can tell immediately whether or not an argument is like an argument that we know to be valid. On the other hand, we might learn how to infer validly by being trained or conditioned to go on in the right way. Now there are two difficulties with the first approach. First of all, an argument must not just be like a valid argument, but it must be like it in relevant respects. In examining two inferences for resemblances, we must have a prior notion of what characteristics of an argument are logically relevant. Again this presupposes a prior knowledge of logic. Secondly, it assumes that we have an innate ability to see with our mind's eye that two arguments resemble each other. It seems highly doubtful that this is so; rather we learn to recognize similarities by conditioning. In any event, either account of how we learn to infer validly is compatible with a conventional view of the logical rules. This is so because both views admit that we can begin to learn the valid inferences only by being shown examples of valid inferences. But if we can learn to infer without knowing the rules, then the rules are unnecessary, for what possible function could they have? If learning how to infer is a matter of being shown cases of valid inferences and learning how to go on in the same way, where do these examples come from? How did we discover in the first place that certain inferences were valid? I suggest that these were discovered simply by reflecting on the regularities in language. Before it was discovered THAT certain inferences were valid, I suggest that people were simply trained to go on in a certain way. To put it in our earlier terminology, human regularities are self-perpetuating. 10 Once we have learned how to infer and have learned how to use rule formulations, it is sufficient in some cases to know the rule formulation to be able to use it correctly. That is, we do not always have to be given instances of rules once we have learned how to use rules.

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While we have not proven that the rules of logic are simply linguistic regularities, nevertheless, we have shown that postulating anything beyond linguistic regularities, such as nonconventional logical rules, will not help us to explain how we learn to infer. Moreover, postulating these nonconventional rules cannot explain how we recognize that certain inferences are valid. But if the rules of logic are simply linguistic regularities, then it is easy to explain how we learn to infer, for learning to infer is nothing more than learning how to go on in the regular way. Also, this can partially explain how we learn to recognize valid inferences, for recognizing valid inferences is a matter of recognizing linguistic regularities. We will argue in Chapter IV that this is a complete explanation of how we learn to recognize LOGICALLY valid inferences, for if a linguistic regularity is a logical rule it is by its very nature valid. The important point that we have proven is that learning how to infer is essentially a LEARNING HOW procedure, and does not involve KNOWING THAT the logical rules are such and such. Since this is the case, we go beyond this to suggest that the rules of logic themselves are nothing but prescriptions to go on in a certain way which arise from linguistic regularities. Learning how to infer, then, is learning how to go on in the regular way, which is therefore the prescribed way. But have we made our case too strong? Surely, to infer in a fully conscious manner we must know the rules. If one conditioned an ape to draw the correct inferences, one would not say that he was truly INFERRING. The point is well taken. There is more to inferring than mechanically coming out with the right answers. But I would suggest that this something extra is simply the knowledge that there are rules governing the inferences. Knowledge OF the rules is something more than is required even for a conscious use of the rules. To play any game by the rules one must be aware that there is a wrong as well as right way to play the game. When one knows that a way of proceeding is the right way, he knows not only that it is the customary way, but also that it is the prescribed way. But to say that one must know or be aware of the rules themselves in order to play by the rules is to ask too much. If logical rules are, as we have been arguing, regular modes of linguistic behavior, then, of course, awareness of the rules is generally present. The mere awareness that people infer in some ways and not in other ways is an awareness of regularity. But it is not an awareness of what the regularity consists in. To know that there are laws of nature is not to know what they are. To be aware of linguistic regularity is to know that there are rules of logic, but it is not to know what they are. To know what the rules are is to formulate the regularity either linguistically or conceptually. And formulating the rule involves forming appropriate concepts, for example, the rule Modus Ponens - "From 'if p, then q' and 'p' to infer ' q ' " - involves the concept of statement of the form 'if p, then q'. Now the concept of statement of the form 'if p, then q' is used to refer to statements which have visually different forms, e.g., "the lead weight will fall if dropped". Therefore, the relationship between the concepts and their instances is not one of simple visual similarity.

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Let us say that we are given several inferences valid according to Modus Ponens, and we are told to go on in the same way making valid inferences; it is likely that at first we would make mistakes. Would it help if we were given the rule formulation of Modus Ponens? Not in the least, for I can do anything or nothing with the rule formulation. Before I can use the rule formulation correctly, I must first learn how to interpret any rule formulations. Then I must learn the concepts involved in this particular rule formulation. In the last analysis what does knowing a concept, such as statement of the form 'if p, then q', consist in? We suggest that this consists in nothing more nor less than knowing how to go on in the right way picking out those statements which have this form and those which do not. And what is the right way of going on? It is the regular way. We choose the concept in formulating our rule to fit a particular class of cases. Given a finite subclass of the class, there is no unique way of going on putting other cases in the class, but there is the conventional way. It is a particular way, but not a unique way. The statement "A lead weight will fall if dropped", falls under the form 'if p, then q' not because it is visually similar, but because that is how we use the concept of statement of the form 'if p, then q \ Such a concept is merely a shorthand notation for this regular way of going on. Therefore, learning how to use the rule formulations of logic is a step beyond merely learning how to infer validly. It involves not only the latter, but also learning a number of new concepts as well. We noted above 11 that logical rules cannot be used as premises in an inference in order to justify the inference. The rules which justify an inference are outside of the inference. We now see that the rules cannot be used as devices which help us to learn how to infer validly, for in order to use a rule formulation to make inferences we must know when an argument is of the form specified in the rule. But knowing this involves knowing the relevant concepts, such as that of statement of the form 'if p, then q'. But learning these concepts is a matter of learning that certain statements are relevantly similar, and learning that a particular rule fits a particular argument is a matter of knowing that this particular argument is relevantly similar to some other particular arguments which were given as paradigm examples. Therefore, in order to learn how to use rules one must already know when arguments are relevantly similar. But if one already knows this, then the rule is unnecessary. We must remember here that we are concerned primarily with the logical rules of natural languages. The point of this argument is to lend credence to the view that the rule formulations of natural languages are merely abbreviations of the fact that we regularly go on in a certain way, and that the rule itself is the regularity with its prescriptive function. We have tried to establish a close relationship between the way in which one learns how to infer and linguistic regularities. We have suggested that learning how to infer is not a matter of learning rule formulations or of getting insight into extralinguistic, nonconventional rules, but is simply a matter of learning how to go on in the regular way, of perpetuating the linguistic 11

See page 37.

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regularity. Rule formulations are useful, not because they help us to learn how to infer, but because they are convenient descriptions of the regularity and because they stand as prescriptions to continue in the same way. The usefulness of rule formulations in regard to artificial languages is somewhat different, for here we are not describing regularities but establishing them. It is assumed in setting up the rules of an artificial language that one knows how to use logical rule formulations. This knowledge can then be transferred to a new set of rules. Therefore, the rules of an artificial language can be communicative in a sense in which the rules of natural language cannot be, for learning how to use any logical rules presupposes knowing how to infer validly. After one knows how to infer, then one can learn how to use logical rule formulations, and, in particular, one can learn how to use those rules which formulate new rules. This account of rules can help us to undercut a critique of conventionalism made by Dummett. He says the following: "It appears that if we adopt the conventions registered by the axioms, together with those registered by the principles of inference, then we must adhere to the way of talking embodied in the theorem; and this necessity must be one imposed on us, one that we meet with." 12 We would argue that the use of the word 'must' here is misleading. Accepting certain rules is a matter of accepting the arguments which fall under the rule, for the rule itself is nothing more than a collection of arguments.13 If I accept a collection of arguments, then I 'must' accept each one, but this 'must' is surely trivial. What makes Dummett's argument seem more plausible than it is is the fact that he refers to 'adopting a convention', which makes it appear as though we are adopting the rule formulations. Conventions are not 'adopted'; conventions 'obtain', for they are linguistic regularities. Even in the case of artificial languages, rule formulations are used to indicate the suggested regularity. The 'conventions' by virtue of which the theorems of artificial languages are necessary are not to be identified with the rule formulations. 3. The Logical Rules and Entailment Statements As we understand it, logic is the study in which we formulate the rules of inference and study them for both pragmatic and theoretical reasons. In everyday life and in the sciences we make valid inferences. It turns out that by using the same rules as we do in making ordinary inferences, we can sometimes get conclusions from the empty set of premises. These 'zero-premise conclusions' are the logical truisms. 12 M. Dummett, "Wittgenstein's Philosophy of Mathematics", reprinted in Philosophy of Mathematics, ed. P. Benacerraf and H. Putnam (Englewood Cliffs, New Jersey, 1964), pp. 494-95. 15 We find a similar view of rules in N. D. Belnap, Jr., et al., "On Not Strengthening Intuitionistic Logic", Notre Dame Journal of Formal Logic IV (1963), 316. "By a rule we shall understand any set of premisses-conclusion pairs, and by an instance of a rule any member of the rule." The phrase 'premisses-conclusion pair' can be taken to mean 'argument'.

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They are byproducts of our linguistic conventions. One might study these logical truisms, for example, in order to discover as many of them as one can. One might also attempt to discover if the truths of arithmetic are likewise byproducts of our language. But in doing this one is not studying logic, but using logic. One is using the rules of inference to see what results can be obtained. On the other hand, logic itself is the study of these rules of inference. It is clear then that we are members of the second school of logic, which, as we recall, identified rules or principles of inference and entailment statements. We noted that this could pose a difficulty for the conventionalist school if entailment statements are true statements about the relationship between propositions. The rules of inference would not then be prescriptive; they would be like directions, a direction being merely a reformulation of a statement of fact. Rules of inference, according to this view, would differ only pragmatically from entailment statements, and they would have the same content. You cannot plant tomatoes before the last frost and still get tomatoes, and, similarly, you cannot argue to a conclusion other than one entailed by the premises and still be guaranteed that it will be true, assuming that the premises are known to be true. With our analysis of rules, however, we can easily distinguish between entailment statements and the rules of inference, retaining the autonomy of each. Entailment statements are statements of the linguistic regularities qua regularities. Rules of inference are statements of the regularities qua prescriptions to continue in the same way. Rules of inference retain their prescriptive character. Entailment statements retain their character of statements which are true or false. The interchangeability between rule formulations and entailment statements is due to their both being aspects of the same empirical fact, the regularity with its prescriptive function. According to this second school of logic there are also particular entailment statements which are the provinces of other sciences. On the one hand there are the entailment relations between universals, such as between fatherhood and manhood; such statements as "Fatherhood entails manhood" would probably be the domain of semantics. There are also the physical entailments of the natural sciences, such as the entailment between the defining properties of iron and the property of being magnetic, assuming that the latter is not a defining property. Now logic is the study of both the purely formal entailments that depend only on the meaning of the logical words and the entailments which are independent of any particular words, formal or material, for example, "A, B entails A." This is a principle of logic involving, besides variables, only the word 'entails'. Other principles of logic involve, besides the word 'entails', only logical words. In studying the principles of logic, we erect a third-order set of principles which are rules for deriving entailment statements from other entailment statements. By using these third-order rules, we can derive any one of the infinite number of principles of logic. But more important for our concern is that these third-order

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principles are applicable equally well to any entailment statement, whether they be general or particular. We have, then, both the general entailment statements of logic and the particular entailment statements of semantics and natural science being treated on a par with respect to the third-order principles. Grouping together the entailment statements of logic with the entailment statements of physics would seem to be damaging to a conventionalist analysis of the former. Most conventionalists want to deny that the latter even exist. We suggest that the logicians of the second school have been correct in assimilating all entailment statements. We further suggest that all entailment statements are like the entailment statements of logic, that is, they are statements about linguistic regularity. This will be discussed in greater detail in Chapter IV. There are, however, some problems in adapting our analysis of the rules of inference to the second school of logic. How are we to interpret their third-order rules of inference? Also, how are we to interpret their position that there are an infinite number of second-order rules of inference? Clearly there cannot be an actually infinite number of linguistic regularities, and, therefore, according to our previous analysis of rules as linguistic regularities, there cannot be an infinite number of rules of inference. This apparent discrepancy can be avoided by distinguishing between direct rules of inference and rules of inference in general, this distinction being analogous to Carnap's distinction between the relation of direct derivability and the general relation of derivability. To distinguish between the direct rules of inference and the rules of inference in general we must first realize that it is, practically speaking, impossible to determine linguistic regularities by direct inspection, especially when one is concerned with the linguistic regularities related to the logical rules of inference. We noted above that the logical rules of language were first determined by reflecting upon the regularities in language. But this reflection cannot be a simple matter of observing how people use language. We would suggest that this reflection is actually a three step procedure. First, one observes in a more or less general way how certain words are used. Then one formulates a set of rules which one hypothesizes to be the logical rules of the language. These rules are then verified against language usage to see whether the way language is in fact used is compatible with the rules. The rules are imbedded in the language, just as the laws of nature are imbedded in nature. But the method which we must use in the first case, and which we generally use in the second case, to discover the regularities is the hypothetico-deductive method. By means of this procedure, an investigator formulates a finite set of rules, hypothesizing them to be the rules of a language. We shall call these the direct rules of inference. This is not to imply, however, that there is a unique set of rules which is THE set of logical rules. The set of rules which we consider to be the direct rules are relative to a particular investigation. For example, the axioms discovered by logical investigators in the past century, while considered as direct rules here,

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cannot generally be identified with linguistic regularities. However, general rules can be derived from these axioms and rules, and these can then be verified against the regularities in language. What is unique, or as unique as it is possible for linguistic regularities to be, is the set of general rules of a language. We must now make the transition from the finite set of direct rules of inference hypothesized by the investigator to the infinite set of general rules, while still retaining the viewpoint that the logical rules are conventions. We note that there are four different types of rules that can be hypothesized by the investigator: (a) rules which govern inferences from a group of statements to another statement, e.g., Modus Ponens; (b) rules which govern inferences from an inference to a statement, e.g., Conditionalization - "From an inference of 'q' from 'p' to infer 'if p, then q'"; (c) rules which govern inferences from a group of inferences to another inference, e.g., Transitivity or, as it is also called, the definition of proof or derivation - "From an inference of 'q' from 'p' and an inference of 'r' from l q' to infer an inference from 'p' to 'r'"; and (d) rules which govern the very structure of proofs, i.e., the structural rules, such as Permutation - "the order of the premises of an inference is irrelevant" - and Weakening - "a repeated premise is irrelevant and may be dropped". The latter rules have a peculiar status in that they are not actually linguistic rules, but rather they are rules which govern the interpretation of the rule formulations of types (a), (b), and (c). That is, in formulating the rules of a language, these rules are not usually formulated, but are assumed as part of the interpretation of the formulations of the other rules. For example, it is assumed that the order of the premises in an argument is irrelevant despite the fact that the premises have a certain order in the rule formulation. Of course, in the rule (regularity) itself, the premises have no particular order and in fact even the conclusion can occur at any point in an argument. What is especially important for our purposes are the rules of type (c). Any investigation of the logical structure of language which makes any attempt at being complete must make provision for the fact that we derive inferences from inferences, that is, that we can construct deductions. Given this we can show how it is possible for language to have an infinite number of linguistic rules. While rules of type (c) are in language second-order rules, from the logical point of view they can be considered as either second-order or third-order rules. The reason for this is twofold: first, there is a close connection between inferences and rules of inference, for rules of inference are, so to speak, collections of inferences, and particular inferences are valid by virtue of being in the collection. Therefore, if we can derive an inference from another inference, then we can derive the first collection of inferences from the second. That is, rules of type (c) are not merely rules for deriving particular inferences from other particular inferences, but are rules for deriving inferences of a certain type from other inferences of a certain type. Secondly, what we call the direct rules of inference are only direct relative to a

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particular investigation. What is a direct rule of inference to one investigator may be a derived rule to another investigator. Therefore, provision must be made by the investigator for deriving rules which, while not taken to be direct, are evidently imbedded in the structure of language. The rules of type (c), operating as thirdorder rules, serve this purpose. By means of these rules the investigator also is able to derive rules which cannot in any sense be considered 'linguistic regularities'. The significant thing to notice about this procedure of deriving rules is that the logical investigator does not have to use any rules which were not discovered in the structure of language itself. The investigator, if his investigation is at all complete, will find rules of type (c), and these very rules can be used by him, as third-order rules, to derive further rules. The procedure of deriving rules here is no different than the ordinary procedure of deriving conclusions fom premises. We have been able to answer the second problem which the conventionalist who treats logic as the study of the logical rules of inference has. There can be an infinite number of logical rules of inference even under a regularity view of rules because among the rules of inference of language there are second-order rules which enable us to derive inferences from inferences, and these same rules can be used as third-order rules for deriving rules from rules. Just as a rule justifies an infinite number of inferences, so the rules of type (c) justify an infinite number of derived rules. But the third-order rules used by logicians who take logic to be the study of the logical rules, usually involve much more than the usual second-order rules of type (c) used as third-order rules.14 In fact the rule of Transitivity, which we take to be a paradigm case of a rule of type (c), is oftentimes treated as a derived third-order rule. The difference between the approach we outlined for obtaining an infinite number of rules of inference from the approach of Curry and Kneale is analogous to the difference in approach between those who take an axiomatic approach to second-order rules and those who take a natural deduction approach. In our approach, the direct rules of inference are analogous to axioms and the rules of type (c), especially Transitivity, are analogous to rules of inference. While I think the Gentzen approach to logic is more 'natural', I think the axiomatic approach to the rules of logic is more straightforward. The intuitive procedure, I believe, is to establish a set of rules of inference, rather than axioms, as the direct rules of inference of a language and which are sufficient for making all inferences within the language. A subset of these rules - the rules of type (c) - are then used to derive from these rules of inference other rules of inference. The direct rules of inference are taken as primitive in deriving the rules in general. But while this procedure is more 'natural', the logician who is concerned with the rules of inference in general can generate these rules in any way that he likes as long as the rules he comes out with correspond to the linguistic regularities. There are theoretical advantages in taking only a single direct rule of inference, e.g., "From 14

See, for example, the set of third-order rules given by Kneale, Development and Curry, Foundations, pp. 192-93.

of Logic, p. 561,

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A to infer A", and generating all of the rules in general from this one direct rule by means of third-order rules. Although there are these theoretical advantages, the point I would like to emphasize is that in order to generate the rules of inference in general, it is not necessary to use anything beyond second-order rules, used as third-order rules. In this way, also, the appearance of an infinite regress of rule levels is avoided. Nothing beyond second-order rules derived from linguistic regularities is needed. With this viewpoint we can also see how logic is the study of not only the logical entailments, but also of entailment in general. The rules of type (c) can be divided into two types. Those rules which concern inferences involving logical constants are logical rules. But those rules of type (c) which concern inferences not involving logical constants essentially are rules regarding entailment in general. Transitivity is a rule of this type. A derivation can include logical inferences, semantical inferences, and physical inferences, but all of the inferences are governed by the overall rule that inferability is transitive. Corresponding to each of the direct and derived rules are statements of entailment. In general, entailment statements DESCRIBE the linguistic regularity, while the rules of inference PRESCRIBE that one go on in the regular way. We noted above, however, that many of the rules and entailment statements do not correspond to anything that can properly be called a 'linguistic regularity', and consequently they cannot in a straightforward sense be interpreted as prescriptions and descriptions. According to our interpretation of rules, we have argued that the regularity precedes the prescription and brings it about, but this interpretation is not applicable to all rules. Once certain rules are brought into effect by virtue of a linguistic regularity, other rules are prescribed by virtue of the nature of the rules of type (c). That is, these other rules derive their prescriptive character not directly from a regularity, but rather from other rules of inference which derive their prescriptive character from a regularity. The entailment statements corresponding to these other rules are to be considered as declarative reformulations of the rules. While all of the entailment statements could be considered as such, we prefer to consider some entailment statements to be actual descriptions of linguistic regularities in order to emphasize their empirical character. The point here is that all entailments and all rules of inference ultimately can be traced back to empirical regularities. Up to this point we have argued that logical rules of inference can be seen as conventions, that is, as regular modes of behavior with a prescriptive force. We have also shown that this view is compatible with the view of logic as the study of rules of inference and with the view of logic as the study of entailment in general. In addition, we have argued that it is of no help to view the rules of logic in any other light, because we have seen that the problems of explaining how one learns to infer validly are multiplied by assuming that rules or entailments can be known a priori by the student. Of course, this argument does not PROVE that logical rules are only linguistic regularities. But as we noted in discussing our position on

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the epistemological status of metaphysics, the best that can be done is to show that such a view is plausible, is internally consistent, and solves the relevant problems. We do not mean to suggest that we have solved all the problems connected with viewing logical rules as linguistic regularities. For instance, logical rules differ markedly from other conventions, such as eating pie with a fork at formal dinners, in that in a valid argument truth is 'transmitted' from the premises to the conclusion. This would seem to be a matter that is independent of convention. We will take up this problem with our interpretation of logical rules in detail in Chapter IV.

D.

QUINE AND THE TORTOISE

At this juncture we would like to discuss a criticism which Quine has made of the conventionalist thesis. Quine has based this criticism on Lewis Carroll's article "What the Tortoise Said to Achilles", which we mentioned briefly at the beginning of our discussion of rules. We noted that Carroll's argument shows that something outside of a valid inference is necessary to justify it; viz., a rule or principle of inference. In connection with Quine's criticism, we will want to look more closely at the import of Carroll's article. Carroll has the Tortoise accept two statements, A and B, and not statement Z which logically follows from A and B. The statements are as follows: (A) Things that are equal to the same thing are equal to each other. (B) The two sides of this triangle are things equal to the same. (Z) The two sides of this triangle are equal to each other. The problem is to get the Tortoise to accept Z since he accepts A and B and the sequence is obviously a valid one. The Tortoise, being of a generous nature, is willing, however, to accept a large number of other statements. For example: (C) If A and B are true, then Z must be true. (D) If A and B and C are true, then Z must be true. While the Tortoise will accept these additional statements, he will still not accept Z. Statements C and D are ambiguous in meaning since they refer to truth. One can, for example, interpret C in either of two ways: (CO If A and B, then Z. (C") The statements (or propositions) A and B entail the statement (or proposition) Z. How does Achilles interpret C? An indication that the Tortoise interprets C as C is given by the fact that the Tortoise says that he will accept C, IF Achilles writes it down in his notebook with A and B. This would seem to indicate that the Tortoise will accept the first-order statement C , but not necessarily the secondorder statement C". It is clear that adding C to the argument as a premise will neither make the argument more valid nor more obviously valid than it already was. This much we can learn from the Carroll article for sure - adding as an

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additional premise to an argument a conditional with the conjunction of the premises as antecedent and the conclusion as consequent will not make an iota of difference to the argument. If one does not accept the conclusion of the argument without the additional premise, he will not accept it with it. If something is going to justify inferring Z from A and B, it must be something other than an additional premise. The problem of inferring Z from A and B and C' is identical with the problem of inferring Z from A and B. Quine patterns his critique of conventionalism on this argument.15 He begins by outlining a conventionalist thesis based on the postulational method or, as it is also called, the method of implicit definition. The truths of logic are true because they are stipulated to be true. But since we have an infinite number of logical truths, we cannot stipulate each of them to be true individually. We avoid this problem by stipulating an infinite number of logical truths to be true all at once. For example with the convention, "(H) Let any expression be true which yields a truth when put for 'q' in the result of putting a truth for 'p' in 'If p then q'", we have stipulated an infinite number of statements to be true. (Strictly speaking, Quine should have written the last two instances of the word 'truth' in (II) as 'stipulated truth'. His defense of conventionalism is at best applicable only to logical truths, and (II) therefore is a general convention for obtaining logical truths from other LOGICAL truths.) After setting up this defense of conventionalism, Quine goes on to criticize it, and we will quote the relevant passage: In the adoption of the very conventions (I)—(III) etc. whereby logic itself is set up, however, a difficulty remains to be faced. Each of these conventions is general, announcing the truth of every one of an infinity of statements conforming to a general description, derivation of the truth of any specific statement from the general convention thus requires a logical inference, and this involves us in an infinite regress. E.g., in deriving (6) from (3) and (5) on the authority of (II) we infer, from the general announcement (II) and the specific premise that (3) and (5) are true statements, the conclusion that (7) (6) is to be true. An examination of this inference will reveal the regress. For present purposes it will be simpler to rewrite (II) thus: (II') N o matter what x may be, no matter what y may be, no matter what z may be, if x and z are true statements and z is the result of putting x for 'p' and y for 'q' in 'If p, then q' then y is to be true. We are to take (II') as a premise, then, and in addition the premise that (3) and (5) are true. 16

If we write 'M' as an abbreviation for 'Time is Money', then we can write statements (3), (5), and (6) as follows: (3) If M then if not-M then M. 15

W. V. O. Quine, "Truth by Convention", reprinted in Readings in Philosophical eds. H. Feigl and W. Sellars (New York, 1949), pp. 250-73. 18 Ibid., pp. 270-71.

Analysis,

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(5) If if M then if not-M then M then if if if not-M then M then M then if M then M. (6) If if if not-M then M then M then if M then M. Quine says that in inferring (6) from (3) and (5) on the AUTHORITY of (II) we are inferring (7) from the statement '(3) and (5) are true' and convention (II'). As he says at the end of the above quotation, (II') is a premise in the inference. Quine fails to distinguish between inferring on the authority of (II') and inferring with (II') as a premise. The distinction between these two inferences hangs on the distinction analogous to the distinction between C' and C". Clearly, if we interpret (II') as a first-order conditional statement and if we add it as a premise along with (3) and (5), we are no further along to inferring (6) than when we had (3) and (5) alone. As long as (II') is interpreted in this manner, we have a situation similar to that with Achilles and the Tortoise. Clearly, if any stipulation is to help us get from (3) and (5) to (6), it must be something outside of the argument, not just another premise in the argument. To compound the difficulty, Quine has confused the inference of (6) from (3) and (5) with the inference of (7) from the statement '(3) and (5) are true'. One might argue that the latter inference does require (II') as an additional premise, and if (II') is an additional premise, then we need logic to get us from (II') and '(3) and (5) are true' to (7). But we are not concerned with this inference; rather we are concerned with the inference of (6) from (3) and (5). Certainly Quine is right, that if (II') must be used as a premise in inferring (6) from (3) and (5), then we are off on an infinite regress, but it is clear that in this case adding (II') as an additional premise is unnecessary. (II') functions in THIS argument as the authority by which we make the inference. It is only by assimilating two very distinct arguments that Quine's critique seems to have an air of plausibility. When setting up a general convention, we must get straight what the relationship is between the convention and the sequences of sentences to which it is applicable. To be more specific we must understand both the logical and the epistemological relationship between the convention and the sentences. First of all, it is clear that if the convention is to function in place of our ordinary justification for an inference, it cannot be a premise in the inference, for then we still would need our ordinary justification for inferring. Secondly, if the convention is not a premise, then it is, so to speak, being used but not mentioned. That is, the convention must be used in making the inference, but it does not actually occur in the inference. What then is the epistemological relationship between the convention and the inference? Generally speaking, the relationship is one of a rule to a rule bound activity, such as a game. When one plays a game, one plays by the rules of the game, but the rules themselves do not occur in the game. A rule is mentioned in a game only if there is a dispute. In this case, the game stops while the dispute is settled, and once settled there is no more mention of the rules. But the game goes on being

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played in accordance with the rules. Once this is realized, we have avoilded the Lewis Carroll type paradox in Quine's critique of conventionalism. The paradox occurs by mistakenly making the rule of a game one of the moves in the game. There is a variant of Quine's critique however which does hit upon a problem. While (II') is not functioning as a premise in addition to (3) and (5) in the inference of (6), one might say that it is functioning as the premise from which one infers that one can infer (6) from (3) and (5). By saying that a general convention is like a rule of a game we solved one problem, but we still have the problem of how one applies a rule to a particular case. Is the relationship between a rule and its application in a particular case one of premise to conclusion? If we need logic in order to apply our general convention (II') to a particular inference, then our general conventions are not functioning in place of the ordinary justification of inference but only in addition to them. That this is possibly what Quine had in mind is indicated by the following passage: In a word, the difficulty is that if logic is to proceed mediately from conventions, logic is needed for inferring logic from the conventions. Alternatively, the difficulty which appears thus as a self-presupposition of doctrine can be framed as turning upon a self-presupposition of primitives. It is supposed that the //-idiom, the noi-idiom, the every-idiom, and so on, mean nothing to us initially, and that we adopt the conventions (I)-(VII) by way of circumscribing their meaning; and the difficulty is that communication of (I)-(VII) themselves depends upon free use of those very idioms which we are attempting to circumscribe, and can succeed only if we are already conversant with the idioms. 17

Let us begin by noting that the criterion by which we judge whether a man understands a rule formulation is not whether he understands the individual words, but whether he uses the rule correctly. Regardless of the idioms used in formulating our conventions, the final criterion of understanding is proper usage. We must rid ourselves of the idea that there is a logical connection between a rule and its proper application such that one infers the application of the rule from the rule formulation. Anyone who has taught logic knows that a student has to be taught how to use a rule. They see that Modus Ponens arguments are valid much more quickly than they learn how to use the rule of Modus Ponens. But if the connection between a rule formulation and its applications were a logical connection one would expect them to learn how to use the rule just as quickly as they learn how to infer any conclusion from certain premises. One might argue that once the student has interpreted the rule correctly, then its correct use comes automatically, that is, there is a logical connection between a rule formulation plus its correct interpretation and its correct use. But a rule formulation together with its correct interpretation is IDENTICAL with its correct use. Of course, there is a logical connection between a rule formulation with its correct "

Ibid., pp. 271-72.

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interpretation and its proper use, but this connection is analogous to the relationship between a bushel of apples and the apples in the bushel. In general the relation between a rule formulation and its instances is much more akin to the relation between a general word and its extension, than to the relation between premises and conclusion of an argument. We will return to this theme on pages 79-82 of Chapter III. Our reply to Quine's criticism, then, is that one does not infer logic from conventions; one does not infer the application of a rule from a rule. One infers in accordance with a rule, and thus one must be taught how to infer; one must also be taught how to interpret correctly a rule formulation, and this is true regardless of the idioms used. In the last analysis, the person who determines whether a rule formulation has been correctly interpreted is the person who formulated the rule, for it is HIS formulation of the rule. While we think that the conventionalist thesis that Quine has set up is deficient, we do not think that Quine has actually hit upon the difficulties. Our first interpretation of his critique is obviously invalid. Our discussion of the second interpretation of his critique gave us a chance to reemphasize our views on the relationship between rules and their instances. Quine is certainly right that if our rules must function as premises with their instances as conclusions, we have not done away with logic. The same argument was used earlier, in fact, to show that knowing how to infer is not simply a matter of knowing or being aware of the rules. But rules do not function as premises. Moreover we have tried to show that the relationship between a rule formulation and its correct application is a tenuous one depending on correct interpretation, and this comes down to correct application. The same point was made earlier in regard to rules themselves. We have tried to show here that setting up rule formulations is really irrelevant to putting logic on a conventional basis. The important thing is the convention, not some arbitrary formulation of it. Similarly we tried to show earlier that the rules of logic (if there are any, taken in a nonconventional sense) are irrelevant. The important aspect of a rule is its correct application, and this must be learned prior to any formulation of it and regardless of any awareness of it. If this is the case, the rule formulation itself performs only nonessential tasks, and we have no need at all for rules (taken in a nonconventional sense). In our natural language, linguistic regularities perform all the jobs we require of rules. In a constructed language, all that is required is training.

E.

IMPLICATION AND INFERENCE

We have argued that entailment statements and rules of inference have the same content - the linguistic regularity - differing from each other only pragmatically. For example, the entailment statement "'if p then q' and 'p' together entail 'q'"

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and the rule of inference "From 'if p then q' and 'p' to infer 'q'" both express the same linguistic regularity. Similarly the following two statements both express an instance of this regularity and do not differ even pragmatically: " 'if Plato is a man then Plato is mortal' and 'Plato is a man' entails 'Plato is mortal'" and " 'If Plato is a man then Plato is mortal. Plato is a man. Therefore, Plato is mortal.' is a valid argument". We have treated implication as an unambiguous concept expressing the relationship between the premises and conclusion of a valid argument or rule of inference. From this point of view there is no denying that material implication is not a kind of implication. What then do we make of decades of use of the term material implication for '=3'? 18 What do we make of the following quotation from Lewis: "If we conceive that the distinguishing mark of an implication relation, plq, is that if p is a true premise and plq holds, q will also be true, then, as we have seen, the number of relations having this property, but distinguished from one another in other respects and answering to different laws, is indefinitely large."?19 There are two distinct cases of prima facie implications to be considered. According to one type of implication, if we are given 'p' and 'plq', we can infer 'q'. According to another type of implication relation, if 'plq' is true, then given 'p' one can infer 'q' immediately without 'plq' as a premise, 'p & q' implies 'p' in this latter sense, for given 'p & q' one can infer 'p' immediately. Now it would appear that according to our characterization of implication the first type of implication noted above reduces to the second, for if 'plq' holds, 'q' is deducible from 'p' without 'plq' as a premise. But one might note that where 'plq' does NOT hold, while 'q' is not deducible from 'p' alone, it is deducible from 'p' together with 'plq'. To put this more generally, where we do not know whether or not 'plq' holds, we do not know whether or not 'q' is deducible from 'p', but we do know that it is deducible from 'p' together with 'plq'. If our characterization of implication is correct, then the two types of implication differ only where 'plq' is false or where we do not know whether it is true. Where 'plq' is known to be true, it is unnecessary as a premise. Therefore, where 'plq' is false or not known to be true, its force must nevertheless be such that if it were true it would be unnecessary. It follows then that 'p => q' does not have the logical force of an implication statement. Now sometimes even when 'plq' is true and unnecessary as a premise in the inference from 'p' to 'q' it is added as a premise anyway, and it is felt that it adds something to the argument, whereas at other times it is felt that it does not add something. For example, one believes that the Lewis Carroll argument "A, B, therefore Z" is valid, and that adding the premise 'C' would not be adding anything. 18

The inspiration for this section comes from Anderson and Belnap, "The Pure Calculus of Entailment", Journal of Symbolic Logic X X I (1962), 19-52, and "Tautological Entailments", Philosophical Studies XIII (1962), 9-24; although the details and particularly the discussion of physically valid arguments are my own. 18 C. I. Lewis and C. H. Langford, Symbolic Logic (New York, 1959), p. 235.

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On the other hand, adding the premise 'If a lead weight is dropped, it falls' to the argument 'I am going to drop this lead weight. Therefore, it will fall.' does add something to the latter argument. I submit that the effect of adding the premise to the argument in the latter case is to turn a physically valid argument into a logically valid argument. Similarly where it is not known whether 'plq' is true, adding 'plq' as a premise turns a physical argument of unknown validity into a logically valid argument. Where 'p, therefore q' is already a logically valid argument there is no point at all in adding 'plq' as a premise. I would suggest that the basic function of 'if . . . then . . .' statements is to turn physical arguments into logically valid arguments. Russell, Whitehead, Lewis, et al. were misled into thinking that 'p => q' was an implication statement because they thought that 'p ^ q, p, therefore q' was a logically valid argument. Regardless of why they thought that the latter was a valid argument, I would argue that it is not a valid argument. The test for whether an argument of the form 'p, plq, therefore q' is logically valid is whether when 'plq' holds, 'p therefore q' is physically valid. This is not true in general for 'p => q', but it is true in general for 'if p then q'. While the argument 'If snow is black, then grass is purple. Snow is black. Therefore, grass is purple.' might have some semblance of being logical, that this is an illusion becomes clear when one tries to consider 'Snow is black. Therefore grass is purple.' as a physically valid argument. It is not. Compare the latter with 'This object is at absolute zero. Therefore, it has no internal molecular motion.' Our view then is that 'if p then q' is the object language counterpart of " p ' implies ' q " and that its function is to make a physical argument into a logically valid argument. The burden of doubt is then shifted from the validity of the argument to the truth of a premise, although the difference here has no cognitive significance. Therefore, the early positivists who wanted to replace universal statements of science with rules of inference, were introducing a distinction which has no cognitive significance. They are interchangeable. What is important is why the rules are what they are. One sometimes conceives of 'if p then q' as 'justifying' the inference from 'p' to 'q'. It is this which makes Achilles' move to add ( Q as a premise seem initially plausible. That it does not work for Achilles should make us wary. The added premise 'if p then q' does not justify an inference; rather it guarantees it by turning it into a logical inference. It is just because Achilles' argument did not need a guarantee that the 'if . . . then . . . ' premise (C) was unnecessary and useless.

F.

RELATED CONCEPTS: SYNTACTICAL RULES,

SEMANTICAL RULES, METALANGUAGE, AND ARTIFICIAL LANGUAGE

To complete our analysis of rules and entailment, we will now discuss a number of related concepts, namely, syntactical rule, semantical rule, metalanguage (versus

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object language), and constructed language (versus natural language). We will make the following points: (a) according to our view of rules the logical rules of a language are syntactical rules; (b) semantical rules are not like logical rules of inference in that they are not prescriptive rules; (c) the metalanguage-object language distinction is applicable in a different way to syntactical rules than it is to semantical rules; and (d) the rules of natural languages are different than rules of constructed languages. First of all, we should note that in the first two topics we are considering the syntactical and semantical rules of natural languages. Therefore, we are concerned with descriptive syntax and semantics, respectively. As Carnap says: "Descriptive syntax is an empirical investigation of the syntactical features of given languages",20 and "By descriptive semantics we mean the description and analysis of the semantical features either of some particular historically given language, e.g., French, or of all historically given languages in general." 21 Our discussion of the logical rules of inference is essentially a development of Carnap's view in The Logical Syntax of Language. He says there that logic is concerned with the formal treatment of sentences, where 'formal' indicates that no reference is made either to the meaning of symbols or the sense of expressions, but only to the kinds and order of the symbols from which the expressions are constructed. We have indicated that we take entailment, in the first instance, to be a relation between sentences based on linguistic regularities. Entailment is for us, therefore, a 'formal' relation, in that it does not depend on the sense of the sentences, but merely on the empirical fact that there are certain transformational regularities in language. The linguistic regularity on which entailment is based is a formal regularity. We can say, then, that the logical rules of inference are syntactical transformational rules, which are independent of the meaning of the sentences. Since we regularly do transform sentences in a certain way, it is possible to hypothesize a set of rule formulations which are compatible with the way in which we transform sentences. The logical rules, in particular, are a subset of these syntactical transformation rules. The particular characteristics of these logical rules will be discussed in greater detail in Sections A.3 and A.4 of Chapter IV. We say, therefore, that the logical rules are conventions. This is Carnap's Principle of Tolerance as applied to natural languages. One apparent difference between Carnap's approach and ours might be noted. Carnap has recently commented that "not only pure syntax and pure semantics but also descriptive syntax and semantics, as I understand them and intend to construct them, do not contain any prescriptive components. . . . But in syntax and semantics I deliberately leave aside all prescriptive factors." 22 As we understand the rules of 20

Carnap, Introduction to Semantics, p. 12. *» Ibid., p. 11. 22 R. Carnap, "Replies and Systematic Expositions", in The Philosophy

of Rudolf

Carnap,

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logic, they do contain a prescriptive component; however, this is merely a verbal difference. What Carnap calls transformational rules are what we call entailment statements, and entailment statements, as we noted, contain only the descriptive aspect of a rule. A rule we remember is a linguistic regularity with a prescriptive function. The entailment statement describes the regularity, and this is what Carnap's transformation rules in descriptive syntax do. Carnap does not consider the rule formulations which contain as well the prescriptive component. In his later work Carnap has found it necessary to restrict his Principle of Tolerance. He still holds that logical rules are syntactical rules of transformation, but he no longer holds that they can be merely a matter of convention in an adequate language. According to Carnap's later beliefs, a language not only has syntactical rules but semantical rules as well. Now if a language is to be 'adequate', the logical rules cannot be merely the result of linguistic regularity, for they must correspond in a certain way to the semantical rules. If we call a language which has only syntactical rules a 'calculus', then if a set of semantical rules form a true interpretation of the calculus, we can say that the calculus together with the semantical rules form an adequate language. It follows then that a historical language may not be adequate in that its rules of transformation may not correspond correctly to its semantical rules. If we accept the semantical rules of the language then we would have to say that the linguistic regularities do not comprise valid rules of inference. This would hardly be compatible with a conventionalist approach to logic, for we want to argue that linguistic regularities are valid rules by virtue of being linguistic regularities. If Carnap is right, then linguistic regularities are valid rules if they correspond in the right manner to the semantical rules. If they do not correspond in the right manner, then linguistic regularities can be invalid rules. While we would agree with Carnap that SOME linguistic regularities can be invalid, we would disagree with him on two counts: (a) the LOGICAL linguistic regularities cannot be invalid, and (b) the validity of logical linguistic regularities is not a matter of their correspondence with semantical rules. These points will be discussed again later. For the time being I would like to look at one aspect of this argument, namely, at the semantical rules on which it is based. In particular, we want to know what kind of rules semantical rules are. Are they conventional rules like the logical rules? Since we are concerned with semantical rules in relation to logic, we are primarily concerned with the semantical rules which Carnap says interpret the logical constants, for example, "A sentence of the form 'nicht . . .' is true if and only if the sentence '. . . ' is not true." We might ask first of all if this type of rule is either prescriptive or based on a prescription. We know that while entailment statements are not prescriptions, they are based on prescriptions, for what makes the entailment statement something more than merely a description of a linguistic regularity is that the linguistic regularity that is described has a prescriptive function. In answering the above question the first thing we might notice is that the

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language in which the rule is expressed is English while the rule is about the German language. Could a semantical rule be expressed so that the metalanguage and object language were the same and still serve its function as a semantical rule? The purpose of semantical rules is "to determine the meaning or sense of the sentences".23 We assume then that the meaning of the sentences of the metalanguage has been determined, that is, that we understand them, and that we are to use them to ascertain the meaning of the sentences of the object language. But if the metalanguage and the object language were the same, the object language would already have to have been interpreted in the first place. Therefore, if the semantical rules are to determine the meaning of the sentences of the object language, the metalanguage must be different than the object language. A certain peculiarity arises, however, if we look at semantical rules as descriptions of linguistic regularities with a prescriptive function. One might say, for instance, that the semantical rule mentioned above describes the linguistic habit of Germans to use the word 'nicht' to mean not, and that this linguistic habit has a prescriptive function which perpetuates the regularity. But if a semantical rule is merely a description of such a linguistic regularity, it might as well have been written, "Der Satz von der Form 'nicht . . .' ist wahr, wenn und nur wenn der Satz ' . . . ' ist nicht wahr." This sentence would be taken to describe the fact that German language users habitually use the word 'nicht' to mean nicht. If semantical rules are merely linguistic regularities with a prescriptive function, then in the statement of the rule it should not be necessary that the metalanguage be different from the object language. On the other hand, if a semantical rule is to DETERMINE the meaning or sense of a sentence, the metalanguage must be different than the object language. I see no plausible sense of 'determine' in which this could be otherwise. It would appear then that we must either give up the view that semantical rules determine the meaning of sentences or the view that they are descriptive of linguistic regularities. In regard to the semantical rules for molecular sentences, I think we must give up the latter view, and along with this we must also give up the view that semantical rules are prescriptive or based on prescriptions. Therefore, following a semantical rule cannot be simply a matter of knowing how to do something as it is in the case of syntactical rules. In fact, we would like to go a step further and say that understanding a semantical rule is not a matter of knowing how at all, but rather a matter of knowing that. We pointed out in the case of prescriptive rules that the essential aspect of understanding these rules is knowing how to go on in the correct maimer. That is, one could operate in accordance with such rules without knowing the rule formulations. Since knowing semantical rules involves understanding some metalanguage, such rules cannot be prescriptive rules. We will return to the discussion of semantical rules in Chapters III and IV with this result in mind. "

Carnap, Introduction

to Semantics, p. 22.

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We have seen that semantical rules involve the metalanguage-object language distinction essentially. In the case of syntactical rules they can be the same, as one can express the syntactical rules for English in English. We can do this in syntax for several reasons: first of all, no claim is made by syntactical rules to give the meaning of the logical words of the object language. They are descriptions of what sentences one can infer from what sentences. Moreover, the essential aspect of understanding syntactical rules is knowing how to follow them, and this is something that can be learned whether or not one understands the rule formulation itself. The rule formulations of syntactical rules are incidental to the rules which exist before their formulation. It is for this reason that the particular language in which they are expressed is irrelevant. Carnap is ambiguous in his use of the term 'pure syntax'. In one of its uses pure syntax is the study of syntactical systems, where a syntactical system is to be contrasted with a natural language. A syntactical system in effect defines a language, an artificially constructed language. Pure syntax, therefore, is the study of artificially constructed languages which are defined by a collection of syntactical rules, i.e., by the syntactical system. Descriptive syntax is the study of historically given languages with their historically given rules. In the latter case we can discover the rules of a language, and our entailment statements are, therefore, descriptive statements about the language. In the case of pure syntax, the syntactical rules are arbitrary stipulations. Carnap would say that these rules DEFINE entailment (or derivability) for a constructed language. I would prefer to say that they STIPULATE the entailments which are to hold in a particular constructed language. There is a sense of the word 'convention' used in philosophy, such that one can say that arbitrary stipulations are 'true by convention' or that they are 'conventions'. We must be careful to distinguish this sense of convention from the sense of convention in which the rules of a natural language are conventions. The rules of a natural language are, we have argued, customs. The rules of a constructed language are arbitrary stipulations. When we argue that logical truisms are true by 'convention', we must be careful to remember that we are talking about 'convention' in the sense of custom, as discussed earlier. We can construct an artificial language parallel to a given historical language, but we must remember that our rules in this case will be conventions in the sense of stipulations, while the rules of the historical language are conventions in the sense of customs.

Ill CONVENTIONALISM, SEMANTICS, AND ONTOLOGY

In the last century various theses of conventionalism in logic have been propounded. All of the theses are not equally tenable, as some can be shown to be inadequate by very simple and straightforward criticisms. On the other hand, we will find that while one of the conventionalist theses is immune to most of these criticisms, a series of criticisms, based on certain semantical theories with their corresponding ontologies, is not so easily rebutted. We will want to look at these semantical theories and ontologies closely to appreciate fully the force of their argument, for a completely adequate thesis of conventionalism in logic cannot be constructed without awareness of the arguments against the thesis and of the conceptual framework in which these criticisms are embodied. In Chapter IV we will attempt to develop an alternative conceptual framework within which the conventionalist thesis is tenable. We will also compare and contrast the usual theses of conventionalism in logic with the theses of conventionalism in geometry and physics and also with Quine's combined thesis of conventionalism in physics and logic. We will see that these four conventionalist theses - conventionalism in logic, in geometry, in physics, and in physics and logic together - as usually conceived, are four distinct theses with fundamentally different rationales. We will then attempt to show in Chapter IV that an adequately developed thesis of conventionalism in any of these fields bears a close resemblance to conventionalism in the others.

A.

NECESSITY, ANALYTICITY, A N D THE A

PRIORI

Since the conventionalist thesis in logic is a thesis about logical truisms, such as 'A or not A', which are considered by many to be necessary, analytic, and a priori truths, it will be well for us to discuss and distinguish these terms and their counterparts: contingent, synthetic, and a posteriori-, for it is the burden of the conventionalist thesis to explain why these logical truisms are necessary, as well as to give a theory about the nature of logical necessity. Let us begin by discussing the a priori-a posteriori dichotomy. It is important to see that these are primarily concepts of epistemology; that is, they are concepts

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regarding how we come to know that a statement is true. A statement is considered to be a priori if we can come to know that it is true without using our five senses, that is, loosely speaking, without observation. Now we could define an a posteriori statement simply as a statement which is not a priori, but it is interesting to attempt to find a positive definition. For example, we might define an a posteriori statement as a statement which we can come to know to be true by means of observation. But there are many statements which most philosophers would say we can never know for certain to be true, such as statements about the past and universal empirical statements. We might weaken our definition of a posteriori to include those statements for which we can at least get evidence by means of observation. Now if we divide philosophers into rationalists and empiricists, we can say that generally the latter hold that observable data is EVIDENCE FOR a universal empirical truth, while the former hold that it is only a SIGN OF the universal truth. Specifically, the rationalists hold that observable data is evidence for a constant conjunction, but they would go on to say that a constant conjunction is only a sign of the universal truth, that is, of the physical entailment. Therefore, while for the empiricists the a priori-a posteriori classification of statements is exhaustive, for the rationalist it is not. For the rationalists there are statements for which we cannot get evidence by observation and which only the most extreme would say could be known to be true without observation. Historically there have been many other characterizations of a priori truth of which I will mention three: (1) universal and necessary truth, (2) truth of which I can be certain, and (3) truth ex vi terminorum.1 The first two definitions have a particular appeal. We do expect a priori truths to be necessary and vice versa, but I do not think that we should by definition obliterate the distinction between apriority and necessity; it seems to me that they are conceptually distinct. With respect to the second definition, while we expect all a priori knowledge to be certain, we do not necessarily expect all certain knowledge to be a priori; for example, some observational knowledge MAY be certain. This latter problem could be overcome if we described an a priori truth as a universal truth of which I can be certain. But then we would have the problem with the imprecision of the notion of certainty. The third characterization above seems to belong more properly in a discussion of analyticity for it has to do with words and their meanings, and we will discuss it in that context. The terms 'necessary' and 'contingent' are often taken as synonyms for 'a priori1 and 'a posteriori, respectively. Kant indicated their close relationship when he said that we could take necessity as a sign of an a priori truth. I do not think, however, that Kant would consider them synonymous.2 Necessity, unlike apriority, is not an 1 See, for example, Wilfrid Sellars, "Is There a Synthetic A PrioriT' reprinted in Science, Perception, and Reality (New York, 1963), pp. 299-300. 1 Kant seems to hold the position that certain truths ATE necessary and universal BECAUSE they are a priori. See, for example, his Critique of Pure Reason, trans. Max Muller, 2nd ed. rev. (Garden City, N e w York, 1961), pp. 26-27.

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epistemological concept. Necessity and contingency are properties of a truth (state of affairs, statement, proposition) independent of the method whereby it comes to be known. If a truth is necessary, and if we can come to know it to be true, then it is a priori, and if a truth is contingent, it is a posteriori. The difficulty with the concept of necessity is that one cannot give a definition of the concept without at the same time giving a theory about the nature of necessity.3 We will therefore refrain from further discussion of necessity until we are also in a position to give a theory about its nature. The terms 'analytic' and 'synthetic' are used ambiguously, sometimes referring to how the statements can be discovered to be true and at other times referring to the cognitive content of the statement. In the first sense they are being used as epistemological terms related to the terms 'a priori and 'a posteriori". To characterize a statement as analytic is to specify in what manner the statement is known a priori. For example, Aune characterizes an analytic statement as one which can be known to be true (or false) by conceptual analysis alone.4 If it is not decidable by conceptual analysis alone, the statement would be characterized as synthetic. While this is certainly a legitimate use of the terms, the more prevalent use of the terms in the literature is to specify the way in which a statement is true. Here we can distinguish at least four different characterizations of analytic statements: (a) a statement which is true by virtue of the meanings of the constituent terms, that is ex vi terminorum; (b) a statement which is true independently of empirical matters of fact; (c) a statement which is either a logical truism or is obtainable from a logical truism by substituting synonyms for synonyms - this use traces its origin back to Kant; 5 and (d) a statement which is empty of factual content.6 We note, on the one hand, that only the first of these characterizations actually tells us why the analytic statement is true. On the other hand, in lieu of a theory of meaning, characterizing a statement as analytic in this sense has the effect of raising questions rather than answering them. We want to know what the meaning of a term is and how a statement can be true by virtue of meanings. The third characterization is the most definite and specific, and also somewhat narrower than the others. A statement, such as "Nothing is red and green all over", would not generally be considered analytic according to this third characterization, although most philosophers would consider it analytic according to the first characterization. Despite the specificity of this characterization, it likewise has the effect of raising questions rather than answering them, for we immediately want to know why a 3

For example, Hume rejected any attempt to define the term 'necessary connection' because it is absurd to define am obscure term in terms of another equally obscure term, such as 'efficacy' and 'force'. He, therefore, went in search of the origin of the idea of necessary connection. See his A Treatise of Human Nature (Garden City, N e w York, 1961), pp. 143-44. 4 Bruce Aune, "Is There an Analytic A Priori?" Jour. Philos. LX (1963), 282. 5 See Kant, Critique, p. 29. • For this characterization of analytic statements see, among others, Alfred Jules Ayer, Language, Truth, and Logic, 2nd ed. (New York, n.d.), p. 79.

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logical truism is true. Why should we give logical truisms special status? Of course, one function of this study is to clarify the status of logical truisms. The second characterization can be understood in two ways. First, a statement is analytic simply if its truth does not depend on empirical matters of fact, although empirical matters of fact may depend on its truth. For example, according to this characterization statements about PHYSICAL entailments between abstract entities would be considered analytic. Clearly, then, taking the second characterization in this manner is too broad, for we definitely want to consider physical entailment statements as synthetic. Secondly, we could consider a statement as analytic if its truth does not depend on empirical matters of fact AND vice versa. I think we can assume here that a statement is analytic in this second sense if and only if it is true ex vi terminorum. The first characterization simply goes a step further and tells us why the statement is true independent of empirical matters of fact. We will be using the term 'analytic' according to this combination of the first and second characterizations. We do NOT take this characterization as equivalent to the fourth. We shall shortly see the reason for denying that analytic statements are always empty of factual content. According to our characterization, then, an analytic statement is a statement whose truth is independent of empirical matters of fact, whereas an a priori statement can be known to be true without observation. We must also be careful not to identify synthetic statements with a posteriori statements. A synthetic statement is one which is not true ex vi terminorum or, taking synthetic in its epistemological sense, which cannot be known to be true by conceptual analysis ALONE. Whichever way we consider 'synthetic', it is possible that a synthetic statement could be known a priori. There may be, for example, a method of obtaining knowledge which does not involve observation or conceptual analysis. One important moral which we should draw from these distinctions is that calling a statement analytic does not commit one to a conventionalist theory. Conventionalism is a theory which begins by assuming that the logical truisms are analytic and which goes on to explain the kind of necessity that logical truisms have. We must be careful to distinguish between what the conventionalist assumes and what he tries to prove, for this assumption would also be accepted by a number of nonconventionalists .7 We denyed above that analytic statements are always empty of factual content. Some analytic statements, specifically the logical truisms, clearly are. A statement such as 'For every p, p or not p' involves ONLY logical constants and bound variables. Such a statement could have empirical content only if the logical constants in some way had empirical content, and this latter possibility seems unlikely. Now what if we consider statements, such as "All bachelors are unmarried" and 7 For example, A. Ewing, "The Linguistic Theory of A Priori Propositions", Proc. Arts. Soc. XL (1940), 231-32.

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"Nothing is red and green all over"? These statements, unlike logical truisms, do contain empirical concepts. Might they not therefore also have empirical content? The question, however, is unclear for we have not specified exactly what it is for a sentence to have factual or empirical content. Let us say that a statement has factual content if we can get evidence for its truth a posteriori. This does not exclude the possibility that the statement could also be known a priori. In fact we shall assume that the two statements above are both analytic, that is, true ex vi terminorum, and a priori, that is, can be known to be true without observation. We are asking whether they might not also be a posteriori. Let us suppose that someone is taught the meaning of 'bachelor' and the meaning of 'unmarried' by ostensive definition on nonoverlapping collections of human beings. Let us also say that he has learned the words well enough to go on correctly classifying human beings as bachelors and as unmarried. Such a person may form the hypothesis that all bachelors are unmarried on the evidence that any bachelor he has seen is also unmarried. Should we not say in this case that he has at least come to believe that all bachelors are unmarried on the basis of observation? While the statement is clearly true independent of experience, there is also a sense in which we would not want to say that it is empty of factual content, for then we could not even form the hypothesis that it is true on the basis of observation. Similar considerations apply to the words 'red' and 'green'. We will return to a discussion of the factual content of analytic sentences in Section C.l of Chapter IV. The nonconventionalist might reply to this ploy by saying that it is the statement "All bachelors must be unmarried" which is empty of factual content and which is a priori and cannot be a posteriori. The statement "All bachelors are unmarried" can then be derived from this by a simple a priori logical step. A similar distinction can be made in the case of laws of nature; for example, between "All metals are conductors of electricity" and "All metals must be conductors of electricity." We can have evidence for the former statement, and if it is true, this is a sign that the latter statement is true also. The difference between this case and the case of "All bachelors are unmarried" is that the statement "All metals must be conductors of electricity" is generally recognized not to be a priori, while "All bachelors must be unmarried" would be considered a priori by the nonconventionalist. Both of these latter statements can be considered to be empty of empirical content when interpreted by the nonconventionalist, since they are statements about abstract, nonempirical entities. The synthetic character of the first and the analytic character of the second is then traced to the fact that while the second is a priori, the first is not. This excursion into nonconventionalist thought is meant as a sign that we have to distinguish carefully between necessary statements and statements of necessity. The considerations relevant to the two might be quite different. The fact that some analytic, a priori statements have factual content while the logical truisms do not indicates that the having or lacking of factual content is not particularly relevant to either the analyticity or apriority of logical truisms. It is other factors which

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determine that a statement is either analytic or a priori. Moreover, the manner in which the nonconventionalist interprets statements of necessity indicates that we must focus our attention on these statements, rather than the necessary statements themselves. It is the content of the statements of necessity which is critical for it is this which tells us something about the analyticity and apriority of necessary statements. It is here that we will find the really important difference between the conventionalist and nonconventionalist. In fact, however, the distinction between necessary statements and statements of necessity has almost universally been either disregarded or denied implicitly by conventionalist authors.8 There is even one conventionalist, Morris Lazerowitz, who explicitly denies the distinction. We will examine the reasons for this explicit disavowal, indicate the difficulties that it gets him into, and criticize it. He argues as follows: H e n c e , the view that the truth of a logically necessary proposition c a n be assured analytically, 'by reference to the meanings' of the words w h i c h express the proposition, and that it also might b e established, by 'generalization f r o m observed instances', entails the c o n s e q u e n c e that necessarily true propositions are not necessarily true. T h e view implies that the s a m e proposition could be true a priori and not o p e n to theoretical falsification and also empirical and o p e n to theoretical falsification. 9

This reasoning leads him to deny that necessary statements can have any factual content, and to go on to assimilate necessary statements to statements of necessity. As he says: "What we know in understanding the first [i.e., a necessary statement] is the same as what we know in understanding the second [i.e., a statement of necessity]."10 He goes on to argue that what we know in understanding these sentences is a fact about language, a verbal fact. He then quickly distinguishes what we know in understanding a sentence from what the sentence EXPRESSES, what it SAYS. This could possibly be tenable if he developed the distinction in greater detail, but he follows this distinction by saying that both the necessary statement and the statement of necessity ENTAIL and are ENTAILED BY contingent facts about how language is used.11 There is a clear-cut criticism which can be made of this view, centering around the strangeness of saying that a necessary statement entails and is entailed by a contingent statement. Regardless of our views about whether or not entailment holds between necessary statements or between contingent statements, it is generally held that a necessary statement cannot entail a contingent statement, and many philosophers would also deny the reverse relation. In lieu of a new theory of entailment, Lazerowitz must say that necessary 8

In particular, there is a failure to distinguish between the following kinds of sentences: "All A's are B's" and "All A's must be B's", as if the considerations relevant to these two sentences were identical. • Morris Lazerowitz, The Structure of Metaphysics (New York, 1955), pp. 260-61. 10 Ibid., p. 268. 11 Ibid., p. 271.

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statements are contingent, for only contingent statements can entail contingent statements. Lazerowitz could have extricated himself from this predicament by distinguishing between statements of necessity, which he could say entail and are entailed by statements about verbal usage, and necessary statements, which are justified by but not entailed by the statements of necessity. An analogous relationship here would be that between a rule and an argument justified by the rule. He could then have argued that because a necessary statement is a posteriori this does not entail it is not also a priori. Moreover, because a statement is a posteriori, it does not follow that it is not necessary. It is only by insufficient analysis that we have characterized a priori and a posteriori as exclusive and as concepts synonymous with necessity and contingency. If a statement is necessary then it may not be open to theoretical falsification despite the fact that it is also a posteriori. (I say "may not be" for I wish to leave open the possibility that there may be necessary statements which are open to theoretical falsification. Whether there are depends at least partially on our analysis of necessity.) I would suggest that Lazerowitz's example, "A flea is an insect", is necessary, not open to theoretical falsification, and a priori, but also a posteriori. My solution of Lazerowitz's problem should not be taken as definitive, but rather as suggestive, for it indicates that we should not identify necessary statements and statements of necessity without being aware of additional difficulties it makes for the conventionalist thesis. The conventionalist who fails to make this distinction finds himself in the absurd position of defending the view that necessary statements are contingent. We will discuss later in the chapter in more detail the exact relationship between statements about verbal usage, statements of necessity, and necessary statements.

B.

EXPOSITION OF T H E CONVENTIONALIST THESIS

With these preliminary remarks behind us, let us go on to discuss the various conventionalist theses. There are three major versions of the thesis: (a) logical truisms are contingent statements about linguistic usage; (b) logical truisms are not statements, but rather they express the implicit rules of language, that is, they are rule formulations in disguise; and (c) logical truisms are not linguistic rules but are true by virtue of linguistic rules. The first version of the thesis is almost universally discredited, even by conventionalists. The only author we have read who approximated saying that logical truisms are statements about linguistic usage is Ayer, who, however, disavows this interpretation of his thesis. What Ayer did say is that analytic statements "simply record our determination to use words in a certain fashion".12 This is the clearest 18

Ayer, Language,

Truth and Logic, p. 84.

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statement of his thesis, and as it stands it could be interpreted in any of the three ways mentioned above. The word 'record' is ambiguous and can be interpreted either as 'describe' or 'depend on'. If one interprets 'record' as 'describe', the thesis is still ambiguous as to whether we are describing the DETERMINATION to use words in a certain way, that is, the rule, or whether we are describing the determination to USE WORDS IN A CERTAIN WAY, that is, the linguistic regularity itself. In the first case one has the second version of conventionalism and in the second case one has the first version. If one interprets 'record' as 'depend on' one has the third version of conventionalism. It is in the latter manner that Ayer interprets his thesis in the Preface to the Second Edition.13 If we were to interpret analytic statements as a description of how in fact we do use words, the thesis is surely open to the objection that then there are no necessary statements, for the necessary statements themselves are thus contingent. Explaining the necessity of necessary statements by saying that they are not necessary statements is no explanation at all.14 According to the second version of conventionalism, analytic statements do not express propositions but are in fact formulations of the implicit rules of language. This is the position which Ayer took in a symposium with Whiteley and Black and later repudiated. He originally said: "They make no statement whose truth can be accepted or denied. They merely lay down a rule that can be followed or disobeyed. Their necessity then, we must say, consists in the fact that it does not make sense to deny them."15 This view has been well defended by Britton who summarizes his argument as follows: "I have claimed that necessary truths are expressions of rules, [and] that it is an empirical fact that we have adopted these rules."16 His argument is based on the well known fact that corresponding to most (if not all) of the laws of logic there are rules of inference. For example, corresponding to the law of logic 'If p and if p then q then q', there is the rule of inference Modus Ponens. Moreover, he argues that we cannot understand a necessary truth unless we know how to use the corresponding rule. Of course, one might accept this and still deny that necessary statements and rules of inference are one and the same, for the distinction between necessary statements and rules corresponds to the distinction which we have made between necessary statements and statements of necessity. One could easily argue, as has Stuart Hampshire, that the "statements or denials of logically necessary connections . . . are formulations of the implicit rules gov-

13

Ibid., pp. 16-17. See the criticism of C. D . Broad, "Are There Synthetic A Priori Truths?" Proc. Aris. Soc. Sup. Vol. X V (1936), 102-17. 15 A. J. Ayer, "Truth by Convention: A Symposium by A. J. Ayer, C. H. Whiteley, and M. Black", Analsis IV (1936-37), 19-20. 19 Karl Britton, "Are Necessary Truths True by Convention?" Proc. Aris. Soc. Sup. Vol. X X I (1947), 100. 14

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erning our use of words".17 As Hampshire goes on to say: "They are statements of logical necessity, or its absence, because they are statements of rules or the absence of rules; the words 'necessity' and 'rule' have the same sense in this context." 18 If Hampshire is correct, then Britton's analysis of rules turns out to be an analysis not of necessary statements, but of statements of necessity, leaving us without an analysis of the necessary statements themselves. Britton's analysis is but a prolegomena to a conventionalist thesis. In this way we come to the third version of conventionalism which asserts that necessary statements are true by virtue of the implicit rules of language. If statements of necessity are rule formulations, then necessary statements are true by virtue of the validity of statements of necessity. It is evident that the burden of the conventionalist argument will rest on making clear the phrase "by virtue of". If statements of necessity are statements about linguistic usage, "by virtue of" cannot mean entailment, for we would thus be back in the position of having contingent statements entail necessary statements. Also, if the statements of necessity are rule formulations, then the relationship between the facts of linguistic usage and linguistic rules must be analyzed, for if the linguistic rules cannot be shown to be conventions, the necessary statements based on them will not be conventional either. What then is the relationship between linguistic rules and necessary statements for the conventionalist? Black claims that necessary statements are NONDEDUCTIVELY 19 CERTIFIED by checking them against the corresponding rule of language. A necessary and sufficient condition for the necessary statement to be true is that the corresponding rule of language be in force.20 If one can empirically confirm that R is a rule of language, then one has certified the corresponding necessary statement S.21 This is interesting, but not very useful, for we want to know how a rule 'certifies' a necessary statement. Again Malcolm says: "What justifies you in putting down 2 + 2 = 4 in your calculation is a certain fact about the way everybody uses the expressions '2 4- 2' and '4'. But you do not assert this fact when in doing the calculation vou write down . . . '2 + 2 = 4'." 22 As in the previous case the exact relation of 'justification' is not specified. Moreover, we note in this case that what justifies the necessary statement is not a linguisic rule but a fact about linguistic usage. Korner has come much closer to the truth in characterizing the relationship between a rule and its instances as one of 'satisfaction'.23 He says that what satisfies a rule is a state of affairs or an empirical proposition which expresses the state of affairs. But how can a rule justify a necessary statement if it is satisfied by an 17

Stuart Hampshire, "Logical Necessity", Philosophy XXIII (1948), 338. Ibid. 19 Black, Models and Metaphors, p. 64. 20 Ibid., p. 68. 21 Ibid., p. 69. 22 N . Malcolm, "Are Necessary Propositions Really Verbal?" Mind IL (1940), 198. 23 S. Korner, "Are All Philosophical Questions Questions of Language?" Proc. Aris. Sup. Vol. XXII (1948), 71. »8

Soc.

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empirical proposition, rather than a necessary statement? As Britton says, a rule governs a procedure. A rule does not justify a necessary statement per se, but rather justifies the inference of the necessary statement from something else. Rules like " '2 + 2' may be substituted for '4' and vice versa" justify not the statement '2 + 2 = 4', but rather they justify inferring '2 + 2 = 4' from '2 + 2 = 2 + 2'. Such an inferring is a procedure, an empirical state of affairs, and it 'satisfies' the rule in that it is precisely the sort of thing which a rule says that one can do. A linguistic rule, like any rule, permits, forbids, or requires a certain action. In the case of the above stated linguistic rule, the permitted action is one of inferring. It likewise becomes clear from this analysis that rules like " ' 2 + 2' may be substituted for '4' and vice versa" will not 'create' necessary truth, since they only justify the inferring of one necessary statement from another. Such substitution rules must be supplemented if we are to have truth by convention. If we have linguistic rules to the effect that a statement of a certain type may be written at any time (in a calculation), these rules would be satisfied by certain states of affairs, namely, when one writes a statement of that type (in a calculation). For example, if I write "John is tall or John is not tall" in the course of a discussion, I am justified in doing this by the rule: "One may write a statement of the form 'p or not p' at any time." We can say that the statement itself is justified by the rule in a derivative sense. Nonetheless, in general I would say that the terminology of 'justification' is unfortunate when applied to the statements themselves. I do not think we should ask what justifies a statement, but rather why is the statement true and why is it necessary? Returning to the arithmetical statement '2 + 2 = 4', we must say in this case that the terminology of justification is not even applicable in an extended sense. What is justified here is an inference and only an inference. We have argued that a linguistic rule justifies - is satisfied by - an inference, and in the case of some rules, one can go on to say that a certain statement has been 'justified'. If we can show that the linguistic rule is a convention, then we have shown that the corresponding inference which satisfies the rule is a matter of convention, and the necessary statement is likewise a matter of convention. If we define a necessary statement as one that can be written at any time (in an inference) and if a statement like 'John is tall or John is not tall' can be written at any time by virtue of a conventional linguistic rule, then we can say that a necessary statement is necessary because of linguistic conventions. The link which we are missing here between linguistic rules and conventions has, I hope, been supplied by our discussion in Chapter II. We indicated that there are linguistic rules, that the logical rules of inference are linguistic rules, and that a linguistic rule is a regular mode of linguistic behavior with a prescriptive function. The link between the facts of linguistic usage and linguistic rules, we argued, was the fact that linguistic rules are just a special kind of linguistic behavioral pattern; namely, one in which the behavioral pattern has taken on a prescriptive function which prescribes that the pattern be continued in the same manner.

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Statements of necessity and entailment statements are related to linguistic rules in two ways, depending on how the former are being used. In some cases one can identify statements of necessity and formulations of linguistic rules, as Hampshire has suggested. When statements of necessity are used in this way to express rules, they are neither true nor false, necessary or contingent. According to our discussion in Chapter II, however, they would express voluntary rather than nonvoluntary rules, where 'voluntary' can be taken to be the rule correlate of 'contingent' for statements. Statements of necessity can also be understood as statements about linguistic regularities. The close affinity between these two uses of statements of necessity arises from the close affinity between linguistic regularities and linguistic rules, both of these being aspects of the same empirical state of affairs. It was the intent of our discussion in Chapter II to fill an almost universal inadequacy in prior conventionalist theories, for previous authors failed to give a sufficient analysis of rules.24 This failure indisposed them to give an adequate analysis of the relationship between linguistic behavior, linguistic rules, and the necessary statements which they supposedly justify. We will now go on to discuss a number of the criticisms which can be handled within the framework of rules and conventions developed in Chapter II. After this we will discuss the more serious criticisms and the semantical and ontological foundation from which they are launched.

C.

CRITICISMS A N D REPLIES

Pap has criticized the conventionalist thesis on the basis that while it does not explicitly assert that necessary statements are contingent statements, the conventionalist is nevertheless committed to this. He argues: [The conventionalist thesis] asserts the synonymity of the following t w o statements: A ) It is necessary that every yard contains three feet; B) "every yard contains three feet" (S) f o l l o w s f r o m the rules governing usage of the constituent terms. But rules, especially linguistic rules (linguistic habits), are n o t propositions f r o m w h i c h any proposition could f o l l o w . H e n c e B should be m o d i f i e d as follows: S f o l l o w s f r o m the proposition asserting the existence of those rules. 2 5

But the proposition which asserts the existence of linguistic rules is a contingent statement, and anything which "follows from" a contingent statement is itself contingent. Consequently, "every yard contains three feet" is a contingent statement, not a necessary statement as A states. The mistake is in misinterpreting the conventionalist, who would not generally 24

Probably the one exception to this is K. Britton, Proc. Aris. Soc. Sup. Vol. XXI, 78-103, who does a detailed analysis of the relationship between a linguistic rule and its instances and especially of the relationship between the rule formulation and its application. 25 Arthur Pap, "Are All Necessary Propositions Analytic?" Phil. Rev. LVIII (1949), 316. See, also, his Semantics and Necessary Truth (New Haven, Conn., 1958), pp. 165-67.

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use the terminology of "follows from" in B. A s we noted above, Malcolm and Black would rather say that S is "justified b y " or "certified b y " the linguistic rules. Lazerowitz would say that S is "entailed by" the proposition that certains linguistic rules exist; that is, S is entailed by certain facts about linguistic usage. This is certainly a mistake, but, as we saw, part of the reason for this is not distinguishing between A and S, for it is A that entails and is entailed by the facts of linguistic usage. When we do distinguish between A and S, we can say that S itself is NOT entailed by the facts of linguistic usage. We would prefer to rewrite B as follows: (B') "every yard contains three feet" (S) the empty set of premises BY VIRTUE OF a linguistic rule. The necessary statement does not follow from a rule or a proposition asserting the existence of a rule, but follows from something else by virtue of a rule. The inference of the necessary statement from the empty set of premises is precisely the sort of thing which a linguistic (logical) rule justifies, the sort of thing which the rule says that I can do. The conventionalist is asserting that B ' is synonymous with A . The important point is that we do not, and need not, assert that a necessary statement FOLLOWS FROM a contingent statement about linguistic habits or the existence of linguistic rules. FOLLOWS FROM

That a necessary statement follows from the empty set of premises, we think, would be generally accepted. The conventionalist novelty is in saying that it follows by virtue of a rule, and in particular, by virtue of a LINGUISTIC rule. A nonconventionalist might argue, for example, that a necessary statement follows from the empty set of premises by virtue of a relation of entailment between the two. Rules of logic, according to this latter interpretation, are not linguistic rules and it is not they that justify the inference. In fact, they would argue that the rules of logic are not rules at all, but entailment relations, and formulations of the rules of logic are merely reformulations of necessarily true statements of entailment. They are 'instructions' or 'directions'. 26 We must see exactly wherein the innovations of the conventionalist thesis lie. Conventionalists are giving a theory of why a statement is NECESSARY, that is, why a statement follows from the empty set of premises. In the case of necessary statements that are conditional statements, they theorize about why the consequent follows from the antecedent and about the kind of relation which exists between them. In the case of necessary statements which can be reduced to truths of logic by substituting synonyms for synonyms, they are giving a theory which accounts for the reducibility of the statement to a statement of logic. I have argued that (second-order) statements of necessity and entailment statements are statements about linguistic usage or rule formulations, but that (firstorder) necessary statements are neither. I must perforce say that statements of necessity do not entail necessary statements. We must be careful, however, for I 26

See above, Chapter II, pp. 40-41.

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am not asserting that the statement "If 'p' is necessary, then p" is false, for it is clearly true. What I am denying is that the statement' 'p' is necessary' ENTAILS the statement 'p'. I defend this by arguing that even the nonconventionalist should not assert this as an entailment.27 The relationship between the statement' 'p' is necessary' and the statement 'p' is much like the relationship between a Platonistic form and the objects which exemplify the form. Likewise, if we can talk about physical entailments existing between abstract entities, then the relationship between these physical entailments and actual cause-effect relationships in the world is one of exemplification. A particular cause-effect relationship is a cause-effect relationship and not an accidental relationship by virtue of exemplifying an entailment between the corresponding abstract entities. Yet it would be very strange to say that the entailment entails the cause-effect relation. An abstract entity or a relation between abstract entities does not ENTAIL what exemplifies it. Part of Plato's problem was to make clear exactly what this relationship of exemplification or participation is. It only confuses the issue to call this relationship an entailment. The same considerations apply directly to the relationship between statements of necessity and necessary statements. We all agree that the latter are true in some sense by virtue of the former. I suggest, however, that calling it an entailment not only gets the conventionalist into trouble, but the nonconventionalist as well. I would further suggest that while the nature of the exemplification relation is still a problem for Platonists, the conventionalist has worked this out. The inference of the consequent of a necessary conditional from its antecedent and the inference of the conditional from this inference are justified by linguistic rules which grow out of entailment relations, the entailment relation being a regular mode of linguistic behavior which has taken on a prescriptive function and become a linguistic rule. This framework also enables us to answer another objection posed by Pap, who makes the criticism in the following words: "A metastatement to the effect that such and such a formula is a theorem in the system that is characterized by such and such postulates and such and such rules of deduction is not a 'rule' of any kind. It is, if true, necessarily true."28 We can take the statement "A is a theorem" as equivalent to "A is necessary", for in natural languages it is not possible to find an essential difference between 'axioms' and 'theorems'. Both are simply necessary. The essential difference is that between the rules and the necessary statements. We distinguish, however, between two types of rules: (a) those rules which allow one to infer a sentence from a group of sentences, e.g., Modus Ponens; and (b) those rules which allow one to infer a sentence from the empty set of premises, e.g., "from the empty set of premises to infer 'p or not p'". Pap is not saying here that the statement "A is a theorem" is necessary, although he would certainly say this as well. His objection is that the statement, "If the rules of a language L are R ^ 27

A nonconventionalist who has asserted that the relation is one of entailment is Donald C. Williams, "The Nature and Variety of the A Priori", Analysis V (1938), 85. 88 Arthur Pap, Introduction to the Philosophy of Science, p. 104.

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R 2 , etc., then A is a theorem" is a necessary statement that cannot be accounted for on the basis of linguistic rules alone, while for the sake of argument, he is allowing that both the antecedent and consequent are contingent statements. That there is a connection between the state of affairs expressed by the antecedent and that expressed by the consequent we must admit; that it is a necessary connection or an entailment we would deny. The link between the existence of certain linguistic rules and the fact that A is a theorem is the inferability of the statement A from the empty set of premises in accordance with the linguistic rules. If the rules of language L are R i ; R 2 , etc., then such and such is a valid inference of A from the empty set of premises, and so A is a theorem. As we have seen, the connection between the rules of a language and inferences which satisfy the rules - which are justified by the rules - cannot be viewed as a necessary connection or an entailment. But then the statement, "If the rules of language L are R 1 ( R 2 , etc., then such and such particular inferences are valid", cannot be construed as an entailment either, for the state of affairs which makes this statement true is the fact that certain inferences are instances of certain rules. The relation here is that between a rule and its instances. One can see this more clearly if one translates 'valid' as "instances of existent rules" in the above sentence. If the rules of language L are R 1 ; R 2 , etc., then such and such particular inferences are instances of existent rules. In other words, if R 1 ; R 2 , etc., are existent rules, then such and such particular inferences are instances of existent rules. But such and such particular inferences are instances of existent rules because R 1 ; R 2 , etc., are existent AND BECAUSE SUCH AND SUCH PARTICULAR INFERENCES ARE INSTANCES OF R

1 ;

R2,

etc.

The link between the existence of rules and the existence of theorems, therefore, depends on the link between rules and their instances. That the relation between a rule and its instances can IN SOME SENSE be construed as 'necessary' is hardly very exciting, although it is important to see that it is at least not necessary in the same way as the relationship between the premises and conclusion of a valid inference. Two more related criticisms have been made against the conventionalist thesis. It is argued on the one hand that the conventionalist thesis must explain "the appearance of necessity which differentiates a priori propositions from rules of a language".29 On the other hand, it is argued that logical truths do not have merely the APPEARANCE of necessity, but they are in fact necessary; there are no alternatives to the truths of logic. Not even powerful dictators or God can alter the laws of logic.30 It is generally held by conventionalists that there are alternative logics, and this fact would seem to operate against their having "the appearance of necessity". A linguistic rule as interpreted by the conventionalist is a convention and not necessary. But if they are not necessary, how can the laws of logic based on these linguistic conventions be necessary? Interestingly enough, while Black was in his preconventionalist period, he wrote 29

s»

E w i n g , Proc. Aris. Ibid., p . 2 1 7 .

Soc.

X L , 214.

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a paper in which he replies to this criticism for the conventionalist.31 Black argues that a statement is held to be 'necessary' because a person will not accept that another person has understood the statement unless the other person does in fact also accept the rule which justifies the statement. One is not considered to have understood the statement "All fathers are male parents" unless one also accepts the rule that the phrase 'male parent' can be substituted for 'father'. But if a condition for understanding a statement is justifying it, then it is no wonder that such statements appear to be 'necessary'. In the case of contingent statements, one can first understand the statement and then verify it to be true or false. In the case of necessary statements, Black argues, the criterion for understanding the statement is verifying it. A necessary statement to be understood must already be shown to be true. The reason for this is that one is NOT ALLOWED TO think that he understands a necessary statement, unless in effect he accepts it to be true. The compulsion to accept as true a necessary statement comes not from the recognition of its truth, but from the fact that one is forbidden not to accept it. Black further argues that a necessary statement is really a rule which is written as a statement, because a statement is oftentimes more pragmatically effective as a rule than a rule formulation itself.32 The function of a linguistic rule, he says, is to promote linguistic uniformity. A necessary statement has the same function, and must therefore be itself a rule functioning more effectively as a rule than the corresponding rule formulation would. While I would not agree with all that Black says, I think he is more or less on the right track. The general technique that the conventionalist must use is to distinguish between the criteria for calling a statement necessary and the cause of the appearance of necessity. This method goes back to the technique used by Hume in Chapter 7 of the Enquiry Concerning Human Understanding. In part I of this chapter, Hume gives us the criteria by which we judge that two kinds of events are causally related. If you will, he gives the truth conditions for cause-effect statements. Since the truth conditions in themselves give no explanation of the peculiar kind of necessity we find in cause-effect relations, in Part II of the chapter he gives a psychological explanation for this experience of necessity. He says that we have in effect been conditioned by the constant conjunction to expect the effect when the cause occurs; our mind jumps to the effect when the cause is presented. This expectation in the mind brought about by conditioning is our experience of 'necessity'. According to Black the criterion for a necessary statement is apriority; that is, once we understand it we know that it is true. We would say that the criterion is unconditional assertability, or inferrability from the empty set of premises, and go on to say that it is inferrability from the empty set of premises by virtue of linguistic rules. As in the case of Hume's analysis of the cause-effect relation, we must also place the source of the appearance of necessity somewhere else than in the truth 31 32

Max Black, "The Analysis of a Simple Necessary Statement", Jour. Philos. X L (1943), 39-46. See above, Chapter II, p. 43, for another discussion of this same point.

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conditions. We would say that the appearance of necessity in the case of necessary statements is occasioned by the fact that the inference of the necessary statement from the empty set of premises, its unconditional assertability, is PRESCRIBED by the linguistic rules, and the prescriptive character of the linguistic rules itself is caused by linguistic regularity in conjunction with society. We take it to be a fact that behavioral regularities can take on a prescriptive character, that there are generalized modes of behavior and that these modes of behavior constrain others to behave in the same way. For example, a woman feels that she ought to wear white at her wedding. We take it that she feels this obligation because it is a custom for women to wear white at their wedding, and a customary mode of behavior causes others to feel that they ought to behave in the same way. We do not propose an explanation for why one feels that one ought to behave in what is the regular way, nor do we propose to explain why society sometimes places sanctions against not behaving in the regular way. This is surely a task for the empirical scientist. What we assume as a matter of empirical fact is simply that SOMETIMES the appearance of necessity which an act has is caused by a generalized mode of behavior. In Sections C.l and C.2 of Chapter III we tried to prove that, in particular, this is the case for some linguistic acts, viz., inferring, and that the reason one MUST infer in one way rather than another is because this way is the regular way, and therefore the prescribed way, that is, prescribed by society. There is a tendency to make the necessary statement itself a prescription to explain the appearance of necessity which it has. I think it is more accurate to say that a necessary statement is necessary by virtue of a prescription. Just as one MAY do certain acts by virtue of a moral rule, so one MAY make certain inferences by virtue of a linguistic rule. It may be the case that sometimes prima facie necessary statements are used as statements of necessity, that is, as formulations of linguistic rules; however, I do not believe that a theory about logical truth should rely on denying that there are necessary statements per se. Inasmuch as it is not the necessary statement itself, but a linguistic rule, which is conventional, we can also explain why it does not seem as though there are alternative logical truths, for there are no alternative logical truths relative to a given set of linguistic rules. A necessary statement is necessary relative to a given set of linguistic rules, for given a set of rules a statement either is or is not unconditionally assertable. There are no alternatives.33 The real problem arises when the nonconventionalist argues that the logical rules of inference themselves do not have alternatives. But if, as I believe, they are only linguistic rules, they have alternatives. I think the explanation here is again a sociological one, which stems from Durkheim's work in the sociology of religion.34 I will give a general outline of how a complete explanation would go, leaving the 33

See Chapter II, page 52, for a further discussion of the necessity involved here. Emile Durkheim, The Elementary Forms of the Religious Life, trans. Joseph W. Swain (Glencoe, 111., 1947). See also his Les Règles de la Méthode Sociologique (Paris, 1907).

34

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details to the empirical scientist. Since linguistic rules have arisen from regularities of societal linguistic behavior, one cannot pinpoint a legislative body which chose the conventions, because there is none. No authority originated the rules of logic. In lieu of a legislative body, one believes that the rule is imposed on one from outside of and beyond society. Another possibility is that a rule may come from society as a whole, and not from a particular segment of it, or it may come from a particular segment of society without there being any specific pronouncement that such and such is to be a rule, without the segment even being aware that they are originating a rule. It seems to me, for example, that the rules of etiquette are nothing more than the regular modes of behavior of the well-to-do. The rules of logic can be analogously viewed as the linguistic regularities of the intelligent, which then impose their way of behaving on other people. It is only when we begin to see that the effects of society are greater than the algebraic sum of the effects of its members that one can begin to see rules as being originated by society, while having the appearance of coming from outside of society. Another condition also operates here, making our logical truths seem inviolable. This is the fact that they play such a basic role in our conceptual framework, in all of our reasoning about the world. To change our logical system would involve changing our knowledge about the world as well. The fact that our logical system plays such a basic role in all of our thinking makes it seem as though there is no alternative. Even in physics, the more basic the role of a law, the more it seems inviolable. In some cases we will not give up a law even in the face of discontinuing evidence until we have something to replace it with, and since the logical truths are never disconfirmed, they are the last statements that we would change. This should not blind us to the fact that our present logical system may at some time prove so inconvenient that we would feel it necessary to change it to promote the growth of scientific knowledge.35 This leads us into another criticism that is made against conventionalism. Quine has argued in his paper, "Truth by Convention", as well as in more recent paper "Carnap and Logical Truth", 36 that the conventionalist thesis is empty, that it has no explanatory force. The conventionalist thesis comes to nothing more than the fact that the logical truths are 'obvious'. Quine's criticism here is based on his earlier criticism that we cannot state the conventions (rules) of logic without using logical constants, and, therefore, we cannot use these formulated conventions without already understanding the logical constants. We discussed this criticism in Section D of Chapter II, noting there that rule formulations are irrelevant, for the important thing is KNOWING HOW to use the rules whether or not they are formulated.37 Let us assume with Quine that we cannot formulate and use the conven35

This has been argued by C. Lewy, "Logical Necessity", Phil. Rev. XLIX (1940), 67; and Hampshire, Philosophy, XXIII, 336-37. 36 W. V. O. Quine, "Camap and Logical Truth", in the Philosophy of Rudolf Carnap, ed. Paul A. Schilpp, Library of Living Philosophers, XII (LaSalle, 111., 1963), pp. 391-93. 37 Above, Chapter II, pp. 60-62.

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tions of logic without already knowing logic. What consequence follows from this? On the other hand it is not clear wherein an adoption of conventions, antecedently to their formulation, consists; such behavior is difficult to distinguish from that in which conventions are disregarded. . . . But when a convention is incapable of being communicated until after its adoption, its role is not so clear. In dropping the attributes of deliberateness and explicitness from the notion of linguistic conventions we risk depriving the latter of any explanatory force and reducing it to an idle label. 38 On these same grounds Quine would say that the axioms of set theory ARE true by convention, because in this case we can more or less pinpoint the time when they were accepted. Clearly the linguistic conventions were not adopted deliberately and explicitly. If they were adopted explicitly and deliberately by a legislative body, much of what we said in Chapter II would have been unnecessary. In the light of this fact, even if our thesis that the rules of logic are linguistic rules is not a metaphysical thesis, it is at least empirically untestable. We are only giving and defending one of several possible explanations of logical rules, but we are not saying that the others are false. We are not suggesting that linguistic behavior depends on whether or not the conventionalist thesis is true. What we have tried to do is to give an explanation which, on the one hand, does not conflict with sociological knowledge and, in fact, uses it and which, on the other hand, makes no use of any metaphysical entities or of any extraempirical source of knowledge. To say that the truths of logic are self-evident or obvious is either to say nothing or to say that they are intuited. Either alternative is repugnant to the conventionalist. In regard to Quine's further point that the conventions are incapable of being communicated until after their adoption, we would reply that this is false on the basis of our discussion in Chapter II. Before we proceed any further with anti-conventionalist criticisms, we will pause to relate our thesis to the notions of necessity, analyticity, and the a priori. We analyze necessity as unconditional assertability or inferability from the empty set of premises (with which even most nonconventionalists would agree) by virtue of linguistic rules (with which nonconventionalists would disagree). We assert that it is statements that are necessary rather than some extralinguistic state of affairs. We do not identify necessity with the impossibility of theoretical falsification. Whether or not a statement is open to theoretical falsification is a question distinct from the question of necessity and is more closely related to the question whether or not a statement has empirical content. We prefer to keep the question of truth distinct from the question of necessity. We also assert that the logical truisms are a priori in the following sense: one who knows the language in which the necessary statement is expressed, and in particular, one who knows how to follow the rules of the language, can know that a logical truism is true independent of observation. While this is only a conditional 38

Quine, "Truth by Convention", p. 273.

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a priori, the condition is perfectly acceptable as long as we are talking about statements. One cannot even know that an a posteriori STATEMENT is true without knowing the language in which it is expressed. As Black has noted, a condition of understanding a necessary statement is accepting the relevant rules. Therefore, an a priori statement is one that one can know to be true once one fully understands it. We merely emphasize the fact that understanding a necessary statement depends on knowing how to follow linguistic rules. We also hold that logical truisms are true by virtue of the meanings of the constituent terms, but what exactly this entails must wait until we have discussed the notion of meaning in some detail. In addition, we hold that logical truisms are empty of factual content and that they are not open to theoretical falsification, yet at the same time there are alternative logics. In sections A. 3 and A.4 of Chapter IV we will try to outline a theory of truth to explain why logical truisms cannot be falsified, and what exactly this means. We will see that there are necessary statements which CAN possibly be falsified, and we will see the manner in which we can distinguish the logically necessary statements from these other necessary statements. In general, the relationship between necessity and truth will be analyzed.

D.

FURTHER CRITICISMS BASED ON SEMANTICAL A N D ONTOLOGICAL THEORIES

1. Truth by Definition We now come to the forms of conventionalism and the critiques of conventionalism which make reference to the meaning or definition of the logical constants. Quine has shown that the usual type of explicit and contextual definitions cannot create truth; they at best transmit truth from a statement containing the definiens to a statement containing the definiendum. 39 Thus Lewis must have had a different kind of definition in mind when he said: "The source of this necessary truth, be it observed, is in definitions, arbitrarily assigned. Thus the tautology of any law of logic is merely a special case of the general principle that what is true by definition cannot conceivably be false." 40 The truth of truth-functional tautologies is truth by definition. The kind of definition that Lewis is talking about is the definition of the sentential connectives by truth tables. It is important to bear in mind that the classical truth table definitions of the sentential connectives can be interpreted in either of two ways.41 The truth table for a sentential connective can be regarded as a formal matrix; the 'T' and 'F' are not taken as abbreviations for truth and falsity but as arbitrary symbols and can be replaced by any pair of symbols such as '0' and '1'. The sentential connective 39 40

41

Ibid., p. 251. C. I. Lewis and C. H. Langford, Symbolic

Carnap, Introduction to Semantics, p. 213.

Logic (New York, 1932), p. 211.

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is then regarded as a function which maps the symbols T and 'F' into 'T' and ' P . This can hardly be considered a method for giving the meaning of the sentential connectives. The truth-functional sentential connective is used to form a sentence out of two sentences such that the truth of the new sentences depends on the truth of the sentences of which it is composed. It is a function which maps truth and falsity into truth and falsity. If 'T' and ' P are taken to be abbreviations for truth and falsity, then the truth table for a sentential connective can be interpreted as a definition of the sentential connective, but the crucial point here is that if a truth table is to be taken as a definition, it must be interpreted. For example, let us look at the truth table for 'and': p q T T T F F T F F

p and q T F F F

For this to give us the meaning of 'and' - for it to be a definition of 'and' - it must be interpreted, or it is just a group of symbols. One way to interpret the truth table is to read it in the following way: "If 'p' is true AND 'q' is true, then 'p and q' is true; otherwise 'p and q' is false." Other readings are possible, but all have one common feature: they have the form of semantical rules of truth with this difference - the semantical metalanguage may be the same as the object language. Can truth table definitions create truth? Now it seems as though Lewis actually interprets the truth tables as formal matrices. When the truth table is used thusly and applied to complex sentences, it in fact turns out that truth functional tautologies are assigned the value T (or '1') in all cases. But in what sense does the method of formal matrices show us that a tautology is a necessary TRUTH? AS long as we are only dealing with formal matrices, the 'T' that is assigned to the tautology in all cases is not to be read as 'True'. Reading the T in the formal matrix as 'True' can be defended, but the point here is that it must be defended. Prima facie one must conclude that Lewis was misled into reading '1' as 'True', and if this mistake is not made, one sees that truth tables, interpreted as matrices, have nothing to do with necessary TRUTH. On the other hand, when the truth tables are interpreted as definitions of the sentential connectives, they have the form of semantical rules. We will leave aside for now the question of whether semantical rules can create truth by convention, and first take up the question of whether there is some other kind of definition which can create truth by convention, returning to our discussion of semantical rules in the next section. Quine has suggested the possibility of implicit definitions, although he prefers to call this method of 'generating' truth the postulational method.42 According to Quine we can determine that a statement is true in either of two ways. Let us 42

Quine, "Truth by Convention", pp. 258-60.

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assume that we have an uninterpreted statement, a well formed series of words one of which does not have meaning. One procedure is to give meaning to this word and then determine whether or not the resulting interpreted statement is in fact true. On the other hand, we can stipulate that the statement is true and by doing this give meaning to the previously uninterpreted word in the statement. As Quine says: "Since all contexts of our new word are meaningless to begin with, neither true nor false, we are free to run through the list of such contexts and pick out as true such ones as we like; those selected become true by fiat, by linguistic convention." 43 There is no question that Quine is critical of conventionalism in logic; but he is not critical of the method of truth by postulation or stipulation. In his more recent discussion of conventionalism, he explicitly states that stipulation can create truth and that mathematicians have created the truths of set theory.44 Quine is happy with the method of truth by stipulation, if he can locate a precise time when the actual legislation has occurred. We can more or less pinpoint the introduction of the symbol ' E '; therefore, we can say that the postulates which introduce the symbol are true by stipulation and that they give the meaning of the symbol. Some philosophers would object to the whole idea of assigning meanings by stipulating truths and would argue that Quine has misinterpreted what one achieves by means of implicit definitions. Quine says that we make a meaningless statement true, and THEREBY assign meaning to the uninterpreted words. It can be argued that what we are actually doing is assigning meaning to a word in such a way that thereby the statements in which it occurs are true. What we are saying when we state a postulate is: "Let the meaning of the uninterpreted words be such that the postulate is true." The uninterpreted words are given meaning by convention, but uninterpreted words always are. The sentence is not true by convention, but by virtue of the meaning of the words. For example, the later Lewis would agree that "the use of linguistic symbols is indeed determined by convention and alterable at will".45 However, "decision as to what meanings shall be entertained, or how those attended to shall be represented, can in no wise affect the relations which these meanings have or fail to have. Meanings are not equivalent because definitions are accepted; definitive statements are to be accepted because, or if, they equate expressions whose equivalence of intensional meaning is a fact." 46 A postulate or implicit definition is true not by convention, but by virtue of the fact that the meanings of the constituent terms have the relationships expressed by the postulate. What is conventional is merely the symbolic language in which we express the relations between meanings. The sentence which we use to express the relations between meanings is analytic, but it is true by convention only in the trivial sense

43 44 45

«

Ibid., p. 260. Quine, "Carnap and Logical Truth", pp. 389-91. C. I. Lewis, An Analysis of Knowledge and Valuation (LaSalle, 111., 1946), p. 51. Ibid., p. 97.

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that we conventionally choose the linguistic symbols by which to express the extralinguistic state of affairs, the relations between meanings. We might put Lewis' objection to Quine in another way. If 'S' is the abbreviation for a particular postulate, we can distinguish between «S» and 'S', where «S» is taken to be the name of the uninterpreted postulate, and 'S' is taken to be the name of the interpreted postulate. «S» is literally meaningless because it has at least one word in it which does not have a meaning. When Quine stipulates the postulate to be true, is he stipulating «S» or 'S'? Lewis would say that Quine is stipulating «S» to be true. By making «S» true by convention, we are determining what we mean by the various uninterpreted symbols in «S». 'S' is then true, and analytically true, but it is not true by convention, for it expresses a relationship between meanings which is independent of 'S'. In effect all we are choosing by convention is what meanings we are going to talk about. By convention we introduce symbols which express these meanings. But the meanings and their relationships are there before we choose to express them. 2. Semantical Rules We can sum up Lewis' objection by saying that an analytic statement is true by virtue of a preexisting relationship between meanings which the analytic statement expresses. We can introduce symbols arbitrarily, but the resulting analytic statements are true by virtue of the relationships between the meanings that the symbols have been given. Lewis' objection is especially interesting because we meet essentially the same objection against conventionalism from a somewhat different direction. We have argued that the logical rules of inference are linguistic rules and there are alternatives to them. Carnap argues that while the logical rules are linguistic rules, as long as we are talking about interpreted languages, such as ordinary language, our alternatives are severely limited. Once our logical constants have meaning we are not free to choose the rules of inference as we wish. The logical rules must be chosen so that those statements which are true ex vi terminorum are deducible from the empty set of premises and only those. Given an interpreted language, the linguistic rules of inference are either correct or incorrect, and this is not a matter of convention.47 We can change the rules of inference, but only if we also change the meanings of the logical words. This kind of critique of conventionalism is very common.48 It is an attempt to reduce the conventionalist thesis to the trivial assertion that we can change our laws of logic only if we also change the meanings of the logical words. We can change true empirical statements

47

Rudolf Carnap, "Foundations of Logic and Mathematics", in International Encyclopedia of Unified Science, ed. Otto Neurath, Vol. I, N o . 3 (Chicago, 1939), 26-29. See, for example, the papers of C. A. Campbell, "Contradiction: 'Law' or 'Convention'?" Analysis XVIII (1958), 73-76; and Williams, Analysis V, 85-94. 48

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into false ones, too, by changing the meanings of the words, but this is not a very philosophically exciting result. Carnap's approach is interesting because he not only TALKS ABOUT the meaning of terms (as Lewis does); he tries to GIVE the meanings of words, and, in particular, the meaning of the logical words, by means of semantical rules; for example, "'rot' means red" and " ' p and q' is true if and only if 'p' is true and 'q' is true". In this context, Carnap goes on to say that a sentence is analytic (L-true) "if it is true in such a way that the semantical rules of S suffice for establishing its truth".49 This is Carnap's translation of "A statement is analytic if it is true ex vi terminorum." Rather than talking about 'meanings' we talk about the semantical rules for a word whereby the word has meaning. If the semantical rules are sufficient to establish the truth of a sentence, then it is analytic, true by virtue of the meanings of the words. Before we can critically examine Carnap's criterion of analyticity, we must decide what he means by the word 'establish'. There are prima facie two senses which the word can have in this context. We can read 'establish' as 'bring about' or as 'prove'. According to the first reading of 'establish', essentially what he is saying is that the truth of the semantical rules is sufficient for the truth of an analytic statement. That is, an analytic statement is true because the relationship between language and the world is such as is expressed by the semantical rules. Consequently, it is not essential that the semantical rules be formulated for them to establish the truth of an analytic statement; it is only essential that the signs of the language be related to certain designata. But is this actually sufficient for the truth of an analytic statement? Can the very fact that the meanings of certain words are such and such be sufficient for the truth of a statement? Surely this is not all there is to the truth of any statement. At the very minimum, a statement is true because it expresses a proposition WHICH IS ITSELF TRUE. Generally we say that a statement is true because it CORRECTLY describes a state of affairs. In any case the truth of a statement is only partly a function of the meaning of its constituent symbols. Semantical rules ALONE cannot establish the truth of analytic statements in the first sense of the word 'establish' which we have been discussing. If we take 'establish' in the sense of 'prove', then what Carnap is saying is that we can prove that an analytic statement is true using only the semantical rules. If the analytic statement is 'A', then we can prove the statement " A ' is true' using only the semantical rules, or, in other words, we can derive the statement " A ' is true' from the semantical rules used as premises. Here it is, of course, essential that the semantical rules be formulated, for they are functioning as premises in an argument. But in point of fact how would we prove that a statement is true using only the semantical rules? For example, how would we show that 'K v ~ K ' is true in S using only semantical rules? According to the semantical rules a statement 'p v q' 49

Carnap, "Foundations", p. 13.

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is true in S if and only if 'p' is true or 'q' is true. Also, ' ~ p ' is true in S if and only if 'p' is not true in S. Accordingly, we can show that 'K v ~ K ' is analytic in S, if we can prove that either 'K' is true in S or ' ~ K ' is true in S (by virtue of the semantical rule for V ) . But we can prove that this is so by proving that 'K' is true in S or 'K' is not true in S (by virtue of the semantical rule for Therefore, showing that 'K v ~ K ' is analytic in S reduces to DEMONSTRATING that' 'K' is true in S or 'K' is not true in S' is true in the metalanguage. If we know that this latter statement is an analytic statement of the metalanguage, then we can deduce from this in the metalanguage using the semantical rules as additional postulates the statement " K v ~ K ' is true in S'. Since we have used only the semantical rules as additional postulates in the logical deduction of ' 'K v ~ K ' is true in S', we are warranted in asserting that 'K v ~ K ' is analytic.50 The difficulty with this analysis arises from the fact that we did not use only the semantical rules of S. In addition, we had to use either the logical rules of inference of the metalanguage or the semantical rules of the metalanguage, as expressed in some meta-metalanguage in order to show that' 'K' is true in S or 'K' is not true in S' is analytic in the metalanguage, and also to show that " K v ~ K ' is true in S' is logically implied by " K ' is true in S or 'K' is not true in S' together with the semantical rules of S. Therefore, we cannot show that the statement is true in S using only semantical rules of S, unless, however, we argue that a statement in the metalanguage can be SHOWN to be analytic without recourse to the semantical rules of the metalanguage. For example, if we argue that the analyticity of ' 'K' is true or 'K' is not true' in the metalanguage is not based on the semantical rules of the metalanguage, but on the meaning of 'or' and 'not' in some extralinguistic sense of meaning, then we can say that what we have done is to explicate the notion of analyticity in S by means of the notion of analyticity in the metalanguage, where the notion of analyticity in the metalanguage is taken as understood, independently of any semantical rules. But then 'analytic in the metalanguage' means something different than 'analytic in S'. In any event, the semantical rules of S ALONE do not suffice for establishing the truth of an analytic statement of S, for we need besides the semantical rules of S either the semantical rules of the metalanguage or some other means of proving the analyticity of a statement of the metalanguage. We have not overlooked the fact that Carnap treats his analysis of 'analytic in S', that is, 'true in such a way that the semantical rules of S suffice for determining its truth', as a CRITERION OF ADEQUACY for a definition of 'analytic in S'.S1 His actual attempts at defining 'analytic (L-true) in S' depend on whether he is giving the definition in special or general semantics. In special semantics, where S is a specific given language, he follows the usual procedure of defining 'analytic in S' in terms of membership in a prescribed class; a statement is analytic in S if and only if it has certain prescribed properties, where these properties do not refer to 50

"

See Carnap, Introduction to Semantics, pp. 78-79. Ibid., pp. 83-84.

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either the semantical rules of S or the analyticity of a statement of the metalanguage.52 He is quite right, however, in noting that not any definition of 'analytic in S' would be adequate. For the definition to be adequate the class of analytic statements so circumscribed must be true ex vi terminorum. Therefore, the criticism above still applies. Even though the definition does not refer to the semantical rules of S, in order to determine whether the definition is adequate we must know what it is for a sentence to be analytic in the meta-metalanguage. We may have a definition of this latter concept, but then in order to determine its adequacy we must know what it is for a sentence to be analytic in the meta-metalanguage, and so on, and we are off on an infinite regress. In sum, Carnap's criterion of adequacy in terms of semantical rules is unsuitable. Now Carnap also has definitions of 'analytic in S' in general semantics, that is, where S is ANY language. Again the definitions cannot make reference to the semantical rules of S, for then the definition would have to be phrased in a metametalanguage, rather than the metalanguage of S. These definitions if successful could be used without reference to the criterion of adequacy. They in effect would serve in place of the criterion of adequacy, and the above criticism would not be valid. As it turns out, however, the critique above is easily transferred directly to the definitions themselves. A definition where this is most clearly applicable is the following: "D16-C1. s , is L-true in S = Df the sentence of ' S i is true in S' is . . . . L-true in M 1 ; " 5 3 where M x is a sublanguage of the metalanguage M. Now we need a definition for 'L-true in M t ' . This definition can be given in M 2 , another sublanguage of M, but it would contain the concept of 'L-true in M 2 ' which would have to be defined in a sublanguage M 3 , and so on to an infinite regress once again. Carnap also suggests another version of D16-C1 in which 'L-true in is replaced by 'C-true in Mj', which is the purely syntactical concept of provability. The latter definition would not lead to an infinite regress but it is of a fundamentally different character than the other definition. This might mean that L-true in M x is equivalent to C-true in M l s but then why is L-true in S not simply equivalent to C-true in S? On the other hand if L-true in M x is not equivalent to C-true in M t , then if C-truth in M x is sufficient to guarantee L-truth in S why is not C-truth in S itself sufficient to guarantee L-truth in S? To put it the other way around, if C-truth in S is not sufficient to guarantee L-truth in S (and Carnap does not think it is) then neither is C-truth in sufficient to guarantee L-truth in S. While Carnap's definition of 'analytic in S' in special semantics does not lead us on an infinite regress, both his criterion of adequacy and his definitions of 'analytic in S' in general semantics do. And it is precisely the latter two which are philosophically interesting and the sort of thing with which we are concerned in this book. We must look elsewhere for an adequate analysis of analyticity. We noted that the word 'establish' could be taken in two senses: 'to bring about' 52 53

Ibid., pp. 81-82. Ibid., p. 86.

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or 'to prove'. This confusion between two senses of the word 'establish' has its root in a more fundamental confusion between two senses of the word 'interpret', whereby the semantical rules are said to 'interpret' the words of the object language. In one sense of 'interpret', we interpret a word which is not understood or which initially does not have meaning by TRANSLATING it into a word or phrase which is understood or which does have meaning. For example, the semantical rule for V in terms of 'or' might very well be understood as a translation of the V of PM into the 'or' of English. This is, of course, quite helpful, but the philosophically oriented student immediately raises further questions. If the 'v' in some sense MEANS 'or', what does 'or' mean? Clearly, the question asked is not: "Can you translate 'or' into a meta-metalanguage for me?" We can assume here that the metalanguage is better understood than any meta-metalanguage would be. For example, I understand what 'if . . . then . . . ' means better than I understand what '—)-' of Reichenbach's probability calculus means. There is a time when it is truly helpful to give semantical rules, and there is a time when it is merely misleading. A similar problem arises when ANALYZING the notion of 'analytic in S', for while it MIGHT be helpful to analyze 'analytic in S' in terms of 'analytic in English', we still want to know what 'analytic in English' means. Now in this case we cannot analyze 'analytic in English' in terms of, for example, 'analytic in three valued logic'. I understand what 'analytic in English' means better than I understand what 'analytic in three valued logic' means. The problem is that I just do not understand it well enough. The real philosophical problem is not with interpreting 'v' or 'analytic in S', but with interpreting 'or' and analyzing 'analytic in English'. Semantical rules as translations are of no use in this connection. We do not need the semantical rules of English to prove that a statement is analytic in English. To say this would be like saying that an English child must understand Russian before he can understand English. Giving the semantical rules of English in terms of Russian as a metalanguage might help a Russian understand English, but they will not help an Englishman understand English. There is, however, another sense of 'interpret' in which we interpret a word not by translating it into a word we already know the meaning of, but rather by relating the word to something extralinguistic, which is its 'meaning'. In this case there is no attempt to help one understand the word. One might quite properly give as a semantical rule " 'red' means red" or even " 'red' means rot", emphasizing not that 'rot' is a word in German but that 'rot' is the name of a certain extralinguistic property. Likewise, we would not emphasize that ' 'K' is true or 'K' is not true' is a statement of the metalanguage, but the fact that this sentence expresses an extralinguistic proposition. We do not use the metalanguage as a language which we assume understanding of; in fact, it might not be well understood. Rather, the metalanguage is used here to formulate a relation which is independent of the metalanguage. Thus, showing how 'analytic in S' depends on 'analytic in the metalanguage is showing how 'analytic in S' depends on extralinguistic facts. We are

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back to Lewis' view that a statement is analytic by virtue of extralinguistic relations between meanings.54 If this is what Carnap had in mind, one wonders why he took the torturous path through semantical rules and metalanguages to get to extralinguistic fact, when he could have proceeded as Lewis does. I think the answer lies in the fact that while Carnap, on the one hand, leans very heavily toward this latter view, he also wants the semantical rules to help us understand the words of the object language. This presupposes that the words of the metalanguage are better understood than the words of the object language. A semantical rule such as "'red' means red" would be useless for helping us to understand the word 'red' since the object language in this case is understood exactly to the same extent that the metalanguage is understood. I think, therefore, that for Carnap the semantical rules 'interpret' the words of the object language in both senses of 'interpret'. They translate the words of the object language into the metalanguage and also relate the words of the object language to an extralinguistic meaning, it being presupposed here that the metalanguage is understood, that one knows the meanings of the words of the metalanguage. His analysis of 'analytic in S' is, therefore, not applicable where S is our natural language. We can also distinguish a third sense of the word 'interpret', a sense which has been very important in the philosophy of science. Here we interpret a word by translating it into or relating it to another word or set of words within the same language; for example, "'bachelor' means unmarried adult male". Implicit definition can be viewed in this way. We are giving the meaning of a word on the basis of words which we presuppose understanding of, but in this case it is not necessary that the metalanguage be a different language. It may be a part of the object language as long as it is understood. It is in this sense of 'interpret' that reduction sentences partially interpret reduced terms. Unfortunately, some philosophers who use the word 'interpret' in this way fail to relate it to the other senses of 'interpret'. 54

We have been reading the semantical rule " 'rot' means red" as " 'rot' means the property red". There is, however, a variant reading, viz., " 'rot' means red things". The two readings of this semantical rule correspond to two of the extralinguistic dimensions of meaning of a term, that is, the intension and extension of a term. While one would seldom see the semantical rule formulation " 'rot' means red things", one might see an equivalent formulation, such as " 'x is red' is satisfied by a or b or c, etc.", where 'a', 'b', 'c', etc., are names of red things. Or one might find the semantical rule formulation " 'x is red' is satisfied if and only if 'x' is replaced by 'a' or 'b' or 'c', etc.", where 'a', 'b', 'c', etc., are names of red things. ( C f r . A. Tarski, "The Semantic Conception of Truth", reprinted in Feigl and Sellars (eds.), Readings, p. 63.) While the rules CAN be used to interpret a term by giving its extensional meaning, it is not clear that they are always used for this purpose. One sometimes finds that an author does not intend to give the meaning of a term in formulating a semantical rule. For example, Tarski is not concerned with giving the meaning of 'white' when he gives satisfaction rules for 'white'; he is concerned with giving a definition of 'true'. Likewise, in discussing interpretations Curry says that no attempt is made to give the meaning of the constituent terms. {Cfr. his Foundations, p. 49.) We are concerned here with interpretation and semantical rules only in so far as they give the meaning of certain terms, for we are considering criticisms of the conventionalist thesis which turn on the meaning of the logical words.

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Is an interpretation in this sense also intended to relate a term to something extralinguistic? We will return to this problem in section E of the present chapter.55 3.

The Modes of Meaning

Since the critiques of conventionalism which we are now discussing depend heavily on distinguishing between a word and its meaning, it might be well for us now to determine what exactly philosophers, such as Lewis, have in mind by THE meaning of a term. We will first distinguish the various modes in which he says a word has meaning, going on then to distinguish the particular mode in which an analytic statement is true by virtue of the meanings of the terms but independent of linguistic convention. There are principally two ways in which a word has meaning, usually called extension and intension or denotation and connotation. In general terms the extension of a word includes the existing particulars to which the word is applied, while the intension of a word goes beyond particulars to capture the sense of the word. One distinguishes between what a word names or refers to and what it means. Historically, one can distinguish two essentially distinct concepts of the extension of a general word; on the one hand, it is oftentimes said the extension or denotation of a general word is the CLASS of objects to which the word is applicable, while, on the other hand, it is also said that a general word names the individuals to which it is properly applicable. In the first case we can say that the denotation of a general word is singular, for a general word names one class; in the latter case we can say that the denotation is multiple, for the general word names each individual thing to which it is properly applicable. In order to differentiate these two senses 55

While we have been discussing the verb "to interpret", it is useful to note here that to EACH of these senses of the verb there are in turn three senses of the noun 'interpretation'. We said that to interpret a word is to relate the word either to something extralinguistic, to words in another language, or to words in the same language. In one sense an 'interpretation' is the relationship between the word and its interpretant. In another sense it is only the interpretent which is called the 'interpretation'. In yet another sense we sometimes say that the semantical rule formulation is the interpretation. In the case of extralinguistic interpretation we can distinguish clearly between these three notions; the designation relation oan be distinguished from the expression of it and from the designata. Distinguishing the three becomes trickier when we consider interlinguistic and intralinguistic interpretations. Should we say that there is a relationship of INTERPRETATION between "red* and 'rot' which is merely expressed by the semantical rule formulation or should we say that the interpretation relationship in this case must be formulated? Certainly there is some relationship between 'red' and 'rot' - they are synonymous - but is synonymy an interpretation relationship? In considering intralinguistic interpretations, and, in particular, implicit definitions, the problem is one of locating the interpretant which is distinct from the interpretation formulation. To what is a term related by means of an implicit definition? Moreover, we have a problem as to how the interpretation should be formulated. Should we say that 'straight line' means Euclid's axioms or should we simply give Euclid's axioms? Surely it is strange to say that Euclid's axioms are the interpretant of 'straight line' but it is equally strange to say that we can give an interpretation without saying that we are doing so - without using a word, such as 'means' or 'designates'. To express the rule as follows: "The meaning of 'straight line' is given by Euclid's axioms", is merely to avoid the problem.

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of denotation, we might say that a general word NAMES the individuals to which it is applicable and DENOTES the class of them. While Mill56 and Martin 57 distinguish between these two possibilities, Lewis68 and Carnap 59 refer only to the denotational sense of extensional meaning. Whichever sense of extensional meaning we prefer, it should be evident that it is not in the sense of extensional meaning that analytic statements are true by virtue of the meaning of the terms. It is in this sense of meaning, however, that analytic statements do have factual content. The word 'bachelor' is applicable to certain particulars as is the phrase 'unmarried adult male'. The factual content of 'All bachelors are unmarried adult males' is the fact that the extension of 'bachelor' is coextensive with the extension of the phrase 'unmarried adult male'. Of course, this is also true for 'All metals are conductors of electricity' which is not analytic. Therefore, coextensiveness of extension does not guarantee analyticity. There are at least four ways in which a general word can be said to have intensional meaning. Lewis distinguishes three: comprehension, signification, and intension. Lewis says that while a general term DENOTES a class of actually existing objects, it comprehends a classification of actual and possible objects. The comprehension of a term is the classification (or class) of both actual objects to which the term is properly applicable and of possible objects to which the term would be truly applicable, if they existed. The primary use of distinguishing between the denotation and comprehension of a term is in discussing singular names. A singular name can fail to name an object and still have meaning in that it at least comprehends a possible object. The signification of a general term is a property; it is the property which all objects have to which the general term is truly applicable. It is that in a thing which indicates that the general term is being correctly applied. This corresponds to the kind of meaning which Mill calls connotation and Carnap calls intension. A property is extralinguistic and extramental, but it is not a particular, concrete object. It is commonly called a universal or abstract entity. The kind of intensional meaning which has special significance for us is that of intension. Lewis distinguishes two kinds of intension, viz., sense meaning and linguistic meaning. Lewis indicates what he has in mind by sense meaning in the following words: The intension of a term represents our intention in the use of it; the meaning it expresses in that simplest and most frequent sense which is the original meaning of 'meaning'; that sense in which what we mean by 'A' is what we have in mind in using 'A', and what is oftentimes spoken of as the concept of A. We shall wish . . . to identify it with

56

"

58

"

See Mill, System of Logic, pp. 16-17. See R. M. Martin, Truth and Denotation (Chicago, 1958), pp. 99-100. See Lewis, Analysis, p. 39. See Rudolf Carnap, Meaning and Necessity, enlarged ed. (Chicago, 1956), pp. 16-23.

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the criterion in mind by w h i c h it is determined whether the term is question applies or fails to apply in any particular instance. 6 0

The general idea goes back to Aristotle, although Lewis gives it a particular twist in order to avoid the difficulties which the realist and conceptualist had. The concept must be clearly distinguished from an image; therefore, Lewis identifies the concept with the 'criterion in mind' for applying the word. It is essential to Lewis' concept of sense meaning that the criterion be mental, and not identifiable with operational definitions or reduction sentences. It is by means of such mental criteria that words ultimately get their meaning; the connection between words and things is affected by means of these criteria, these concepts. This is a semantical idea that was developed in some detail by Scholastic logicians. The sense in which a word has linguistic meaning is the sense in which a word means other words. One can identify the linguistic meaning of a word with the conjunction of all other words, each of which must be truly applicable to any thing to which the given word is applicable.61 This is the mode of meaning in which 'bachelor' means 'unmarried adult male'. Lewis also gives us a somewhat more general interpretation of linguistic meaning when he says: "It may be taken as constituted by the pattern of definitive and analytic relationships of the word or expression in question to other words and other expressions."62 It is in this sense of meaning that one can say that the meaning of the word 'straight line' is given by Euclid's five postulates or the meaning of the logical constants is given by the logical truisms.63 The assymetry between linguistic meaning and both sense meaning and signification suggests two other modes of meaning. Just as one might associate the meaning of a word with a conjunction of words which are analytically related to the given word, so one might also associate the meaning of a word with the conjunctions of properties and the conjunction of concepts which are analytically related to the property and concept which are signified and meant, respectively, by the given word. These additional kinds of meaning, as well as linguistic meaning, are best applicable to words like 'bachelor' rather than words like 'red'. One might argue, for example, that the 'criterion in mind' for applying the word 'bachelor' is nothing more than the conjunction of the properties of being unmarried, of being 60

Lewis, Analysis, p. 43. Ibid., p. 39. 82 Ibid., p. 131. 83 The concepts of sense meaning, signification, and linguistic meaning are closely related to two of the senses of the word 'interpret' which we distinguished on pages 93-96 above. In fact, sense meaning and signification are two ways in which a term can have an extralinguistic interpretation. It is by virtue of the linguistic meaning of a term that the term has an intralinguistic interpretation. N o w Lewis subordinates linguistic meaning to sense meaning in that the linguistic meaning of a term depends on its having sense meaning but not vice versa. I think Lewis is generally on the right track and we should be cautious in saying that a word has meaning when it has only been intralinguistically interpreted. On the other hand, we should not assume a priori that every term necessarily must have a signification or sense meaning. 61

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an adult, and of being a male. While the assymetry between linguistic meaning and signification and sense meaning suggests a further division of the latter two, additional analysis might even suggest dropping the original kinds of signification and sense meaning, at least in certain cases. We will leave this merely as a suggestion, returning to it again in Section C.l of Chapter IV. A noteworthy point about all of these kinds of meaning is that they take meaning to be a relation between a word and an object or objects. In some cases the object is an abstract entity, in other cases a mental entity, and in still other cases concrete particulars. Likewise, Carnap takes the word 'means' or 'designates' to be a relation word, where the relata are a word and an entity, respectively. In accordance with this concept of meaning as a relation, he should have written his semantical rules with a noun both on the left and the right. For example, instead of "'rot' means red", he should have said "'rot' means redness" or "'rot' means the property redness". The fact that these latter locutions are peculiar is one reason why a number of philosophers dispute this theory of meaning as a relation between a word and a thing.64 They would emphasize the idea that 'rot' means red, not redness, and therefore deny that means is a relation. They would interpret the semantical rule "'rot' means red" as "'rot' has the same use as the word 'red' in OUR language, i.e., English". Since the semantical rule is itself written in our language, i.e., English, there is no need to specify the language in which the word 'red' occurs. It is also assumed that the hearer understands English, and already knows the use of the word 'red'. If we assume the latter, we can say that the semantical rule gives the use of the word 'rot'. We will return to this alternative theory of meaning in Chapter IV. 4. The 'Meaning' Critique of Conventionalism According to Lewis, analytic statements are true by virtue of the relations between the criteria which we have in mind in applying words to the world. The extralinguistic facts which underlie analytic statements are for Lewis relations between SENSE meanings, i.e., concepts which he treats as rules or schemata for applying a word to the world. If a schema for applying one word includes the schema for applying another, we can say that the schemata and, therefore, the words are analytically related. For example, if part of the schema for ascertaining that a certain object is a bachelor is using another schema to ascertain that it is unmarried, then the concept of bachelor, the rule for applying the word 'bachelor', includes the concept of 'unmarried', the rule for applying the word 'unmarried'. If the concept of bachelor is treated as distinct from the conjunction of the concepts of 64 See Wilfrid Sellers, "Is There a Synthetic A Priori?" and "Some Reflections on Language Games", reprinted in his Science, Perception, and Reality, pp. 289-320 and 321-58, respectively; A. P. Alston, "The Quest for Meanings", Mind LXXII (1963), 79-87; and Gilbert Ryle, "Meaning and Necessity", Philosophy X X I V (1949), 69-76.

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unmarried, adult, and male, clearly there is a relation between the concept of bachelor and the conjunction of concepts which is extralinguistic. Accordingly, one can say that the statement 'All bachelors are unmarried adult males' is true because of the relationship between concepts, independent of linguistic convention. As long as one accepts the theory of meaning involved, or at least accepts the existence of concepts and their particular relation to words which Lewis outlines, one has a critique of the conventionalist thesis as applied to statements such as "All bachelors are unmarried adult males." Our prime concern in this study, however, is with logical truisms, which have the singular characteristic that they are true by virtue of the meaning of the LOGICAL words involved. Looking at the various senses of meaning which Lewis has described, except for linguistic meaning, we find that logical words have meaning in none of these senses. Clearly, 'or' does not name either particular objects or a class of particular objects. Neither does 'or' name a classification of possible objects; the problem is not just that we do not have 'ors' in the world either. If 'or' has neither comprehension nor signification, it is difficult to see how there can even be the concept of or in Lewis' sense of concept, since we cannot have a criterion in mind for applying the word 'or' to things in the world. It would seem then that logical truisms are not analytic in the sense of being true by virtue of the meanings of the logical words, inasmuch as logical words do not have meaning in the crucial mode of sense meaning. The difficulty arises because according to classical concepts of meaning to which Lewis is adhering, logical words are syncategorematic terms, that is, they do not have meaning independently of other words. It has been generally recognized that the traditional classifications of meaning do not apply to logical words, although the consequences of this have not been fully developed. Unfortunately, Lewis does not recognize the problem. He in fact says that logical truths are no different than any other analytic statements. They are all true by virtue of extralinguistic relations between the sense meanings of the constituent words. What do we then make of Lewis' attempt in Chapter 5 to show that the syllogism Barbara is analytic?65 In particular, Lewis attempts to show that the following statement is analytic: "(A) If all M is P and all S is M, then all S is P." Lewis says that if we know the intensional meaning of ' a l l . . . is — ', then we know that the relation between X and Y expressed by 'All X is Y' is transitive. It is obvious that Lewis has in mind here by intensional meaning something other than sense meaning, that is, criterion in mind for applying the concept, for the expression 'All . . . is ' is syncategorematic. There are no criteria for applying it to objects in the world. Nor is there any criterion for applying the concept of transitivity to the world. Objects in the world are not transitive; relations are. Relating the concept of 'all . . . is — ' t o the concept of transitivity is like relating one 45

Lewis, Analysis,

pp. 116-19.

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concept I do not have to another concept that I do not have; that is, if concept is taken as criterion in mind for applying the word to the world. Lewis must face this question: "Is there anything more to a relation R being transitive than that 'If xRy and yRz, then xRz' is true?" One might reply that the fact that this statement is true is part of my criteria for calling a relation transitive. But when the concept must depend on sentences, it is no longer bridging the gap between the world and words. Is such a concept really a mental concept? In showing that (A) is analytic, Lewis relies on an ambiguous notion of intensional meaning, and not on the more specific notion which he has developed of sense meaning; it is on the latter that analyticity is based. One could, of course, save Lewis' critique by a more thorough-going Platonism, arguing, for example, that analytic statements are true by virtue of the significations of the constituent words. If the significations were not limited only to properties and relations, but instead all words had meaning in the mode of signification, then the logical truisms could be analytic by virtue of the relations between meanings (significations). One would then have to admit that 'All . . . is — 'or', etc., have signification; they signify some abstract entity which is not exemplified by particular objects. One would further have to admit that these abstract entities bear some kind of relation to each other which underlies the analyticity of logical truth. On the other hand, the Platonist would have to work out the exact nature of this relationship. Many philosophers have criticized conventionalism in logic on the ground that it amounts to nothing more than the trivial thesis that we can change the truths of logic if we change the meanings of the logical words. For this critique to be plausible, however, it must be accompanied by a theory about what the meaning of logical words is and how the truth of logical truisms depends on these meanings. Lewis, as we saw, did not give us an adequate theory for the meaning of logical words, and the outright Platonist does not show us how the truth of logical truisms depends on his abstract entities. Until we have located a theory of meaning that can answer both of our questions, we can not tell how disastrous, if at all, this critique is to the conventionalist. 5. Necessity and Propositions There is a variant of the meaning critique which takes as its point of departure not the meaning of logical words but the meaning of logical statements. Pap develops this critique by first stating what he takes to be a version of the conventionalist thesis: "The existence of certain linguistic habits relevant to the use of a sentence S is a necessary and sufficient condition for the necessity of the proposition meant by S."66 His critique hinges on the argument that while the existence of linguistic conventions may be necessary and sufficient for the fact that 66

Pap, Semantics

and Necessary

Truth, p. 164.

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S expresses a necessary proposition, they are irrelevant to the necessity of the proposition itself, which is an extralinguistic entity. A sentence may at one time be necessary and at another time contingent by virtue of a change in the meaning of certain terms. If the meaning of the word 'yard' changes, then the sentence 'Every yard contains three feet' ceases to be necessary, for the sentence is now expressing a different proposition. The original PROPOSITION, however, is still necessary. Necessity is a time-independent predicate of propositions; once a proposition is necessary it is always necessary. While our assessment of this critique must await the next chapter, to understand the full force of it we must understand what Pap has in mind by 'propositions' and why he thinks that they are in some sense distinct from sentences. Inasmuch as the distinction is of historic importance, we will also note its usual development. The distinction is usually most fully developed in the context of a theory of meaning which differentiates between intensional and extensional meaning. Frege, for example, distinguishes between the sense and nominatum (intension and extension) of sentences. The nominatum of a sentence is its truth value, while the sense of a sentence is the proposition expressed by it. It is by virtue of its nominatum that a sentence is either true or false; thus, a sentence must have a nominatum. The sense of a sentence is the Gedanke, the objective content of a sentence "which is capable of being the common property of many".67 Carnap makes essentially the same distinction in Meaning and Necessity. Lewis' approach involves several intriguing novelties, although utilizing the same basic distinction. Lewis takes a proposition to be a linguistic entity, a more historically justified usage than that of Frege, Carnap, and Pap, although the latter usage is more prevalent today. Lewis' novelty is in identifying a proposition not with a sentence, but with a that-clause or participial phrase. "Mary is making pies now" is a statement or assertion. The proposition that Mary is making pies now or Mary making pies now is what is asserted by the statement. The utility of this approach is that we can then distinguish between questions, suppositions, imperatives, and so forth, by what they do to the proposition. While a statement asserts a proposition, a question questions it. As a consequence of this, Lewis' second novelty is to treat a proposition, i.e., a that-clause as a predicate. This contrasts with the customary approach today, which is to say that a sentence means a proposition, while a that-clause names it. A that-clause is taken to be a proper name of the proposition, just as the 'ness' ending on a predicate turns it into a name of the property. A that-clause is to a proposition as a sentence in single quotes is to the sentence without quotes. The result of Lewis' treating that-clauses as general terms is that he can give a theory of their meaning which is perfectly parallel to his theory of meaning for any other general term. The extension of a proposition is the actual world if it is 67

G. Frege, "On Sense and Nominatum", reprinted in Feigl and Sellars (eds.), Readings, p. 89.

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true, and nothing if false. The comprehension of a proposition is the classification of possible worlds in which it would be true. The signification of a proposition is a state of affairs, where by 'state of affairs' Lewis means approximately what Frege means by Gedanke and Carnap means by 'proposition'. Lewis goes a little further still. Since a proposition is a kind of general term, he talks about a state of affairs as being exemplified by the world, just as a property might be exemplified by a particular object. While Lewis does not discuss the sense meaning of propositions, he does say that the linguistic meaning of a proposition is all the deducible consequences of a proposition taken together.88 An aspect of this view, as well as of Carnap's, is that truth is taken to be a concept applicable, if not primarily, at least properly to propositions (states of affairs) as well as to sentences. In fact, one might go so far as to say that sentences are true by virtue of the fact that they express propositions which are true. The logical extension of this is to say that two sentences entail each other by virtue of the fact that the propositions entail each other, and a statement is necessary by virtue of the fact that the proposition it expresses is necessary. Truth, entailment, and necessity all hang together. This becomes especially clear when one reflects on the fact that if 'A' ENTAILS 'B' then if 'A' is TRUE, 'B' must also be TRUE. The view that truth is primarily a property of propositions and that entailment is a relation between propositions is explicitly held by Pap, who says: "It is a property of a sentence that it belongs to the English language, that it is composed of so many words, that it is of the declarative type. . . . But entailing such and such consequences is, like the property of being true, not a property of sentences as such but only of sentences as meaning such and such propositions (thus ' 'p' is true' is elliptical for 'the proposition expressed by 'p' is true')." 69 He brings up the so-called 'translation argument' against the view that truth and entailment are applicable to sentences. This argument is based on the supposition that the following two sentences are translated into another language in different ways: (1) 'A' entails 'B'. (2) That A entails that B. In the translation of sentence (1), " A " would not be translated while in the 68

While Lewis' view is compatible with the approach that talks about "making a statement" as "asserting a proposition", one rather odd consequence is that the assertion "Mary is making pies now" is not a statement about Mary. The sentence "Mary is making pies now" can be analyzed by distinguishing between the prepositional component and the assertional component. The propositional component is 'that Mary is making pies now'. When we rephrase the sentence so as to exhibit its propositional component we have the 'The World is that Mary is making pies now'. What we are talking about in this sentence is evidently the world and we describe it as a that-Mary-is-making-pies-now kind of world. It seems then that if we confine our use of language to full sentences, we have no need to distinguish proper names, common nouns, etc. All we need are terms which signify propositions, and a set of words to distinguish between asserting, questioning, postulating, etc. I think this is the result of treating asserting on a par with questioning, postulating and the like. Is asserting something which is done to a proposition by a statement? Or is asserting a matter of meaning (expressing) a proposition? 69 Pap, Semantics and Necessary Truth, pp. 199-200.

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translation of (2), 'that A' would be translated. This is so because "the name of a sentence formed by quoting the sentence does not belong specifically to the language to which the named sentence belongs but is rather a meta-linguistic name common to all languages that use the quoting device."70 This leads us to the conclusion that sentences (1) and (2) are not synonymous since they are translated into another language in different ways, indicating that (2) is not just a translation of (1) into the object language. If (2) is independent of (1), we must talk about entailment relations between propositions as well as between sentences. In fact Pap would go so far as to say that (1) is in fact parasitic on (2) and should be read "the proposition expressed by 'A' entails the proposition expressed by ' B " \ Where Carnap (Logical Syntax of Language) would have called (2) a pseudo-object sentence, Pap might very well call (1) a pseudo-syntactical (or semantical) sentence in the formal mode of speech. Before we go on we must note again that this argument depends heavily on the supposition that under a translation 'A' in (2) gets translated but " A " in (1) does not, which in turn depends on a particular use of quotation marks. It also depends on the view that a sentence in one language can be translated into a sentence of another language. It is a short step from this to saying that there is some common thing that each of the sentences means, this meaning being commonly called the proposition. The very possibility of translation from one language to another leads to the supposition of propositions and to a positive critique of the view that there are no propositions. Pap also criticizes the view that propositions are only classes of synonymous sentences. As I see it, his primary criticism is based on the supposition that there can be unexpressed propositions just as there can be unnamed flowers. Simply because we have not expressed a truth about flowers does not mean that the truth does not exist. Similarly, because we have not expressed a logically necessary truth does not mean that the truth does not exist. But if a proposition is unexpressed, according to the class of sentences view of propositions such a proposition is identical with the null class! Undoubtedly Pap is right that roses would be red even if no one were around to utter the sentence: "Roses are red." The truth of the proposition that roses are red does not depend on the sentence, "Roses are red", being expressed. But one wonders whether the same is true of the sentence, "If A and if A then B then B." Would it be true that if A and if A then B then B if the sentence were never expressed? It seems as though the fact that roses are red is quite a different thing than the fact that if A and if A then B then B. I am willing to admit for the sake of argument that there are unexpressed atomic propositions, but the fact that there are unexpressed atomic propositions is not a sufficient justification for arguing that there are unexpressed logically necessary propositions. To justify the latter position requires an indepen™ Ibid., p. 200.

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dent argument because of the evident difference between atomic facts and logically necessary facts. To assimilate these two kinds of facts is to beg the very question at issue. We must make two other points about Pap's view of propositions. The approach to propositions which we have outlined so far is a semantic approach; propositions are postulated in the process of giving a theory of meaning for sentences. While Pap recognizes this approach he also gives a nonsemantic justification for the existence of propositions. According to this view propositions are the "objects of propositional attitudes", in particular, of belief. A proposition is what can be believed. An interesting sidelight of this approach is that certain sentences are not taken as expressing propositions; for example, statements of identity and contradictory statements. Also logically equivalent sentences do not necessarily express the same proposition. This view of propositions is much closer to what Lewis might call the sense meaning of a sentence rather than its signification. The proposition here seems to be something mental which is believed, doubted, and so forth, and which both an Englishman and an Italian could have in common despite their differing languages. The translation argument is used once more to show that the sentence 'He believes that A' cannot be translated as 'He believes the sentence 'A''. An Italian and an Englishman might both believe that A is true without having a disposition to assent to the same sentence 'A'. The second point concerning Pap's view of propositions is that he disavows Platonism. Propositions, according to Pap, are not abstract entities. It is unclear, however, what he does think they are. He says, for example, that they are like sense data, and that while it is incorrect to talk about the ontological status of either propositions or sense data, it is correct in a nonreductionist sense to talk about their existence!71 This distinction is similar to Carnap's distinction between external and internal questions of existence.72 The latter concern questions of existence within a linguistic framework and are meaningful, whereas the former concern the validity of the entire linguistic framework; these are meaningless ontological questions. The advantage of Carnap's distinction is that it permits one to do ontology without calling it that. Moreover, it calls for a more tolerant attitude toward other ontologies, for after all they are only alternative linguistic frameworks. Pap, however, goes a step further than Carnap, and suggests a method for deciding between alternative linguistic frameworks (or ontologies). "The question whether propositions and other abstract entities exist is not, indeed, decidable empirically, not even in the indirect empirical way in which scientists decide whether atoms and electrons exist, but it is nonetheless a cognitive question: it is decidable the way questions of semantic analysis are decidable, by examining whether proposed translations into a language with a specified primitive vocabulary preserve 71 72

Ibid., pp. 189-90. R. Carnap, "Empiricism, Semantics, and Ontology", Revue Inter, de Phil. IV (1950), 20-40.

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the meanings of the translated statements." 73 I agree with Pap that if we are to DEFEND a conventionalist view of logical truth, we must account for the manner in which we talk about propositions; in particular, we must show how proposition talk can be eliminated in favor of that concerning sentences and rules, the latter, of course, taken in my regularity sense. This we will attempt to do in the next chapter. However, I would object to interpreting my results, if successful, as having PROVED that logic is true by convention and that propositions do not exist. I would hesitate to say that because we do not need propositions, that, therefore, they do not exist, just as I would hesitate to say that because we do not need harmful viruses, therefore, they do not exist. It is my belief that the truth of a metaphysical thesis does not depend on the cleverness of its exponents, any more than the truth of a scientific thesis does. We will assume in the following that propositions are language independent entities, although their exact nature will not be directly relevant to our discussion. 6. Linguistic Rules and Validity There is one more critique of conventionalism which we must mention; the considerations which it occasions will play a leading role in the next chapter. We have regarded the matter of knowing the logical rules of inference as being a matter of KNOWING HOW to infer. Gellner has pointed out that we must distinguish between two distinct aspects of 'knowing how'. 74 We may separate them by specifying the first aspect as "knowing how to follow the linguistic rules" and the second aspect as "knowing how to infer VALIDLY". 'Knowing how' in English is often taken to mean not only knowing how to follow a specified set of rules but also knowing how actually to achieve the goal of the game. If the rules of logic are linguistic conventions and the goal of logic is to achieve valid arguments, what guarantee is there that the linguistic rules will lead to valid arguments? After all, these are such that when their premises are true, their conclusions are also true. Because one is permitted by linguistic rules to infer a sentence from a given group of sentences and because one knows how to obey these rules, it does not follow that the inferences will be valid. The premises and conclusions of most valid arguments are contingent, synthetic sentences about the world. Their truth seemingly is independent of convention, for the conventionalist theory is traditionally a theory about necessary statements, not contingent statements. If, however, the truth of the premises and conclusion of a valid inference are independent of convention and the logical rules of inference are merely linguistic conventions, it could be the case that the logical rules of inference are not valid. For example, there might be a linguistic rule which allows us to infer 'x is ugly' from 'x is a Martian', and some Martians may not be ugly. Such a rule, however, is obviously not a logical rule of inference, and the 73 74

Pap, Semantics and Necessary Truth, p. 194. Ernest Gellner, "Knowing How and Validity", Analysis

XII (1951), 32-33.

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conventionalist does not claim that all linguistic rules are valid. He does claim that linguistic rules are sufficient to explain all NECESSITY; however, they are sufficient to explain the validity only of the logical rules of inference and the TRUTH only of logical truisms. What does one then make of the rules suggested by Prior: "From 'A' to infer 'A-tonk-B'" and "From 'A-tonk-B' to infer 'B'"?75 From these rules, which have the appearance of logical rules, we can derive the rule: "From 'A' to infer 'B'", a nonlogical rule, and, moreover, an invalid rule. If these were actually conventions of language, one might know how to follow the logical rules of inference and yet not know how to infer validly! This is almost a reductio ad absurdum of the conventionalist thesis. We have thus far discussed several versions of the conventionalist thesis in logic, having chosen to defend the version which says that necessary statements are necessary by virtue of the fact that they are inferrable from the empty set of premises, where the logical rules of inference are taken to be linguistic conventions, in the sense of 'convention' set forth in Chapter II. We have defended this view against a number of opponents; however, there are two criticisms and one problem which still remain for the conventionalist thesis. One criticism is based on the supposition that it is extralinguistic propositions which are necessary, and that linguistic conventions can in no way guarantee the truth or necessity of an extralinguistic proposition. The second criticism is based on the fact that logical rules of inference permit the inference of true contingent conclusions from true contingent premises. The critic here argues that while linguistic conventions might guarantee the necessity of necessary statements and possibly even the truth of logical truisms, they cannot guarantee the truth of contingent statements. If a logical rule of inference is merely a linguistic rule, what guarantee is there that the linguistic rule is valid, that is, that it never permits an inference from a true contingent premise to a false contingent conclusion? The assumption here is that the truth of contingent statements is independent of linguistic conventions, while it allows that the truth of necessary statements may not be. The problem which the conventionalist faces is to give some kind of theory of meaning for logical constants. To a certain extent he is also faced with the criticism of the outright Platonist. Although we have seen that Lewis' criticism based on the sense-meaning of terms is inapplicable to the logical words, one might very well hold a Platonistic theory of meaning for the logical words. The conventionalist must give a satisfactory nonPlatonistic theory of meaning for logical words which is compatible with his thesis. He does not have to show that the Platonist is wrong, but only that the Platonistic abstract entities are unnecessary.

75

A. N . Prior, "The Runabout Inference-Ticket", Analysis

X X I (1960), 38-39.

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CONVENTIONALISM I N GEOMETRY A N D PHYSICS

Before we discuss these difficulties, we would like to complete this chapter with a short discussion of three other types of conventionalism; conventionalism in geometry and physics and Quine's logico-physical conventionalism. These will at first sight seem to be quite diverse theses and independent of the conventionalist thesis in logic which we have propounded. Nevertheless, after discussing the meaning of logical words in the next part we hope to show that all of these conventionalist theses actually bear quite a bit in common. Conventionalism in geometry has been defended by Poincaré, Reichenbach, and Griinbaum, based on the work of Riemann. Here we will give Griinbaum's defense. He makes a point of distinguishing conventionalism in geometry from what he calls trivial semantic conventionalism. For example, he says: "It is a serious obfuscation to identify the Riemann-Poincaré doctrine that the ascription of the congruence or equality relation to space or time intervals is conventional with the platitude that the use of the unpre-empted word 'congruent' is conventional."78 According to Griinbaum, even though the word 'congruent' has been given a semantical interpretation, this interpretation is not sufficient to pick out a unique class of spatial intervals. The reason for this is the metric amorphousness of space, i.e., because space is a continuum. The term 'congruent' is given a semantical interpretation via the axioms of congruence by the method of implicit definition. An infinite class of spatial intervals are still permitted under this interpretation of the word, and one can specify a unique class of intervals as THE congruence class only by an arbitrary conventional choice. Therefore, the conventional choice in geometry arises AFTER the word has been given a semantical interpretation. If the conventional choice arose only before the word had been given an interpretation, then conventionalism in geometry would not be distinguishable from the conventional choice one always has in giving meaning to a 'new' word. This latter kind of conventionalism is what Griinbaum calls trivial semantic conventionalism. It is Griinbaum's claim that conventionalism in geometry is not a semantic conventionalism because the conventional choice arises despite the fact that the term 'congruent' has been interpreted. If space consisted of discrete points (as opposed to being a continuum), one would not have this conventional choice. Equality of spatial intervals would in this case reduce to the equality of the number of intervening points. But since in our case space is a continuum, the number of intervening points is the same between any two distinct points, for there is an infinity of intervening points. I have used the term 'interpret' in the last two paragraphs as Griinbaum uses it. It is quite evident that Griinbaum is not using the term in the sense of a translation of the word 'congruent' into a synonymous term of another language. Also, he does not mean by an 'interpretation' relating a word to an extralinguistic entity or group 76

Adolph Griinbaum, Philosophical

Problems

of Space and Time (New York, 1963), p. 27.

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of entities. This is what the axioms of congruence - the interpretation of 'congruent' - fail to do.77 Griinbaum apparently means by an interpretation relating a word to other words of the language. Lewis would say that this is giving the linguistic meaning of the word 'congruent'. Griinbaum does succeed in relating the word 'congruent' to extralinguistic classes of spatial intervals by means of a conventional choice, but one might very well wonder about the appropriateness of considering this conventional choice nonsemantical, for in one sense of 'semantical interpretation' this is precisely what his conventional choice accomplishes. I think Griinbaum is right in refusing to label conventionalism in geometry as a trivial semantic conventionalism. On the other hand, one might very well question whether it is entirely a nonsemantical conventionalism. For a defense of conventionalism in physics we go to Duhem.78 Duhem distinguishes between common sense physical laws and the laws of physics as a developed science. The distinction arises because physics uses terms which are not directly observable, such as 'mass', 'pressure', and 'temperature'. While these clearly refer to properties of observable things, the properties themselves are not observable. In the jargon of philosophy of science such terms are considered to be operationally defined inasmuch as we have instruments for measuring these properties. Duhem introduces his conventionalist thesis into physics by denying that these terms are introduced into physics by DEFINITION. According to Duhem these terms are introduced into physics by means of physical laws, for the use of instruments in measuring these properties implies the belief in certain physical theories; viz., the theories of the instruments themselves. One accepts the results of a measurement with an instrument as the correct measurement of the property because one believes that there are valid physical laws relating the property to the results of measurements. In point of fact, however, the results of measurements are not always accepted as correct; it is sometimes claimed that outside influences affected the measuring process resulting in an incorrect reading. For example, a scientist would not accept the result of temperature measurement with a mercury thermometer if the measurement was obtained in a strong magnetic field. Nor can a definition be reinstated by observing that the operational definition contains cautions against measuring temperature under these influences, for some of these influences are not known when the method of measurement is devised and accepted by the scientific community. One can, of course, avoid this difficulty with operational definitions by adding a clause to the definition not mentioning particular cautions but 77

One might say that the axioms of congruence partially interpret the word 'congruence', for they delimit the possible sets of spatial intervals that can be called 'congruent'. In general, implicit definitions cam be said to partially interpret words. But whether we say that the axioms of congruence merely linguistically interpret or partially extralinguistically interpret the term 'congruence', the fact remains that the axioms of congruence do not give a complete semantical interpretation of 'congruence'. 78 Pierre Duhem, The Aim and Structure of Physical Theory, trans. Phillip Wiener (Princeton, 1954), pp. 165-79.

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mentioning all disturbing influences in general, that is, a ceteris paribus clause. This will turn the operational definition into an analytic statement at the expense of its no longer being operational, as we never know for sure when ceteris paribus. Duhem avoids this entire problem by admitting that one must rely on physical laws in making measurements; an inductive risk is involved when one measures. It is laws rather than definitions which introduce these terms into science. Duhem would also say that the meaning of these terms is given by the laws of the instruments for measuring them. Again, as with geometry, crucial terms are introduced by means of implicit definitions. The difference in this case is that the implicit definitions are physical laws while in geometry they are admittedly analytic statements. Also, and this is an important point, the terms of physics cannot be given an extralinguistic interpretation as can the term 'congruent' of geometry, for properties like temperature are unobservable, while congruence is observable. An immediate consequence of this is that individual statements of science which contain these unobservable terms essentially cannot be observationally tested. We can only test groups of statements of science together, because from several statements which involve this unobservable term we can deduce consequences which do not contain the term, and which can then be observationally tested. In the case of a disconfirmation of the group of laws, which law should we reject? Conventionalism enters here, for Duhem says that we can reject any law which eliminates the undesirable consequence. We must, therefore, make a conventional choice as to which law we are going to drop. Is this a case of trivial semantic conventionalism? Clearly not. It is not a simple matter of crudely changing the meaning of a term. Is it a form of semantic conventionalism? To a certain extent it is, for if we change the laws of the instruments, there is a sense in which we have changed the meaning of the introduced term, as we have changed its implicit definition. We might further ask, have we changed the extralinguistic interpretation of the term? Extralinguistic interpretations and implicit definitions can very well be out of kilter with one another, and changing implicit definitions - laws of the measuring instruments - could be viewed as an attempt to bring the two into line. The difficulty with pinpointing the exact nature of conventionalism in physics is that philosophers of science have not generally agreed on how to regard the extralinguistic meaning of these words, if in fact they have any. If one denied that the terms had any extralinguistic meaning, the conventionalism would be thorough-going, for one would not only deny that individual statements are testable, but also that they are either true or false individually. If, on the other hand, the terms are held to have an extralinguistic meaning, the conventionalism is merely epistemological, involving only our inability to ascertain the truth rather than the denial of its existence. There may be a middle of the road position which asserts that these terms have extralinguistic meaning, but of the kind which Lewis calls sense meaning, criteria in mind. With this position one might very well deny that individual statements of science are true or false. The

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question is not so much whether or not there really is a property in things which we call temperature, but whether, even if there is, science has anything to do with it. The same problem arises with theoretical entities, the only difference being that in this case we are dealing not only with unobservable properties but with unobservable things as well. Quine has extended the Duhemian thesis to include logic, as well as science. Quine states his thesis in the following passage: "Réévaluation of some statements entails réévaluation of others, because of their logical interconnections - the logical laws being in turn simply further statements of the system, certain further elements of the field. Having reevaluated one statement, we must reevaluate some others, which may be statements logically connected with the first or may be statements of logical connections themselves."79 According to Quine, in the face of disconfirming evidence we not only have a choice in rejecting a physical law, but we might reject a law of logic as well. It is not simply that the laws of physics must face observational evidence as a group but it is the laws of physics together with laws of logic that must face observation evidence. And one can reject one of the laws of logic just as easily as reject one of the laws of physics. Surprisingly enough, Quine's defense of his thesis is less adequate than Duhem's original defense fifty years earlier. Quine gives examples of cases in which we must conventionally choose to reject one statement or another, for the statements jointly are falsifiable, but not individually. But he gives no philosophical underpinning for his thesis. Clearly laws of logic are involved in testing the laws of physics, for in order to test groups of physical laws we must logically deduce observational consequences. The fact that they are involved in the process of testing does not mean that they themselves are being tested. One might very well admit the Duhemian part of Quine's thesis and yet deny his extension of it to logic. Quine must show that he is not involved in a trivial semantic conventionalism in changing the laws of logic. That we can change them is clear; that it is not a trivial matter is not.

78

Quine, "Two Dogmas of Empiricism" in his From a Logical Point of View, 2nd ed. (New York, 1961), p. 42.

IV THE SPIRIT OF CONVENTIONALISM

The conventionalist thesis must account for several problems concerning meaning left outstanding in Chapter III. A consideration of these problems and rebuttal of relevant criticisms will enable us to elicit the ontological, semantical, and epistemological commitments of conventionalism. We will then be in a position in the second section of this chapter to formulate the complete statement of conventionalism which has thus evolved in response to the various criticisms. With the conventionalist thesis in logic completely developed and defended, in the third and final section we will consider related conventionalist theses in order to place in perspective the conventionalist thesis in logic. And while the thesis in logic is independent of the other theses because of the unique character of logical constants, we will see that all the conventionalist theses depend on the peculiar character of certain words. The foundation of all conventionalism is semantical.

A.

PROPOSITIONS, MEANING, AND TRUTH

1. Quotation Marks and the Translation Argument

If the translation argument is correct, then the following two sentences are not synonymous: "the sentence 'p' is necessary" and "the proposition that p is necessary". The translation argument rests on the assumption that in translating the first sentence into another language, 'p' does not get translated while in the second sentence it does. This is sometimes justified by saying that 'p' does not occur in the first sentence, while it does in the second. Of course, " p " occurs in the first sentence, but it is argued that 'p' does not occur in ' 'p''. What is the justification for treating ' 'p'' and 'that p' in such radically different ways? I do not think there is any justification, and I suggest that they are synonymous expressions.1 In particular, I would like to point out that according to the meaning of " p " which Carnap in fact espouses, although he is not consistent in this usage, " p " is indisguishable from 'that p' except where 'p' is actually a variable 1

I am indebted in this section to Wilfrid Sellars, "Grammar and Existence: A Preface to Ontology", reprinted in his Science, Perception, and Reality, pp. 247-81.

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which is existentially quantified. This latter case must be treated separately. When Carnap actually uses quotation marks, he uses them in such a way that " p " is radically different from 'that p'. He believes that they are so different that we must use a nonextensional metalanguage when we are dealing with absolute concepts, such as 'that p', but an extensional metalanguage is sufficient in dealing with semantical concepts, such as " p " . The reason for this, according to Carnap, is that one can substitute for 'p' in 'that p', but one cannot substitute for 'p' in " p " . Carnap makes two distinctions which are important for our purposes. He distinguishes between sign events and sign designs, and he distinguishes between syntactically equal and syntactically unequal expressions. Now we must ask what is the effect of putting a symbol or expression in quotation marks? The first reaction is to say that we have formed a NAME of THE symbol or expression. If it were correct to say "THE symbol or expression", it would also be correct to say that we have formed a proper name of the symbol or expression. If the symbol in quotation marks is a proper name for a unique particular, one might be able to justify not translating 'p' in " p " , for a proper name oftentimes does not get translated. For example, the former leader of North Viet Nam was called "Ho Chi Mhin" even in English newspapers, and Dostoevsky's character Ivanovitch is called "Ivanovitch" even in the English translation. On the other hand, the present Pope is called "Pope Paul" rather than "Pope Paulo". But is " p " a proper name of a unique symbol or expression? To be more specific, according to Carnap, what is ' 'Chicago is large'' a name of, and is it a proper name? The answer is that in some cases it is a proper name, but when it is functioning as a proper name, it is the proper name of a sign EVENT rather than a sign design. In syntax and semantics, we are rarely concerned with sign events. A type of sentence in which " p " occurs as a proper name is the following: "The word 'following' which occurs just before the last colon is printed with black ink." This is not, however, the type of sentence which interests us in syntax and semantics. We are generally concerned with a type or class of signs. In particular, when we make statements, such as " 'p' is true" or " 'p' entails 'q'", we are rarely making a statement about a particular concrete sentence or pair of sentences. Rather, we are making a statement which is applicable to all sentences of a certain type or 'design'. Once we realize this, we see that an expression, such as " p " , rarely occurs as a proper name. In fact, expressions like " p " are much like general terms, and sentences such as " ' p ' is true" are really shorthand versions of "All sentences of the type 'p' are true" or, if you will, "All 'p's are true." If one wants to insist that " p " is a singular noun rather than a common noun (general term), one must be prepared to admit that it is also an abstract rather than a concrete noun. If " p " in the sentence " 'p' is true" is a singular noun, it must be referring to the sentence TYPE itself. Whether we treat " p " as an abstract singular noun or a concrete common noun, we do not have the translation problem which we have with singular concrete nouns, i.e., proper names. There is no prima facie reason why we should

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not translate either common nouns or abstract nouns, and in the case of nouns of the form ' 'p'' the correct translation is ' ' q ' w h e r e 'q' is the translation of 'p'. While agreeing with most of the above, one might still deny that " p " can be translated as " q " , where 'q' is the translation of 'p', since one might concede that " p " is not a proper name, yet still object that the extension of " p " (assuming " p " is a common noun) only includes the sign events of a particular language. For example, one might argue that a particular instance of the word 'rot' is not included in the extension of " r e d " . To take a more relevant example, one might say that a particular instance of the symbol 'v' is not included in the extension of " o r " . The reasoning here would be that the extension of a SYNTACTICAL common noun like " o r " must RESEMBLE what is inside the quotation marks of the common noun. Since syntax is concerned with FORM there must be a similarity of form, where 'form' in the second instance is taken to mean shape. But when we say that syntax is concerned with form, do we mean shape? When we say, " 'und' begins with 'u'", we ARE concerned with the visual shape or form of 'und'; the extension of " u n d " in this case includes only those words which are visually similar to 'und'. Is syntax then exclusively concerned with form in the sense of shape? Some syntactical theorists, such as Martin, 2 do seem to have this in mind, although it is not quite clear whether they really intend this since they restrict their discussion to constructed object languages. It is evident, however, that Carnap does NOT have this in mind, as he distinguishes between syntactically equal and unequal symbols. When two symbols are equal, we say that they have the same design. "But that does not in any way prevent their having different visual shapes; . . . or differing in colour, or any other characteristics that are syntactically irrelevant." 3 This aspect of Carnap's theory has generally been overlooked, and he does not dwell on it himself. Most logicians consider syntax to be a theory of the order and arrangement of symbols. In this context all symbols and sentences which have a different visual shape are syntactically unequal. Even Carnap says that two EXPRESSIONS are equal only if the corresponding symbols are equal. But prima facie there is no reason we could not consider two expressions as equal regardless of the constituent symbols. For example, if Carnap will allow that ' = ' and ' = ' are equal, there is no reason why we could not allow that 'and' and 'but' are equal. The difficulty is the same in both cases; namely, what criteria can we use to determine whether or not two symbols or expressions are equal? Carnap says simply that this is to be determined by the syntactical rules. Nevertheless, it is one thing to establish in pure syntax that two symbols are equal and quite another to ascertain this in descriptive syntax. The difficulty, however, is not fundamentally different than that experienced in discerning the transformational rules of a natural language, as long 2

See, for example, R. M. Martin, Truth & Denotation: A Study in Semantical Theory (Chicago, 1958), Chap. III. 3 Rudolf Carnap, The Logical Syntax of Language, trans. Amethe Smeaton (Paterson, New Jersey, 1959), p. 15.

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as we restrict our concern to the logical words. For if we find that the transformational rules involving one logical word are isomorphic to those involving another, we can say that there is only one rule involved. For example, if I discover the rules "From 'p and q' to infer ' p ' " and "From 'p;q' to infer 'p'", I can say that there is only one rule, and 'and' is syntactically equal to ';'; that is, they both have the same design. We can retain something of the common notion of syntax, if we treat 'and' as a symbol, rather than a word. We can say that 'and' is a symbol of the design ';' and ';' is a symbol of the design 'and'. Furthermore, we can correctly say that 'and' occurs in 'p;q', if we read this as "a symbol of the design 'and' occurs in 'p;q'", as Carnap suggests. Syntax can go on as usual if we are careful regarding how we read the syntactical rules. In actually doing descriptive syntax, however, I suggest that it is more profitable to distinguish between logical syntax and grammatical syntax, where the latter is concerned with form as the order and arrangement of symbols, while the former is concerned only with 'logical form'. It is pointless for the logician to distinguish between 'if p, then q' and 'q, if p', while it is a matter of concern for the linguist. One might justifiably argue that in going this far we have given up syntax and progressed to semantics; however, I do not think that in considering 'logical form' we are necessarily doing semantics, although I would agree that it is not syntax in the accepted sense of the word 'syntax'. The sentences 'if p, then q' and 'q, if p' are not syntactically equal, for the arrangement of the symbols is different. On the other hand, we hold that merely by a study of linguistic regularities we can determine that the sentences have the same logical form; that is, that the same logical rules are applicable to both. It is not NECESSARY for us to see WHEN the sentences are used, but merely HOW they are used in inferential contexts. If this is the case, we are not doing semantics. To avoid confusion we will call the study of logical form, formalistics. This in itself does not justify our arguing that symbols from DIFFERENT languages can have the same design. Since we usually take a single language as our object language in doing formalistics, the formalistic equality is generally applied only between symbols in a given language. There seems to be no reason, however, why we cannot combine natural languages to form a single object language. For example, we could consider a combination of English and German as an object language. In this case 'und' would be formalistically equal to 'and'. In fact, for the purpose of discussing the conventionalist thesis in regard to natural languages, it is perhaps best to consider all natural languages together as one language, inasmuch as the conventionalist thesis is not a thesis about any one particular language, but about all natural languages.4 In particular, when we consider the genesis of a rule from a linguistic regularity, it is quite likely that the transition in many cases occurred in one language and the rule was transferred to another language as the second language evolved out of the first. If we regard as one language a collection of natural languages, we can use the relation of formalistic equality even between symbols of languages usually considered to be distinct. 4

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We have argued that' 'p'' is a common noun in practically all semiotical contexts, and, furthermore, the extension of "p", as it is used in formalistic contexts, includes not only expressions with the visual form 'p' but all expressions which are formalistically equal to 'p'. Therefore, we see no objection to translating 'p' of " p " by replacing 'p' with a formalistically equal expression when we are in a formalistic context. When we are in a strictly syntactical context, the extension of " p " is limited, and the free translation suggested here is not permissible. We are, however, concerned with the contrast between 'that p' and " p " , and in a strictly syntactical context, expressions such as 'that p' do not occur. Since these arguments for denying a difference between " p " and 'that p' come from Carnap himself why did he treat the two as quite different? He says that the 'p' in 'that p' can be replaced or substituted for, while he implies that this is not the case for the 'p' in "p". 5 This, I think, is a mistake. Whatever rules are laid down for replacement, it would seem that they should treat that-clauses and quoted expressions equally. Both that-clauses and quoted sentences serve the same purpose; they are functions which turn sentences into nouns. In fact, when one considers the interpretation of quoted sentences which treats them as names for the abstract sentence types, one has a perfect parallel between that-clauses and quoted sentences. They are both names for abstract entities. Apparently Carnap did not realize this fact about propositions for he gives the following definition, which is clearly grammatically incorrect. "D17-C1. © ; is true in S =DF there is a (proposition p such that (B => A))'. By our definition of necessity, a statement which is not deducible from the empty set of premises is contingent. We have said that a nonatomic sentence is true either if it is deducible from true premises or if only true sentences are deducible from it in conjunction with other true premises. According to this definition, logically true statements are true. We suggest, however, that there is a fundamental distinction between logical truisms and other true nonatomic sentences. We put forward the following definition of logical truth: Def. A is a LOGICAL TRUTH if and only if either A is a logically necessary statement, or whenever an atomic sentence B is deducible from A together with other premises, B is deducible from the other premises without A, or both.

The definition says, in effect, that a contingent logical truism is irrelevant to our reasoning regarding the world. While the statement '(A => (B => A))' is not necessary in the modification of system TA, it is clearly a logical truth for any true atomic sentences which are deducible from it are deducible from the additional premises necessary for the deduction. This suggests that the distinction between logical truths and physical truths is of more fundamental importance than the distinction between necessity and contingency, for the latter distinction is dependent on the vagaries of language. Actually this is only a difference of degree, as a nonatomic sentence could not be either a logical truth or a physical truth if it were 10

Curry, Foundations, p. 178.

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not for the existence of linguistic rules. Consequently, we suggest the following criterion of completeness: A set of L-rules is complete if all the logical truths are necessary truths in the system. One might ask why we are rejecting the criterion of logical truth which Quine, among others, has suggested; that is "a logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical particles". 11 From my point of view, the difficulty with this approach is that it assumes a common notion of truth for logical and physical truths. We have distinguished three basically different kinds of truth: (a) the truth of atomic sentences, which is independent of rules of inferences; (b) the truth of nonatomic sentences, which depends on the rules of inference and the truth of atomic sentences; and (c) the logical truisms, which have the second kind of truth, but which are also true in a way which is independent of both of the first two kinds of truth. Those sentences which have the second kind of truth but not the third we call PHYSICALLY TRUE sentences, and those sentences which have the third kind of truth as well as the second we call LOGICALLY TRUE sentences. Among both the logically true and the physically true sentences we distinguish two types - the necessary truths and the contingent truths. The logical truths are not simply a special kind of physically true sentence for they are true in a fundamentally different way. The truth of logical sentences is not dependent on the truth of atomic sentences nor of physically true nonatomic sentences, while the truth of physical sentences depends partially on the truth of atomic sentences. That a logical sentence is true can be determined not only independently of knowledge of the empirical world, but also independently of reinterpreting the components other than the logical particles. Both Quine's and my criteria suggest that logical truths are independent of the empirical world, but it is my criterion which points out why they are independent of the empirical world. The only atomic sentences which are logically deducible from logical truths are deducible from the additional premises necessary for the deduction. In other words, they are irrelevant to our reasoning about the empirical world. Moreover, we would argue that Quine's characterization is quite misleading, for no definition of truth is given for nonatomic statements. If we assume, as is quite natural, a semantic definiton of truth for nonatomic statements and truth under all reinterpretations is asserted to be a sufficient condition for logical truth, we have the undesirable consequence that the truth or falsity of each in a series of contingent atomic statements is sufficient for the truth of a necessary logical statement. Consider the logical truth: John is tall or not tall (D). According to the semantic definition of truth (D) is true if 'John is tall' is true or 'John is tall' is false. In turn a sufficient condition for the latter would be that 'John is tall' is true. Consider a reinterpretation of (D), viz., Peter is short or not short (D'). A sufficient condition 11 Quine, "Two Dogmas of Empiricism", pp. 22-23. See also the treatment of logical truth in R. M. Martin, The Notion of Analytic Truth (Philadelphia, Univ. of Pennsylvania Press, 1959), pp. 25-26.

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for the truth of (DO is that 'Peter is short' is false. This can be repeated for each reinterpretation of (D). Let us assume furthermore that 'John is tall' is true, 'Peter is short' is false, etc. These atomic statements are individually contingent, and their conjunction is a contingent physical truth, but according to Quine's characterization of logical truth together with the semantic definition of truth, this contingent statement is sufficient for the logical truth of the necessary statement (D). But surely we do not want to say that (D) is a logical truth because a contingent statement or even an infinite number of contingent statements happen to be true or false. As Wittgenstein has pointed out: "The mark of a logical proposition is not general validity. To be general means no more than to be accidentally valid for all things." 12 A logical truth does not depend for its truth upon the truth of nonlogical statements, and any definition of logical truth should make this clear. One might ask whether the definition which we have given of logical truth is extensionally equivalent to the definition given by Quine; do the two definitions pick out the same class of statements? The answer is yes. We shall first prove that a logical truth is true and remains true under every reinterpretation. We will then prove the reverse, using in the course of the proofs our definition of truth for nonatomic sentences, as well as our definition of logical truth. If a statement is deducible from the empty set of premises, it is deducible from true premises, and if a statement B is deducible from a statement A together with other true premises only if B is deducible from the other premises without A, then B is true. Then any conclusion deducible from A together with other true premises is true, and by our definition of 'true', A is true. Therefore, if a statement A is logically true it is true. That a logically true statement is also true under all reinterpretations follows from the fact that the L-rules are formal, that is, they refer to sentence forms rather than to particular sentences. Therefore, anything which can be said about a particular logically true statement A can be said about any statement with the same form as A, that is, for any reinterpretation of A. To prove that a statement A which is true and remains true under every reinterpretation is logically true we note that a statement which satisfies the hypothesis is such that for every statement A* of the same form as A either (1) there is a set of true statements from which A* is deducible, or (2) from any set of true premises B* together with A* only true conclusions C* are deducible. In case (1) for every A* either (a) A* is deducible from a set of statements containing only logically true statements, in which case A* is also logically true, or (b) A* is deducible only from premises which are true, but not logically true. The latter case, however, is possible only if A* is deducible from any statements whatsoever, for if A* is deducible from only a particular set D* of true statements, then there will be another reinterpretation of A, e.g., A**, such that the set D** will be false under that interpretation. Since the deduction of A* from D* depended only on the form 12

Wittgenstein, Tractatus Logico-Philosophicus, (New York, 1961), 6.1231.

trans. D. F. Pears and B. F. McGuinness

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of the statements A* and D*, A** will be deducible only from D**, all of which are false, and, therefore, A** will not be deducible from some set of true premises. But if A* is deducible from any statements whatsoever it is deducible from the empty set of premises and is, therefore, logically true. To deal with case (2) we first note that any statement which is true under all reinterpretations will of necessity be nonatomic. The true premises B* and the true conclusion C* will either (a) be all atomic, or (b) contain at least one nonatomic sentence. In case (a) C* must be one of B* for if it is not then it would be possible to give an interpretation to A, e.g., A**, such that all of B** are true and C** is false, and C** will be deducible from A** and B**. Under this interpretation A** would be false, contrary to our hypothesis. Case (b) in turn reduces to case (a) by virtue of our definition of the truth of nonatomic sentences in terms of their inferential connections with atomic sentences. But in either case then C** is deducible from A** without B**, and accordingly A is logically true. Therefore, if a statement is true and remains true under every reinterpretation it is logically true. B.

THE CONVENTIONALIST THESIS REVIEWED

1. Final Statement

of the Conventionalist

Thesis in Logic

As we noted in Chapter III Pap states the conventionalist thesis as follows: "The existence of certain linguistic habits relevant to the use of a sentence S is a necessary and sufficient condition for the necessity of the proposition meant by S." 13 By modifying this statement an adequate version of the thesis can be achieved. Instead of 'habits' we would prefer to say 'rules', interpreting this in the manner outlined in Chapter II. 'Habit' implies something ingrained in the individual which can be determined by observation of the individual; however, we believe that the regularities can be determined at best by observations on a society. Even here it is a quite complex operation with many pitfalls. In accordance with our discussion in section A.l of this chapter, we rewrite Pap's statement of the thesis as follows: The existence of certain linguistic rules relevant to the use of a sentence of type S is necessary and sufficient for the necessity of any sentence of the same form as S. It must be remembered that sentences from different languages can have the same form, as we showed in the extension of Carnap's notion of syntactical equality which thus includes not only visually different symbols within the same language but also visually different symbols from different languages. To a great extent we consider the rules to be interlinguistic. If the rule as formulated is held to apply to only one language, there is clearly a sense in which this rule is merely a manifestation of something more general; linguistic rules, as conceived by us, are interlinguistic at least among languages which have a common etymological basis. 13

Pap, Semantics and Necessary

Truth, p. 164.

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We must also remember that whether or not a sentence has a particular form depends not only on its visual shape but also on the linguistic rules relevant to it. When the rules relevant to a sentence of a certain visual shape change, the sentence is no longer of the same form. Formalistics is concerned with logical form, and this is not to be confused with visual form. Logical form is a function of the logical rules. Learning how to follow a linguistic rule is not merely a matter of recognizing visual shapes, as it is in the case of most artificial languages, for a single linguistic rule may be relevant to many different visual shapes. This obviates the objection that the same sentence may mean different things at different times, and, therefore, there are propositions which they mean. We would say that only the visual shape has remained constant while the type of the sentence has changed. This is a sufficient account of the necessity of sentences of the same form from related languages, e.g., the Indo-European languages. In these cases we can speak of a single set of linguistic regularities transmitted etymologically from language to language and, correspondingly, of a single set of necessary statements. While this argument is, I believe, interesting and relevant, it cannot account for the necessity of different sentences which "express the same proposition" from nonrelated languages, e.g., Chinese and English. The linguistic regularities here cannot be considered etymologically as one, yet we want to consider two necessary statements which express the same proposition as being merely statements of the same form, that is, as governed by the same set of linguistic rules. A basic assumption of the entire argument is that sentences from nonrelated languages can express the same proposition, the justification of which requires finding such sentences. This is no mean feat and it involves certain paradoxes, as Quine has pointed out.14 Our problem is to explain it, assuming that it is possible. If it is not possible to produce such a 'Radical translation', our previous defense for related languages is sufficient. We must first define what we mean by saying that two linguistic rules are the if they do not have a common etymological root. We are saying that they are formalistically identical; that is, they can be FORMULATED by a single rule formulation, where in the rule formulation an expression, such as 'p & q', is read as "statement of the DESIGN 'p & q ' " . If the rules relevant to showing that a statement A of language P is necessary can be expressed by the same formulation which expressed the rules relevant to showing that statement B of language Q is necessary, we say that A and B are of the same form, and the same rule (-formulation) is necessary and sufficient for the necessity of both. SAME

It has been argued by Pap that according to the conventionalist thesis the term 'necessary' is not a time-independent predicate, where "a time-independent predicate is defined as a predicate P such that sentences of the form 'x is P at time t' are meaningless".15 This, however, is only partially applicable to the conventionalist thesis as developed here, the extent of its applicability depending on the range of values for the variable 'x'. If 'x' were to range over names of concrete instances of 14

W. V. O. Quine, Word and Object (Cambridge, Mass., 1960), Sec. 13.

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sentences considered as a series of blotches of printer's ink on a piece of paper, then the critique is not applicable, inasmuch as the conventionalist thesis is about sentences in so far as they are elements with a certain form, that is, as elements concerning which there are rules. When we consider the range of 'x' to be either names of particular sentences considered as sentences of a certain form, the situation is not so simple. When thought of in this way, 'necessary' is a time-independent predicate, for as long as a particular sentence is of the same formalistic type it is always necessary. We must constantly bear in mind here that the conventionalist thesis is not a thesis about visual shapes. It is quite true that a sentence of a particular visual shape may at one time be necessary and at another time contingent, but this is not true when we consider sentences of a particular formalistic type. If a sentence of a particular type is necessary at one time, it is always necessary, and if it is contingent at one time, it is always contingent. There is one sense, however, in which the predicate 'necessary' is time-dependent, for until the sentence comes into existence it is not necessary, and for that to occur there must also come into existence both the linguistic rules of sentence formation as well as the linguistic rules of inference. As long as the sentence type is in existence, as long as the rules are extant, a sentence remains necessary. Pap further objects that according to the conventionalist thesis the NN-thesis, which states that necessary truths are necessarily necessary, is not correct.16 The import of the NN-thesis depends on whether or not one accepts the existence of propositions. Let us distinguish the following four kinds of sentences: (a) "It is necessary that p"; (b) "It is necessary that it is necessary that p"; (c) "'p' is necessary"; and (d) " " p ' is necessary' is necessary". Both sentences (a) and (b) are about propositions. The first ascribes necessity to a proposition that p; the second ascribes necessity to the proposition that p is necessary. There is no fundamental difference between these two sentences. There is, however, a fundamental difference between sentences (c) and (d), for sentence (c) is a sentence in a metalanguage, while (d) is a sentence in a meta-metalanguage. (d) is a sentence about a sentence in a metalanguage. Since we analyze sentences (a) and (b) as equivalent to sentences (c) and (d), we will direct our attention to the truth of (d) assuming the truth of (c) rather than to sentence (b). If sentence (c) is true, is (d) also true? The answer, according to the conventionalist thesis, is that it depends. Since (c) and (d) are sentences in two different languages, it may be that in some cases (d) is true and in other cases not. We interpret (c) as saying that 'p' is deducible from the empty set of premises. Whether or not (c) in turn is deducible from the empty set of premises depends on the linguistic rules relevant to the predicate 'necessary', this predicate being a metalanguage predicate rather than a predicate or sentential connective of the object language. The difficulty with deciding whether or not the NN-thesis is true is similar to the 15 Pap, Semantics and Necessary Truth, p. 120. 1» Ibid.

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difficulty encountered with the concepts of entailment and implication. For example, does 'p' entail 'q or not q'? Natural languages are for the most part silent on this question. Linguistic rules do specify that 'q or not q' is unconditionally assertable. But they do not assert or deny that 'q or not q' is logically deducible from a contingent sentence 'p'. What is required here is a proposal for the use of the word 'entails'. The situation is similar in the case of sentence (d) above. The rules of natural language do not specify how we are to use the predicate 'necessary' as applied to sentences of necessity. We have rules for the use of 'necessary' as applied to object language statements, but natural languages are not concerned with predicates as applied to statements of a metalanguage. What is required here is a proposal for the use of the word 'necessary' in a meta-metalanguage. In particular we require linguistic rules for the metalanguage, a statement of necessity in the meta-metalanguage being, according to our view, a transcription of a rule for the metalanguage. One might suggest that we deny the NN-thesis in order to account for the fact that the statements are necessary only by virtue of the contingent fact that the language has certain rules. On the other hand, we could incorporate a linguistic rule for the metalanguage which specifies that one can unconditionally assert any direct or indirect rule of inference in its entailment or statement of necessity form, for a rule is to a certain extent independent of the linguistic regularity on which it it based. There is also the psychological fact that the rule does not seem to be a convention, and appears as something imposed from outside of society. In any case one could, if one desired, make a necessary statement necessarily necessary. A middle of the road position might notice that there is a great difference between logically necessary and physically necessary truths. A physically necessary truth is, after all, open to theoretical falsification; its truth depends on implying true atomic sentences. On the other hand, a logically necessary statement is immune to falsification, since it is completely independent of the empirical world. One might reserve the necessarily necessary attribution for logically necessary truths and call physically necessary truths only contingently necessary. One could propose a linguistic rule which allows one unconditionally to assert the entailment statements and statements of necessity corresponding only to L-rules. In any case, we suggest that each of these must be discussed on its own merits as a proposal for rules for the metalanguage. The question of whether the NN-thesis holds generally, in some cases, or in no cases is not a true-false question, but requires a decision. When the predicate 'necessary' occurs in sentences of the form "It is necessary that if p then q", we can interpret this in two ways: (a) " ' p ' entails ' q ' " ; and (b) "if p then q " where 'if . . . then . . . ' is an intensional, not a truth functional, sentential connective. Under interpretation (a), 'necessary' is a metalinguistic predicate. Under (b), where it is specified that 'if . . . then . . . ' is a sentential connective, 'necessary' occurs merely to indicate that 'if . . . then . . . ' is not to be construed as truth functional. It has generally been assumed that the use of intensional connectives

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in the object language commits us to necessary connections. This occurs only if we fail to distinguish between sentences (a) and (b). 'If . . . then . . .' is a logical constant, and as such it has only a linguistic meaning, given completely by the logical rules for their manipulation. One cannot draw any essential difference between different kinds of connectives (modal, extensional, etc.) as long as it is remembered that they are all logical symbols and their role is given by the linguistic rules. The only difference is the degree of manipulation possible; more manipulation is possible with extensional connectives than with modal connectives. I see no objection, therefore, to using modal connectives in an empiricist language. Such usage does not land one in rationalism any more than the use of only extensional connectives eliminates it, because the difference between rationalism and empiricism depends on the interpretation of 'entails', not of 'if . . . then . . .'. If our object language has nonextensional connectives, then one could introduce 'necessary' into the object language as a unary logical connective, definable in terms of the nonextensional logical connectives. For example, one might give the following definitions: Np = Df (p ->- p) -v p or Np = Df ~ p -=*" p, where '->' is a nonextensional 'if . . . then'. In this case one would expect the NN-thesis to hold, but one must not confuse this case where 'necessary' is an object language connective, with the previous case in which 'necessary' was a metalinguistic predicate. These two distinct cases must be considered separately with regard to the NN-thesis. It may hold in one case without holding in the other. It is interesting to note that according to our definition of truth for nonatomic sentences, the truth of both extensional and nonextensional sentences DEPENDS ON the truth or falsity of their atomic components AND the logical rules of inference. The difference is that the truth of an extensional nonatomic sentence can be inferred directly from the truth or falsity of its component atomic sentences, whereas the truth of nonextensional sentences depends on whether only true statements can be inferred from them (together with other true statements as premises). Moreover, a nonextensional sentence which contains an intensional connective with the force of 'logical implication' will be, if true, logically true, and its truth will be completely independent of the truth of atomic sentences. 2. Logical Constants vs. Descriptive Predicates The conventionalist thesis which we have developed and defended rests on the assumption that logical constants, which have only linguistic meaning, are distinguisable from descriptive constants, which have meaning in the linguistic mode and the extralinguistic modes as well. This enabled us to meet the proposition and validity critiques. Nonatomic sentences, including logical truisms, do not express propositions, for they have only linguistic meaning, while there is a sense in which atomic sentences do express propositions for they have more than just linguistic meaning. The validity critique was answered by appealing to the fact that all logical

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rules, except Repeat, have either a premise or a conclusion which is nonatomic, and whose truth, therefore, depends on the logical rules. Our definition of L-rule, in turn, depends on distinguishing atomic from nonatomic sentences. The unique semantical character of a special class of terms is the nucleus of our thesis' defense. Pap has raised serious objections against this distinction, focusing his attack on Quine's definition of logical truth, although it is equally applicable to our definition. We recall that Quine's definition of logical truth depends on the distinction between logical and descriptive constants, for a logical truth, according to Quine, is a sentence which remains true under all reinterpretations of the terms OTHER THAN the logical constants. Pap poses the problem as one of constructing a definition of 'logical constant' which is independent of the notion of logical truth. He notes that while the usual procedure of logicians of enumerating the logical constants does "serve the function of criteria of application, [it] clearly cannot be regarded as an analysis of intended meanings".17 Pap also indicates why he does not believe that the term 'logical constant' can be defined negatively as 'non-descriptive term'. We have asserted that a logical constant is a term which has only linguistic meaning; however, it may suffice in defining the term 'logical constant' to say that it does not have extensional meaning. There is another class of terms also which does not have extensional meaning; namely, the operators. These are functions which turn nouns into nouns, as distinct from predicates, which turn nouns into sentences, and the logical constants, which turn sentences into sentences. Examples of operators are 'father of and 'mass of'. Operators can be converted into predicates by certain additions: for example, 'father of' can be made into a relation by the addition of 'is' - 'is the father of' - or into a one place predicate by the addition of a referring expression, such as 'John' - 'is the father of John'. I would suggest that the operators differ from logical constants in being truly syncategorematic terms - co-predicates; that is, while they do not in themselves have extensional meaning, when suitably combined with additional expressions, the composite expression has extensional meaning. Taking this additional group of expressions into consideration, we suggest as our definition of 'logical constant' the following: a logical constant is a symbol or expression which neither has extensional meaning nor is syncategorematic in the sense described. It is important for us to exclude operators from the class of logical constants for while logical constants cannot occur in atomic sentences, operators can. Pap might very well be justified here in replying that this definition is useless until we have given a clear criterion for deciding whether or not a term or composite expression has extensional meaning. Plainly, there are no absolute criteria which we can give. But at this point the objection begins to appear similar to Quine's criticism of the notion of synonymy.18 It is difficult to give criteria for 17 18

Arthur Pap, "Logic and the Concept of Entailment", Jour. Philos. XLVII (1950), 379. Quine, "Two Dogmas of Empiricism", pp. 20-27.

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determining whether or not a term in a natural language has extensional meaning, and it is just as formidable a task to give criteria for determining whether or not two terms in natural languages have the same meaning. The difference between these two is that in the former case we generally do not have difficulty in determining in individual cases whether or not a term is a logical constant, while in the latter case we often do have a great deal of difficulty determining that two terms are synonymous. The solution, however, in both cases is the same. We have indicated throughout that we regard the conventionalist thesis to be one of several credible theses about the nature of logical truth. We have given a possibly true theory about the relationship between logical language and the world. We further suggest that some terms of language are not used to describe the world; rather they have two distinctive functions. On the one hand, they enable us to condense a great deal of information about the world in a small amount of language. One must recognize the difference between a condensation of a great deal of knowledge and the knowledge itself, for in order to get explicit information about the world from a condensation one must be able to derive atomic sentences and this involves the use of L-rules. On the other hand, logical constants are also used when we want to state hypothetical knowledge about the world, that is, possible descriptions of the world. What we have in mind are statements from which we can derive descriptions of the world, but only if we add as additional premises other descriptions of the world. Quite often both of these functions, condensation and hypothetical knowledge, are combined in a single statement, as is typical of the scientific laws. These are statements which have been devised in order to condense a great quantity of hypothetical information - information whose truth depends on other information being true. All of these statements only potentially describe the world. In order to derive actual information from them we must know how to use them, that is, we must know how to deduce atomic sentences from them, or these sentences would be completely meaningless. To say that all terms of language have meaning in the same way is as open to objections as our thesis, for this would imply that prepositions, such as 'of' and 'on', articles, such as 'a' and 'the', and parenthetical expressions, such as 'after all' should all be treated in the same way. In talking about their relation to the empirical world we do not, however, want to say that they all have the same relation. Different kinds of terms have different functions, and it is one of the philosopher's tasks to determine what the kinds of terms are and how their functions differ. We have differentiated what we consider to be a distinct class of terms and their peculiar function. Other philosophers may want to distinguish classes of terms in diverse additional ways. This is well and good. But, surely, the philosopher of language commits an injustice if he treats all terms as if they were proper names or adjectives or prepositions. A rational reconstruction of language is often involved because some terms cannot be clearly placed in one category or another. This is certainly a problem for anyone who wishes to assert that there are synonymous

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terms in natural language. The problem is lessened for one who distinguishes a class of logical constants, for we have little problem in deciding whether or not a certain term is properly applicable to things or pairs of things, i.e., has extensional meaning, either by itself or as part of a composite expression. There is an interesting sidelight to Pap's critique of the distinction between logical constants and descriptive predicates. Pap asks if it would be clarifying to define a descriptive term as one that refers to an observable feature of the world. While we would not want to consider this a necessary condition for a term's being descriptive, we would consider it to be sufficient. Pap argues that it is not even sufficient, for do not numbers refer to observable features of the world - "counting is surely a mode of observation"19 - and yet are they not shown to be logical terms by the logistic reduction of arithmetic? I will not attempt here to resolve this difficulty, except to point out that while counting may be a mode of observation, it is surely a more complex type of observation, perhaps involving deductive and inductive inference as well. On the other hand, I agree with Stephen Korner when he says: "The logicist account of applied mathematics implies an illegitimate conflation of mathematical number-concepts and corresponding empirical ones."20 I would expect, therefore, that number statements of applied mathematics and ordinary usage would turn out to be nonatomic, but nonlogical, statements; however, a development and defense of this view would take us far afield the philosophy of mathematics.

C.

THE CONVENTIONALIST THESIS E X T E N D E D

1. A nalytic Sentences In this section we will be concerned with sentences which can be turned into logical truths by substituting synonyms for synonyms; that is, with those sentences that are analytic in the narrow sense; for example, "All bachelors are unmarried." Some philosophers characterized as conventionalists have directed their attention to this type of sentence, but have ignored the nature of the underlying logical truisms.21 We will see how the conventionalist thesis in logic can be extended to cover this further class of sentences. We recall that the meaning critique of Lewis was most effective against those sentences in which descriptive predicates occurred essentially because Lewis' position was that such a sentence is true by virtue of the extralinguistic meanings of the terms involved. The sentence itself was true by convention only in the trivial sense »

Pap, Jour. Philos. XLVII, 380. The Philosophy of Mathematics: An Introduction (New York, 1962), p. 61. 21 This is true, for example, of Max Black. See his "Necessary Statements and Rules", Phil. Rev. LXVII (1958), 313-41.

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that words possess their meanings by virtue of a convention. Once, however, the words have been assigned a meaning, the sentence is true by virtue of correctly describing this relation between meanings. On the other hand, when a term does not have extralinguistic meaning, the process of assigning meaning is the same process as relating the term to other terms, relating the sentences involving the term to other sentences. In this case one cannot distinguish conventional and nonconventional aspects of the truth of a sentence involving the term essentially. These sentences are true by convention because the only truth that they have is that which has been given to them by convention. What we would suggest is that analytic sentences (in the narrow sense) can also be viewed as a kind of logical truth, and as such they share in the properties of logical truths. For example, they are not open to theoretical falsification, and their truth does not depend on the empirical world. Lewis would agree with both of these characterizations of analytic truths, yet still deny that they are true solely and completely by convention, as are logical truths. After all, even if we cannot assign signification and sense meaning to logical terms, we can for descriptive terms, and analytic sentences are true not only by virtue of the linguistic meaning of the descriptive terms but also by virtue of their extralinguistic meaning. According to Lewis, analytic sentences are merely linguistic expressions of extralinguistic relations between properties and concepts. Lewis tells us that the sense meaning of a term is the criteria which we have in mind in applying the term. He also says that the criteria in mind for applying one term may include the criteria in mind for applying another term. In this case we shall say the former concept is complex as it includes another concept. There are many types of complex concepts, such as bachelor, sibling, and soluble, depending on how the simpler concepts are united to form the complex concept. The concept bachelor is formed from a conjunction of concepts; the concept sibling is formed from the disjunction of the concepts brother and sister; and the concept soluble is a hypothetical concept formed from the concepts of being placed in water and dissolving. Lewis considers only the first type of complex concept, the conjunction of simpler concepts, in discussing analyticity. For example, he says: "We know that 'All squares are rectangles' because in envisaging the test which a thing must satisfy if 'square' is to apply to it, we observe that the test it must satisfy if 'rectangle' is to apply to it is already included."22 The concept of square is clearly a conjunctive complex concept. It is not clear that Lewis could extend his analysis to nonconjunctive complex concepts and we will, therefore, concern ourselves only with the conjunctive type, in particular with the relationship between the complex concept and the simpler concepts conjoined to form it. Let us begin by looking at the concept of bachelorhood. What is the relationship between this concept and our concept of unmarried? Is our concept of bachelorhood distinct from and only externally related to our concept of unmarried? Or is 22

Lewis, Analysis,

p. 152.

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it that our concept of bachelorhood is nothing more than a combination of concepts, among which is the concept of unmarried. Is knowing how to apply the concept of bachelor to an individual a matter of knowing how to apply the concept of unmarried, and so forth? We note two exceptions. One might have learned the concept of bachelor ostensively by having bachelors pointed out to him; however, if an individual's concept of bachelor is learned ostensively, the sentence "All bachelors are unmarried" would not then be analytic FOR HIM. The sentence would not be true by virtue of the meanings of the terms as HE understands them. For this individual, the fact that all bachelors are unmarried would be a physically true sentence; it would have factual content. But clearly one who derives factual information from an analytic sentence does not fully understand the relevant concepts. Another exception would be that of someone who did not have a distinct concept of unmarried but whose concept of bachelor included the concept of unmarried. This would be the case for someone who had learned the full meaning of bachelor, but who never bothered to form a separate concept for the unmarried aspect of the concept of bachelor. Again knowing how to apply the concept of bachelor would not involve knowing how to apply the concept of unmarried, for this individual does not have the latter concept. In this case the individual would not even understand the sentence, "All bachelors are unmarried." Returning to our original question, we find that one who understands the sentence "All bachelors are unmarried" and who also knows that it is a priori must know how to apply the concept of unmarried in applying the concept of bachelor. If the concept of bachelor is in reality merely a collection of concepts considered together and applied together, what can we say about the relation between the two concepts? We have here a relation of entailment which is no more exciting than the relation of entailment between the premises and conclusion of an argument which is valid by virtue of the rule Repeat. What we are suggesting is that even if we consider the sense meaning of descriptive terms, we can still have truth by convention, for I see no reason to deny that it is we who have created the complex concept by considering together a number of simpler concepts. If this is so, the relationship between the complex concept and one of the simpler concepts is a logical entailment of the most trivial sort. We still have to contend with the signification of descriptive terms. Is it not the case that the property of being a bachelor entails the property of being unmarried? Again we must ask: "Is there a distinct property of bachelorhood, or is the property of bachelorhood nothing more than a collection of simpler properties?" Once more the theme is to attribute to the world the least number of entities which are absolutely essential. It can plausibly be argued that individuals actually only have simple properties, and that complex properties are a mental creation. If we can ascribe to the world only simple properties and attribute to man only simple concepts, then this is the path chosen by the conventionalist. If complex properties do not exist independently in the world, then they cannot entail simpler properties, and similarly

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in the case of complex concepts. If a complex property and a complex concept are merely collections of simpler properties and concepts, respectively, the former entail the latter only in the way that a collection of premises entails one of its members, and this is the essence of logical entailment. While complex properties and concepts, as such, do not exist, we do have terms, such as 'bachelor', which indicate our decision to consider collections of properties and concepts together. This decision is manifested linguistically by the acceptance of a linguistic rule to the effect that the collection of words 'unmarried, adult, human male' can be replaced by the single word 'bachelor'. It is by virtue of this linguistic rule that analytic sentences reduce linguistically to logical truths. This rule cannot be invalid by virtue of the fact that the word 'bachelor' in its extralinguistic modes of meaning does not refer to a distinct concept or property. Analytic truths, therefore, like logical truths, are essentially linguistic entities. The linguistic rule effects the transition from a collection of words to a single word, a transition which does not exist extralinguistically.23 2. Synthetic A Priori Truths

The second class of analytic sentences is the class of sentences which many would agree are not reducible to logical truisms, and yet which are true ex vi terminorum. These sentences, while they are a priori, are not analytic in the narrow sense, and are therefore sometimes characterized as synthetic a priori truths.24 An example is: "Nothing is red and green all over." We will assume that no acceptable definition of 'red' or 'green' has been found which when substituted in the sentence would turn it into a logical truism, although this is controversial.25 Nevertheless, the sentence is clearly both a priori and necessary. Achieving a conventionalist theory about this type of sentence involves one in a nominalism which goes beyond logical nominalism. While we would not deny that words like 'red' have extralinguistic meaning, we would not identify this meaning with a property or abstract entity. We would identify the extralinguistic meaning of a term with its extension, and then argue that while it is essential that a descriptive word have both extension and intension, the latter is given by its linguistic meaning, where the linguistic meaning of a term is ultimately determined by linguistic rules. If we abstract from the intension of the terms 'red' and 'green', our statement that nothing is red and green all over is merely a physically true statement. It is 23

We recall that we have considered analytic truths involving complex concepts of only the conjunctive type, as these are the only types of analytic truths which Lewis considers. 24 The inspiration for, if not the content of, this section comes from Wilfrid Sellars, "Is There a Synthetic A Priori?" reprinted in his Science, Perception, and Reality, pp. 298-320. 25 See H. Putnam, "Reds, Greens, and Logical Analysis", Phil. Rev. LXV (1956); A. Pap, "Once More: Colors and the Synthetic A Priori", Phil. Rev. LXVI (1957); and H. Putnam, "Reds and Greens Again: A rejoinder to Arthur Pap", Phil. Rev. LXVI (1957).

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equivalent to the statement that we never call something red that we have called green or vice versa. If we abstract from the extension of these terms, the statement appears as a necessary statement; it is deducible from the empty set of premises by virtue of the linguistic rules governing the words 'red' and 'green'. The statement is a priori because it is necessary. One who knows the language knows that the statement is true merely by reflecting on the (linguistic) meanings of the terms. All necessary statements are necessary in the same way; they are necessary by virtue of the linguistic rules. Synthetic a priori statements, therefore, have factual content even for one who knows the language, in contrast to analytic sentences. It is necessary to resort to a more thoroughgoing nominalism here in order to avoid extralinguistic entailments. If the word 'red' really referred to a property of an object, then the fact that an object could not have both the properties red and green at the same time could not be simply a matter of linguistic rules. If we had a linguistic rule which permitted us to say that an object was both red and green all over, it would be clearly invalid, and the full impact of the validity critique would be realized. On the other hand, if the extralinguistic meaning of a term consists solely of its extension, what is included within the extension of a word is a matter of convention (custom). The relevance of this fact comes to the fore when we consider that synthetic a priori sentences are open to theoretical falsification. Linguistic rules might permit us to assert that nothing is red and green all over and yet the extension of the words 'red' and 'green' may overlap. In a sense, we can say that the necessary statement has been falsified, which is possible because the extralinguistic and intralinguistic meanings of these terms are independent of each other. The linguistic rules may not be valid in relation to the extension which has been assigned to the terms. On the other hand, because of the conventional nature of the extension of these terms, we can always retain the rules and modify the extension. Neither the extension nor the intension of these terms can be said to be more essential. It is by virtue of their extension that these terms have extralinguistic reference. On the other hand, it is because of the linguistic rules that these terms have intension or sense. When a synthetic a priori truth is falsified, we must modify either the extension or the intension of the relevant term or terms. But whichever we modify, we change the meaning of the words. One often hears the charge of triviality brought against conventionalist theses. "Of course you can change a logical truth, but only at the expense of changing the meanings of the terms." This is an obviously true, but not very important, observation. It would be significant only if terms had meaning in the way that is assumed by these critics. Of course, if we change the relationship between logical terms, their meaning has been changed for the only meaning that they have is given by their interrelationships. This is similarly the case with complex terms as occur in analytic sentences. The irrelevance of this triviality argument becomes even more striking if we take a nominalistic stand in regard to simple descriptive predicates. If we change

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linguistic rules we are changing meanings and if we change extensions we are changing meanings, but in some cases we MUST change one or the other. The difference between synthetic a priori truths and the other two cases is that in these latter cases we are never in a position where we MUST change something. In the former case a necessary sentence can be false in a nontrivial way and yet to remove the inconsistency we must change meanings. Surely if we remove a nontrivial falsehood by changing meanings, this change cannot be trivial. It is in this way that scientific laws can be necessary and yet open to falsification. They are necessary by virtue of linguistic rules. Moreover, the linguistic rules give part of the meaning of the relevant terms, their intension. On the other hand, scientific laws can turn out to be false. When this occurs, we sometimes change the extralinguistic meaning of the terms, as in the case of the law of conservation of energy, and sometimes the linguistic rules - we reject the law. There are some linguistic rules and some extensions which we are not willing to give up, but for the most part it is a relative matter. In either case we have made an important change in order to correct a falsehood. We have already seen how closely knit are the notions of meaning and truth in regard to logical truths and analytic sentences. We see now that these two are likewise bound together in regard to physical laws. The type of conventionalism we have discussed here is applicable even to directly observable descriptive predicates. It arises not because these terms lack extralinguistic meaning, as does logical conventionalism, nor because these terms lack a distinct extralinguistic meaning, as does the conventionalism in regard to analytic sentences, but because of the nature of extralinguistic meaning itself - it is solely extensional - and the concommitant importance of the linguistic dimension of meaning. 3. Conventionalism in Geometry We have already indicated in section E of Chapter III that conventionalism in geometry obtains because the axioms of congruence while giving the intralinguistic meaning of 'congruent' are not sufficient to pick out a unique class of spatial intervals which are equal in length, for space is a continuum. According to conventionalism in geometry, the extralinguistic meaning must be given to the term 'congruent' by a conventional choice. In line with our agreement with Lewis' assertion that without extralinguistic meaning a descriptive term is merely a symbol, we would argue that until the term 'congruent' has been extralinguistically interpreted, it has not been interpreted at all. If we change the extralinguistic interpretation of a term like 'congruent', we are changing its meaning, and if we must change the meaning of a term, we have a kind of semantic conventionalism. Noting that conventionalism in geometry is semantic does not, however, do justice to the thesis. After all, even conventionalism in logic is a kind of semantic conventionalism. In changing logics we change the linguistic meaning of the logical

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constants; in changing our congruence standard in order to change geometries we are changing the extralinguistic meaning of 'congruent'. A logic is never falsified, but a geometry can be, and in the face of a falsification of a geometry we can change either the geometry or the extralinguistic interpretation of the term 'congruent'. Moreover, we can change the latter without changing the linguistic meaning of 'congruent'; 'congruent' remains a spatial equality predicate. Is this not the same situation that obtains in regard to the so-called synthetic a priori sentences? In the face of the admittedly unlikely falsification of "Nothing is red and green all over" can we not, if we wish, retain this sentence as true and change the extralinguistic meaning of 'red' and/or 'green'? And if we retain the sentence "Nothing is red and green all over", are we not preserving the linguistic meaning of 'red' and 'green'? This similarity between the two conventionalisms can give us a clue to the implications of the thesis in geometry. We recall that the conventionalism of synthetic a priori statements depended essentially on a nominalistic approach even to simple (primitive) descriptive predicates. This was why changing the extralinguistic meaning of 'red' and 'green' was not a trivial matter. Something similar to this is the underlying reason why changing the extralinguistic meaning of 'congruent' is not a trivial matter. When we assert that space is a continuum, we are saying that spatial intervals do not have the relation of congruence. The word 'congruence' does not have a signification. While the word 'congruent' must have extension in order to have extralinguistic meaning, if it has only extension, changing its extension is not a trivial matter. If there were a relation of equality, changing its extension would not even be permissible. A change of congruence standard, involving as it would a change in all modes of meaning, would be completely a semantical change of the trivial sort. The essence of conventionalism both in geometry and of synthetic a priori truth lies in the fact that one mode of meaning can change without another mode changing. This in turn depends on the modes of meaning being independent, which depends on a nominalistic approach to descriptive terms. While logical terms and complex descriptive predicates have only linguistic meaning, other descriptive terms, while they have extralinguistic meaning as well, have it only in the mode of extensional meaning. I believe that Griinbaum would agree at least in part with the above paragraphs. When Putnam attempted to trivialize Griinbaum's thesis by showing that the same reasoning was applicable to color predicates as to the phrase 'spatially congruent', Griinbaum replied as follows: "The term 'spatially congruent' has the same (nonclassical) intension and only a different extension in the context of alternative space metrics; but if the words 'same color' were used neologistically in Putnam's alternative description to mean same gruller rather than same color, then the original and new intensions would be avowedly incompatible."26 What is significant about the phrase 'spatially congruent' is that its extension can be changed without changing its intension, while this is not true for 'same color'. But why does Griinbaum think that the intension of 'same color' would be dif-

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ferent if its extensions were different? Part of the reasoning is, I believe, found in the following sentence: "I take it to be a matter of objective visual fact whether or not a given color quale inheres in a given color patch at a certain time, and hence a matter of objective visual fact whether it is the same as (or color congruent to) the quale present in another color patch." 27 But that this cannot be the whole reason is evident from the fact that from the VISUAL point of view it is an objective fact whether or not two spatial intervals are of the same length. Moreover, geometrical conventionalism rests on the thesis that space is a continuum, which is clearly not a visual matter at all. Therefore, Griinbaum must believe that it is not merely an objective VISUAL fact whether or not a given color quale inheres in a given color patch, but rather an objective THEORETICAL fact about the color patches. Therefore, when Griinbaum talks about the 'original and new intensions' of 'same color' he must have in mind theoretical facts about colors, which theoretical facts are not forthcoming with regard to 'same length'. 'Same length' has no intension except that given intralinguistically, whereas wave theory, for example, gives us the 'intension' of 'same color'. Since Griinbaum's intensions are apparently physical facts about objects, they are quite unlike what would ordinarily be construed to be an 'intension'. This, however, does not matter if all that one wishes to show is that 'spatial congruence' is conventional in a way in which 'color congruence' is not. I believe that Griinbaum has established that fact. On the other hand, if one wishes to show that geometrical conventionalism is not semantic, one is not at liberty to use 'intension' as one wishes. One must show that in some classical sense of intension color words have intension while spatial words do not. If the considerations of the previous section are correct, even color words have intension only in the sense of linguistic meaning. Moreover, even if that were accomplished it would not show that geometrical conventionalism was not semantic; it would show that color conventionalism was trivially semantic, but that geometrical conventionalism was nontrivially semantic - 'semantic' because in changing the extension of 'congruent' one changes its meaning; 'nontrivially' because in changing the extension one is not changing an extralinguistic intension of which there is none. 4. Conventionalism in Physics We have already indicated in section E of Part III that conventionalism in physics, as in geometry and logic, depends on the special semantical characteristics of a certain group of terms. These are the so-called operationally defined terms and theoretical terms of the sciences. Griinbaum, however, has made a point of distinguishing conventionalism in geometry not only from trivial semantical conventionalism but also from Duhemian physical conventionalism. Griinbaum realizes that 26

Adolph Griinbaum, "Reply to Hilary Putnam's 'An Examination of Griinbaum's Philosophy of Geometry' ", Boston Studies in the Philosophy of Science V (1968), 82. " Ibid., p. 81.

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conventionalism in geometry is a type of semantical conventionalism when he makes the following statement: "An alternative metrization in the sense of Poincaré affects only the language in which the facts of optics and the coincidence behavior of a transported rod are described: the two geometric descriptions respectively associated with two alternative metrizations are alternative representations of the same factual content."2* Griinbaum is saying that two different geometries with two different extralinguistic interpretations of the term 'congruent' (or alternatively of 'straight Une') may have the same factual content, and so the two formulations differ only linguistically. A geometry alone does not have factual content, but only a geometry with an extralinguistic interpretation. One can correctly describe the same set of spatial facts with one geometry or another as long as one uses the appropriate interpretation in each case. It is useful also to draw an analogy between geometry and physics, considering geometry itself as a law formulation and the definition of congruence as its interpretation. These two together make up a physical geometry, a scientific system; neither alone has factual content. Griinbaum contrasts this situation in geometry with what he understands to be Duhem's claim regarding physics, which is that we can have two pairs of physical laws, all four of which have factual content, such that each pair has the same factual content, but such that the analogous laws of each pair have a different factual content. For example, if we consider our pairs to be composed of a physical geometry (G) and a theory of optics (O), he takes Duhem to be asserting that while G x of the first pair has a different factual content than G 2 of the second pair, and similarly for Ox and 0 2 , that nevertheless the pair G j - O i has the same factual content as the pair G 2 - 0 2 . Therefore, the G 2 - 0 2 is not merely a linguistic reformulation of the other pair, for the factual commitments of G 2 differ from those of G x and likewise for 0 2 and CV This differs from the situation in geometry where the two elements of the analogous pair do not have factual content independently of each other. I do not think that Duhem would assent to this characterization of his thesis. He would agree that both statements of the pair are physical laws, but I believe he would deny that they have factual content independently of each other. For the statements to have factual content all of the descriptive terms in each statement must have meaning, and, in particular, they must have extralinguistic meaning. Concerning crucial terms, such as 'mass', 'temperature', and 'pressure', which occur in physical laws Duhem says: "These ideas are not only abstract; they are, in addition, symbolic, and the symbols assume meaning only by grace of the physical theories." 29 The terms of physics are not only abstract according to Duhem, they are also symbolic, where by an abstract idea Duhem means what we mean by a general concept or general term. The abstract ideas of physics are not extracted from concrete reality, they are constructed. "They are abstractions produced by slow, complicated, and conscious work, i.e., the secular labor which has elaborated 28

Adolph Griinbaum, Philosophical

Problems

of Space and Time, p. 125.

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physical theories. If we have not done this work or if we do not know physical theories, we cannot understand the law or apply it." 30 My interpretation of these comments of Duhem is that these terms of physics have only linguistic meaning, and this meaning is given to the terms by the laws (theories) of physics themselves. There is an analogy between nonatomic sentences and these symbolic ideas of physics. While they both have only linguistic meaning, they share equally in extralinguistic meaning by virtue of being related to other terms (sentences) which have extralinguistic meaning. The symbolic terms of physics differ from the logical constants in the same way that physical sentences differ from logical truisms. Again, to avoid confusion it is best to consider these terms as having only linguistic meaning, while remembering that they are related to terms which have extralinguistic meaning. If I have interpreted Duhem's conventionalism correctly, it differs from the thesis in geometry because both of the statements of a pair are physical laws. It is similar in that both pairs of statements must be combined if they are to have factual content. Moreover, the laws together give the meaning of the crucial terms, such as 'temperature'. This is similar to the situation in geometry where one of the statements is necessary to give an interpretation to the term 'congruent'. The difference is that in physics both of the laws give part of the meaning of the symbolic term. As in the case of geometry two pairs of statements which have factual content are merely linguistic reformulations of each other, for the individual statements do not have factual content independently. Moreover, the crucial symbolic term(s) have a different meaning in each pair as with the term 'congruent' in geometry. Conventionalism in physics is ultimately, therefore, a kind of semantic conventionalism as are all the other conventionalist theses which we have discussed. Conventionalism in physics is, therefore, no more trivial than conventionalism in logic, depending as they both do on the fact that certain terms cannot have an extralinguistic interpretation. The roles of the logical constants and the symbolic terms of physics differ only in the fact that the logical constants are not even linguistically related to terms which have extralinguistic meaning while the symbolic terms of physics are. Although this makes it appear as if the individual statements of physics have factual content independently, the laws of physics depend on each other for meaning, and a fortiori for factual content. Again we find that meaning and truth, definition and synthetic statement, are inextricably combined. We can separate the two by rational reconstruction of language as we did in treating the conventional nature of analytic sentences; however, as we approach closer to the actual milieu of the scientist, we find that any such distinction appears more and more arbitrary. Almost every statement of science functions partly as a statement of meaning and partly as a statement of fact. As indicated in section E of Part III, we can give operational 29

Pierre Duhem, Aim and Structure of Physical Theory, reprinted in Arthur Danto and Sidney Morgenbesser (eds.), Philosophy of Science (New York, Meridian Books, 1960), p. 183. »« Ibid., p. 184.

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definitions for terms of physics, such as 'temperature', only by doing an injustice to the way that science actually proceeds. The meanings of the terms in science are not fixed once and for all; they are not even partially interpreted once and for all by reduction sentences. As science proceeds, sometimes it modifies the laws of the instruments and sometimes it modifies the laws determined with the instruments, its aim being an overall simplicity and adequacy. 5. Conventionalism in Physics and Logic Combined We now turn our attention to Quine's thesis which extends Duhemian conventionalism to include logic as well. We have seen that in the face of falsifying evidence we must change some law of physics yet having a choice within limits as to which law to change. The laws are not individually testable, for they do not have factual content individually. In order to test even a collection of laws of physics we must use the rules of logic to deduce atomic sentences which can be directly checked against the world. Since this is the case, Quine argues that if the deduced atomic sentences are false we can, if we wish, change the rules of logic rather than any of the laws of physics. He further argues that not even collections of laws have factual content but only collections of laws together with logical rules of inference. We would agree with Quine, but only because of the kind of justification which we have given for conventionalism in logic. We have argued that any nonatomic sentences have factual content only indirectly by virtue of the possibility of deducing atomic consequences by means of the logical rules of inference. Since the scientific laws, both individually and collectively, are by their very nature nonatomic sentences, it follows that they have factual content only indirectly. Since the logical rules are merely linguistic rules which give the only meaning that the logical constants have we are perfectly free to vary the rules if we wish, thus varying the indirect factual content of the laws also. Therefore, in the face of discontinuing evidence we can change the linguistic rules or the laws themselves. In either case the deducible atomic consequences will be different and we have removed the inconsistency. It must be noted, however, that if we change the L-rules we will find ourselves in the position of having to change all the rest of the laws of physics, for the L-rules apply to all sentences of a certain form. If we change only the laws, the consequences will be much less drastic, although even here the extent of the disruption that occurs in science depends on the particular law(s) that one chooses to change.

APPENDIX AN IMPOSSIBILITY P R O O F O F THE CONVENTIONALIST THESIS

John L. Pollock in an excellent and ingenious article 1 has attempted to prove that the conventionalist thesis is impossible. We will attempt to show that his argument has failed. We will proceed in three steps: (1) we will briefly sketch the core of his argument without going into details where it is unnecessary for our purpose; (2) we will sketch a variant of his argument which does not have the undesirable consequence; and (3) we will show that only that second argument is applicable to the conventionalist. According to Pollock's argument the Conventionalist Thesis (CT*) is committed to the following three premises: (1) h - P ; (2) There exists a number-theoretic predicate J 1 of T such that for every sentence cp of T, if |— cp then | - ® ( # 9 ) ; and (3) For every sentence cp of T, (J 1 (# cp), where 'P' stands for Peano's axioms, which are assumed to be consistent; ' j— * is the syntactical predicate 'is a theorem of T', where T is formalized classical mathematics; and #cp is the Goedel number of 'cp'. Pollock then constructs a sentence 0 in T such that 0 se ^ ( # ~ 0 ) is a theorem of T, i.e., (4) H 0 = ^ ( # ~ 0 ) . From (3) we get (5) =» ~9). From (4) and (5) we get (6) 1 - 9 => From (6) we get (7) | 0. From (2) and (7) we get (8) b ^ ( # ~ 0 ) . From (4) and (8) we get (9) i - e . From (7) and (9) we conclude (10) T is inconsistent. 1

"Mathematical Proof", American Philosophical

Quarterly IV (1967), 238-44.

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If T is weakened to prevent its being proved inconsistent, then it is too weak to do what the conventionalist expects of it. Let us now look at a variant of Pollock's argument. We will retain premises (1), (2), and (4), but will modify (3) as follows: ( 3 + ) For every