a) The Doctrine of “Empty Tautologies”

In our century, the Kantian challenge was accepted by the logical positivists and some outstanding philosophers who influenced them. Among the latter, Russell and Wittgenstein come to mind. Allow me to speak of “analytic philosophers” in order to group together those who professed to being positivists and also those who merely influenced the positivists or were influenced by them. In the hands of analytic philosophers, the Kantian challenge took a “linguistic turn,” a turn every bit as fertile as Kant’s “Copernican turn.” Language became the medium through which philosophical problems are to be viewed.

Kant had argued this way: (1) arithmetic truths are necessary; (2) but they are not analytic; (3) therefore, there must exist a necessity not derived from analyticity. Analytic philosophers, by contrast, applied modus tollens: Since there is no necessity other than that of analyticity, either (1), or (2), or both, must be false. And since they could not convince themselves of the falsity of (1), they concluded that (2) must be false: Arithmetic truths must be analytic. This confronted them with a problem. They now had to find a notion of analyticity, according to which arithmetic could plausibly be said to be analytic. And this notion, ideally, should also cover logic. What they came up with was a “linguistic” or “semantic” notion of analyticity. This conception was never clear. Nor should we really talk about one notion, since there were several of them, which were often identified or even confused with each other. It is a hopeless task to try to untangle all of the themes that enter into the concept of “semantic analyticity.” I shall have to be content with mentioning some of the main ingredients.

There is no necessity other than that of analyticity! This is the rallying cry of analytic philosophy. We could call it the “Humean dogma.” Empirical matters of fact, according to the Humean tradition, are never necessary. This implies that arithmetic, since it is necessary, cannot be factual. But if it is not factual, then what does it say? The search for a satisfactory notion of analyticity is also a search for a satisfactory answer to this question. The dialectic revolves around two distinctions: Firstly, around the analytic-synthetic dichotomy and, secondly, around the factual nonfactual distinction. The synthetic is of course identified with the factual; there is no problem here. The analytic must then be the nonfactual. But what sense can the nonfactual have? Does it not merely comprise nonsense? Metaphysics, many analytic philosophers held, is meaningless because it makes no factual statements. Now, if arithmetic and logic are analytic and, hence, nonfactual, how can they possibly avoid the fate of metaphysics?

We can therefore distinguish three interrelated problems that vexed analytic philosophers: (1) What kind of necessity is the necessity of arithmetic (and logic)? (2) What notion of analyticity characterizes arithmetic (and logic)? and (3) What meaning can nonfactual statements have?

The early Wittgenstein proposed a radical answer to these questions; an answer which influenced the later discussions of the logical positivists and their admirers. Let us briefly consider this answer, pretending, as we always must in the case of Wittgenstein, that there is a view to be studied rather than a mere web of loosely connected thoughts.

In the famous dialogue between Achilles and the Tortoise, Lewis Carroll has the tortoise argue that a valid inference can be drawn only after infinitely many tasks have been completed. Consider, for example, the familiar inference modus ponens:

Carroll argues that unless we also assume:

we cannot infer Q. But even this will not do. We must further grant that (4) and (5) entail Q. And so on, ad infinitum. Now, it has often been pointed out that Carroll’s formulation of the argument involves a temporal sequence of mental acts of granting certain premisses and that it is spurious for that reason alone. But it has also been claimed that if this appeal to time and psychology is removed, there remains a problem. Here is how Coffa sees the problem: “If the justification of the inference from (4) to (3) requires an appeal to the logical law (5), why don’t we need to appeal to further logical laws in order to justify the inference from (4) and (5) to (3); and so on, ad infinitum” (J. A. Coffa, To the Vienna Station, chap. VIII).

The obvious solution to the puzzle is to point out that the inference is justified without an appeal to the logical law as an added premise. The inference involves premises (1) and (2) and nothing else. From (1) and (2) follows (3), and that is the end of the story. However, and this ‘however’ is all-important, we may ask why we accept this inference as valid and not, say, the inference from Q and If P, then Q, to P. In answer to this question, we cite the logical law:

We point out in defense of modus ponens that (6) is true or, equivalently, that it is a fact: It is a fact that all states of affairs are such that if p is a fact, and if p then q is a fact, then q is a fact. That is the way the world is! On the other hand, the corresponding sentence for our inference from Q to P is not true. There is no such fact. In short, the valid rules of inference are those which are based on logical laws, but these laws are not premises of the corresponding inferences.

All of this is fairly obvious. What it leads back to is the question after the nature of logical laws. I just claimed that logical laws are facts about the world. And this, of course, raises the question of how we discover and verify such facts. We could have saved ourselves the detour through Carroll’s infinite regress argument and confronted the problem directly: What is the nature of the laws of logic? But there seems to persist a view among philosophers that the problem of the nature of logical laws is somehow infected by the regress argument. Wittgenstein, for example, seems to have thought that Carroll’s argument showed something important about the nature of logic. What it shows, presumably, is that logic needs no justification. If I understand Wittgenstein correctly, he may have reasoned from the insight that the inference modus ponens needs no further premise to the conclusion that it needs no justification in terms of a logical law. Quite to the contrary. When we understand the premises of modus ponens, he holds, we also “see” the alleged logical law. The process is reversed: it is not the logical law which justifies the inference, but it is the inference which somehow calls our attention to the so-called logical law. Since we accept the inference, we accept the law; and not the other way around. Furthermore, the so-called logical law is not really a law, a fact, at all. Nor is its verbal expression a declarative sentence. If it were such a sentence, then it would say something; it would state a fact about the world. But this is not what logic is. Rather, logic “shows itself’; we recognize it as soon as we understand language. Logical sentences, contrary to what we have assumed, do not say anything about the world. In this respect, they are fundamentally different from other sentences:

If p follows from q, I can conclude from q to p. The mode of inference is to be understood from the two propositions alone. Only they themselves could justify the inference. “Laws of inference” which—as in Frege and Russell—are supposed to justify the inference, are senseless and would be superfluous.

(L. Wittgenstein, Tractatus Logico-Philosophicus, 5. 132)

From the correct premise that modus ponens does not require the relevant logical law as an added assumption, Wittgenstein concludes fallaciously that this law is “superfluous” and even “senseless.” Of course, it is “superfluous” in the sense that it is not needed as an added premise, but it is not superfluous if we sort out valid inferences from invalid ones. And the logical law is certainly not “senseless.” I think that we can now see more clearly how the infinite regress argument fits into Wittgenstein’s view. As he conceives of the situation, the argument allows him to get rid of the logical law; first, by denying its importance, then by denying its very existence. Logic is dispensed with in favor of inferences, and what justifies the inferences is not logic, but our understanding of language.

The stubborn fact remains that logicians, and even ordinary people, talk about the laws of logic. No matter what Wittgenstein says, it is a fact that all states of affairs are such that if p is a fact, and if p then q is a fact, then q is a fact. Wittgenstein may try to sweep the laws of logic under the rug, but he can hardly hope to succeed using such a small rug. At the very least, he has to expand his view so that he can explain what these sentences of logic are about.

“My fundamental idea,” Wittgenstein says, “is that the logical constants’ do not represent anything” (Wittgenstein, 4.0312). It follows that there are no negative, no molecular, and no quantified facts. It also follows that logical sentences must connect with the world in a peculiar fashion. According to Wittgenstein, they show things, but do not say anything. It is not clear to me, even after reading a number of informed commentaries, how this distinction between showing and saying is to be understood. But we may possibly be enlightened if we consider what Wittgenstein has to say about identity.

Wittgenstein remarks: “Parenthetically, to say of two things that they are identical is nonsense, and to say of one that it is identical with itself says nothing” (Wittgenstein, 5.5303). Nothing could be farther from the truth. To say of two things that they are identical is not nonsense, but false. And to say of one thing that it is identical with itself is to say, not nothing, but that it is self-identical. It comes as no surprise, therefore, that Wittgenstein should arrive at the amazing conclusion that “expressions like ‘a = a’, and those derived from them, are neither elementary propositions nor otherwise meaningful expressions” (Wittgenstein, 4.243).

What moves Wittgenstein to the strange assertion that a false identity statement is nonsense and that a true one says nothing? He seems to think that this follows from the alleged fact that the false identity statement is a contradiction, and that the true identity statement is a tautology. He holds that tautologies, that is, the propositions of logic, say nothing (Wittgenstein, 5.142 and 5.43). Contradictions, on the other hand, have presumably no sense because they do not allow for any possible situation (Wittgenstein, 4.462).

From our point of view, a = a is not a logical truth. It is an instance of the ontological law of self-identity. (This law and its instances must be carefully distinguished from identity statements of the form: a is identical with the F.) Since it is an instance of an ontological rather than a logical law, it is true, not as a matter of logic, but as a matter of ontology. But this is a minor point. Our important disagreement with Wittgenstein concerns his claim that tautologies say nothing. The statement ‘a = a’ may be said to be trivial, and obvious, and even useless, but it cannot be denied that it says something, namely, something trivial, obvious, and useless. And there is no difficulty at all in stating what it is that the sentence says: It says that a is identical with itself. What holds for this statement of self-identity holds also for many instances of logical laws: They may be obvious and trivial, but they do say something. The tautology ‘P or not-P’ is obvious, trivial, and useless, but it says something, namely, that either P is a fact or else not-P is a fact. Similarly for contradictions. The contradiction ‘P and not-P’ may be saying something that is obviously and trivially false, but it does say something, namely, that both P is a fact and not-P is a fact. Tautological and contradictory statements, contrary to Wittgenstein’s claim, make sense, are meaningful, and make true and false assertions, respectively.

We saw that Wittgenstein claims that some things cannot be said but show themselves. A case in point, and a relatively clear case, is identity. “Identity of object,” he says, “I express by identity of sign, and not by using a sign of identity. Difference of objects I express by difference of signs” (Wittgenstein, 5.53). In his “ideal” or “perspicuous” language, only one name occurs for every named thing, and every name is assigned to just one entity. One could therefore tell by the shape of an expression whether or not it represents the same thing as another expression, even if one does not know what thing the expression represents. Identity of objects thus “shows itself’; it is not stated in the “ideal” language. But what philosophical difference does this make? What philosophical insight follows? Whether the fact that a thing is different from another thing is stated in the language or merely indicated in some fashion or other, it remains a fact to be reckoned with. No linguistic operation can make this fact vanish from the world. There is Wittgenstein’s Tractatus on my desk to the left of my typewriter. That is a fact. And it remains a fact if I do not state it as I just did, but merely “show it” by drawing a picture of the Tractatus to the left of a picture of my typewriter. Most importantly, the picture does not prove that there is no such thing as the relation of being to the left of, just as using different names for different things does not prove that there is no such thing as the relation of being different from each other (of being non-identical).

(Wittgenstein and many other philosophers fail to appreciate that there are facts involving identity which are not instances of the law of self-identity. For example, there are such “informative” facts as that Mozart is the composer of The Magic Flute. Not all identity statements, therefore, are trivial, obvious, and useless. Thus, even if we could argue from triviality to non-existence, it would not follow that there is no identity relation.)

Let us now return to logic proper. How does a tautology “show” what it does not say? And what, precisely, is it that is shown? I think that there is no answer to these questions in the Tractatus. But there is a theme that floats like a mist over the philosophical landscape inhabited by Wittgenstein and his admirers: Grammar somehow reflects the structure of the world. Language somehow shows by its structure what cannot be described by means of it, namely, the structure of the world. However, the negative side of Wittgenstein’s view is relatively clear: Tautologies are not statements about the world; they say nothing about the world; they are not factual.

There is a third strand to Wittgenstein’s early view. It centers around the notion of a truth table. Truth tables are thought to define the logical connectives out of existence, so that they vanish from the ontological inventory of the world (see Wittgenstein, 5.4 to 5.441). There seem to be two parts to Wittgenstein’s argument against logical things; one explicit, the other merely hinted at.

In the first part, Wittgenstein argues that the “interdefinability” of the logical signs shows that these signs do not represent logical things. For example, the fact that ‘if P, then Q’ can be “defined” in terms of ‘not’ and ‘or’ is supposed to prove that the expressions ‘if-then’, ‘not’, and ‘or’ do not represent anything. If they represented logical things, so the argument goes, then what ‘If P, then Q’ represented would have to differ from what ‘Not-P or Q’ represented, since the former would contain the relation if-then, while the latter would not. But the “interdefinability” shows that what these two expressions represent is the same. Wittgenstein’s point can be made most persuasively for negation, (1) If ‘not’ represented a logical entity, then ‘P’ and ‘not-not-P’ would have to represent different states of affairs, since only the latter represents a state of affairs which contains the thing not. (2) But these two sentences represent the same state of affairs. (3) Therefore, ‘not’ does not represent anything.

Wittgenstein’s argument rests on a mistaken assessment of what “interdefinability” can yield. He assumes mistakenly that the interdefinability of, say, ‘if-then’ with ‘or’ and ‘not’ means that ‘If P, then Q’ and ‘Not-P or Q’ are merely two different expressions for the same thing (for the same state of affairs). But we are not dealing here with mere abbreviations, with mere linguistic conveniences. Rather, we are dealing with equivalences. The so-called definition is not of the form:

but is of the form:

And (B) follows, as an instance, from the logical law:

Thus we have, firstly, a logical law (a logical fact), and, secondly and automatically, an instance of this law. But neither from (B) nor from (C) does it follow that ‘If P, then Q’ and ‘Not-P or Q’ represent the same state of affairs: and only if one could show that they do, could Wittgenstein’s argument be sound.

Since I hold that the conditional relation is not the same thing as the combination of negation and disjunction, I conclude that the two states of affairs are not the same. In the case of negation, it seems to me to be obvious—no doubt, as obvious as it seemed to Wittgenstein that the opposite was true—that P is not the same state of affairs as not-not-P. (That these two states of affairs are not the same does not imply, of course, that there must be infinitely many more states of affairs of the form not-not-not-not-P, etc.) Why am I so sure that there are “logical relations”? Well, if there were no such things, then one should be able to do logic without them, just as one is able to do zoology without unicorns. If the expressions for negation and the connectives were mere “syncategorematic signs,” then one should be able to eliminate them. Otherwise, what is the point of calling them “syncategorematic”? Has anyone ever really seriously tried to get along without ‘not’, ‘neither-nor’, ‘and’, ‘if-then’, etc.? And yet, so many philosophers seem to believe that these expressions stand for nothing, are mere artifacts of language. Of course, we shall not permit any other way, straightforward or round-about, of depicting those relations or of “showing” what their expressions commonly represent. Nor shall we be impressed by the usual practice of “reducing” some of them to others. One single connective is as ontologically significant as a thousand of them, for it proves the existence of the sub-category of connective.

What these considerations show is that even if Wittgenstein had not mistaken true equivalences for abbreviations, his argument would still be unsound. Assume, for the sake of my point, that all connectives “reduce” to neither-nor. Even then we would be left with “irreducible” neither-nor sentences and the corresponding complex states of affairs. This is where the second part of Wittgenstein’s argument makes its appearance. In order to eliminate the neither-nor relation and the corresponding complex states of affairs, one invokes truth-tables. The idea is to “define” ‘neither-nor’ in the familiar fashion by the truth table:

T, T: F

T, F: F

F, T: F

F, F: T (cf. Wittgenstein, 4.442).

This table does not contain the expression ‘neither-nor’. Nevertheless, what it is supposed to convey is a certain proposition, namely, the proposition: The statement Neither P nor Q is true if and only if P is false and Q is false. (Equivalently: Neither P nor Q is a fact if and only if P is not a fact and Q is not a fact.) And this proposition clearly concerns the relation neither-nor. The truth table, we must keep in mind, is merely a convenient way of expressing a certain proposition, nothing less, and nothing more. It is not a definition, neither of the connective, nor of anything else. In general, truth tables are not definitions of the connectives, but are convenient ways of stating general truths about the existence of molecular facts as a function of the existence of their constituent states of affairs.

Wittgenstein’s early conception of logic is rather obscure. When it is not obscure, so I have argued, it is mistaken. The laws of logic can be stated as any other laws can. The instances of logical generalities, tautologies, and contradictions are not empty and senseless, respectively. A tautology, far from saying nothing, says something that is necessarily (logical necessity!) true. A contradiction, far from being nonsensical, says something that is necessarily false. Finally, there is no truth to Wittgenstein’s claim that there are no “logical things.” The quantifiers and connectives are just as much part of the furniture of the world as elephants and spatial relations.

Since Wittgenstein’s early views about the nature of logic are obscure if not clearly mistaken, one cannot but wonder why they had such a tremendous influence on the logical positivists. One explanation is that they promised to lift a heavy philosophic burden from their shoulders. Wittgenstein’s views promised to exorcise the synthetic a priori from philosophy. As Schlick and others saw it, empiricism—their blend of empiricism—consisted in the rejection of the synthetic a priori. Carnap, in his intellectual biography puts it this way:

Wittgenstein formulated this view in the more radical form that all logical truths are tautological, that is, that they hold necessarily in every possible case, therefore do not exclude any case, and do not say anything about the facts of the world.

. . . he did not count the theorems of arithmetic, algebra, etc., among the tautologies. But to the members of the Circle there did not seem to be a fundamental difference between elementary logic and higher logic, including mathematics. Thus we arrived at the conception that all valid statements of mathematics are analytic in the specific sense that they hold in all possible cases and therefore do not have any factual content.

What was important in this conception from our point of view was the fact that it became possible for the first time to combine the basic tenet of empiricism with a satisfactory explanation of the nature of logic and mathematics.

(The Philosophy of Rudolf Carnap, pp. 46-47)

In Carnap’s words: The true (valid) statements of mathematics are analytic because “they hold in all possible cases and therefore do not have any factual content.” Might one not with equal force argue that the laws of zoology are analytic and without factual content, since they, too, hold “in all possible cases”? The only difference between (a) the laws of “sentential logic,” (b) the laws of arithmetic, and (c) the laws of zoology is that the laws of (a) hold for all states of affairs, those of (b) hold for all numbers, and the laws of (c) hold for all animals.

b) Wittgenstein and Rules of Grammar

According to Kant, the a priori is the necessary. Arithmetic (and logic), everyone agreed, is necessary. But what kind of necessity attaches to arithmetic (and logic)? This is the most fundamental question of the Kantian challenge. Kant claimed that it is not the kind of necessity that arises from analyticity. The logical positivists and their friends rebelled: There is only one kind of necessity, they chanted, and analyticity is its source. But we must constantly keep in mind that, unless they were logicists in the Fregean mold, their notion of analyticity was suspect. We just saw that Carnap at one point thought of analyticity in terms of tautologies, taking a leaf from the Tractatus. The verbal bridge between the Kantian notion and this modified version of analyticity consists of the phrase lack of factual content’. One may mistakenly think that the new notion is the same as the old one, because according to either notion, analytic statements are supposed to lack factual content. From our point of view, the phrase ‘lack of factual content’ is much too vague to advance the philosophical dialectic.

However, analytic philosophers soon discovered a new kind of necessity and, therewith, a true alternative to Kant’s doctrine of the synthetic a priori nature of arithmetic. This necessity is based on meaning, and this meaning emanates from rules (of grammar). In one of his lectures at Cambridge, Wittgenstein reputedly said that “To a necessity in the world there corresponds an arbitrary rule in language” (see Coffa, chap. XIV). As with most of Wittgenstein’s aphorisms, this one allows for several interpretations. It seems not too far fetched, in the light of other things Wittgenstein said, to take it to mean that there is no independent necessity in the world, but only a kind of necessity which is a projection of grammar. The basic problem with this new notion is obvious. Either the rules of grammar are arbitrary, or else they are not. If they are, then the necessity of arithmetic and logic, such as it is, is merely a matter of convention. If they are not, then the question arises of what objective features of the world are reflected by these rules of grammar. And in this case, there are two plausible answers: The rules are either determined by the structure of the mind or else by the structure of the (nonmental) world. The first alternative is the Kantian view; the second yields our form of realism.

The basic idea of the semantic theory of the a priori is simple: A priori propositions are not really propositions at all, but are rules that determine the meanings of words. But though the idea may be simple, its development runs into a string of difficulties. Let us once again take a slight detour and return to Wittgenstein’s early view on negation. He held that there is no such thing as negation; for if there were, then Not-not-P would have to be different from P. Around 1930, he argues that the law of double negation, to the effect that all p are such that not-not-p if and only if p, is not really a law, but a rule that constitutes the meaning of negation:

There cannot be a question of whether these or other rules are the correct ones for the use of ‘not’ (that is, whether they accord with its meaning). For without these rules the word has as yet no meaning; and if we change the rules, it now has another meaning (or none), and in that case we may just as well change the word too.

(See Coffa, chap. XIV)

And Ambrose’s notes for 1932 contain the following remark: “The objection that the rules [of grammar] are not arbitrary comes from the feeling that they are responsible to the meaning. But how is the meaning of ‘negation’ defined if not by rules? not-not-p = p does not follow from the meaning of ‘not’ but constitutes it” (Coffa, chap. XIV). This is a relative of the view that certain sentences do not really represent states of affairs (propositions), but somehow “define” terms that occur in them. The difference is that Wittgenstein does not think of them as “implicit definitions,” but as rules. But the strategy is the same: Certain perfectly normal sentences are separated from the rest and declared to be fundamentally different from other sentences in that they provide the meanings for their terms. These sentences are precisely those which, according to tradition, formulate a priori truths. The problem with this strategy is the obvious one that we need a principle of separation: How are we to distinguish between “ordinary” sentences and sentences which, though they look perfectly ordinary, are really definitions or rules in disguise? Why, for example, is the law of double negation not a true statement about (all) propositions, just as the law that whales are mammals is a true statement about all whales? Or is this latter law also a rule which constitutes the meaning of the word ‘whale’? If so, what about the statement that John has blue eyes; does it constitute the meaning of ‘John’?

The usual reply is that one can disagree about the color of John’s eyes without disagreeing about the meaning of ‘John’, but one cannot dispute the law of double negation without disagreeing about the meaning of ‘not’. But if one cannot dispute the law without a different meaning, then the law must somehow constitute the meaning of ‘not’ rather than rest on it. Acceptance of the law signals that one will follow the traditional rules for the use of ‘not’. Let us cast this reply in the form of an argument:

I think that it is obvious that (3) does not follow from the true premises (1) and (2). Nor can I think of any true assumptions from which (3) follows. But, then, I am of course convinced that (3) is false. Consider the law that all whales are mammals. Surely, one can find some persons who believe that whales are fish. Assume that Mary actually believes this. She disagrees with John about whether whales are mammals or fish. Do we have a factual or a semantic disagreement? Well, do they mean the same by ‘whale’ (or/and ‘mammal’ and ‘fish’), or don’t they? We ask Mary, showing her a picture of a whale, or taking her to the ocean. All her answers indicate that she means by a whale the same animal we have in mind. May we not conclude that she means by ‘whale’ the same as John does, but that she factually disagrees with him about whether or not these creatures are mammals? But what happens if an eager Wittgensteinian insists that there is really a semantic disagreement and, hence, that the law of zoology is not at stake? He brushes aside all of our tests with Mary and keeps insisting that she could not possibly mean the same by ‘whale’ as we do, because she believes that whales are fish. What she means by ‘whale’ is something that, among other things, is a fish and, hence, cannot be the same as what we mean by it. The apostle of Wittgenstein thus uses the existence of the disagreement between Mary and John as proof that it must be semantic in nature. But in doing so, I submit, he reduces his view to absurdity: Upon his criterion, any disagreement about the truth of a proposition becomes a semantic disagreement and there are no factual disagreements at all.

But let us assume that the Wittgensteinian concedes that the law about whales is a statement of fact rather than a rule for the use of ‘whale’. Why does he believe that the law of double negation is different in character? Of course, the former is about whales, while the latter is about states of affairs; the subject matter is different. And of course, the law of double negation is much more general than the law about whales. And of course, we are much more certain, perhaps, that the law of logic holds. And of course, we cannot even imagine exceptions to the law of double negation, while we have no trouble imagining whales that lay eggs. But all of these differences do not add up to the difference between a factual and a semantic disagreement.

A direct defense of Wittgenstein’s position is equally implausible. Is it really impossible for two logicians to disagree about the truth of a logical proposition? For example, must two logicians really disagree about the meaning of ‘or’ when they disagree about the law corresponding to disjunctive syllogism? I do not think so. The recent history of logic, as well as of set theory, has shown beyond the shadow of a doubt that factual disagreements in these fields of inquiry do exist.

The plausibility of Wittgenstein’s view rests on a specific notion of meaning, namely, meaning as use. Furthermore, ‘use’ must be understood in a very wide sense. This explains why Wittgensteinians so ferociously attacked the so-called “reference theory of meaning.” Only by discrediting this theory could they hope to make room for their own method of philosophizing. Recall our example of the disagreement about whales. Mary and John, we shall assume, look at the same animal in the ocean, and they both agree that they are talking about that same animal “over there.” John asserts that it is a mammal, while Mary insists that it is a fish. Surely they are talking about the same whale; they mean the same thing by ‘that whale over there’. They are referring to the same whale. But meaning, the Wittgensteinian holds, is not reference. Meaning is use, and Mary uses ‘whale’ differently from John, for she adamantly asserts that this whale is a fish. This meaning of ‘meaning’ leads straightforwardly to the absurd conclusion that no two people can mean the same thing by an expression as long as they disagree about the truth of a sentence in which the expression occurs.

But the really important fallacy of Wittgenstein’s line of reasoning hides somewhere else. Assume that two logicians disagree about the truth of the proposition associated with disjunctive syllogism:

(DS) All states of affairs p and q are such that: If (p or q and not-p), then q. Logician A believes that (DS) is true, while logician B believes that it is false. May not A say to B: “Well, I see what you are driving at. But, perhaps, we do not mean the same thing by ‘or’. The ‘or’ I have in mind is such that (DS) is true”? Assume also that B agrees that they may have two different notions in mind. In that case, they should stop arguing whether (DS) is true or false; for, upon one notion it is true, and upon the other, it is false. The situation is similar to the one where one person insists that Paris is in France, while another insists that it is in Indiana. As soon as they discover that they are talking about different towns, they stop arguing. They now agree that a certain town named ‘Paris’ is in France, and that another town by the same name is in Indiana. Our two logicians could agree, similarly, that (DS) is true as long as ‘or’ means or₁, and false if it means or₂. But it does not follow from their agreement, and this is the important point I wish to make, that the respective sentences about Paris are rules rather than declarative sentences. From the fact that there was semantic disagreement, it does not follow that the sentences in question cannot represent states of affairs. Quite to the contrary. After the semantic confusion has been cleared up, both parties to the dispute will declare that the sentences are true and false, respectively. Of course, there is the other possibility that they agree that there is one and only one connective or and that, therefore, one of them must be mistaken about (DS). In this case as well, though, the disagreement is about a matter of fact.

What do the alleged rules of use have to do with grammar? Surely, someone who denies that whales are mammals, someone who says: “Whales are not mammals,” does not speak ungrammatically. Nor is it a violation of grammar to assert that green is not a color or that a pencil is both blue all over and also not blue all over at the same time. We may say that the first is a denial of an a priori truth, and that the second is a contradiction, but neither is ungrammatical. Still, according to some passages in Wittgenstein, that green is a color is due to the “grammar of color words.” It seems to me that just as his view rests on a vague notion that meaning is use, so does it depend on a diffuse notion of grammar.

But why should we not try to improve on Wittgenstein’s idea and see where it may lead us? In order to have a relatively precise notion of grammar, let us consider the so-called formation rules for the artificial languages philosophers have constructed. These rules describe which strings of signs are well-formed or, as we may wish to say in our context, which strings of signs are grammatical. They correspond to the implicit rules of English grammar. For example, while the expression ‘It is not the case that whales are mammals’ is well-formed (grammatical), the expression ‘It is not the case that apple’ is not. The relevant formation rule may read like this: “A combination of ‘not’ with a whole sentence is well-formed, while a combination of ‘not’ with anything else but a sentence is not.” The procedure is so well known that I shall not go into detail. What interests us is the origin of such rules of grammar. Why is ‘Not-P’ well-formed, while ‘Not-F’ is not (where ‘F’ is a property word)?

Our philosophy has an obvious and plausible answer: The rules of well-formedness reflect the ontological laws of the world. For example, it is a law of ontology that negation attaches to states of affairs and to nothing else. This is the reason why expressions of the form ‘Not-P’ are well-formed, while the negation of a property is not. If the world were different from what it is, although we cannot imagine such a difference, negation could combine with a category other than the category of state of affairs. This possibility must be distinguished, it should be added, from possible explications of the nature of negation within the framework prescribed by the just mentioned law of ontology. One must agree that negation “attaches” to states of affairs rather than to properties in isolation, for example, but that does not as yet imply that negation cannot be a relation between things and their properties. According to this view, negation exists in the form of negative exemplification. There are two indefinable, unanalyzable, relations between things and their properties, namely, positive and negative exemplification. I think that this view is mistaken, but its falsehood does not subtract from its intelligibility (see my The Categorial Structure of the World, pp. 356-57). And there are other possible categorizations of negation. The important fact is that the rules of grammar are not arbitrary, that they are determined by the laws of ontology, and that these laws lie even deeper than the laws of logic. The sentence ‘This pencil is both blue and not blue all over at this moment’ is perfectly grammatical, as I already pointed out, even though it states a contradiction, that is, a logical falsehood. In the artificial language of philosophers, both ‘P and not-P’ and ‘P or not-P’ are well-formed, even though the first represents a contradiction, while the second represents a logical truth (tautology).

Of course, Wittgenstein rejects this view. In regard to our example of negation and the apple, as we mentioned earlier, he is reported to have said:

You want to say that the use of the word ‘not’ does not fit the use of the word ‘apple’. The difficulty is that we are wavering between two different aspects:

(1) that apple is one thing or idea which is comparable to a definite shape whether or not it is prefaced by negation, and that negation is like another shape, which may or may not fit it:

(2) that these words are characterized by their use, and that negation is not completed until its use with ‘apple’ is completed. We cannot ask whether the uses of these two words fit, for their use is given only when the use of the whole phrase ‘not apple’ is given. For the use they have they have together.

The two ideas between which we are wavering are two ideas about meaning (1) that a meaning is somehow present while the words are uttered (2) that a meaning is not present but is defined by the use of the sign. If the meaning of ‘not’ and ‘apple’ are what is present when the words are uttered, we can ask if the meanings of these two words fit; and that will be a matter of experience (i.e., of fact). But if negation is to be defined by its use, it makes no sense to ask whether ‘not’ fits ‘apple’; the idea of fitting must vanish.

(Coffa, chap. XIV)

I do not waver between these two views. The first view seems to me to be correct. I would not describe it as Wittgenstein does, but in essence it is true that since ‘not’ represents negation rather than, say, the property of being sweet, and since ‘apple’ represents the property of being an apple rather than, say, the fact that there are apples, ‘not apple’ is not well-formed. It is not grammatical. It does not represent something that is ontologically possible. Why does Wittgenstein think that this view is to be rejected in favor of the second one?

‘Not’ and ‘apple’ have certain uses. For example, ‘not’ is used with sentences, if I may put it so, not with words. This is the very reason why we reject ‘not apple’ as ungrammatical: ‘Not’ is not used that way. But Wittgenstein goes on to argue that we cannot ask whether or not ‘not apple’ is ungrammatical, for whether or not it is ungrammatical depends, in part, on whether or not ‘not apple’ is ungrammatical. Thus the notion of something’s being ungrammatical “must vanish.” But I think that we have no difficulty in determining whether or not ‘not apple’ is ungrammatical. This is an open and shut case. And in general, the notion of well-formedness or of grammaticality makes perfect sense. Wittgenstein’s argument must be fallacious. Specifically, it is not true, as Wittgenstein contends, that we can only decide whether or not ‘not apple’ is ungrammatical after we have already made up our minds whether or not it is ungrammatical. It is clear in what direction Wittgenstein’s contention leads: If the issue can only be decided after it has been decided, then it can only be decided, if it can be decided at all, by fiat. We can only cut off the threatening infinite regress of having to find a reason for our decision before we find a reason for our decision, if no such reason is required in the first place, that is, if we can simply decree that the expression is ungrammatical. Wittgenstein’s line of reasoning leads to the conclusion that whether or not an expression is ungrammatical is a matter of our choice, no laws of whatever sort restrict us.

What is the crucial step in this line of reasoning? Does the use of ‘not’ fit the use of ‘apple’? Well, what is the use of ‘not’? Surely, so Wittgenstein seems to argue, the use of ‘not’ involves, in part, the use of ‘not’ in connection with ‘apple’. Thus to know the use of ‘not’, I must know how ‘not’ is used in combination with ‘apple’. This means that I must know whether or not ‘not’ fits the use of ‘apple’, in order to know the use of ‘not’. But now we are running around in circles: In order to know whether or not ‘not’ fits ‘apple’, I must know the use of ‘not’; and in order to know the use of ‘not’, I must know whether or not ‘not’ fits ‘apple’. The only way of breaking out of this circle, according to Wittgenstein, is to decree whether or not ‘not’ fits ‘apple’. Only by means of a convention can we escape from the circle.

This line of reasoning resembles Poincaré’s vicious circle accusation against impredicative descriptions. As I interpret Poincaré, he objects to the following description (“definition”) of the square root of two: the least upper bound of the set of numbers whose square root is at most two. He objects to it because he thinks that it is circular. It is allegedly circular, because it uses the notion of number in general. The notion of the square root of two must be excluded from the notion of number in general before the latter can be used to describe (“define”) the former. It is assumed that the notion of number in general somehow consists of the notions of individual numbers. It is assumed that one can have the general notion of number only if one already has the notion of the particular number which is the square root of two. Similarly, in Wittgenstein’s argument it is assumed that one can have a notion of the general use of ‘not’ only if one already has the notion of a particular use. The use of ‘not’, it is assumed, somehow consists of all of the particular uses to which the word is put.

Ultimately, as I pointed out earlier, Poincaré’s objection rests on the mistaken belief that we cannot acquire the notion of number without first acquiring the notion of every individual number. What is true is, rather, that we come by the notion of number, not by acquiring the notions of individual numbers, but by recognizing that something or other is a number. We come by the notion of the property of being an elephant, not by having met all of the individual elephants in the world, but by having seen one or two elephants. In summary, in order to acquire the notion of a property, it is not necessary to be acquainted with all of the things that have the property. Wittgenstein’s argument contains the same mistake as Poincaré’s. He assumes that the use of ‘not’ somehow consists of all of the particular uses to which the word is put. What is truly amazing is not that Wittgenstein should have arrived at a false conclusion, but that he should have made this particular assumption, for he talks constantly about the rules of grammar. The use of ‘not’, he ought to have realized, is given by such a rule. For example, the rule that ‘not’ can only occur within the context of a sentence decides that ‘not’ and ‘apple’ do not fit. Similarly, the formation rule that ‘or’ is a connective between sentences determines that ‘or 7’ is ill-formed. Wittgenstein has got it wrong when he argues that “if negation is to be defined by its use, it makes no sense to ask whether ‘not’ fits ‘apple’; the idea of fitting must vanish.” To the contrary, if negation is defined by its use, through the grammatical rules of English, then it makes perfect sense to ask whether or not ‘not’ fits ‘apple’, and the answer is then obvious.

The real question is, as I mentioned before, whether the rules of grammar conform to the laws of ontology or are arbitrary. I believe that the laws of ontology determine the rules of grammar, just as the laws of logic determine the rules of inference. Wittgenstein, on the other hand, seems to hold that the rules of grammar are somehow derived from the linguistic behavior of people, and that there can be no further question as to whether or not this behavior has to conform to anything. ‘Not apple’ is ungrammatical, not because of any feature of the world, but because people do not say “not apple”, and we are not allowed to ask why they do not speak that way. The rules of grammar, after everything has been said, turn out to be conventions.

Wittgenstein presents an argument to the effect that nothing could justify the rules of grammar (see Coffa, chap. XIV). It goes like this: (1) A statement that justifies the grammatical rule governing ‘not’ would have to be of the form: Since negation only attaches to states of affairs, ‘not’ cannot be combined with anything but a sentence; (2) But for this statement of the rule to make sense, the part ‘negation only attaches to states of affairs’ must make sense; (3) It follows from bipolarity that the phrase ‘negation does not only attach to states of affairs’ must make sense; (4) But this is precisely what we are denying by our attempt to justify the grammatical rule; (5) therefore, if the rule could be justified, it could also be violated; (6) but it cannot be violated; (7) Therefore, it cannot be justified.

If this is indeed Wittgenstein’s argument, then it clear that he goes wrong in step (4): We justify the grammatical rule, not by claiming that the sentence ‘negation does not only attach to states of affairs’ is meaningless, but by claiming that it is false. We justify the grammatical rule by claiming that the statement “Negation attaches only to states of affairs” is true, just as we justify modus ponens, not by claiming that a certain sentence is senseless, but by claiming that its negation is a law of logic.

I must hasten to add, as always when discussing Wittgenstein, that there are also numerous passages in which Wittgenstein seems to contradict the view that the rules of grammar are arbitrary. This inconsistency is most obvious in his discussion of mathematics. On the one hand, Wittgenstein, in agreement with our interpretation of him, says such things as: “The axioms of mathematics are sentences of syntax . . . The axioms are postulations of forms of expressions.” On the other hand, he also states flatly that “set theory is false” (see Coffa, chap. XIV).

c) Analyticity and the Vienna Circle

I believe that it is fair to say that the logical positivists never developed a coherent and clear notion of necessity. Most of them agreed, however, on three things. Firstly, they insisted on a sharp distinction between analytic and synthetic statements. Secondly, they identified the necessary (the a priori) with the analytic. Thirdly, they adopted Wittgenstein’s dogma that the factual coincides with the synthetic. It must be said to their credit that Wittgenstein’s distinction between what can be said and what can only be shown did not find wide acceptance. As far as the necessity of logic, arithmetic, and set theory is concerned, it was explained in a potpourri of ways. Nothing reveals their confusion about the nature of necessity and, hence, of the a priori, better than Ayer’s treatment of this topic in his famous Language, Truth, and Logic, where he states the dilemma of the “empiricists” with admirable clarity:

Where the empiricist encounters difficulty is in connection with the truths of formal logic and mathematics. For whereas a scientific generalization is readily admitted to be fallible, the truths of mathematics and logic appear to everyone to be necessary and certain. But if empiricism is correct no proposition which has a factual content can be necessary or certain. Accordingly the empiricist must deal with the truths of logic and mathematics in one of the two following ways: he must say either that they are not necessary truths, in which case he must account for the universal conviction that they are; or he must say that they have no factual content, and then he must explain how a proposition which is empty of all factual content can be true and useful and surprising.

(A. J. Ayer, Language, Truth and Logic, quoted from

Philosophy of Mathematics, P. Benacerraf

and H. Putnam eds., p. 290)

As so often is the case, there is a way out of the dilemma. Our form of empiricism rejects Ayer’s choice between necessity and factual content. The truths of mathematics and logic, we hold, have factual content and are necessary. Their necessity consists, not in their being devoid of factual content, but (a) in their being most general and (b) in the fact that their negation is unimaginable. Ayer chooses to be impaled by the second horn of the alleged dilemma. He cannot let go of the Wittgensteinian doctrine that a necessary truth must be devoid of factual content. Nor does he doubt that the truths of mathematics and logic are necessary. He must therefore explain how something so empty of all factual content can nevertheless be true, useful, and surprising.

But first Ayer tries to discredit the first of his two alternatives; the first of the two horns. He attributes this view to Mill, but we shall pretend that his objections are raised against us. The first objection is that “We may come to discover them [the truths of logic and mathematics] through an inductive process; but once we have apprehended them we see that they are necessarily true, that they hold good for every conceivable instance.” (Ayer, quoted in Benacerraf and Putnam, p. 292.) Empirical generalizations, of course, are not of this sort. After we have discovered them, we do not see that they “hold good for every conceivable instance.” If we substitute for Ayer’s term ‘conceivable’ the word ‘imaginable’, we are in complete agreement with him. It is indeed true that the truths of mathematics and logic (as well as many other truths) are such that we cannot imagine them to be false, while the truths of science can be imagined to be false (if they can be imagined at all!). This is precisely one side of their necessity. But the fact that these truths are such that we cannot imagine them to be false does not imply that they are devoid of factual content. Rather, factual propositions can be divided into two large classes: Those that can be imagined to be false and those that cannot.

Next, Ayer argues that the truths of logic and mathematics must be necessary, and hence devoid of factual content, by giving examples of situations in which a law of mathematics or logic seems to be violated, but for which we decide, on second thought, to save the law. His point is that the laws of logic and mathematics are immune against falsification:

And this indicates that Mill was wrong in supposing that a situation could arise which would overthrow a mathematical truth. The principles of logic and mathematics are true universally simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and mathematics are analytic propositions or tautologies.

(Ayer, quoted in Benacerraf and Putnam, p. 293)

In this way, Ayer establishes a connection with analyticity: The truths of logic and mathematics are necessary because they are analytic. To be analytic is to be tautologous. To be tautologous means to be devoid of factual content: a tautologous proposition “holds for all cases and, therefore, says nothing in particular,” as the incantation goes. By calling the truths of logic and mathematics analytic, Ayer really tells us no more than that they are devoid of factual content, whatever that may mean. But he also mentions the rules of language use. What then is it to be: Necessity explicated in terms of the “empty tautologies” of the early Wittgenstein, or necessity explicated in terms of the “grammatical rules” of the later Wittgenstein?

Ayer gives three reasons why we cannot abandon the truths of logic and mathematics: (1) We would be contradicting ourselves, (2) we would sin against the rules of the use of language, and (3) our utterances would become self-stultifying.

Assume that we gave up the truth (T) that two plus two is four and accepted that it is five instead. How would we be contradicting ourselves? Of course, if we believed both (T) and not-(T), we would believe a contradiction. But this is not the case. We have given up (T) and now believe not-(T) instead. Is not-(T) itself a contradiction? This question becomes: Is not-(T) a logical falsehood? I have argued in detail that arithmetic truths are not logical truths. Hence, while not-(T) is certainly an arithmetic falsehood, it is not a logical falsehood. I do not see, therefore, how we would be contradicting ourselves if we gave up (T).

Let us take a look at the second reason: If we accepted not-(T), we are told, we would sin against the rules of use of language. What are these rules? I cannot think of any plausible candidates that would not also either prohibit us from accepting that whales are not mammals, or else beg the question of why we never discard the truths of arithmetic. Moreover, if we really violated some rule of the use of language and did nothing else in our rejection of (T), then it must appear more rather than less mysterious that we do not easily give up arithmetic truths. For, nothing seems to be easier than to change our linguistic rules, and such changes occur all the time. On the other hand, if it were a matter of fact rather than of the rules of language that two plus two is four rather than five, then we should expect the most horrendous repercussions from any attempt to repeal this truth of arithmetic.

Ayer goes on to give a more precise explanation of analyticity and why analytic statements are incorrigible:

Thus, the proposition “There are ants which have established a system of slavery” is a synthetic proposition. For we cannot tell whether it is true or false merely by considering the definitions of the symbols which constitute it. We have to resort to actual observation of the behavior of ants. On the other hand, the proposition “Either some ants are parasitic or none are” is an analytic proposition. For one need not resort to observation to discover that there either are or are not ants which are parasitic. If one knows what is the function of the words “either”, “or”, and “not”, then one can see that any proposition of the form “Either p is true or p is not true” is valid, independently of experience.

(Ayer, quoted in Benacerraf and Putnam, pp. 294-295)

Here then we have the test for analyticity: A proposition is analytic if and only if we “need not resort to observation to discover” that the proposition is true. (Notice that Ayer speaks of “validity,” but I shall only say that arguments are valid (or invalid).) It is true that we need not consult experience to establish that the analytic proposition about ants is true: That it is true follows from the fact that it is an instance of a law of logic. But the same holds for synthetic propositions which are instances of the laws of nature: They also can be seen to be true without further observation. I know, for example, that the whale Walter is a mammal without “resorting to observation.” We must therefore turn to the logical law itself. Ayer maintains, it seems to me, that we can see that this law is true, independently of experience, if we know the function of the connective words. But this claim is false. That all states of affairs (all propositions) behave in a certain fashion, that every one of them is such that either it is a fact or its negation is a fact, is just as much a fact about states of affairs and, hence, the world, as it is a fact about whales that every one of them is a mammal. In either case, the case of states of affairs and the case of whales, knowing the “function” of ‘or’, ‘not’, ‘if-then’, etc. is not to know the truth of the law. We know the function of ‘not’ and ‘if-then’ perfectly well without knowing on this basis alone whether or not it is not the case that all whales are mammals. We must know, in addition, something about whales in order to know whether or not they are mammals. And similarly in the case of states of affairs: We must know something about states of affairs before we can know whether every one of them is such that either it is a fact or else its negation is a fact.

We do not deny, of course, that there is a tremendous difference between a law of biology and a law of logic. Some laws are much more fundamental than others; they are much more general than others; they are much more pervasive than others; they are much more important than others. But they are facts, nevertheless. And it is a grave mistake to believe that they are devoid of factual content or that they tell us nothing about the world. The opposite is true: The laws of logic and mathematics have so much factual content and tell us such important things about the world that they serve as the very foundation for our more esoteric inquiries into the social structure of ant colonies.

Ayer argues that analytic propositions cannot be refuted because they have no factual content. Since they have no factual content, facts cannot touch them. We come to a different conclusion. The laws of logic and mathematics have factual content. They can therefore be refuted. But since they concern the most general features of reality, they can only be refuted by the most general facts about reality. This means that the tautology mentioned by Ayer cannot possibly be refuted by facts about ants. How could it be, since it is not about ants? But it would be refuted if states of affairs were other than they are.

Ayer argues that analytic propositions, even though they are devoid of factual content, can nevertheless give us new knowledge. They can be “useful” and “surprising” because they call attention to linguistic usage (Ayer, quoted in Benacerraf and Putnam, p. 295). Such propositions are not nonsense like the propositions of metaphysics. Where does this usage come from? It seems to be fixed by definition, as far as I can make out. The truth of an analytic proposition, Ayer says, follows simply “from the definition of the terms contained in it” (Ayer, quoted in Benacerraf and Putnam, p. 297). Does this mean that the truth of the law of excluded middle follows from the definitions of ‘or’ and ‘not’? And how does the truth of a law follow from definitions? Let us be guided by a familiar example. Assume that we agree to use ‘bachelor’ as a convenient abbreviation for the much longer ‘unmarried male of marriageable age’. Now, all sorts of truths will follow from our agreement. For example, it will be true by definition that all bachelors are males; and it will also be true that all bachelors are unmarried. These truths follow, it must be emphasized, from the abbreviation plus certain laws of logic. For example, that all bachelors are males follows from our abbreviation together with the law that if anything has properties F and G, then it has property F.

Can we make a parallel case for the law of excluded middle? What are the abbreviational agreements (definitions) for ‘or’ and ‘not’? What is ‘or’ an abbreviation of? What is ‘not’ an abbreviation of? I cannot think of any plausible answers. The two cases are completely different. We may suspect that Ayer thinks of the truth-tables as definitions of the connective words, but we have already seen that the truth-tables do not represent linguistic conventions (abbreviations). Rather, they depict certain laws of logic. The truth-table for ‘not’, for example, depicts the law that all states of affairs p are such that if p is a fact, then not-p is not a fact, and if p is not a fact, then p is a fact.

We can see how precarious Ayer’s grasp of the nature of definition is when we turn to his discussion of mathematics. He starts out with the standard positivist interpretation of geometry:

We see now that the axioms of a geometry are simply definitions, and that the theorems of a geometry are simply the logical consequences of these definitions . . .

(Ayer, quoted in Benacerraf and Putnam, p. 297)

There is no sense, therefore, in asking which of the various geometries known to us are false, and which are true. Insofar as they are all free from contradiction, they are all true.

(Ayer, quoted in Benacerraf and Putnam, p. 298)

We conclude, then, that the propositions of pure geometry are analytic.

(Ayer, quoted in Benacerraf and Putnam, p. 298)

Take Poincaré’s notion that the axioms of geometry are implicit definitions, combine it with Hilbert’s identification of consistency with truth, and, presto, geometry is analytic! Of course, we can also convince ourselves along the same line that botany is analytic. Things are getting easier and easier. We may have thought that in order to decide whether or not geometry is analytic, we would have to decide whether or not the axioms of geometry are based on experience. Well, do we need observation in order to decide whether or not the axioms of geometry are true? What are these axioms about? Are they about chalk points, pencil lines, and triangles formed by light beams? If they are, then we must of course observe such points, lines, and triangles, just as we must observe whales in order to decide whether or not they are mammals. However, Ayer holds that these axioms are “not in itself about physical space” (Ayer, quoted in Benacerraf and Putnam, p. 297). And he hints that this follows from the discovery of non-Euclidian geometry. But how could it follow? How could the fact that a non-Euclidian set of axioms is consistent prove that a Euclidian set is not about “physical reality”? How could the discovery of non-Euclidian geometries possibly show that Euclidian geometry has no factual content? But we need not even use the discovery of non-Euclidian geometry in order to show that geometry is devoid of factual content and, hence, analytic. A shortcut is presumably available, as we saw a moment ago. We merely have to decree that the axioms of geometry are “definitions.” What follows from these definitions is then true, not as a matter of fact, but as a matter of definition. What follows from these definitions is therefore analytic.

Things are no less confused when we turn from geometry to arithmetic:

We see, then, that there is nothing mysterious about the apodeictic certainty of logic and mathematics. Our knowledge that no observation can ever confute the proposition “7 + 5 = 12”, depends simply on the fact that the symbolic expression “7 + 5” is synonymous with “12”, just as our knowledge that every oculist is an eye-doctor depends on the fact that the symbol “eye-doctor” is synonymous with “oculist”.

(Ayer, quoted in Benacerraf and Putnam, p. 299)

No more talk about axioms being definitions. Ordinary numerical equations are now claimed to be true by definition. More precisely, they are said to be true because of the (fortuitous?) circumstance that language contains more than one expression for the same thing. One wonders why Ayer did not wave the magic wand of synonymy over geometry as well. That the inner angles of a triangle add up to two right angles could be claimed to be true “by definition” because the expression ‘the sum of the inner angles of a triangle’ is synonymous with ‘two right angles’. But why stop with geometry? History, for example, is apodeictic, that is, devoid of factual content, that is, analytic, that is, a matter of synonymy. That Salzburg is the birthplace of Mozart is surely due to the fact that ‘Salzburg’ is synonymous with ‘the birthplace of Mozart’; that Napoleon is the vanquished of Waterloo follows from the fact that ‘Napoleon’ is merely short for ‘the vanquished of Waterloo’. Of course, I am jesting. Salzburg is correctly described as the birthplace of Mozart, and the number 12 is correctly described as the sum of 7 and 5, but neither the statement that Salzburg is the birthplace of Mozart nor the statement that 12 is the sum of 7 and 5 is analytic. And just as little as ‘Salzburg’ is an abbreviation of ‘the birthplace of Mozart’, just as little is ‘12’ an abbreviation of ‘the sum of 7 and 5’.

Nor is Ayer the only one who meets Kant’s challenge by definition, so to speak. Lest I be accused of being unfair to Ayer, let me quote a passage from Hempel:

For the latter [the proposition 3 + 2 = 5] asserts nothing whatsoever about the behavior of microbes; it merely states that any set consisting of 3 + 2 objects may also be said to consist of 5 objects. And this is so because the symbols “3 + 2” and “5” denote the same number: they are synonymous by virtue of the fact that the symbols “2”, “3”, “5”, and “+” are defined (or tacitly understood) in such a way that the above identity holds as a consequence of the meaning attached to the concepts involved in it.

(C. G. Hempel, “On the Nature of Mathematical Truth,” p. 368)

A little later (on p. 369), Hempel even claims that the transitivity of identity follows from the definition of identity.

Let us return to Ayer. It seems that Ayer has some misgivings about his short proof that arithmetic is analytic. There is the obvious objection that this proof leaves no room in arithmetic for surprises, for invention or discovery. He answers that “The power of logic and mathematics to surprise us depends, like their usefulness, on the limitations of reason. A being whose intellect was infinitely more powerful would take no interest in logic and mathematics.” (Ayer, quoted in Benacerraf and Putnam, p. 300.) And in a footnote, Ayer quotes Hans Hahn: “An all-knowing being needs no logic and no mathematics.” But Ayer’s answer does not jibe with his analysis of arithmetic. If what he says about the equation 7 + 5 = 12 is correct, then our limitations of reason have nothing to do with our interest in arithmetic. Nor is it the case that God can dispense with logic and arithmetic because he is all-knowing. No, if Ayer is correct, then our interest in logic and arithmetic is entirely due to our verbosity. If we could just stick to the practice of using no more than one expression for one thing, logicians and mathematicians would be unemployed. God’s lack of interest in logic and mathematics would not be due to his superior intellect, but merely to the fact that he is laconic.

I have dwelt on Ayer’s view because it reflects accurately, in my opinion, the logical positivist’s response to the Kantian challenge. This answer, I submit, consists of a hodgepodge of unconvincing ideas. Arithmetic is claimed to be analytic, contrary to what Kant says, because (1) it consists of tautologies without factual content, (2) it follows from the rules of language (rules of grammar?), (3) it follows from definitions (it is a matter of synonymy), and (4) it consists of definitions. The unifying idea behind this conception of analyticity, if there is one, is that analytic propositions are trivially true because we make them true, and we make them true by manipulating language.

d) Conventionalism

The manipulation of language is clearly a conventional affair. Thus analyticity is ultimately a matter of convention. Ayer says quite explicitly that analytic propositions “simply record our determination to use words in a certain fashion. We cannot deny them without infringing the conventions which are presupposed by our very denial, and so falling into self-contradiction” (Ayer, quoted in Benacerraf and Putnam, p. 299; my italics). Conventionalism in logic and mathematics, we may say, is the unofficial view of logical positivism. I cannot discuss the many articles which have been devoted to this view, but it may shed light from yet a different angle on our own position, if we briefly consider Carnap’s version of conventionalism and Quine’s criticism of it.

What leads to conventionalism is the view that analytic statements are somehow true as a consequence of certain linguistic practices. The general idea is to view logic and arithmetic as games, played according to certain rules. Wittgenstein described it this way to his Vienna audience:

Frege rightly opposed the view that the numbers of arithmetic are signs. The sign ‘0’ does not have the property that when added to the sign ‘1’ it gives the sign ‘1’ as a result. In this critique Frege is right. But he did not see what is justified in formalism, that the symbols in mathematics are not the signs, and yet they have no ‘meaning’ (Bedeutung). For Frege the choice is as follows: either we are dealing with ink-marks on paper, or else these ink-marks are signs of something, and what they represent is their meaning. That this is not a correct alternative is shown by the game of chess: here we are not dealing with the wooden pieces, and yet they do not represent anything—in Frege’s sense, they have no meaning. There is a third possibility, the signs can be used as in a game.

(Coffa, chap. XVII)

I do not think that Frege overlooked this third alternative. For him and for us, the crucial distinction is between ink-marks that represent something and ink-marks that do not; there is no third alternative. But we do not deny that ink-marks may be studied from two different points of view. Firstly, we may study their properties as ink-marks, for example, their chemical composition. Or, secondly, we may study the rules which someone has adopted for writing these marks down in a certain order. For example, it may be agreed never to write down a shape like ‘o’ followed by a shape like ‘+’. By adopting such rules, one may design an interesting game of writing down strings of ink-marks. But while Wittgenstein (and Carnap) think that arithmetic is a game of this sort, Frege and I do not. We do not overlook a third possibility; we are not being misled by false analogies; we are not being seduced by language. We straightforwardly deny Wittgenstein’s (and Carnap’s) contention that arithmetic is like chess: Chess is a game, arithmetic is not!

Carnap worked on Wittgenstein’s idea with his customary technical skill and love of detail. The rules of the game are christened “syntax.” The shapes of the signs of syntax matter as little as the shape of the rook matters in chess. All that matters are the rules for the use of the shapes. But if arithmetic is syntax, we protest, what happens to truth? Truth plays no role in chess, but it is of the essence of language. Well, analytic sentences are recognizable by their shapes alone, according to arbitrary rules of composition, and these sentences are supposed to represent the truths of logic and arithmetic. Why we select those particular strings of signs and these specific rules for their formation, we must not ask. This is a mere matter of convention. But why stop at the “logical” transformation rules? Why, indeed! Carnap clearly saw that the “syntactic” notion of analytic truth leads straightforwardly to a “syntactic” notion of truth in general:

Wittgenstein continues: “And so also it is one of the most important facts that the truth or falsehood of non-logical sentences can not be recognized from the sentences alone”. This statement, expressive of Wittgenstein’s absolutist conception of language, which leaves out the conventional factor in language construction, is not correct. It is certainly possible to recognize from its form alone that a sentence is analytic; but only if the syntactical rules of the language are given. If these rules are given, however, then the truth or falsity of certain synthetic sentences—namely, the determinate ones—can also be recognized from their form alone. It is a matter of convention whether we formulate only L-rules, or include P-rules as well; and the P-rules can be formulated in just as strictly formal a way as the L-rules.

(R. Carnap, The Logical Syntax of Language, p. 186)

(But see also Carnap’s later remark that “truth and falsehood are not proper syntactical properties; whether a sentence is true or false cannot generally be seen by its design, that is to say, by the kind and serial order of its symbols” Carnap, 1959, p. 216).

In this fashion, we can make any “synthetic sentence” true by virtue of its form alone and, hence, can turn it into an analytic sentence. The sentence ‘All whales are mammals’, for example, can easily be made true by convention and turned into an analytic sentence. Not everyone, however, was impressed by Carnap’s surprising result. Schlick clearly saw the trick:

When Carnap explains . . . that one can construct a language with extra-logical transformation rules by, for instance, including natural laws among the principles (i.e., they are considered as grammatical rules), then this way of putting things seems to me to be misleading in the same sense as is the thesis of conventionalism. It is true that a sentence (a sign sequence) which under the presupposition of ordinary grammar expressed a natural law can be made into a principle of language simply by stipulating it as a syntactical rule. But precisely by this device one changes the grammar and, consequently, interprets the sentence in an entirely new sense, or rather, one deprives the sentence of its original sense. It is then not a natural law at all; it is not even a proposition, but merely a rule for the manipulation of signs.

(Coffa, chap. XIX, p. 30)

It is interesting that Schlick does not draw the next and obvious conclusion that as little as Carnap’s P-rules are laws of nature, so little are his L-rules laws of logic. The syntactic transformation rules are also “merely rules for the manipulation of signs”. We establish a connection with logic only if we acknowledge that these rules are not conventional, that they are not picked willy-nilly, but that they reflect the laws of logic.

It may be possible to interpret Carnap in such a way that he gives a coherent reply to Kant’s challenge (see Coffa, pp. 32-33). As far as the object language is concerned, conventionalism rules: Whether or not one accepts, say, the axiom of choice, is not a matter of truth or of fact, but merely a matter of proposing the kind of language one wishes to use. There is no fact, no truth, that could be called “the axiom of choice.” There really is no such thing, therefore, as accepting or rejecting the axiom of choice as being true. All one can do is either to incorporate or not to incorporate a certain string of signs into one’s language, namely, that string which ordinarily is thought to express the axiom of choice. But, and this is where we have to interpolate, it is a fact, and not merely a matter of convention, that the so-called laws of logic, arithmetic, and set theory are a matter of convention in the manner just described. On the second level, therefore, there are facts. Now, if this is approximately Carnap’s view, then our objection is that it is not true that the laws of logic, arithmetic, and set theory are a matter of convention.

As obvious as the failure of conventionalism is, to many analytic philosophers it seemed to be the only alternative to the dreaded Kantian doctrine of the synthetic a priori. Quine is a case in point. In his paper “Truth by Convention,” he describes in detail how logic, mathematics, and even physics can be made “true by convention.” One merely has to axiomatize the respective set of laws and treat the axioms as implicit definitions. The implicit definitions are said to be “true by convention.” Quine’s point is that logic and mathematics cannot be sharply distinguished from the sciences by claiming that the former are true by convention while the latter are not. He is taking the pragmatic point of view:

If in describing logic and mathematics as true by convention what is meant is that the primitives can be conventionally circumscribed in such a fashion as to generate all and only the so-called truths of logic and mathematics, the characterization is empty: our last considerations show that the same might be said of any other body of doctrine as well. If on the other hand it is meant merely that the speaker adopts such conventions for those fields but not for others, the characterization is uninteresting; while if it is meant that it is a general practice to adopt such conventions explicitly for those fields but not for others, the first part of the characterization is false.

(W. V. O. Quine, “Truth by Convention,” p. 341)

In a style reminiscent of Carnap himself, Quine here rejects Carnap’s claim that he can sharply distinguish between the a priori and the a posteriori or, what for the logical positivists is the same, between the analytic and the synthetic, between the merely conventional and the factual.

Quine’s criticism is well taken. But there remains an obvious question: Is there then no difference at all between the a priori and the a posteriori? Is there no necessity attached to logic and mathematics? Are logic and mathematics completely on a par with the empirical sciences? Conventionalism, we agree with Quine, cannot capture the felt difference. But must we conclude that there is no difference at all? Quine cannot deny that there is a difference, but his aversion to metaphysics does not permit an ontological account of it. He turns it behavioristically into a “contrast between more and less firmly accepted statements” (Quine, 1964b, p. 342). “There are statements”, he says, “which we choose to surrender last, if at all, in the course of revamping our sciences in the face of new discoveries; and among these there are some which we will not surrender at all, so basic are they to our whole conceptual scheme” (Quine, 1964b, p. 342). But this answer merely raises a further question: Why do we so stubbornly cling to the truths of logic and mathematics? A pragmatist has no answer to this question. Perhaps it makes no sense to him. However, Quine sees a way of combining his pragmatism with the Carnapian conventionalism:

Now since these statements are destined to be maintained independently of our observations of the world, we may as well make use here of our technique of conventional truth assignment and thereby forestall awkward metaphysical questions as to our a priori insight into necessary truths. On the other hand this purpose would not motivate extension of the truth-assignment process into the realm of erstwhile contingent statements.

(Quine, 1964b, p. 342)

Quine admits that there is a difference between the truths of logic and mathematics, on the one hand, and the truths of science, on the other. In contrast to Carnap—and in conformity to his pragmatism—he holds, however, that this difference is gradual. But the difference, though it may be gradual, exists, and we must ask in what it consists. Quine, as I said a moment ago, never addresses this question.

Carnap, we must emphasize, held fast to the analytic-synthetic distinction. However, in response to Quine’s criticism of his conventionalistic explication of the distinction, he can do no better than point to his earlier distinction between changes in language and changes in truth-value:

I should make a distinction between two kinds of readjustment in the case of a conflict with experience, namely, between a change in the language, and a mere change in or addition of, a truth-value ascribed to an indeterminate statement (i.e., a statement whose truth-value is not fixed by the rules of language, say by the postulates of logic, mathematics and physics). A change of the first kind constitutes a radical alteration, sometimes a revolution, and it occurs only at certain historically decisive points in the development of science. On the other hand, changes of the second kind occur every minute. A change of the first kind constitutes, strictly speaking, a transition from a language L_n to a new language L_n+I.

(Coffa, chap. XX)

After all is said and done, the distinction between analytic and synthetic truths turns out to be based on nothing more than a blind belief that if we reject instances of the first kind, we are changing our language, while if we reject instances of the second kind, we are not. That this is not just the judgment of one who finds much of Carnap’s philosophy unclear and even evasive can be seen from an interesting letter by Tarski to Morton White (see A. Tarski, “Letter to Morton White”). Tarski very cautiously endorses an empiricistic conception of logic and mathematics: “I would be inclined to believe (following J. S. Mill) that logical and mathematical truths don’t differ in their origin from empirical truths—both are results of accumulated experience.” Then he goes on to say: “I think that I am ready to reject certain logical premises (axioms) of our science in exactly the same circumstances in which I am ready to reject empirical premises (e.g., physical hypotheses) . . .” And Tarski concludes with this comment on Carnap’s view:

Whether this description is true and adequate—I don’t know. I have the impression that many people would agree with it. I think, e.g., that Carnap would agree. Nevertheless, he would claim that there is a fundamental difference between a change of logic and a rejection of a physical theory; for, he would say, in the first case we change the language, we begin to use the words in a new meaning, while in the second no such change occurs. Why? Well, this follows from his definition of meaning (which in turn is based on a definition of logical terms and logical truth). Of course, he is permitted to do so; he can accept whatever definition of meaning he wants; and perhaps I would find it also convenient to accept his definition. However, I don’t see how such a definition can really affect the situation; if you define logical axioms as those which cannot be changed without the change of your whole language, then of course you cannot change them without changing the language—this is a truism (at least, in our present logic).

(Tarski, p. 32)