JODY AZZOUNI

A PRIORI TRUTH*

1.

Over forty years ago, writing of the “totality of our so-called knowledge or beliefs”, Quine claimed that “no statement” contained therein “is immune to revision” (Quine, 1980, pp. 42-43). Naturally philosophers were suspicious. No statement? ‘All bachelors are unmarried’? ‘All vixens are female foxes’? In a friendly amendment Quine applauded, Putnam (1975) exempted these examples and others like them, using his notion of one-criterion words. The amendment applied only narrowly, leaving the epistemic force of Quine’s claim untouched.

I think, however, philosophers were far more disturbed by the (ap- parent) implication of this thesis that the principle of noncontradiction is revisable, than they were by the potential loss of the above analytic truths.’ There is something, one thinks, so terribly basic about this principle that it is a sine qua n~lz of rationality itself. This thought directly emerges in attempts to show that Quine’s picture is made incoherent by applying his claim to the principle of noncontradiction.’

But intuitions about the grip the principle of noncontradiction has on rationality are disturbed in turn by the existence of deviant logics in which strong forms of the principle of noncontradiction are false.’ Worse, these deviant logics are not merely uninterpreted calculi; they are serious attempts at constructing self-referential languages, and con- ceivably could turn out to be the best way to carry out this project.

Putnam, in a series of important papers,’ explores the relationship between the a prioricity of the principle of noncontradiction and ratio- nality. Choosing not a strong version of the principle of noncontradic- tion, but the weakest: ‘Not every statement is both true and false,’ a principle compatible with the above mentioned deviant logics, he gives what I think are the strongest arguments available for why this claim should be regarded as a priori true.

Nevertheless I think Quine’s original claim is the right one: no state- ment is a priori true.’ This does not mean, however, that it is possible to revise the truth-value of any statement. Rather, as we will see, for

Erkenntnis 37: 327-346, 1992. @ 1992 Kluwer Academic Publishers. Printed in the Netherlands.

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some statements revision must take the form of being ‘revised out’ of a conceptual scheme altogether: being ‘inexpressible’ there.6

Up to this point I will have been defending Quine against Putnam’s amendments. But, ironically, in so doing I will not have exactly pre- served Quine’s epistemic insights. Traditionally, a priori truths were supposed to be stability points in a conceptual scheme: held constant in truth-value come what may, they were truths one could rely on. But the (a priori) Zogical truths served a second (and distinguishable) pur- pose in addition: they were to be descriptions of ways in which concep- tual schemes would not change, they were to be markers of the limits of epistemic possibility,’ the borders beyond which only irrational be- lievers could wander. In dropping a priori truths, therefore, Quine apparently eliminated constraints on conceptual change - anything (nearly enough) becomes (epistemically) possible.8

Perhaps therefore it is no surprise both that Quine has found himself with “little to say of [the scientific method] that was not pretty common knowledge” (Quine, 1986c, p. 493), and that principles of charity loom so large in his methodology and that of fellow travelers such as David- son. Descriptions of rational practice emptily become only the descrip- tion of our own practices.’

Contrary to this, we will find that the two roles of a priori truths described above separate, and consequently the rejection of a priori truth does not simultaneously empty epistemic possibility of content: although there are properties that every rationally accessible conceptual scheme must have, and such properties can be described, they are not described by a priori truths. Neatly put, not every rationally accessible conceptual scheme has the resources to describe what is required in any rationally accessible conceptual scheme.

We start by briefly noting that mathematics, logic, and semantics (if the latter is not sterile) each breed its own kind of possibility. There is the logically possible: all those possible states of affairs that do not violate logical laws. If one manages a crisp distinction between logic and mathematics, then a notion of the mathematically possible, a kind of possibility with a range narrower than that of logical possibility becomes available. lo Finally, if semantic theory can yield a notion of

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analyticity - truth by virtue of meaning alone - this notion too will breed a notion of the analytically possible, a notion possibly independent of the others. I’

A natural question is whether any of these notions are relevant to epistemic possibility. Quine’s answer, even when he finally concedes a limited notion of analyticity, has been to disparage the idea. Rather, the truth-value of alzy statement is still up for grabs. i2

There are reasons to be suspicious of this position when it comes to logic. In the locus cZassicus of the position (Quine, 1980), Quine spends most of the article on analyticity, and indeed, most of the subsequent literature has followed suit by worrying about his objections to fhat notion. When Quine finally applies his revisability claim to logic, he merely invokes quantum logic as an alternative to classical logic. But that, or intuitionism (another respectable alternative) have rather a lot of sentences in common with classical logic. l3 Surely such examples do not show that the truth value of every logical truth is open to revision. Still available is the possibility of a core collection of logical truths (the intersection of the alternative logics) being Llnrevisable. Indeed, it is one such truth that Putnam, 1983b. attempts to fasten upon (namely, ‘Not every statement is both true and false’,) as an example of an a priori truth.

We put off evaluating Putnam’s claim, and for now tackle the elemen- tary question of what a logical law is. Logieal laws are puzzling in a way that mathematical laws seem not to be. Statements such as ‘2+2= 4’, ‘A normed linear space X is complete if and only if every absolute summable series is summable’, and others like them are about mathe- matical objects and the relationships among them.lJ

On the other hand, it is often claimed that (first order) logic has no particular existential commitments - it has, in a word, no subject. But there is something problematical about this. Nearly everyone agrees that a statement such as, “If John runs, then John runs”, is not a logical law, but an insfalzce of a logical law. We could understand logical laws e.utensionally, that is, just as collections of sentences of the above sort. But to do that would be to obscure their epistemic properties, in particular, the centrality Quine is so fond of. We must describe (some- how) the mechanisms we use to gerzerate instances of the laws.

I have nothing surprising to offer here as a solution. For reasons largely parochial to the first order predicate calculus (i.e.. complete-

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ness), we can mark out the desirable class of sentences syntactically, utilizing one proof procedure or another, or equivalently, semantically, by one method or another utilizing the notion of truth.15

The point of the observation just made is that logical laws (although not their instances) are about (in a perfectly robust sense of ‘about’) the language we apply the laws to. What this doesn’t mean, I hasten to add, is that a notion of ‘truth’ is necessarily required - the fact that we can characterize instances of logical laws syntactically in the first order case shows that we need never invoke ‘truth’ if the notion disturbs us. It also doesn’t mean that a metalanguage/object language distinction is required, even if we decide to use a truth predicate. The logical laws can be about the statements of the very same language they are couched in.

But there is an important qualification that must be made here, although it is not an easy one to explain. Consider the project of describing all the logical laws used to police the totality of our knowl- edge. Pertinent to this project are an indeterminate collection of seman- tic and syntactic ‘concepts’. Given a language, we can have at our disposal all the concepts necessary to do the job if we describe the language from outside, from the vantage of a metalanguage, as it’s commonly put. Doing so, however, leaves the principles of the metalan- guage uncharacterized. On the other hand, there are strategies on the market for doing the job from within the vantage point of the language under study. But on pain of paradox such approaches must leave one or another aspect of the object language uncharacterizable.16 As we shall see shortly, this fact bears directly on the question of whether there are a priori truths.

Let us now turn to Putnam’s claim that (*) ‘Not every statement is both true and false’, is an a priori truth. As I read Putnam, 1983b, he starts by assuming that any theory which negates (*) will have to be a theory that consists of every statement and its negation. Clearly such a theory is useless. i’

So far so good. But couldn’t we adopt (*) while restricting the appli- cation of universal instantiation to it? Putnam’s response is that in doing so, “we are playing verbal games: we simply don’t mean what is

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normally meant by ‘every statement is both true and false”’ (Putnam, 1983b, p. 102).

In his ‘Note’ to 1983b, Putnam recognizes that his reliance here on (some sort of) theory of meaning allows us to define two ways in which a statement can be revised. As before, we can simply negate some statements (or change their truth-values). But others can be revised by challenging the concepts they contain. Putnam suggests that in- tuitionism exemplifies this move: it replaces the classical notions of truth and falsity with something else. Thus Putnam now finds himself upholding a (slightly modified) version of the Quinean thesis that every statement is open to revision.

But in his ‘Note to supersede (supplement?) the preceding note’, Putnam reverses his stand again. For even if we concede that certain concepts have been rejected, that doesn’t show that no statement is a priori. Consider the statement, ‘If the classical notions of truth and falsity do not have to be given up, then not every statement is both true and false’. That statement is a priori true regardless of whether we reject the classical concepts of truth and falsity or not.

This won’t do, and there are two ways to see this. First the quick and dirty explanation: the above hypothetical mentions the notion of truth in the antecedent and uses it in the consequent. But use of a notion is verboten in a context where we have rejected it (a recipe for embarrassment: assert the above hypothetical in an intuitionistic con- text and watch what happens). We may be able to describe the classical notion of truth in a conceptual scheme from which the concept is absent, but that doesn’t imply we can use it (or a satisfactory translation of it). This is not just a technical error: there is no way to do what Putnam needs the hypothetical for unless we retain the notion so re- jected.

Here is the deep reason behind the failure just cited. Truth really is a problematical notion. (See the items cited in footnotes 3 and 16.) On some formal models of its operation, one simply cannot use the notion to say things like, ‘Not every sentence is both true and false’, as Putnam and other philosophers - myself included - intend its interpretation, either because there is no single truth predicate with sufficient scope (e.g., on Tarski’s approach), or because quantification is restricted (e.g., on Parson’s approach). Other solutions which allow something like the above remark to be made are restricted in other ways.

We saw in Section 2 above that the statement of the logical laws calls

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for some description of the language the laws apply to. But ‘descriptive capacity’ cannot be too rich on pain of paradox, and it is precisely which way is best to limit such capacity that is in question: for all we know, part of the best solution will cost us our capacity to express (*I.

In Putnam (1983~) I detect a somewhat different argument. Consider the Absolutely Inconsistent Rule (AIR): Infer every statement from every premise and from every set of premises, including the empty set. Here are Putnam’s claims:

(1) (2)

The AIR cannot be accepted by any rational being. A fully rational being should see and be able to express the fact that the AIR is incorrect.

Two qualifications. (2) is not meant to commit us to the a priori status of (*), but rather to the a priori status of the statement best expressed in our conceptual scheme by (*). That is, we are not a priori committed to the logical form of (*), since the statement expressed by (*) may be expressible in an alternative conceptual scheme where quantification (for example) is absent. (2) thus means that some statement in the conceptual scheme of said rational being should translate as a rejection of the AIR. Our most reasonable candidate is (*), but that is not an a priori requirement.

Second, (2) is not meant to imply that any rational being whatever should be able to express beliefs about the classical notion of truth. Rather, Putnam claims that (*) is generic, i.e., that it does not rely on a particular notion of either ‘truth’ or ‘statement’. The notions it uses are pretheoretic (and presumably compatible with a wide array of theoretical elucidations: intuitionistic, classical, epistemic, whatever).

Certainly, claiming that the notions of ‘true’ and ‘statement’ em- ployed in (*) are pretheoretical detours around the roadblock that disabled Putnam’s hypothetical. But, alas, it doesn’t really come to grips with what I called ‘the deep reason’ for the failure of the hypothe- tical. Putnam points out quite correctly that an objection to generic notions on the grounds that they are not regimented falls afoul of the fact that most of our working language is not regimented. This is true: we rely on unregimented notions all the time, and since, presently, no approach to the truth predicate is recognized as the right one, our ordinary use of ‘true’ is clearly unregimented also.

But this doesn’t give us carte blanche in our employment of ‘true’.

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In particular, when issues arise about what requirements rationality places on conceptual schemes, we cannot assume that it must be that ‘a rational being should see and be able to express the fact that the AIR is incorrect’. For suitable regimentations of the notion of truth may exclude just that very ability.

Unregimented truth clearly gives us (or some of us, anyhow) the impression that natural languages are universal. But this impression cannot be used to supply a requirement on rationality: not with paradox looming around the next corner. Notice this is the case even if we decide that the best regimentation of our conceptual scheme does allow the expression of the unacceptability of the AIR.18 For, currently, whatever logic we take as the best candidate for us does not exclude substituting something else later. More is called for than being our best choice if a principle is to be a priori.

4.

Intuitions that illegitimately rely on the purported universality of our conceptual scheme are not the only ones that contribute to the sensation that there is something a priori about principles of noncontradiction. Putnam (1983b) writes:

We can imagine all of the predictions of non-Euclidean physics coming true, even if we happen to be Euclidean physicists. But we don’t know what it would be like for all the predictions of the theory that consists of every statement together with its negation to come true. (p. 103)

This intuition, which is a strong one, turns on our apparent inability to understand what it would mean for contradictory instances to be true, say, ‘John is running’, and ‘it is not the case that John is running’. Considering such examples gives the impression that I pulled a sleight of hand when I invoked the rather theoretical universality considerations above, and applied them to the laws of logic. For the irrationality of contradiction, one might argue, lies not in the mere asserting of a law of logic which allows contradictory instances, but in the instances themselves allowed by that law. What experiences could one possibly have that would even tempt one to think that both a statement and its contradictory should be inferred?

Once the matter is put this way it is rather easy to flush out tempting possibilities. Imagine that we have a rather well-developed theory about

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a certain area. The theory predicts one thing, say, but observation yields the opposite; and so we are tempted to retain both claims.

Well, maybe not (temptations are notoriously matters of taste). We don’t settle for such contradictions - we tinker with theory and/or methods of observation until they no longer arise; and I have no argu- ment with this. But we could decide that, in fact, certain pairs of contradictory statements are true, and avoid vacuity by simply re- stricting our inference rules from operating on such instances. (See appendix for details.) Our not doing so thus seems to be a top-down methodological consideration, l~dt a matter of the incomprehensibility of a statement and its negation both being true.

But, the objection might arise, surely we couldn’t observe a contradic- tion, could we? We couldn’t observe, upon opening a box, that a sheet of paper is red and that same sheet is not red, could we? In fact, we know a priori that such a thing couldn’t happen, right? And this apart from general desires to avoid contradiction because, say, they make our inference schemes overrich.

Alas, invoking unimaginability here is simply a red herring. Certainly I can’t imagine seeing that a sheet of paper is red and that that same sheet is not red, but I can’t imagine seeing a sound either. Nevertheless, it isn’t a priori that sounds can’t be colors: one can imagine theory changing in such a way that we come to claim that certain colors are sounds. I don’t mean to suggest that something unimaginable would then happen (i.e., that au Churchland, we would then come to see sounds), but certainly we can’t take the mere unimaginability of some- thing to indicate its impossibility: theory is more subtle than that. This means that if we take the fact that we can’t imagine observing contradictions as indications that they are impossible, that is already the application of a methodological decision to possible experience: it really isn’t further evidence that such intuitions indicate the presence of something a priori.

5.

So far our analysis has been narrow in scope: we have concerned ourselves with the specific claim that (some version of) the principle of noncontradiction is a priori true. But even if what I have claimed above is convincing, it doesn’t explain why logical necessity fails to have a

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priori force, in the sense that I have not made explicit what it is about our evidential procedures for establishing logical truths that blocks their a priori status.

Of course logic is not the only source for purported a priori truths. Mathematics and semantics have been taken to be sources as well. And so a general examination of the epistemic basis for these subjects is in order too if one is to be (reasonably) convinced that there are no a priori truths to be found.

But first I want to draw the reader’s attention to a distinction that will shortly prove significant, namely that between truths which hold in every rationally accessible conceptual scheme, and truths which hold in our conceptual scheme but by virtue of methods independent of the empirical sciences. As we will see, although I think the former category is empty (which is what I mean here by saying ‘there are no a priori truths’), I think there are lots of truths in the latter category.”

Let us turn then to logic and mathematics. I have discussed else- where*” what I take the practice of logic and mathematics to be so I shall be more sketchy here: they are the study of algorithmically gen- erated systems (such as Peano Arithmetic or ZF) which are grouped together by truthlike predicates. Furthermore, application within a con- ceptual scheme to nonmathematical and nonlogical domains marks out which systems we take to be true: in the classical approach, we take as true a class of systems large enough to supply us with all the applications we need in science - e.g., the set of systems we take to refer to the standard model of ZF.

Truthtalk, having generated talk of the standard model, generates in turn a notion of implication which transcends any proof procedure tied to a particular system (except in nice cases where the proof procedure is complete). Taking the sentences Tarski-true in the standard model to be true, yields a class of true sentences that are ‘approximated’ by the systems we have in hand but never captured by them.

The algorithms used to generate these systems have two pertinent epistemic properties. The first is that such methods do not depend (epistemically) on the empirical sciences. By this I only mean that when we derive proofs for our mathematical claims, we do not rely on the empirical sciences to justify our beliefs that the proofs are good ones. Thus, although empirical scientific results may affect what mathematical statements we take to be true (because they may change which systems

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it is most convenient to apply to nonlogical and nonmathematical do- mains), they will not affect our perception of which mathematical sen- tences belong to which systems.

The other pertinent property is that the results of algorithmic pract- ices are always corrigible. By this I simply mean that we can always fail to apply an algorithm correctly: we can be wrong in our claim that a statement belongs to a system.

Now just as talk of truth forces us to transcend particular systems to the standard model they are systems of, similarly, algorithmic neces- sity - the independence of mathematical practice from the empirical sciences, is transcended by a notion of necessity tout court. Anyone who wishes to can think of mathematical objects as not in space and time, as unchanging and eternal. But this is just a pictorial representa- tion of the independence of mathematical practice from empirical science. For example, it is merely talk of the standard model and that model’s necessity - notions generated by our talk of truth and Godel’s theorem from systems and algorithmic derivation.*’

Now here is one possible objection to the claim that some mathema- tical statements are a priori: we have conceded that results of algorithms are corrigible. But this means that any a priori truth generated on the basis of an algorithm must be corrigible too. But what epistemic justification is there for a notion of ‘a priori true’ which allows state- ments with such properties to be ones we could be wrong about? Putnam (1983b, pp. 135-136) claims that such a notion of a priori truth is a consistent position, and one which is sane and modest. Maybe so, but if we are unsure of all statements, a priori and not a priori, what is the (epistemic) difference between them supposed to be? Degrees of sureness? Incidentally, this objection is meant to fault the suggestion that either of two classes of statements can be a priori true: mathema- tical statements, or metamathematical statements that describe which mathematical statements belong to which systems.

Here’s my counterargument. The way one establishes there are a priori truths (if one can) is by analyzing our methods of gathering evidence, and showing that such methods force the existence of certain true statements.*’ Then we are sure of a priori truths subject to two qualifications. First, that we don’t change our methods of gathering evidence. And second, that we haven’t made a mistake in our claims about how we do gather evidence. In other words, statements we are unsure of because our methods of gathering evidence for them do not

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yield conclusive verdicts are different from statements we are unsure of (despite the fact that our methods do yield conclusive verdicts) because we may subsequently decide to modify our methods or because we decide that we were mistaken about what our methods yield.23

So, to sum up, my claim is that the objection above is invalid because it conflates the different ways we can be unsure of the truth of a statement, ways of being unsure which are epistemologically distin- guishable and epistemologically significant.

Nevertheless, despite my dislike of this objection to the denial that algorithms can be a source of a priori truths, I don’t think they can, anyway. First the easy part: in conceding the possibility of changing the mathematical and logical systems we apply to the empirical sciences, we have ruled out a priori true status for any mathematical or logical statement. This leaves the metamathematical statements to the effect that certain statements belong to certain systems. In arguing, first, that these claims are not affected by (empirical) scientific considerations, and second, that issues of corrigibility are irrelevant, what avenue is left to me to deny their a priori true status?

This. As we saw in our discussion of the principles of noncontradic- tion above, we cannot guarantee that any rationally accessible concep- tual scheme must have the resources to describe any particular hypothe- tical to the effect that statement A belongs to system B.

6.

I want to draw the same conclusion about analyticity and a prioricity as I have drawn about logical and mathematical necessity and a priori- city. But the argument must go a little differently. Let us view analytic truths as sentences that are true by virtue of the properties of the language they are sentences of. Furthermore let us assume that one mark of understanding a language just is knowing these analytic truths. Then, there are two versions of the claim that such truths must be a priori as well.

Version 1: There is a set of properties P such that any language L utilizable by a rational being must have P, and consequently any rational being must know the set of analytic truths A that hold in L by virtue of P.”

Version 2: There is a set of properties P such that any language L utilizable by a human being must have P, and consequently any human

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being (who knows a language L) must know the set of analytic truths A that hold in L by virtue of P.

Version 1 is simply wrong for reasons we have already seen. We can concede that a possible language must have certain properties in order to be utilizable by a rational being, but this only bears on the a prioricity of analytic truths, if such truths are about possible languages (I am assuming that there are no other necessary facts about rational beings that such beings must know). But we have already seen that universality is not a requirement in any case (since it is achieved only at the expense of paradox), and consequently it is wide open in what ways (and how well) a language can describe its own properties.

Version 2, on the other hand, is an appealing gloss on what it means for an analytic sentence to be a priori for several reasons. First, it can be argued that pure rationality is so unconstrained and vague a notion that we lack a clear understanding of the limitations it offers.

Second, it can be argued that to shift to considerations of pure rationality is to lose sight of the significant epistemic constraints that we humans must labor under. For example, perhaps some rational beings have no need of algorithms or sense experience in order to gather knowledge (one purported rational being is rumored not even to have the need to gather knowledge). But, it could be argued, to invoke a notion like this is to practice supernaturalized epistemology: rather than do this, we must take account of the limitations of biology and technology insofar as these bear on our epistemic practices. Limi- tations of biology, for example, could trap us in the genetically neces- sary utilization of natural languages which have peculiar semantic prop- erties - if such peculiar properties implied the existence of certain analytic truths occurring in all natural languages, for all practical (natu- ralized) purposes, such truths would be a priori.

There is very natural objection to version 2 that is a companion objection to one we’ve seen before. It goes this way: whatever the results of science are, they must be corrigible. But this means that any a priori truths generated on the basis of scientific investigation must be corrigible too, and so not a priori after all. As before, we defuse this objection by noting that it depends on a conflation of two different levels: the level at which we study our methods of evidence-gathering, and the level at which we apply them.

Nevertheless, version 2 still won’t do. I have conceded its vision of

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epistemology in the sense that we must rule out powerful means of evidence-gathering if they are beyond our biological and technical pow- ers. But to use version 2 to establish the existence of a priori truths, one must do something rather different: one must argue that the presence of certain statements is required in every natural language. But what statements could these be? There are two possibilities: statements about the natural languages themselves, or statements about something else.

Let’s take the first possibility. It is hard to see what requirements natural languages (on their own) place on self-reference. One reason is that natural languages seem to demand too much, so much so that it has occurred to more than one thinker that they are simply semantically inconsistent.*’ What strengthens this impression is the very slight re- sources needed to generate paradox. Most likely, whatever restrictions placed on natural languages to avoid paradox will be generated by analysis of optimal formal models, and not by nativist linguistic con- siderations. This makes version 2 inapplicable here.

What then, about statements about other things? This is also deeply implausible. To show it, one would have to exhibit a vocabulary re- quired in every natural language. Then any terms definable from that vocabulary would be a source not only of analytic truths but a priori ones as well. But what could this vocabulary be? It cannot be terms of sense-experience because, (1) none of those terms are required of human languages (humans could be blind or deaf, but still speak a language in which terms referring to visual or aural phenomena are absent), and (2) other terms are not definable in such terms anyway.26

I stress again that I am not arguing here against (or for) the claim that there are analytic statements; I am arguing that no such statements are a priori true, in the sense that such statements must be expressible or true in every natural language.

It is worth pointing out that this doesn’t mean that the epistemic status of such statements is empirical in the sense that scientific laws are empirical. Just as I have argued that the application of algorithms to recognize that mathematical sentences belong to systems is, although a corrigible procedure, independent of results in the empirical sciences, so too is the recognition of analytic truths, if there are any. Epistemol- ogy must get away from the simple sorting of statements into those knowledge of which ‘depends’ on ‘experience’ and those knowledge of which is not so ‘dependent’ or the simple rejection of such a distinction.

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CONCLUSION

There are several epistemic distinctions among truths that I have argued for in this paper. First, there are those truths which hold of every rationally accessible conceptual scheme (class A truths). Second, there are those truths which hold in every rationally accessible conceptual scheme (class B truths). And finally, there are those truths whose truth- value status is independent of the empirical sciences (class C truths). The last category broadly includes statements about systems and the statements they contain, as well as statements true by virtue of the rules of language itself.

At the risk of anachronism, I’ll describe the positions of Carnap (1956); Quine (1980); Grice et al. (1956); the various Putnam’s and myself in terms of the above distinctions: both Carnap and Quine (pretty much) think there are no class A or class B truths. Both Putnam (1975) and Putnam (1983~) think there are class A and class B truths, and that these classes overlap. I deny there are class B truths but affirm the existence of class A truths (although I haven’t given explicit examples of the latter here). Finally, everyone here but Quine (1980) thinks there are class C truths (of one sort or another). Putnam (1975) attempts to show that certain class C truths are simultaneously class A and class B truths. Grice et al. (1956) take pains to distinguish the claim that there are class C truths from the claim that there are class A truths, and claim (against Quine, 1980) that no argument showing there are no class A truths shows there are no class C truths.

On my interpretation of Quine (1980) he thinks that the nonexistence of class A truths shows there are no class C truths - given the extra bit of argument that a notion of ‘true by convention’ or ‘true by virtue of meaning’ without epistemic content, is a distinction without signifi- cance. But that issue, which is the one Grice et al. (1956) are concerned with, has not been the focus in this paper - and so in a sense I have shifted the terms of the original debate.

Here I have been primarily concerned to distinguish epistemic notions and sort out how and in what ways they relate to each other. A primary tool in this exercise has been the explicit recognition that formal models of truth make universality an unlikely property of our conceptual schemes. If I have not convinced anyone that the epistemic notions sort out the way I think they do, I hope at least that some burden-shifting has occurred: that philosophers do not either take it for granted that

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certain notions must be expressible in any conceptual scheme or treat the fact that conceptual schemes must be (in some sense) limited as of little (philosophical) moment.

On the other hand, if I am right about the epistemology, it follows that previous attempts to mark out the necessary structures in rationally accessible conceptual schemes via a priori truths is hopeless. What I think must replace their role, what I call globally incorrigible sets of sentences, is a topic for another time.

APPENDIX

A (very) simple method of changing the truth value of any sentence without (very much) impairing the utility of the conceptual scheme it is in:

(1) (2)

Reverse the sentence’s truth-value. Rewrite inference rules so that none of them apply to the sentence.

A caveat. Clearly this method is generalizable to any syntactically characterizable class of sentences. If the class of sentences is small enough, the conceptual scheme is only minimally impaired (and one cannot rule out an overall gain in simplicity, given changes elsewhere in the conceptual scheme). That is, the resulting conceptual scheme is rationally accessible. On the other hand, change enough sentences and the rational accessibility of the result cannot be guaranteed.

Another caveat. In pressing this example, one must not confuse a conceptual scheme with a postulate system. A postulate system may have a fixed number of sentences as premises and one or two inference rules (including, possibly, substitution). Looked at this way, the sugges- tion that the truth value of any sentence is up for grabs may seem absurd, since it could be a postulate in such a system (and cost us too many sentences). But what must be always considered is what the total class of sentences being excluded is. Generally speaking, we can use any number of equivalent postulate systems to describe the same conceptual scheme and care must be taken not to let the parochial character of a particular postulate system fool us into overestimating the centrality of a particular instance of a logical truth merely because it is central to that postulate system.

Yet another caveat. Whether we regard the propositions expressed

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by the sentences outright excluded from expression in our subsequent conceptual scheme will depend on whether we can find a substitute which ‘expresses the same proposition’.

One final caveat. In claiming that the above method yields rationally accessible conceptual schemes when applied to sufficiently small sets of sentences, I am not obliged to actually describe situations where the optimal conceptual scheme would be one of this sort. But it is worth pointing out that the one thing which prevents this sort of method as a solution to liar’s paradoxes in self-referential languages is that the class of viciously self-referential sentences is not syntactically charac- terizable.

NOTES

* My thanks to Arnold Koslow and Mark Richard for their helpful suggestions. I also want to thank the City University of New York Graduate Center for inviting me to be a visiting scholar academic year 1989-90, during which time this paper was written. While there I was partially supported by a Mellon fellowship from Tufts University, for which I am grateful. i This at least was my experience wherever I heard philosophers express informal criti- cisms of the views expressed in Quine (1980). One also sees the same intuitions at work in (some) objections to the claim that logical truths are the results of convention. Of course this is not so of Quine’s objection to the latter doctrine. ’ For example, Katz (1979). 3 See, e.g., Priest (1979) and the Journal of Philosophical Logic 13 (1984), for other articles. ’ I have in mind Putnam (1983a, 1983b and 1983~). These articles depict several ‘flip- flops’ in position. Putnam’s explicit reason for exposing his vacillation in print is the ‘metaphilosophical’ one that philosophers rarely expose their indecision this way. But he does eventually settle down on the claim that there are, indeed, absolutely a priori truths. ’ In Azzouni (1990) I introduced a notion of ‘a priori’ which roughly meant ‘algorith- mically generated and independent of the empirical sciences’. This notion was a terminol- ogical innovation based on my attempt to capture something of what traditional theorists were after when they called mathematical truths ‘a priori’. Since I regarded the traditional notion as bankrupt, I thought such an innovation appropriate. I now think this was rash because my current attempt to discuss various threads of the traditional notion is complicated by my previous terminology. So, in this paper, I will use the term ‘a priori’ pretty much as, say, Putnam (1983~) does. Other notions will be described explicitly (e.g., ‘algorithmically generated and independent of the empirical sciences’). 6 It may seem that Davidson (1974) using broadly Quinean principles, has ruled out the coherence of the kind of revision I describe. On the contrary. See my Ineffability, forthcoming. I should also add that my claim does not presuppose a notion of ‘meaning’ robust enough to apply across possible conceptual schemes (although it is compatible with such a possibility). ’ I understand rationality (vaguely enough) as the set of epistemic practices one should

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have. We picturesquely capture rationality ‘extensionally’ via the notion of an epistem- idly possible conceptual scheme. This class is identified from the vantage point of the conceptual scheme we currently hold. Two caveats: first. these conceptual schemes must not be idealized by taking them to be closed under implication. Doing so obscures significant facts, e.g., that there are mathematical truths we don’t know. But the concep- tual schemes aren’t finite eirher. for we ‘know’ an indefinite number of implicit but ‘obvious’ truths. Second. in modal logic, generally, one distinguishes between the entire set of possible worlds, and that set of possible worlds accessible from the actual world. A similar distinction is pertinent here. Since we are concerned with ‘what we can and cannot be wrong about’. and not general considerations about epistemic logic, we consider only those epistemically possible worlds accessible from ours. I call them the rational1.y accessible corzcepmal schemes. A notion of epistemic possibility close to mine may also be found in Putnam (1983~). ’ This was seen as an implication of Quine’s claims pretty early on. Grice et al. (1956) write: “The point of substance (or one of them) that Quine is making, by this emphasis on revisability. is that there is no absolute necessity about the adoption or use of any conceptual scheme whatever.. .“. (p. 211) But Grice et al.. do not see this claim as incompatible with analytic truths. where they gloss the latter (somewhat as Carnap does) as necessities within conceptual schemes. See Sections 5 and 6 below where I interpret such ‘necessities’ as truths whose evidence-procedures are independent of the empirical sciences. ’ Putnam (1983d) claims. .‘. the ‘normative’ becomes for Quine the search for methods that yield verdicts that one oneself would accept“. I” I draw the boundary between logic and mathematics. as Quine does. letting the former extend as far as the first order predicate calculus takes us. This allows the description of the various set theoretical possibilities compatible with the standard predicate calculus as ‘logically possible’. But this dividing line is controversial and nothing I will say here turns on it. ” A notion of analyticity like this is explored by Smith et al. (199?). Putnam (1975) contemplates a much more limited version of the same notion. ” Quine takes a dim view of notions of logical and mathematical possibility in any case. See Quine (1986b). And. for him. what corresponds to the sensation of a prioricity is rermdiry, the latter always a matter of degree, and never absolute in its impact. Finally. Quine stresses that even if some truths are learned in the process of learning words (i.e., are analytic), they “are fitted into a system without regard to individual pedigree” (Quine, 1986a. p, 95). ” They may have a lot of sentences in common. but does that imply that they have

statements in common? On one view, at least, all the connectives in intuitionism have different meanings from those in classical logic - and this means that none of the statements involving intuitionistic connectives are expressible in the classical context and vice versa. Of course Quirze would look askance at using meaning this way to defend his claim. Good enough: in what follows I try to show that views about the legitimacy of rnearktg (and consequentlv arza/yficiiy. understood linguistically) are irrelevant to views ;,bout the existence of a priori truths (understood epistemologically).

Well. that is at least what we are prone to say before epistemological worries about how we gain access to mathematical objects drives us to positions such as Structuralism. I am only offering an intuitive contrast between mathematical laws and logical laws and so won’t get further into this topic now.

344 JODY AZZOUNI

” For details, see Quine (1970, Chap. 4). I6 The literature is this area is enormous. Apart from the citations in note 3 above, and the absolutely fundamental Tarski (1983) much of the literature may be found in Martin (1984). More recently, there is Barwise et al. (1987) and McGee (1989). For an explicit discussion of the impossibility of a ‘universal’ language, see Herzberger (1980-81). See also my 1991.

I should stress that the truth predicate is usually blamed for the threat of paradox. But, contrary to this impression, there is a global problem here which may be solved by tinkering with the truth predicate, or the logic, or the capacity of the language to describe its own syntax. At present it is simply wide-open which strategy or combinations of strategies is the best way to go. I7 Why? Not for any particularly deep reason. Such a theory makes no distinctions at all - it is pragmaticully useless to us. This is not a property peculiar to theories which house contradiction. A theory which has no implications (imagine the only inference rule admitted is the null one) will be equally useless. Nor do all theories which countenance contradiction have this property as the literature cited in note 3 illustrates. ‘* Does the Tarskian approach allow “a fully rational being to see and be able to express the fact that the AIR is incorrect”? Superficially, at least, it doesn’t seem to. At any stage in the hierarchy, one’s truth predicates only range over a portion of the statements pertinent. But. perhaps a schematic understanding of how to iterate the Tarskian hier- archy suffices for “being able to see” the truth of (*) - although I must glumly point out that there are technical problems with iterating said hierarchy. Being able to express the truth of (“), however, is a different matter. Perhaps reflection principles of some sort are available. perhaps not. I9 Furthermore. I suspect the tenacity of the traditional notion of the ‘a priori’ is due to its being (at times) a fuzzy amalgam of both notions. “I Azzouni (1990). ” McGee (1989,) seems to have a different picture. He argues that we should treat logical necessity as derivability in some explicitly specifiable system of rules. Doing so makes the notion amenable to analysis in modal terms. He goes on to add, “[tlhere is not, in fact, a single forma1 system such that our informal notion of logical necessity coincides with derivability in that formal system. Instead there is a whole family of formal systems derivability in each of which is a reasonable candidate for how to make our informal notion precise” (p. 45).

McGee essentially invokes vagueness to explain the gap incompleteness breeds between e.g., any formal system of arithmetic and the standard model. By contrast, I see G as representing derivation (which is generally how it is taken). I distinguish logical and mathematical necessity, and argue that the latter derives from the standard model.

I think the differences described here are more than terminological. McGee’s view treats the informal notion of ‘arithmetically true’. say, as something to be ‘precisified’ by one or another forma1 derivational system. But this seems to overlook how the formal systems are limited: they are incomplete, not just vague, and our notion of trmh plus GGdelian methods enable us to pick out sentences the informal notion covers which the formal system does not. We can thus augment these systems without limit. This, to me at least, suggests a deep disanalogy with genuinely vague terms as they occur in natural languages. In particular, the use of ‘true’ to generate instances of mathematical truths which are not derivable (in a particular system) is mathematically quite precise.

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I’d rather say that mathematical necessity is unformalizable than try to reduce it to formal notions of derivability; 2x For example, suppose a ‘method’ of gathering evidence for truths is determining whether they are intuitively irrefutable (e.g., ‘clear and distinct’). Presumably, this method forces the truth of ‘I think. therefore I am’, although not ‘there are material objects’. ” For example, many proofs in mathematics are proof-sketches. They are arguments to the effect that certain proofs exist. Such sketches are (alas) sometimes wrong: no such proof exists.

On the other hand, we could decide that the Axiom of Choice is unacceptable. ” For example, a pertinent property might be that certain ‘concepts’ must be expressible in the language L. and the analytic truths therefore would be sentences expressing the meaning of those concepts. ” This is not an outlandish idea. Languages can be semantically inconsistent but still utilizable pragmatically. Consider a language with the inference rules of the standard predicate calculus, plus the rule: infers, where $ is an inconsistent sentence. This language is quite utilizable provided we judiciously avoid the last inference rule. Arguably, natural languages are similar: we judiciously avoid applying inference rules to self-referential sentences which will yield contradiction. ” Thus the failure of the various species of phenomenalism.

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Manuscript submitted November 6, 1990 Final version received October 24, 1991

Dept. of Philosophy Tufts University Medford, MA 02155 USA