Perhaps the most pervasive conviction within the Western epistemological tradition is that in order for a belief to constitute knowledge it is necessary (though not sufficient) that it be epistemically justified: that the person in question have a reason or warrant which makes it at least highly likely that the belief is true. Historically, most epistemologists have distinguished two main sources from which such justification might arise. It has seemed obvious to all but a very few that many beliefs are justified by appeal to one’s experience of the world. But it has seemed equally obvious to most that there are other beliefs, including many of the most important ones that we have, which are justified in a way which is entirely independent of any appeal to such experience, justified, as it is usually put, a priori, by reason or pure thought alone.

In his 1970 book Philosophy of Logic, Quine propounds what he calls ‘the deviant logician’s predicament’: when a reformist logician tries to deny a law of classical logic, he succeeds only in changing the subject. This position, summed up in the aphorisms’ deny the doctrine and change the subject’ and’ an illogical culture is a mistranslated one,’ has struck many of Quine’s readers as backsliding. The old Quine denied that any statements whatever are analytic in the sense of being true solely in virtue of what they mean; the new Quine holds that certain laws of logic cannot be denied without changing the meanings of the logical connectives. Does not the position of the new Quine invest logical laws precisely with the status ‘true solely in virtue of what they mean’?

The analytic/synthetic distinction has played a central role in discussions of a priori knowledge throughout the twentieth century. One of the primary reasons for the prominence of this distinction is the widespread influence of the tradition of logical empiricism which endorsed the following principles:

(LEl) All analytic propositions are knowable a priori

and

(LE2) All propositions knowable a priori are analytic.

Hence, proponents of the a priori often argue in support of the contention that the propositions of a particular discipline, say mathematics or logic, are knowable a priori by arguing that it consists solely of analytic propositions. On the other hand, detractors of the a priori often reject such knowledge on the grounds that the analytic/synthetic distinction is not cogent. My primary goal in this paper is to challenge the prevalent acceptance of (LE1).

Mathematics has become so sophisticated that it is easy to lose touch with the mathematical realities out of which it arises. Many nowadays think of mathematics as complicated games with forests of symbols. Yet this is an image which could not have got a grip on the ancient mathematicians, like those of the Pythagorean brotherhood, who stood nearer to the roots of those forests. Now we are lost in the higher branches of mathematics, and we sometimes forget that all these growing branches are nourished ultimately by roots far below in down-to-earth reality.

The early Pythagoreans discovered things which are thus and so, and which could not have been otherwise. Yet although these things could not have been otherwise, the Pythagoreans who discovered them could have done otherwise. Hence the truths which the Pythagoreans discovered are things which would still have been so even if no one had ever discovered them.

If α and α’ are distinct variables and ϕ and ϕ’ are open sentences of some language, where ϕ’ is the result of replacing one or more free occurrences of a in α with free occurrences of α’ in ϕ’, then a universal closure of ⌜(α=α’ → (ϕ → ϕ’))⌝, is an indiscernibility principle of that language. For instance, (1) is an indiscernibility principle.

The existence of opaque constructions falsifies the familiar unrestricted principle of substitution which affirms that co-referential expressions are intersubstitutable in all contexts without change of truth-value. But indiscernibility principles are another matter. Not every counter-example to the unrestricted principle of substitution is a counter-example to some indiscernibility principle. Indeed, it is likely to be thought that there is no counter-example to any indiscernibility principle, and that the semantics of variables and objectual quantification ensures that all indiscernibility principles are true.

The modem rationalist tradition initiated by Descartes has as one of its central tenets the independence of the human understanding from the senses. Regardless of the different ways in which independence from experience is understood, there is much common ground among the modem views on the a priori. Yet Kant, culminating this tradition, introduces an entirely new conception of the a priori never before articulated in the history of philosophy. This is the notion of elements in knowledge which are independent of experience but nevertheless closely connected, in a special way, with experience.

Although for Kant the a priori has a privileged position in the structure of knowledge - as it has for other modem rationalist philosophers - one of the most striking, and often neglected, aspect of his conception of the a priori is the great extent to which it is opposed to foundationalism.

In the second edition, Kant summarized the question behind the Critique of Pure Reason this way: ‘How are a priori synthetic judgments possible?’ (B19) We can easily understand his interest in synthetic judgments; he thought that analytic ones could not tell us anything new (A5-6=B9). There are only two ways to get judgments that are analytic: by drawing out what is contained in our concepts and by combining the resulting propositions inferentially into arguments. Neither could ever tell us anything not already ‘thought in [the concepts we have used], though confusedly’ (A7=B10-ll), and even if either could, it could not give us anything against which to test it for truth or falsity. ‘In the mere concept of a thing no mark of its existence is to be found’ (A225=B272; cf. Bxvii-xviii). In the search for knowledge, analytic judgments get us nowhere.

Contemporary empiricism is closely allied with naturalism. Not only do empiricists hold that all our knowledge is based upon sensory experience, but they also tend to offer some sort of causal account of how this experience comes about. The causal ingredient in knowledge seems very plausible — after all, my knowing that there is a tea cup on my desk is based on sense impressions which are caused by the cup itself. Photons come from the cup to my eye; a signal is then sent down the optic nerve into the visual part of the brain, and so on. And without that causal process, I likely wouldn't have the knowledge that I do have.

Consider the following sentence schema:

This sentence entails that ϕ.

Call a sentence which is obtained from this schema by the substitution of an arbitrary, contingent sentence, s, for ϕ, the sentence CS (for ‘Curry’s Sentence’). Thus,

(CS) This sentence entails that s.

Now ask the following question: Is CS true?

One sentence classically entails a second if and only if it is impossible for both the first to be true and the second to be false. Thus ‘Xanthippe is a mother’ entails ‘Xanthippe is female’ if and only if it is impossible for both ‘Xanthippe is a mother’ to be true and ‘Xanthippe is female’ to be false. CS makes a claim about a purported entailment. Thus, CS is true if and only if it is impossible for both the sentence it mentions as entailing a second to be true and the sentence it mentions as being entailed by the first to be false. In other words, CS is true if and only if it is impossible for both CS to be true and s to be false. In yet other words, CS is false if and only if it is possible for both CS to be true and s to be false.

At the one extreme stands Hume, who -at least according to common conceptions - rules out entirely the possibility of material or non-logical necessity, and who therefore rules out also the possibility that we might enjoy that sort of certain knowledge that earlier philosophers had assumed as a matter of course to be correlated therewith. At the other extreme stands Adolf Reinach, the hero of our present story, who defends the existence of a wide class of material necessities falling within the domain of what can be a priori known. More precisely, Reinach holds that there are certain categories of entity whose factual instantiation brings with it as a matter of necessity the instantiation of certain other correlated categories. The instantiation of the category color necessitates in this fashion the instantiation of the category visual extension.