A Priori and a Posteriori
A Priori and a Posteriori
A PRIORI AND A POSTERIORI
The distinction between the a priori and the a posteriori has always been an epistemological one; that is to say, it has always had something to do with knowledge. The terms a priori and a posteriori are Scholastic terms that have their origin in certain ideas of Aristotle; but their use has been considerably extended in the course of history, and their present use stems from the meaning given to them by Immanuel Kant. The terms literally mean "from what is prior" and "from what is posterior." According to Aristotle, A is prior to B in nature if and only if B could not exist without A; A is prior to B in knowledge if and only if we cannot know B without knowing A. It is possible for these two senses of "prior" to have an application in common; substance, for example, is prior to other things in both of these senses and in others. It follows that to know something from what is prior is to know what is, in some sense, its cause. Aristotle believed that it is possible to demonstrate a causal relationship by means of a syllogism in which the term for the cause is the middle term. Hence, to know something in terms of what is prior is to know it in terms of a demonstrable causal relationship. To know something from what is posterior, on the other hand, can involve no such demonstration, since the knowledge will be inductive in form.
The transition to Kant's conception of the matter is evident in Gottfried Wilhelm Leibniz. According to the latter, to know reality a posteriori is to know it from what is actually found in the world, that is, by the senses, by the effects of reality in experience; to know reality a priori is to know it "by exposing the cause or the possible generation of the definite thing" (Nouveaux Essais, Bk. III, Ch. 3). It is also possible to speak of a priori proofs. As a general consequence of this, Leibniz could distinguish between "truths a posteriori, or of fact," and "truths a priori, or of reason" (ibid., Bk. IV, Ch. 9); for a priori truths can be demonstrated in terms of their being based on identical propositions, while a posteriori truths can be seen to be true only from experience. Thus the distinction between the a posteriori and the a priori comes to be a distinction between what is derived from experience and what is not, whether or not the notion of the a priori also has the notion of demonstration in terms of cause or reason associated with it. Such is the distinction in Kant, and it has remained roughly the same ever since. Since in Kant there is no simple opposition between sense experience and reason (there being also the understanding), it is not possible to express the distinction he laid down as one between what is derived from experience and what is derived from reason.
The distinction, then, is roughly equivalent to that between the empirical and the nonempirical. Kant also connected it with the distinction between the necessary and the contingent, a priori truths being necessary and a posteriori truths contingent. But to assume without further argument that the two distinctions coincide in their application is to assume too much. The same is true of the distinction between the analytic and the synthetic; this too cannot be assimilated without argument to that between the a priori and a posteriori. Whether or not these distinctions coincide in their applications, they certainly cannot have the same meaning. The distinction between the a priori and a posteriori is an epistemological one; it is certainly not evident that the others are.
The Distinction Applied to Concepts
The distinction between the a priori and the a posteriori has been drawn not only in connection with truths or propositions but also in connection with concepts. Indeed, some truths are doubly a priori; not only is their truth knowable independently of experience but the concepts that they involve are similarly independent of experience. The distinction between a posteriori and a priori concepts may seem a perspicuous one, for it may be thought to be a distinction between concepts that we derive from experience by building them up therefrom and concepts that we have independently of experience. It has sometimes been said also that the latter concepts are innate ideas, with which we are born, so that we have no need to acquire them. But the question whether ideas are innate or acquired seems to be one of psychology, as is the question how we acquire ideas if we do. The distinction under consideration, being an epistemological one, has no direct connection with psychology. A concept that is independent of experience may or may not be innate; and although it cannot be acquired directly from experience, it may still be that experience is in some way a necessary condition of our having the concept. What then does it mean to say that a concept is independent of experience? The answer must be in terms of the validation of the concept.
It may be assumed for present purposes that a concept is what is meant by the corresponding term (although this may not be a fully adequate view and bypasses the question whether concepts are independent of words). To have a concept will thus at least be to understand the corresponding term. Perhaps, then, an a posteriori concept is one expressed by a term understandable purely in terms of experience, and an a priori concept one that does not satisfy this condition. The point has sometimes been made by saying that an a posteriori, or empirical, concept or term is one that is cashable in terms of sense experience. This is of course a metaphor, and what it means is that the meaning of empirical terms can be given by definitions that must ultimately depend on ostensive definitions only. Ostensive definitions are those which provide the definition of a term by a direct confrontation with experience. To define a term ostensively it is necessary only to repeat the expression together with some form of pointing to the object or phenomenon in question. It is highly questionable, however, whether any performance of this kind could ever constitute definition as such. For the meaning of a word to be taught in this way there would have to be (as Ludwig Wittgenstein in effect pointed out at the beginning of his Philosophical Investigations ) a previous understanding that the noise made was a word in a language and in a language of a definite sort. Furthermore, it would have to be understood what sort of term was being defined—whether it was descriptive and, if so, what range of phenomena it was being used to describe. If all this must be understood, it can scarcely be said that the term in question is defined purely by reference to sense experience.
Nevertheless, there is some distinction to be made here. Even if such terms as "red" cannot be defined purely by reference to experience, they could not be understood fully without experience, for example, by someone who does not possess and never has possessed sight. There is a sense in which the blind can, up to a point, understand terms such as "red," in that they can know that red is a color and even a color of a certain sort related to other colors in certain ways. But since they cannot know when to apply the term in fact, there is an obvious sense in which they do not have a full understanding of it—and the same applies to the notion of color itself. A posteriori terms and concepts may thus be defined as those that directly require our having experience in order for us to apply them or those that can only be fully understood by reference to terms that directly require our having experience to apply them. Whether or not a creature without experience could ever come to have a concept such as, for example, validity, it is clear that being able to apply the concept does not directly require experience. This may afford the basis of a distinction between a posteriori and a priori concepts. There may be various views about a priori concepts, concerning, for example, whether they are to be restricted to concepts of, or concepts involved in, mental operations on a posteriori concepts. Empiricists have in general held that the only a priori concepts are those that express relations of ideas. The field is thus restricted to the concepts of logic and mathematics.
The Distinction Applied to Propositions
In a sense, the distinction between concepts presupposes the distinction between propositions, since concepts can be applied only in propositions. According to the rough criteria already mentioned, an a priori proposition will be one whose truth is knowable independently of experience. It may be questioned, however, whether there are any truths that can be known if the subject has no experiences whatever. Hence, the matter is better put in terms of the validation of the proposition in question, in terms of its verification or falsification. It has sometimes been suggested that a proposition is a priori if its truth is ascertainable by examination of it alone or if it is deducible from such propositions. An a priori proposition would thus be one that provides its own verification; it is true in itself. This account is too restrictive, since there may be propositions whose truth is ascertainable by argument that makes no reference to empirical matters of fact, but that may not be deducible from any propositions of the kind previously mentioned. That is to say, there may be circumstances in which it is possible to validate propositions by argument that makes no reference to matters of fact discoverable by experience. Empiricists have generally denied this, but the possibility of what Kant called "transcendental arguments" cannot be so lightly dismissed. Aristotle's argument for the truth of the principle of contradiction would be a case in point, namely, that a denial of it already presupposes it.
On the other hand, to say simply that a priori propositions are those whose truth can be discovered without reference to experience is too wide a definition. For it may be argued that the terms in which many such propositions are expressed could only be fully understood by reference to experience. A proposition may be a priori without its involving terms that are without exception a priori. It was for this reason that Kant distinguished between a priori and pure a priori judgments; only in the latter are all the terms a priori. In view of this, an a priori proposition may be defined as one whose truth, given an understanding of the terms involved, is ascertainable by a procedure that makes no reference to experience. The validation of a posteriori truths, on the other hand, necessitates a procedure that does make reference to experience.
can analytic propositions be a posteriori?
It has already been mentioned that Kant superimposed upon the a priori–a posteriori distinction the distinction between the analytic and the synthetic. There are difficulties involved in defining this latter distinction, but for present purposes it is necessary to note that Kant assumed it impossible for analytic judgments to be a posteriori. He does this presumably on the grounds that the truth of an analytic judgment depends upon the relations between the concepts involved and is ascertainable by determining whether the denial of the judgment gives rise to a contradiction. This latter procedure is surely one that makes no reference to experience. Kant is clearly right in this. As already seen, it is not relevant to object that since analytic judgments, propositions, or statements need not involve purely a priori terms, evaluation of the truth of some analytic propositions will involve reference to experience; for in determining whether a proposition is a priori, it is necessary to take as already determined the status of the terms involved. It is similarly irrelevant to maintain that it is sometimes possible to come to see the truth of an analytic proposition through empirical means. It may be possible, for example, for a man to realize the truth of "All bachelors are unmarried men" as an analytic proposition as a consequence of direct experience with bachelors. But this consequence will be an extrinsic one. That is to say that while the man may attain this insight in this way, it will be quite accidental; the validity of the insight does not depend upon the method by which it is acquired. That is why the definition of an a priori proposition or statement involves the idea that its truth must be ascertainable without reference to experience. As long as a nonempirical procedure of validation exists, the proposition in question will be a priori, whether or not its truth is always ascertained by this procedure. It is quite impossible, on the other hand, for an a posteriori proposition to be validated by pure argument alone.
must a posteriori propositions be contingent?
Given that all analytic propositions are a priori, it is a further question whether all synthetic propositions must be a posteriori. This is a hotly debated question, with empiricists maintaining that they must be. But first it is necessary to consider the relation between the a priori–a posteriori dichotomy and the necessary–contingent one.
Kant certainly associated the a priori with the necessary, and there is a prima facie case for the view that if a proposition is known a posteriori, its truth must be contingent. For how can experience alone tell us that some thing must be so? On the other hand, it might be maintained that we can learn inductively that a connection between characteristics of things holds as a matter of necessity. Some philosophers maintain that natural laws represent necessary truths, and they do not all think this incompatible with the view that natural laws can be arrived at through experience. What is sometimes called intuitive induction—a notion originating in Aristotle—is also something of this kind; we see by experience that something is essentially so and so. An even greater number of philosophers would be willing to assert that, in some sense of the word "must," experience can show us that something must be the case. Certainly the "must" in question is not a logical "must," and empiricists have tended to maintain that all necessity is logical necessity. This, however, is just a dogma. It seems plausible to assert that an unsupported body must in normal circumstances fall to the ground.
Yet it must be admitted that the normal philosophical conception of necessity is more refined than this, and to say that an unsupported body must in normal circumstances fall to the ground need not be taken as incompatible with saying that this is a contingent matter. Similarly, there is an important sense in which natural laws are contingent; they are about matters of fact. If we also think of them as necessary, the necessity in question stems from the conceptual framework into which we fit them. It is possible to conceive of empirical connections in such a way that, within the framework of concepts in which we place them, they are treated as holding necessarily. It is still a contingent matter whether the whole conceptual framework has an application. If propositions expressing such connections are a priori, it is only in a relative sense.
must a priori propositions be necessary?
It seems at first sight that there is no necessity for nonempirical propositions to be necessary, or rather that it is possible to construct propositions which, if true, must be true a priori, while they apparently remain contingent. These are propositions that are doubly general. They may be formalized in such a way as to contain both a universal and an existential quantifier, for example, (x ) · ∃ y · ϕ xy. Such propositions have been called by J. W. N. Watkins (following Karl Popper) "all and some propositions." Because they have this kind of double generality, they are both unverifiable and unfalsifiable. The element corresponding to the universal quantifier makes them unverifiable; that corresponding to the existential quantifier makes them unfalsifiable. Under the circumstances they can hardly be said to be empirical. An example of this kind of proposition is the principle of universal causality, "Every event has a cause," which is equivalent to "For every event there is some other event with which it is causally connected." It has been claimed by some philosophers, for instance, G. J. Warnock, that this proposition is vacuous, since no state of affairs will falsify it. But the most that can be claimed in this respect is that no particular state of affairs which can be observed will falsify it. It is clearly not compatible with any state of affairs whatever, since it is incompatible with the state of affairs in which there is an event with no cause. It remains true that it is impossible to verify that an event has no cause.
Watkins does not claim that the proposition is necessary, although the principle of causality has been held by many, for instance, Kant, to be an example of a necessary truth, and it could no doubt be viewed as such. But it is also possible to treat it as a contingent truth, one that holds only in the contingency of every event being causally determined. How we could know that such a contingency held is a further question. It is clear that nothing that we could observe would provide such knowledge. Such propositions certainly could not be known a posteriori; if true, they must be known a priori if they are to be known at all. The difficulty is just this—how are they to be known at all? Thus, it may be better to distinguish between a priori propositions and nonempirical propositions of this kind. A priori propositions are those which can be known to be true and whose truth is ascertainable by a procedure that makes no reference to experience; nonempirical propositions of the kind in question are not like this, for their truth is, strictly speaking, not ascertainable at all. If we accept them, it must be as mere postulates or as principles whose force is regulative in some sense.
This does not exclude the possibility that there are other propositions whose truth can be ascertained by a nonempirical procedure but that are less than necessary. It has been argued by J. N. Findlay that there are certain propositions asserting connections between concepts that are only probable, as opposed to the commonly held view that all connections existing among concepts are necessary. He maintains that our conceptual systems may be such that there are connections between their members that are by no means analytic; the connections do not amount to entailments. Perhaps something like the Hegelian dialectic is the prototype of this. Findlay argues, for example, that if one has likings, there is the presumption that one will like likings of this sort; on this sort of basis one could move toward the notion of a community of ends. It is difficult to speak more than tentatively here. Given, however, that the propositions stating these conceptual connections are, if true, then true a priori (as they surely must be), it is not clear that it is necessary to claim only that what one knows in relation to them is probable. Certainly the connections do not constitute entailments; but this of itself does not mean that what one knows is only probable. The fact that the argument for a certain position is not a strictly deductive one does not mean that the position cannot be expressed by truths that are necessary and can be known to be so. For the argument may justify the claim to such knowledge in spite of the fact that the argument is not deductively valid in the strict sense. If such a necessary proposition does not seem to have universal application, this may be due to the fact that it holds under certain conditions and that its necessity is relative to these conditions. This was Kant's position over the principle of universal causality. He held that the principle that every event has a cause is necessary only in relation to experience. If propositions of this sort lack absolute necessity, they need not lack necessity altogether. The tentative conclusion of this section is that while some propositions may in a certain sense be both nonempirical and contingent, it nevertheless remains true that if a proposition is known a priori, it must be necessarily true in some sense or other.
must a priori propositions be analytic?
It has been suggested in the previous section that there may be a priori propositions that are not analytic. They depend for their validation on a priori argument but cannot be given a deductive proof from logical truths. The question of the synthetic a priori is one of the most hotly debated topics in philosophy and has, indeed, been so ever since Kant first stated the issues explicitly. Empiricists have always vehemently denied the possibility of such truths and have even tried to show that a proposition that is a priori must be analytic by definition. Most attempts of this sort rest on misconceptions of what is meant by these terms.
Kant's synthetic a priori
Kant claimed that synthetic a priori truths were to be found in two fields—mathematics and the presuppositions of experience or science—although he denied that there was a place for them in dogmatic metaphysics. He maintained that although mathematics did contain some analytic truths (since there were propositions which summed up purely deductive steps), the main bulk of mathematical truths were synthetic a priori; they were informative, nonempirical, and necessary, but not such that their denial gave rise to a contradiction. These characteristics were in large part due to the fact that mathematical knowledge involved intuitions of time (in the case of arithmetic) and space (in the case of geometry). Kant's conception of arithmetic has not found much support, and his view of geometry has often been considered to have been undermined by the discovery of non-Euclidean geometries. It is doubtful, however, whether the situation is quite so simple as this, for what Kant maintained was that an intuition of space corresponding to Euclidean geometry was necessary at any rate for creatures with sensibility like ours. That is to say, what we perceive of the world must conform to Euclidean geometry, whether or not it can be conceived differently in abstraction from the conditions of perception. Whether or not this is true, it is not obviously false.
The main attack on the Kantian view of arithmetic, and thereafter on that view of other branches of mathematics, came from Gottlob Frege and from Bertrand Russell and Alfred North Whitehead. Frege defined an analytic proposition as one in the proof of which one comes to general logical laws and definitions only; and he attempted to show that arithmetical propositions are analytic in this sense. The crucial step in this program is Frege's definition of "number" roughly in terms of what Russell called one-to-one relations. (Russell himself gave a parallel definition in terms of similarity of classes.) Given Frege's definition of number, arithmetical operations had to be expressed in terms of the original definition. It is at least an open question whether this attempt was successful. The definition has been accused of being circular and/or insufficient. This being so, the most that can be claimed is that arithmetic, while not reducible to logic, has a similar structure. Nevertheless, Gödel's proof that it is impossible to produce a system of the whole of formal arithmetic that is both consistent and complete may be taken to cast doubt even on this claim. At all events, the exact status of arithmetical truths remains arguable.
Other synthetic a priori truths claimed by Kant were the presuppositions of objective experience. He tried to demonstrate that the truth of such propositions as "Every event has a cause" is necessary to objective experience. These propositions indeed express the necessary conditions of possible experience and of empirical science. As such, their validity is limited to experience, and they can have no application to anything outside experience, to what Kant called "things-in-themselves." According to Kant, these principles—which are of two kinds, constitutive or regulative in relation to experience—are ultimately derived from a list of a priori concepts or categories, which he claims to derive in turn from the traditional logical classification of judgments. These principles, in a form directly applicable to empirical phenomena, are also established by transcendental arguments. In the "Second Analogy" of the Critique of Pure Reason, for example, Kant sought to show that unless objective phenomena were irreversible in time, and therefore subject to rule, and therefore due to causes, it would be impossible to distinguish them from merely subjective phenomena. Causality is therefore a condition of distinguishing phenomena as objective at all. The cogency of this position depends upon the acceptability of the arguments, and it is impossible to examine them here. It is to be noted, however, that what the arguments seek to show is that certain necessary connections between concepts must be accepted if we are to give those concepts any application. The connection between the concepts of "objective event" and "cause" is not an analytic one, but it is a connection that must be taken as obtaining if the concepts are to have any application to empirical phenomena.
Another instance of this kind of situation, perhaps more trivial, can be seen in such a proposition as "Nothing can be red and green all over at the same time in the same respect." This proposition has sometimes been classified as empirical, sometimes as analytic; but it has been thought by empiricists a more plausible candidate for synthetic a priori truth than any of Kant's examples. There is clearly some kind of necessity about this proposition. It may be possible for something to appear red and green all over, but to suggest that something might be red and green all over or that one might produce examples of such a thing has little plausibility; for in some sense red excludes green. The question is, In what sense? Since "red" does not mean "not-green" and cannot be reduced to this (for terms such as "red" and "green" do not seem capable of analysis), the proposition under consideration cannot, strictly speaking, be analytic. How can red and green exclude each other without this being a logical or analytic exclusion? It is not merely a contingent exclusion, since it is clearly impossible to produce something that is red and green all over (shot silk, for example, although it appears so, does not conform to the conditions of being two-colored all over), and we cannot imagine what such a thing would be like.
It may be suggested that red and green are different determinates of the same determinable—color. We distinguish colors and use different terms in order to do so. To allow, then, that something might be described by two such terms at the same time would be to frustrate the purposes for which our system of color classification was devised. However, this may sound too arbitrary. After all, given two colors that do in fact shade into each other, we might be less reluctant to allow that something might be both of them at once. It is no accident that we distinguish colors as we do. For creatures of our kind of sensibility, as Kant would put it, colors have a definite structure; it is natural to see them in certain ways and to conceive of them accordingly. We then fit them to a conceptual scheme that reflects those distinctions. If we will not allow that something may be red and green all over, it is because the mutual exclusion of red and green is a necessary feature of our scheme of color concepts. Yet the whole scheme has application to the world only because we see colors as we do.
the relative and absolute a priori
Because of the empirical preconditions for our scheme of color concepts, if we maintain that we can know a priori that something cannot be red and green all over, it cannot be absolutely a priori. For the truth that something that is red cannot also be green at the same time and in the same respect can scarcely be said to be ascertainable without any reference whatever to experience. The same is true of the principle of universal causality discussed earlier. It might be maintained that the truth that every event has a cause is necessary because "cause" and "event" are so definable that there is an analytic connection between them (implausible as this may be in fact). In that case the proposition in question would be true in all possible worlds (to use a Leibnizian phrase), since its truth would not depend on what is. In a world in which no events occurred, it would be true, in this view, that every event (if there were any) would have a cause. We can know the truth of this proposition absolutely a priori. However, if the principle is not analytic (and it is clearly not, in its ordinary interpretation) but is still thought to be necessary, this can be so only because the connection between cause and event is necessary to our conception of the world as we see it. Similarly, the mutual exclusion of red and green is necessary to our conception of colors as we see them. These propositions are not true in all possible worlds, and while their truth can be known a priori, it is not known absolutely a priori.
On the other hand, the so-called laws of thought, such as the principle of contradiction, while not analytic, must again be known absolutely a priori, whatever the kind of necessity they possess. The truth of the principle of contradiction is necessary to the possibility of thought in general, including the thought of the principle itself. It is not possible even to deny the principle without presupposing it. It cannot be maintained that its truth is in any way ascertainable by a procedure that makes reference to experience. Its truth is a necessity of thought, not of experience, and is not relative to experience. Hence it may be said to be known absolutely a priori.
Of those propositions that are absolutely a priori there are two kinds—analytic truths and the principles of logic themselves. (It is perhaps not surprising that these have sometimes been classified together, even if wrongly so.) On the other hand, there are some truths that are necessary but known only relatively a priori—truths such as the principle of causality and the principle of the incompatibility of colors. Finally, of course, there is that large class of truths which can only be known a posteriori. But for philosophers these are naturally much less interesting than truths of the first two kinds—those which are a priori in some sense or other. And over these there is still much argument.
See also Analytic and Synthetic Statements; Aristotle; Empiricism; Frege, Gottlob; Gödel, Kurt; Gödel's Theorem; Hegelianism; Innate Ideas; Kant, Immanuel; Knowledge, A Priori; Laws of Thought; Leibniz, Gottfried Wilhelm; Logic, History of; Mathematics, Foundations of; Popper, Karl Raimund; Propositions; Russell, Bertrand Arthur William; Whitehead, Alfred North; Wittgenstein, Ludwig Josef Johann.
For Kant's distinction between a priori and a posteriori, see the Norman Kemp Smith translation of the Critique of Pure Reason (London, 1953), especially the introduction but also the chapters on the aesthetic and the analytic of principles. For the precedents to Kant's distinction, see Aristotle, Posterior Analytics, especially Bk. I.2; Gottfried Leibniz, Nouveaux Essais, translated by A. G. Langley as New Essays concerning Human Understanding (Chicago: Macmillan, 1916), especially III.3 and IV.9. See also Arthur Pap, Semantics and Necessary Truth (New Haven, CT: Yale University Press, 1958), Pt. 1.
For the application of the distinction to concepts or terms, see H. H. Price, Thinking and Experience (London: Hutchinson, 1953); for criticisms of the notion of ostensive definition, see the opening sections of Ludwig Wittgenstein, Philosophical Investigations (Oxford: Blackwell, 1953); and Peter Geach, Mental Acts (London: Routledge and Paul, 1957), especially sections 6–11; and for discussion of the a priori in relation to analyticity, see G. H. Bird, "Analytic and Synthetic," Philosophical Quarterly 11 (1961): 227–237; L. J. Cohen, The Diversity of Meaning (London: Methuen, 1962), Chs. 6 and 10; and Ch. 5 of the work by Arthur Pap cited above.
For discussion of nonempirical propositions that are not necessary, see J. N. Findlay, Values and Intentions (London: Allen and Unwin, 1961); two articles by J. W. N. Watkins: "Between Analytic and Empirical," Philosophy 32 (1957): 112–131, and "Confirmable and Influential Metaphysics," Mind 67 (1958): 344–365; and compare G. J. Warnock, "Every Event Has a Cause," in Logic and Language (Oxford: Blackwell, 1953), Vol. II.
For Kant's views on the possibility of the synthetic a priori, see the Kemp Smith translation cited above. For discussions of examples, see: D. W. Hamlyn, "On Necessary Truth," Mind 70 (1961): 514–525; S. N. Hampshire, "Identification and Existence," in Contemporary British Philosophers (3d Series) (London, 1956); D. J. O'Connor, "Incompatible Properties," Analysis 15 (1955): 109–117; D. F. Pears, "Incompatibility of Colours," in Logic and Language (Oxford: Blackwell, 1953), Vol. II; and compare Aristotle's discussion of the principle of contradiction in Metaphysics IV.4.
other recommended titles
Boghossian, P., and C. Peacocke, eds. New Essays on the A Priori. Oxford: Clarendon Press, 2000.
BonJour, L. In Defense of Pure Reason: A Rationalist Account of A Priori Justification. Cambridge, U.K.: Cambridge University Press, 1998.
Casullo, Albert, ed. A Priori Knowledge. Aldershot, U.K.: Ashgate, 1999.
Coffa, J. A. The Semantic Tradition from Kant to Carnap. Cambridge, U.K.: Cambridge University Press, 1991.
DePaul, M. R., and W. Ramsey, eds. Rethinking Intuition: The Psychology of Intuition and Its Role in Philosophical Inquiry. Lanham, MD: Rowman & Littlefield, 1998.
Foley, Richard. A Theory of Epistemic Rationality. Cambridge, MA: Harvard University Press, 1987.
Fumerton, Richard. "A Priori Philosophy after an A Posteriori Turn." Midwest Studies in Philosophy 23 (1999): 21–33.
Gendler, T. S., and J. Hawthorne, eds. Conceivability and Possibility. Oxford: Oxford University Press, 2002.
Goldman, Alvin. Epistemology and Cognition. Cambridge, MA: Harvard University Press, 1986.
Hanson, P., and B. Hunter, eds. Return of the A Priori. Calgary, AB: University of Calgary Press, 1992.
Hawthorne, John P. "Deeply Contingent A Priori Knowledge." Philosophy and Phenomenological Research (September 2002): 247–269.
Kitcher, Philip. The Nature of Mathematical Knowledge. New York: Oxford University Press, 1983.
Kripke, Saul A. "Identity and Necessity." In Naming, Necessity, and Natural Kinds, edited by S. P. Schwartz, 66–101. Ithaca, NY: Cornell University Press, 1977.
Kripke, Saul A. Naming and Necessity. Cambridge, MA: Harvard University Press, 1980.
Kripke, Saul A. Wittgenstein on Rules and Private Language. Cambridge, MA: Harvard University Press, 1982.
Moser, Paul, ed. A Priori Knowledge. Oxford: Oxford University Press, 1987.
Moser, Paul. Knowledge and Evidence. Cambridge, U.K.: Cambridge University Press, 1989.
D. W. Hamlyn (1967)
Bibliography updated by Benjamin Fiedor (2005)