1 Cf. discussion by Alquie, Ferdinand ed., Descartes, Oeuvres Philosophiques, Tome II, (Paris: Editions Gamier Freres 1967), 582Google Scholar, n. 1; Graciela De Pierris, (this volume), n. 2, Leibniz, G. in Remnant, and Bennett, eds., New Essays Concerning Human Understanding (Cambridge: Cambridge University Press 1981), 293-4Google Scholar, and 475.

2 Cf. Leibniz, New Essays, Preface, 49-50, and Kant, I.Critique of Pure Reason, Smith, Kemp ed. (London: Macmillan 1968)Google Scholar, B3. What’s needed to warrant them is non-empirical insight into their universal necessity, perhaps from ‘a priori’ reasons which explain why they are universally and necessarily so. An equally traditional, ‘Platonic,’ variation is that mathematics concerns ‘abstract’ objects that need not, perhaps indeed cannot, be exemplified in the world we experience, and thus cannot be known on the basis of that experience.

3 New Essays, 445. Cf. also, ‘The senses can hint at, justify, and confirm these truths, but can never demonstrate their infallible and perpetual certainty’ (80).

4 Leibniz, New Essays, 434, and Monadology, sec. 33 from Selections, Wiener, ed. (New York: Scribners 1951); Tait, W.A. “The Concept of A Priori Truth’ in Proof and Knowledge in Mathematics, Detlefson, Michael ed. (London: Routledge 1992)Google Scholar argues that Leibniz’s notion of a priori truth as truth whose reason is discoverable by analysis shows it is a non-epistemic notion of truths derivable from primitive identical truths and definitions. Albert Casullo (this volume) points out that many recent philosophers, e.g. Ayer, do work with some such nonepistemic notion of ‘a priori’ truth. But Leibniz is more problematic, since he also claims that ‘a reason is a known truth whose connection with a less well known truth leads us to give our assent to the latter’ and that a reason isn’t an a priori reason (a known truth which is ‘a cause not only of our judgment but of the truth itself) unless it and its connection with its consequence are known, or at least knowable, non-empirically (New Essays, 475).

5 For example, if only certain types of truth can be certain a priori, or follow from a priori certain truths in a priori certain ways, perhaps they can be called ‘a priori truths,’ even if they are susceptible to some degree of a posteriori warrant. If certain types of truth can be known only a posteriori, whereas others can be warranted either a priori or a posteriori, then perhaps the latter can be called ‘a priori’ truths.

6 Cf. Leibniz, : ‘the truth of sensible things is established by the links amongst them; these depend upon intellectual truths, grounded in reason, and upon observations of regularities among sensible things themselves’ (New Essays, 444).Google Scholar

7 Laurence Bonjour (this volume) endorses an argument of this sort, acknowledging its ‘internalist’ presuppositions, and the opposition of ‘externalist’ critics for whom warranted conclusions may require warranted and relevant premises, and thus valid principles of relevance, but doesn’t require warrant for the validity of the principles themselves, or even the availability of such warrant by reflection alone. See James van Cleve, ‘Reliability, Justification, and Induction,’ Midwest Studies in Philosophy 9 (1984).

8 That’s so even though our knowledge of theorems often depends on recalling steps in proofs, and thus isn’t entirely a priori.

9 Leibniz, New Essays, 406.Google Scholar Leibniz isn’t endorsing the ‘scholastic’ claim wholeheartedly since he thinks it doesn’t properly appreciate the possibility and epistemic significance of demonstrating what are commonly taken as axioms.

10 Aristotle, Posterior Analytics, trans. Barnes, J. (Oxford: Clarendon Press 1975)Google Scholar, Book I, ch. 31

11 The characterization of ‘induction’ follows Barnes’s notes, op. cit., 259-60. As Barnes notes, this process mirrors, or perhaps is identical with, the process by which we acquire the concept of A (so as to understand and know what is included in its definition) by induction or abstraction from the several images of As we retain in memory from our various perceptions of As. The universal is ‘grasped from seeing’ and so is grasped ‘from the posterior.’ But whereas ‘seeing occurs separately for each’ (particular instance), ’comprehension grasps at one time that it is thus in every case.’ The Aristotelian view contrasts with the Platonic, in which sense experience and conversation concerning A prompt, or serve to recall from a prior disembodied life, a pure intellectual perception or intuition of a perfect A (form) to which physically perceived and discussed As may approximate, but which nonetheless transcends and is separate from these As. It also contrasts with the Leibnizian, according to which the content of our fundamental mathematical and metaphysical ideas can’t be explained as arising from features of experience or our interaction with the world. Ideas with such content must be innate, or our capacity to have them must be innate, with sense experience providing the ’occasion’ for its activation and for reflection on our ideas. Although knowledge of mathematical and metaphysical truths requires only examining these ideas, the ideas aren’t innately distinct or adequate, and our knowledge isn’t innate. Barnes thinks the innatist claim is either the dubious one that we have innately occurring ideas or knowledge, or the trivial one that we have an innate capacity to know things or have ideas upon external sensory stimulation, apparently overlooking the issue of the content of ideas (250-1).

12 Cf. Barnes, op. cit., 256-60. For Barnes, knowing that As are necessarily F by understanding A and F, and their connection, isn’t a matter of grounding knowledge in some distinctive act of understanding terms or a distinctive intuitive grasp of necessary connections. It’s just a matter of understanding or comprehending that A necessarily includes B — by accepting this generalization by induction from perceived and remembered particulars. For a more traditional interpretation, see Chisholm, R.Theory of Knowledge, 2nd ed., (Englewood Cliffs: Prentice-Hall 1977)Google Scholar, reprinted in Moser, Paul ed., A Priori Knowledge (Oxford: Oxford University Press 1987).Google Scholar There is also some dispute about the extent to which Aristotle’s own most mature views concerning our knowledge of first principles differ from the historically influential views presented in Posteriori Analytics. See Irwin, TerenceAristotle’s First Principles (Oxford: Oxford University Press 1988).Google Scholar

13 Wilfrid, Sellars ‘Is There A Synthetic “A Priori“?’ Philosophy of Science (1953)Google Scholar, reprinted in Science, Perception, and Reality (London: Routledge and Kegan Paul 1963), 300 (bracketed insertion ours). He adds ‘But if the reasonableness of asserting “All A is B” does not rest on experience, on what does it rest? … The reasonableness, we are told, rests on a correct understanding of the meanings of the terms involved. In short, a priori truth is truth ex vi terminorum.’

14 This isn’t to say that degree of empirical warrant can’t be affected by other factors as well, e.g. coherence with other empirically warranted propositions, or that empirical warrant for a hypothesis must be a matter of confirmation by instances rather than a function of the number and variety of observed phenomena and kinds of phenomena it explains.

Nor does it mean hypothetical cases that are neither perceived nor remembered can’t be relevant to the empirical confirmation and disconfirmation of generalizations as well. In thinking about an hypothesis, we might realize that it implies that if circumstances C were to occur, Y would occur. If we have independent empirical evidence for thinking that it is likely that Y would occur, or that it wouldn’t occur, if C were to occur, we can respectively confirm or disconfirm the hypothesis without any empirical evidence that Y actually does or doesn’t occur. Moreover, a single observation can greatly confirm a hypothesis when we are empirically warranted in thinking the observed event unlikely if the hypothesis is false. In both cases, the empirical confirmation or disconfirmation of the hypothesis is nonetheless ultimately a function of the number and variety of perceived and recalled events.

15 Indeed many philosophers sympathetic with this tradition endorse them. Chisholm, Theory of Knowledge, 2nd ed., (Englewood Cliffs: Prentice-Hall 1977), ch. 3Google Scholar, and the various philosophers he cites, e.g. Duns Scotus. Though Aristotle suggests the need for several experienced particulars in several places, (e.g. Post. An. A 31, N. Ethics, VI, 8), he doesn’t suggest that the degree of warrant is in any way proportional to the number and variety of instances. Eleanore Stumpf, in her valuable ‘Aquinas on the Foundations of Knowledge’ Aristotle and his Medieval Interpreters, Bosley and Tweedale, eds., CJP Supplementary Volume 17, presents a view of Aristotle and Aquinas similar to Barnes’s but neither she nor Barnes really addresses the issues that Sellars and (a)-(d) bring to the fore.

16 Philip, KitcherThe Nature of Mathematical Knowledge, (New York: Oxford University Press 1982), 15.Google Scholar Hereafter ‘NMK.’

17 In the second edition of Theory of Knowledge (Englewood Cliffs, NJ: Prentice Hall 1977), Chisholm defends ‘intuitive induction’ as a method whereby propositions become axiomatic for one by understanding universals and grasping what they include and exclude, but drops his discussion from the third edition (Englewood Cliffs, NJ: Prentice Hall 1989). What remains from the second is the account of a priori knowledge above in terms of the inherent certainty of certain propositions for anyone who accepts them. The shift may reflect a recognition that the latter view makes reflection on instances epistemically irrelevant, however psychologically relevant it may remain to producing acceptance.

18 And why should we restrict a priori knowability to necessary truths, by definition rather than by philosophical argument. Philip Kitcher (see Mathematics, below) provides a definition which leaves out Chisholm’s stipulation but implies that some self-knowledge, e.g., ‘I exist’ is a priori (NMK, 24,27) Leibniz, on the other hand, thinks that ‘I am’ and other ‘first experiences’ are axioms in the sense that they are immediately known without proof and are certain to anyone who accepts them, no matter what her experience. But he classifies them as a posteriori, because in accepting them we don’t see why they are true. That allegedly distinguishes them from those necessary truths known a priori without proof, the ‘first illuminations’ (New Essays, 434, 411). This either seems just as ‘stipulative’ as Chisholm or appeals to the ‘original’ sense of ‘a priori’ mentioned above.

19 Despite their demonstrability, despite their apparent acceptance by some persons simply because experience confirms them, indeed despite the apparent rejection of some propositions we once took to be certain without demonstration. Leibniz seemed to have thought that they gained in epistemological status since whatever degree of evidence or illumination they had before being demonstrated from more primary truths was less than what they had after being demonstrated. Cf. New Essays, 414.

20 Our knowledge of the axioms would then (contra Aristotle, Post. An. Book A, 2) be as much based on our knowledge of the theorems as the latter is on the former, since what would then matter for the warrant of axioms is their interconnection with theorems.

21 Cf. Firth, Roderick ‘Coherence, Certainty, and Epistemic Priority,’ Journal of Philosophy (1964).CrossRefGoogle Scholar

22 These conclusions are resisted by Sosa, Ernest in ‘Knowledge and Intellectual Virtue,’ Monist (1985).CrossRefGoogle Scholar

23 When we can’t conceive A without conceiving B, our warrant might be undermined when various other propositions we can or can’t conceive false imply, in ways we can’t conceive as not holding, that A may occur without B. Or because we are also unable to conceive of the impossibility of A without B without thinking that there must be some reason why, and yet be unable to conceive of any way to prove the impossibility. As Pap, Arthur notes (Semantics ami Necessary Truth [New Haven: Yale University Press 1958])Google Scholar, inconceivability can’t be a criterion of impossibility if what is relevantly inconceivable is defined by what is logically impossible. However, much of the force of Pap’s criticisms of conceivability and inconceivability as criteria is lost when the criteria are construed, not as definitions of possibility and impossibility, but as giving marks of defeasible or prima facie warrant for ascribing possibility and impossibility. When we try to conceive p and are consciously unable to do so, that doesn’t entail that p is impossible.

24 Whitehead, A.N. and Russell, B.Principia Mathematics to *56 (Cambridge: Cambridge University Press 1973), 59;Google ScholarRussell, ‘On “Insolubilia” and Their Solution by Symbolic Logic,’ in Essays in Analysis, Lackey, D. ed. (New York: Brazilier 1973), 193. Cf. Cioffa, AlbertThe Semantic Tradition from Kant to Carnap: To the Vienna Station (Cambridge: Cambridge University Press 1991)Google Scholar, ch. 7. In 1912, Russell also suggested that some logical principles at least (e.g. the law of non-contradiction) were absolutely self-evident or certain, e.g. The Problems of Philosophy (London: Oxford University Press 1912), ch. 7.

25 Cf. Pollock, JohnKnowledge and Justification (Princeton: Princeton University Press 1974), 329.Google Scholar Pollock provides the most comprehensive account of the structure of defeasible a priori warrant.

26 Contra Kitcher, Philip ‘The Naturalists Return’ in Philosophical Review (1992), 72.Google Scholar

27 After all, the warrant we have for thinking there is a tree before us when we ostensibly see one may be undermined when a reliable authority tells us our vision is unreliable or when we also reach out and don’t feel a tree. That doesn’t make our knowledge really testimonial or tactile knowledge when, in the absence of undermining evidence, we ostensibly see and take there to be a tree before us.

28 Thus, though there may a priori knowledge, there might not be any a priori truths in the sense of propositions that can have a warrant that isn’t empirically defeasible or to which empirical evidence can’t even in principle be relevant.

29 Pollock, however, argues that logical intuitions of possibility and necessity and implication constitute prima facie warrant for assertions of possibility and necessity, because it’s in terms of these intuitions that we learn to make and correct such assertions and their meaning is determined.

30 Cf. the challenges concerning the efficacy of laws of nature from Fraasen, B. vanLaws and Symmetry (Oxford: Clarendon Press 1989).CrossRefGoogle Scholar

31 Putnam, H. ‘Analyticity and Apriority: Beyond Wittgenstein and Quine,’ Realism and Reason (Cambridge: Cambridge University Press 1983)CrossRefGoogle Scholar

32 Leibniz, New Essays, 434.Google Scholar Non-identical propositions like ‘A is 6’ are evident only by experiencing instances of A and B together, or by demonstration, e.g. via a middle term ‘C linking A and B.

33 Falsehoods that are reducible to express contradictions by analysis are analytic falsehoods; those that are not are synthetic.

34 Leibniz, ‘Reflections on Knowledge, Truth, and Ideas’ in Selections, 286Google Scholar

35 ‘Our certainty regarding universal and eternal truths is grounded in the ideas themselves, independently of the senses’ (New Essays, 392). Leibniz’s distinction between truths of reason and truths of fact is justly often cited alongside David Hume’s distinction between relations of ideas and matters of fact. Relations of ideas don’t depend on the existence of anything in nature corresponding to the ideas and thus can be discovered without empirical evidence for such things, and there is no contradiction in denying matters of fact. However, for Hume even God’s existence isn’t a relation of ideas. More importantly, ideas for Hume are always (ideas of) imaginable, recallable, or introspectible particulars. Relations of ideas are thus relations between ideas of particulars, rather than truths about what is universally and necessarily the case.

36 The classic full length treatment of this charge and similar ones remains Pap’s Semantics and Necessary Truth.

37 Kant, Critique of Pure Reason, Introduction, B 10-11Google Scholar

38 New Essays, 293. ‘Something which is thought possible is expressed by a definition; but if this definition does not at the same time express this possibility then it is merely nominal, since in this case we can wonder whether the definition expresses anything real — that is, possible — until experience comes to our aid by acquainting us a posteriori with the reality (when the thing actually occurs in the world.) This will do when reason cannot acquaint us a priori with the reality of the thing defined by exhibiting its cause or the possibility of its being generated. So it is not within our discretion to put our ideas together as we see fit, unless the combination is justified either by reason, showing its possibility, or by experience, showing its actuality and hence its possibility.’

39 Leibniz, ‘Reflections on Knowledge, Truth, and Ideas’, Selections, 288Google Scholar

40 Camap, R.Meaning and Necessity, 2nd ed. (Chicago: University of Chicago Press 1967), Appendices; Sellars, W.Google Scholar ‘Is There a Synthetic A Priori?’ in SPR. See also A. Cioffa, ch. 7.

41 Cf. Sellars, ‘Is There a Synthetic A Priori?’ Empirically warranted assertions, Sellars adds, may even give us reasons for adopting or modifying our rules, including those unconditionally licensing assertion, without these assertions thereby being warranted by empirical evidence for their truth.

42 Besides concerns arising from Godel’s theorem, an elementary statement of which can be found in Pollock op. cit.

43 Kant, B 756

44 She doesn’t elaborate but cites Henrich, Dieter ‘Kant’s Notion of a Deduction’ in Kant’s Transcendental Deductions, Forster, ed. (Stanford: Stanford University Press 1989).Google Scholar

45 Philip Kitcher, NMK. ‘Affordances’ is a technical term borrowed from the psychologist J.J. Gibson, referring to what the environment offers, provides, or furnishes animals, either for good or ill. For example, krill affords nourishment to baleen whales, a wetland affords a habitat to ducks, and the African savanna affords predatory cats to gazelles. According to Gibson and his followers, it is the affordance that is perceived. Cf. NMK, 12 and Gibson, J.J.Vie Ecological Approach to Visual Perception (Boston: Houghton Mifflin 1979).Google Scholar

46 It is less clear what Kitcher wants to say about the modal status of mathematical truths. He never outright denies their necessity. However, he has denied that our knowledge of them as necessary need involve an a priori warrant (or, conversely, that our knowing them a priori would imply their necessity). Cf. Kitcher, Philip ‘Apriority and Necessity,’ Australasian Journal of Philosophy 58 (1980) 89-101.CrossRefGoogle Scholar

47 Bigelow, J.The Reality of Numbers (New York: Oxford University Press 1988); cf. 3-4, 110-17Google Scholar

48 Cf. the tradition of Armstrong, D.M.Universals (Boulder, CO: Westview Press 1989);Google Scholar and Lewis, D. ‘New Work For a Theory of Universals,’ Australasian Journal of Philosophy 61 (1983) 343-77.Google Scholar

49 NMK, 17. We have added the temporal indices; Kitcher acknowledges often leaving these implicit (cf. NMK, 22). Kitcher also leaves the notion of ‘warrant’ officially unanalyzed, but it is significant that he cites Alvin Goldman’s recent ’externalist’ treatment of the notion (cf. NMK, 18, n. 6; and Alvin Goldman’s ’What is Justified Belief?’ in Pappas, G., ed., justification and Knozvledge (Dordrecht: Reidel 1979); and his Episteniology and Cognition (Cambridge, MA: Harvard University Press 1985). What externalists like Goldman deny is the more traditional ‘internalist’ claim that the warrant producing process need involve the internal possession — in principle available to conscious reflection — of a reason for thinking the belief true. Of course, from the internalist’s standpoint, such extemalism opens the door to skepticism, since it allows that we may have no reasons for our beliefs. Larry Bonjour argues (this volume) that externalists avoid traditional skeptical problems only by changing the subject. For instance, on its externalist construal, Kitcher’s initial assumption of mathematical knowledge would be worth little in the eyes of an internalist, since it does not imply that we have any (good) reasons for the putative items of mathematical knowledge, which is what was originally at issue. An externalist conception of warrant would seem to be a particular affront to the traditional conception of a priori knowledge as knowledge gained by conscious reasoning, thought, or reflection alone, although we note below that on a certain construal the issue of a priority can nevertheless be seen to arise even within the externalist’s framework.

50 NMK, 55. Cf. also Pollock.

51 Kitcher, Philip ‘The Naturalists Return,’ 53-144;Google Scholar cf. esp. 77 n. 72. Insertion in brackets ours.

52 Other authors in this collection who share this perspective include Andrew Irvine, James Brown, and perhaps Barry Smith and Albert Casullo.

53 And for a. The lives must be sufficient for X to, e.g., comprehendingly follow a proof of p, if that is what o involves.

54 Internalists and externalists may disagree about whether the epistemically relevant sort of likelihood is subjective or objective.

55 Harman, GilbertThought (Princeton, NJ: Princeton University Press 1973)Google Scholar

56 One such life should be sufficient for an externalist. Internalists might insist that L represent a substantive proportion of such lives, so that warrant isn’t undermined by mere reflection on the probabilities of defeaters occurring.

57 Bonjour (this volume) makes this point forcefully. We would surmise that, in thinking of warrants as the externalist does, one tends to undervalue the role of conscious reasoning, thus making it easier to lose sight of its contribution entirely. Although it is worth pointing out that to the extent an externalist takes unconscious but purely internal information processing to play a distinctive role in a certain class of warrants, she is in effect countenancing a priori warrants a la Bigelow.

58 The pattern includes the consideration that, relative to a given assignment, each free occurrence of ‘x’ in a given open sentence picks out the same object. We take that to be a ‘semantic’ aspect of the pattern.

59 Kazmi may actually disagree with this construal, since at one point he suggests that the theory of propositional attitude ascriptions in question is, perhaps, false on other grounds that he does not mention, and at another point he seems to link a priority more with metaphysical than with logical necessity. But then he owes us his account of a priority and its objects.

60 Cf. e.g. Kripke, SaulNaming and Necessity (Cambridge, MA: Harvard University Press 1980).Google Scholar

61 Cf. Putnam, Hilary ‘There is at Least One A Priori Truth’ and ‘Analyticity and A Priority’: Beyond Wittgenstein and Quine’ in Realism and Reason.Google Scholar

62 This qualification, here and below, signals that the warrant for the predicted result is not simply a matter of the warrant we have for accepting the taxonomy; or isn’t just warrant that accrues to all our beliefs, including those that warrant us in accepting the taxonomy, by increasing coherence among all our empirically warranted beliefs and theories.

63 Colin McGinn once proposed that we simply define ‘a priori’ truth as one that can be known without causal interaction with the subject matter of some justifying statement, and ‘a posteriori’ truth as one that can only be known by such causal interaction. See ‘Modal Reality’ in Healy, Richard ed., Reduction, Time, and Reality (Cambridge, MA: Cambridge University Press 1981); and also ’A Priori and A Posteriori Knowledge,’ Proceedings of the Aristotelian Society (1976) 195-208.Google Scholar Such a definition would of course rule out saving the causal theory.

64 Brown, James ‘π in the Sky,’ in Irvine, Andrew D. ed., Physicalism in Mathematics (Dordrecht: Kluwer 1990), 95-120;CrossRefGoogle Scholar cf. esp. 116; but also 112, n. 47, in which he hints at the position he here adopts.