2 The paradox originally appeared in Curry, Haskell B. ‘The Inconsistency of Certain Formal Logics,’ Journal of Symbolic Logic 7 (1942), 115-17.CrossRefGoogle Scholar It is not difficult to see that the paradox can be formed using other connectives or relations (e.g. relevant entailment), in addition to both material implication and classical entailment.

3 This relation is made explicit in Goldstein, Laurence ‘Epimenides and Curry,’ Analysis 46 (1986), 117-21.CrossRefGoogle Scholar

4 Of course, since s is completely arbitrary in both Curry’s paradox and the Epimenides paradox, it also follows (by a series of arguments analogous to the ones just given) that not just s (⁓s), but every contingent sentence is true. From this it follows in both cases that s and ⁓s are true and, thus, that the sentence s & ⁓s is true.

5 See Church, Alonzo ‘Review of A. Koyné’s The Liar,’ Journal of Symbolic Logic 12 (1946), 131.Google Scholar For additional discussion, see Cohen, L.J. ‘Can the Logic of Indirect Discourse be Formalized?’ Journal of Symbolic Logic 18 (1957) 225-32;CrossRefGoogle Scholar as well as Prior, A.N.’s two articles, ‘Epimenides the Cretan,’ Journal of Symbolic Logic 23 (1958) 261-6;CrossRefGoogle Scholar and ‘On a Family of Paradoxes,’ Notre Dame Journal of Formal Logic 2 (1961) 16-32.

6 There are exceptions. For example, see Gupta, Anil ‘Truth and Paradox,’ Journal of Philosophical Logic 11 (1982) 1-60.CrossRefGoogle Scholar

7 See Russell, Bertrand ‘Mathematical Logic as Based on the Theory of Types,’ American Journal of Mathematics 30 (1908) 222-62CrossRefGoogle Scholar, reprinted in Logic and Knowledge, Marsh, Robert C. ed. (London: Allen & Unwin 1956), 59-102.Google Scholar Also see Russell, BertrandThe Principles of Mathematics (Cambridge: Cambridge University Press 1903)Google Scholar, Appendix B.

8 At least this is the standard interpretation of Russell. For a contrary interpretation in which Russell is understood as being committed to orders or types of propositional functions, rather than to a hierarchy of entities, see Landini, Gregory ‘Reconciling PM’s Ramified Type Theory with the Doctrine of the Unrestricted Variable of the Principles,’ in Irvine, A.D. & Wedeking, G.A.Russell & Analytic Philosophy (Toronto: University of Toronto Press forthcoming).Google Scholar

9 Or perhaps equivalently, that no collection can be definable only in terms of itself. See Russell, Bertrand ‘Mathematical Logic as Based on the Theory of Types,’ in Logic and Knowledge, Marsh, Robert C. ed. (London: Allen & Unwin 1956), 63;Google Scholar and Whitehead, A.N. & Russell, BertrandPrincipia Mathematica, Vol. 1 (Cambridge: Cambridge University Press 1910), 37.Google Scholar

10 Ramsey, Frank P. ‘The Foundations of Mathematics,’ Proceedings of the London Mathematical Society 25 (Series 2, 1925)CrossRefGoogle Scholar, reprinted as ch. 1 of Ramsey, ’s The Foundations of Mathematics, Braithwaite, R.B. ed. (London: Routledge & Kegan Paul 1931), 20.Google Scholar

11 See Tarski, Alfred ‘The Semantic Conception of Truth,’ Philosophy and Phenomenological Research 4 (1944) 341-75CrossRefGoogle Scholar, and ‘The Concept of Truth in Formalized Languages,’ in Logic, Semantics, Metamathematics (New York: Clarendon Press 1956) 152-278.

12 Kripke, SaulJournal of Philosophy 72 (1975), 691f.CrossRefGoogle Scholar

13 The paradox is similar to the postcard paradox (viz., that which arises when on one side of a postcard is written ‘The sentence which is written on the other side of this postcard is true’ while on the other side of the postcard is written ‘The sentence which is written on the other side of this postcard is false’) and to Buridan’s 9th Sophism (viz., that which arises when Socrates says, ‘What Plato is saying is false’ while at the same time, Plato says, ‘What Socrates is saying is true’). See Hughes, G.E.John Buridan on Self-Reference (Cambridge: Cambridge University Press 1982), 79-83; 173-83.Google Scholar

14 For example, see Tarski, Alfred ‘The Concept of Truth in Formalized Languages,’ in Logic, Semantics, Metamathematics (New York: Clarendon Press 1956), 165, and passim.Google Scholar

15 Kripke, Saul ‘Outline of a Theory of Truth,’ Journal of Philosophy 72 (1975), 691.CrossRefGoogle Scholar (Emphasis removed.) Gupta makes much this same point when he comments that although we have fairly clear intuitions about both paradoxical and nonparadoxical sentences which contain the truth predicate, ‘the division between these two classes of sentences is not a straightforward one. It cannot be said, for example, that the first class consists of all and only those sentences that involve one form of self-reference or another.’ See Gupta, Ani! ‘Truth and Paradox,’ Journal of Philosophical Logic 11 (1982), 1.CrossRefGoogle Scholar

16 Kripke, Saul ‘Outline of a Theory of Truth,’ 692Google Scholar

17 Herzberger, Hans A. ‘Paradoxes of Grounding in Semantics,’ Journal of Philosophy 67 (1970) 145-67CrossRefGoogle Scholar

18 Or that it is false, or of a class of sentences that they are all true, or mostly true, or all false, or mostly false, etc. Details of such generalizations are not difficult to construct. For example, a sentence which asserts that all sentences of a certain class are true will be false but grounded whenever at least one sentence in the class is false and grounded, irrespective of the groundedness (or lack of groundedness) of other members of the class.

19 Formally, this basic idea is exhibited within a hierarchy of interpreted languages in such a way that a sentence is grounded if and only if it obtains a truth-value at the smallest fixed point of the hierarchy, and ungrounded otherwise. A paradoxical sentence turns out to be a sentence which cannot consistently be assigned a truth-value at any fixed point. It follows that although all paradoxical sentences are ungrounded, not all ungrounded sentences need be paradoxical. For example, unlike the paradoxical ‘This sentence is false,’ the sentence ‘This sentence is true’ is not paradoxical even though it is ungrounded. In other words, it can be assigned a single truth-value, but only arbitrarily. This basic idea of Herzberger’s appears to have been anticipated at least by Ryle. See Ryle, Gilbert ’Heterologicality,’ Analysis 11 (1951) 61-9, esp 67f.CrossRefGoogle Scholar

20 Sainsbury, R.M.Paradoxes (Cambridge: Cambridge University Press 1988), 117Google Scholar

21 As Kripke’s Watergate example shows, groundedness is not, in general, an intrinsic property of a sentence, but typically depends instead upon the empirical context of the sentence’s utterance.

22 The term is introduced in Fraasen, Bas van ‘Presupposition, Implication and Self-Reference,’ Journal of Philosophy 65 (1968), 147.Google Scholar

23 Yet another way of reaching much this same conclusion is as follows: consider a sentence, s, which is neither true nor false; then the biconditional ’s’ is true iff s fails since ‘“s” is true’ is false but’s’ is neither true nor false. Thus either T must be rejected, or a theory of truth-value gaps abandoned. (See Haack, SusanPhilosophy of Logics (Cambridge: Cambridge University Press 1978), l0lf.)CrossRefGoogle Scholar

24 See Priest, GrahamIn Contradiction (Dordrecht: Martinus Nijhoff 1987), 11CrossRefGoogle Scholar and passim. Much of the groundwork for this book was laid by Priest in earlier articles, including ‘Logic of Paradox,’ Journal of Philosophical Logic 8 (1979) 219-41; ’Sense, Entailment and Modus Ponens,’ Journal of Philosophical Logic 9 (1980) 415-35; ‘To Be and Not To Be: Dialectical Tense Logic,’ Studia Logica 41 (1982) 249-68; ‘Logical Paradoxes and the Law of Excluded Middle,’ Philosophical Quarterly 33 (1983) 160-5; ‘Logic of Paradox Revisited,’ journal of Philosophical Logic 13 (1984) 153-79; and ‘Semantic Closure,’ Studia Logica 43 (1984) 117-29.

25 Priest, GrahamIn Contradiction, 94ff.;Google Scholar ‘Logic of Paradox,’ 226ff.; To Be and Not To Be: Dialectical Tense Logic,’ 254ff.

26 See Lukasiewicz, Jan ‘On 3-valued Logic’ (1920) and ‘Many-valued Systems of Propositional Logic’ (1930)Google Scholar, in McCall, S.Polish Logic (Oxford: Oxford University Press 1967)Google Scholar, and Kleene, S.C.Metamathematics (Amsterdam: North Holland 1952), 332ff.Google Scholar

27 See Parsons, Terence ‘True Contradictions,’ Canadian journal of Philosophy 20 (1990), 336.CrossRefGoogle Scholar Of course, as Parsons points out, the formal similarities between theories of truth-value ‘gaps’ and theories of truth-value ‘gluts’ mean that criticisms of one account can typically be translated into criticisms of the other. Both accounts, for example, regularly have expressibility problems: the ‘gapper’ believes that some sentences are neither true nor false, yet typically has difficulty saying this truly; in contrast, Priest believes that some views (including those of the ‘gapper’) are not true, yet generally has difficulty saying this truly. See 344ff., especially n. 9 and 10 of Parsons.

28 Priest, GrahamIn Contradiction, 91Google Scholar

29 Ibid., 91

30 Ibid., 90

31 Similar arguments can be given that are analogous to the Epimenides. For discussion about how Curry’s paradox relates to naive set theory, see Meyer, Robert K.Routley, Richard and Dunn, J. Michael ‘Curry’s Paradox,’ Analysis 39 (1979), 124-8.CrossRefGoogle Scholar

32 In fact, Priest himself criticizes other purported solutions to the paradoxes for lacking just such motivation.

33 Priest, GrahamIn Contradiction, 108Google Scholar

34 Kripke, Saul ‘Outline of a Theory of Truth,’ 692Google Scholar

35 Recall, not just Kripke’s Watergate example, but Buridan’s 9th Sophism and the postcard paradox mentioned inn. 13 above.

36 This general strategy not only leads to the resolution of the barber paradox. It is also the same strategy, stripped of all its formal technicality, which lies at the heart of most modem solutions to the set-theoretical paradoxes. Just as in the case of the barber paradox, the conclusion we draw from Russell’s paradox is that the Russell set (which both is and is not a member of itself) does not exist. For example, see Boolos, George ‘The Iterative Conception of Set,’ journal of Philosophy 68 (1971), 215-32CrossRefGoogle Scholar, reprinted in Benacerraf, Paul & Putnam, HilaryPhilosophy of Mathematics, 2nd ed. (Cambridge: Cambridge University Press 1983), 486-502.Google Scholar Also compare Barwise, Jon & Etchemendy, JohnThe Liar (New York & Oxford: Oxford University Press 1987).Google Scholar

37 Peter Apostoli has pointed out to me that, strictly speaking, a conclusion this strong is not essential. Even if one accepts the meaningfulness of SL as a metatheoretic report, R, that an object language sentence, SL, fails to be assigned the value True, what the contradictions show is that this fact cannot be expressed in the object language of the model, at least not by the sentence SL itself. Indeed, as Apostoli points out, one merely has to inspect the weak-Kleene truth function for negation to see that the composite predicate ‘not true’ as it occurs in the object language is not co-extensive with the same predicate as it occurs in the metalanguage. Hence, contrary to our assumption, we have failed to achieve semantic closure: there are truths that this language cannot report. Whether this argument should be understood as assuming the meaningfulness of SL in actual fact, or merely for the purpose of reductio is, I think, a moot point.

38 In this respect, the proposal being offered here mirrors that of Buridan. By assigning truth values to concrete sentence tokens, Buridan was able to assign different truth values consistently to distinct tokens of the same type. However, unlike Buridan’s account, this proposal does not assign the value ‘false’ to the sentence token SL. For a lively discussion of whether Buridan is successful in avoiding the semantic paradoxes, see Hazen, Allen ‘Contra Buridanum,’ Canadian Journal of Philosophy 17 (1987) 875-80CrossRefGoogle Scholar, and Hinckfuss, Ian ‘Pro Buridano; Contra Hazenum,’ Canadian Journal of Philosophy 21 (1991) 389-98.CrossRefGoogle Scholar

39 For this reason, this solution has similarities with other ‘non-statement’ or ’non-propositional’ accounts, such as those of Bar-Hillel, Goldstein, Kneale, Prior, and Sobel. It is also motivated by D.C. Stove’s ‘case by case’ view concerning formal inference in general. See Bar-Hillel, Y. ‘New Light on the Liar,’ Analysis18 (1957) 1-6;Google ScholarGoldstein, Laurence ‘Epimenides and Curry,’ Analysis 46 (1986) 117-21;CrossRefGoogle ScholarKneale, W.C. ‘Russell’s Paradox and Some Others,’ British Journal for the Philosophy of Science 18 (1971) 321-38;CrossRefGoogle ScholarPrior, A.N. ‘Epimenides the Cretan,’ Journal of Symbolic Logic 23 (1958) 261-6;CrossRefGoogle ScholarSobel, Jordan Howard ‘Lies, Lies, and More Lies: A Plea for Propositions,’ Philosophical Studies 67 (1992) 51-69;CrossRefGoogle Scholar and Stove, D.C.The Rationality of Induction (Oxford: Clarendon 1986), ch. 9.Google Scholar

40 Of course, this sketch will have to be modified slightly for statements which make reference to language itself.