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Consider the following list of sentences, named ‘The List’:

- Tasmanian devils have strong jaws.
- The second sentence on The List is circular.
- If the third sentence on The List is true, then
*every sentence*is true. - The List comprises exactly four sentences.

the third sentence of The List is trueis true. By substitution, it follows that

If the third sentence of The List is true, then every sentence is trueis true. But, then, Modus Ponens on the above two sentences yields that

every sentence is trueis true. So, by conditional proof, we conclude that

If the third sentence of The List is true, then every sentence is trueis true. By substitution, it follows that

the third sentence of The List is trueis true. But, now, by Modus Ponens on the above two sentences we get that

every sentence is trueis true. By naive truth theory we disquote (or, in this case, dis-display, as it were) to conclude: Every sentence is true! So goes (one version of) Curry’s paradox.

- 1. Brief History and Some Caveats
- 2. Curry’s Paradox: Truth- and Set-Theoretic Versions
- 3. Significance, Solutions, and Open Problems
- Bibliography
- Other Internet Resources
- Related Entries

There are basically two different versions of Curry’s paradox, a truth-theoretic (or proof-theoretic) and a set-theoretic version; these versions will be presented below. For now, however, there are a few caveats that need to be issued.

Caveat 1. *Loeb’s Paradox*. Prior’s version is (in effect)
rehearsed by Boolos and Jeffrey (1989), where neither Prior nor Curry
is given credit; rather, Boolos and Jeffrey point out the similarity of
the paradox to reasoning used within the proof of Loeb’s Theorem; and
subsequent authors, notably Barwise & Etchemendy (1984), have
called the paradox *Loeb’s paradox*. While there is no doubt
strong justification for the alternative name (given the similarity of
Curry’s paradox to the reasoning involved in proving Loeb’s Theorem)
the paradox does appear to have been first discovered by Curry.

Caveat 2. *Geometrical Curry Paradox (Jigsaw Paradox)*. This
is *not* the same Curry paradox under discussion; it is a
well-known paradox, due to *Paul* Curry, having to do with
so-called geometrical dissection. (The so-called Banach-Tarski
geometrical paradox is related to *Paul* Curry’s
geometrical paradox.) See Gardner 1956 and Fredrickson 1997 for full
discussion of *this* (geometrical) Curry paradox.

where ‘[ ]’ is a name-forming device. Assume, too, that we have the principle calledT-Schema: T[A]A,

(NB: We could also use the principle calledAssertion: (A& (AB))B

By diagonalization, self-reference or the like we can get an
arbitrary sentence, *C*, such that:

whereC= T[C],F

T[Again, using the same instance of the T-Schema, we can substituteC] (T[C]),F

Letting

1. C(C)F[by T-schema and Substitution] 2. ( C& (C))FF[by Assertion] 3. ( C&C)F[by Substitution, from 2] 4. CF[by Equivalence of CandC&C, from 3]5. C[by Modus Ponens, from 1 and 4] 6. F[by Modus Ponens, from 4 and 5]

Moreover, assume that our conditional, , satisfies Contraction (as above), which permits the deduction ofUnrestricted Abstraction:x{y|A(y)}A(x).

(fromssA)

In the set-theoretic case, letss(ssA).

So, coupling Contraction with the naive abstraction schema yields, via Curry’s paradox, triviality.

1. xC(xx)F[by Naive Abstraction] 2. CC(CC)F[by Universal Specification, from 1] 3. CC(CC)F[by Simplification, from 2] 4. CCF[by Contraction, from 3] 5. CC[by Modus Ponens, from 2 and 4] 6. F[by Modus Ponens, from 4 and 5]

Setting negation aside (for purposes of Curry), we assume a
propositional language with the following connectives: conjunction
(&), disjunction
(), and entailment
(). (For purposes of resolving Curry’s paradox, negation
may be set aside; however, the current semantics allow for a variety of
approaches to negation, as well as quantifiers.) An interpretation is a
4-tuple, (*W*,*N*,[ ], *f*), where *W*
is a non-empty set of worlds (index points), *N* is a non-empty
subset of *W*, [ ] is a function from propositional
parameters to the powerset of *W*; we may, for convenience, see
the range of [ ] as comprising propositions (sets of worlds at
which various sentences are true), and so call the values of [ ]
*propositions*. We let *NN* be the set of so-called
non-normal worlds, namely *NN* =
*W**N*. In turn,
*f* is a function from (ordered) pairs of propositions to
*NN*. Now, [ ] is extended to all sentences (*A*,
*B*, ...) via the following clauses:

[The value of an entailment is the union of two sets:A&B] = [A][B][

AB] = [A][B]

With all this in hand, validity is defined in the usual way: namely, as truth-preservation at allN=W, if [A][B]; otherwise,N=.

NN=f([A],[B]).

Why restrict the definition merely to normal worlds? The explanation goes hand-in-hand with the informal interpretation of non-normal worlds; according to Priest’s suggestion, non-normal worlds should be understood to be worlds where the laws of logic are different — different from the actual laws, where such laws are expressed by (true) entailment claims. Accordingly, since our definition of validity is an attempt to capture our (actual) logical laws, we need not, and should not, worry about worlds where the logical laws are different, at least not in our definition of validity. Such worlds, however, are otherwise very important; as one can easily verify, such worlds afford the usual logical laws (within the positive fragment at issue) but do not sanction the unwanted "laws" — e.g., Assertion and the like. In this way, one can enjoy naive truth theory (or naive set theory) without tripping into triviality as a result of Curry sentences.

Priest (1992) gives a sound and complete proof theory for the given semantics, but this is left for the reader to consult.

One philosophical issue confronting the given semantics is the very nature of such non-normal worlds. What are they? As intimated, Priest’s suggestion is that they are simply (impossible) worlds where the laws of logic are different. But is there any reason, independent of Curry’s paradox, to admit such worlds? Fortunately, the answer seems to be ‘yes’. One reason has to do with the common (natural language) reasoning involving counter-logicals, including, for example, sentences such as ‘If intuitionistic logic is correct, then double negation elimination is invalid’. Invoking non-normal worlds provides a simple way of modelling such sentences and the reasoning involving them.

Another objection also arises. Notice that, on the foregoing
semantics, there are (non-normal) worlds where the law of
simplification, i.e., *A*&*B*
*B*, is false;
however, there is no world (normal or otherwise) at which we have a
false *B* but true *A*&*B*. Likewise for all
other worlds where the logical laws differ; the worlds themselves, as
it were, do not break the laws, even though the laws are false at such
worlds. What explains this "lack of supervenience" at non-normal
worlds? Priest himself offers no explanation, and the problem remains
an open one. None the less, here is a suggestion (which has yet to be
explored in print): What would it take for logical laws to fail? Most
philosophers will agree that it is hard to imagine worlds in which
there are events that contravene logical laws. My suggestion is that
the only way for logical laws to fail is via arbitrary "fiat", as it
were. No world (possible or otherwise) comprises events that refute,
contravene, or otherwise show the actual logical laws to be false; what
is required to falsify logical laws is mere arbitrariness; and such
arbitrariness is precisely what one gets from the function, *f*.
The suggestion, then, is simply this: For logical laws to fail at any
world (and, hence, at non-normal worlds) one requires arbitrariness and
thereby a lack of the supervenience at issue. Whether this suggestion
solves the (philosophical) problem at hand is an (other) open
problem.

There are other philosophical (and logical) problems that remain open. One of the most important recent papers discussing such problems is Restall’ "Costing Non-Classical Solutions to Paradoxes of Self-Reference" (see Other Internet Resources). Restall shows that the sorts of non-classical approach discussed above must give up either transitivity of entailment, infinitary disjunction or distributive lattice logic (i.e., an infinitary disjunction operator distributing over finite conjunction); otherwise, as Restall shows, Curry’s paradox arises immediately and triviality ensues. The importance of Restall’s point lies not only in the formal constraints imposed on suitable non-classical approaches to Curry; its importance lies especially in the philosophical awkwardness imposed by such constraints. For example, one (formal) upshot of Restall’s point is that, on a natural way of modelling propositions (e.g., in familiar world-semantics), some classes of propositions will not have disjunctions on the (given sort of) non-classical approach; the philosophical upshot (and important open problem) is that there is no known explanation for why such classes lack such a disjunction. (Needless to say, it is not a sufficient explanation to note that the presence of such a disjunction would otherwise generate triviality via Curry’s paradox.)

The foregoing issues and open problems confront various
non-classical approaches to paradox, problems that arise particularly
sharply in the face of Curry’s paradox. It should be understood,
however, that such problems may remain pressing even for those who are
firmly committed to classical approaches to paradox; for one might be
interested not so much in *accepting* or *believing* such
non-classical proposals but, rather, merely in using such proposals to
model various naive but non-trivial theories — naive truth
theory, naive set theory, naive denotation theory, etc.. One need not
believe or accept such theories to have an interest in modeling them
accurately. If one has such an interest, then the foregoing problems
arising from Curry’s paradox must be addressed. (See Slaney 1989, and
the classic Meyer, Dunn, and Routley 1979, and also Restall 2000 for
further discussion.)

- Barwise, J., and Etchemendy, J., 1984.
*The Liar*, New York: Oxford University Press. - Boolos, G., and Jeffrey, R., 1989.
*Computability and Logic*, 3rd edition, New York: Cambridge University Press. - Burge, T., 1979. "Semantical Paradox",
*Journal of Philosophy*76:169-198. - Curry, H., 1942, "The inconsistency of certain formal logics",
*Journal of Symbolic Logic*7, pp. 115-117. - Frederickson, G., 1997,
*Dissections: Plane and Fancy*, Cambridge: CUP. - Gardner, M., 1956.
*Mathematics, Magic and Mystery*, New York, Dover Publ. - Gaifman, H., 1988. "Operational pointer semantics: Solution to
self-referential puzzles I", in Vardi, M., ed,
*Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge*, Morgan Kaufmann. - Goldstein, L. 1986. "Epimenides and Curry".
*Analysis*463:117-121. - Gupta and Belnap, 1993.
*The Revision Theory of Truth*, Cambridge, MA: MIT Press. - Kripke, S., 1975. "Outline of a theory of truth",
*Journal of Philosophy*72:690-716. - Meyer, R. K., Routley, R. and Dunn, J.M., 1979, "Curry’s paradox",
*Analysis*39, pp. 124- 128. - Myhill, J., 1975. "Levels of Implication". In A. R. Anderson, R. C.
Barcan-Marcus, annd R. M. Martin, editors,
*The Logical Enterprise*, pp. 179-185. Yale Univ. Press. - Myhill, J., 1984. "Paradoxes".
*Synthese*, 60:129-143. - Priest, G., 1987.
*In Contradiction*, Martinus Nijhoff. - Priest, G., 1992. "What is a non-normal world?",
*Logique & Analyse*139-40:291-302. - Prior, A. N., 1955. "Curry’s Paradox and 3-Valued Logic",
*Australasian Journal of Philosophy*33:177-82. - Read, S., forthcoming, "Self-Reference and Validity Revisited",
in
*Medieval Logic and Language*, Mikko Yrjonsuuri (ed). (online PDF preprint) - Restall, G., 2000,
*An Introduction to Substructural Logics*, Routledge. (online précis) - Simmons, K., 1993.
*Universaliity and the Liar*, New York: Cambridge University Press. - Moh Skaw-Kwei. "Logical Paradoxes for Many-Valued Systems".
*Journal of Symbolic Logic*, 19 (1954), pp. 37-39. - Slaney, John. 1989. "RWX is not Curry Paraconsistent", in G.
Priest, R. Routley, and J. Norman (eds.),
*Paraconsistent Logic: Essays on the Inconsistent*, Philosophia Verlag, 472--480.

- Restall, G., "Costing Non-Classical Solutions to Paradoxes of Self-Reference".
- Restall, G., 1994,
*On Logics Without Contraction*, (Ph.D. dissertation, University of Queensland)

beall@uconn.edu

*First published: January 10, 2001*

*Content last modified: January 10, 2001*