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1. The term “Curry’s paradox” appears to originate inFitch 1952; other influential earlyformulations include Moh 1954, Geach1955, and Prior1955. These will be discussed insection 2. Precursors to Curry’s paradox are also found in the work ofmedieval and late scholastic logicians: for references and discussion, see Ashworth1974: 125, Read2001 and Hanke2013.

2. Curry’s paper is entitled “The Inconsistency of CertainFormal Logics” (1942b). By“inconsistency”, he meansabsolute inconsistency,i.e., triviality.

3. Typically, a theory’s consequence relation will be aclosure relation, in the sense that \(\Gamma\vdash_{\mathcal{T}} \alpha\) if and only if \(\alpha\) is among thesentences in \(Cn(\Gamma)\), where \(Cn\) is a closure operation. Seeentry onalgebraic propositional logic. However, Ripley (2015b) shows that theconsequence relations of many of the theories that yield“contraction-free” responses to Curry’s paradox (seesection 4.2.1) fail to be closure relations.

4. The term “Curry sentence” is sometimes used in a narrowersense, namely for a sentence that says of itself (only) thatif itis true, then \(p\) (or, alternatively, an absurdity) is true(e.g., Beall 2009: 33; Zardini2011: 503). Some discussions (e.g.,Restall 1994: vii; Humberstone2006) demarcate the relevantnotion using abiconditional rather thanintersubstitutability, namely as any \(\kappa\) such that\(\vdash_{\mathcal{T}} \kappa {\leftrightarrow}(\kappa{\rightarrow}\pi)\). The disadvantage here is that the behavior of the\({\rightarrow}\) is tied to that of the \({\leftrightarrow}\), eventhough they play very different roles in generating paradox. Note thatthe mere existence of a Curry sentence for a sentence-theory pairposes no paradox, nor is it an objection to the theory in question.For example, \(\pi {\rightarrow}\pi\) will be a Curry sentence for\(\pi {\rightarrow}\pi\) and the theory that consists of the theoremsof classical logic, since \(\pi {\rightarrow}\pi\) will beintersubstitutable with \((\pi {\rightarrow}\pi) {\rightarrow}(\pi{\rightarrow}\pi)\). The authors thank Lorenzo Rossi for raising thisissue.

5. If the consequence relation \(\vdash_{\mathcal{T}}\) is assumed to betransitive, condition (ii) can be replaced with the condition that\(\pi' \vdash_{\mathcal{T}} \pi\). However, one of the responses toCurry’s paradox considered below rejects transitivity.

6.Section 6 will consider a somewhat broader family of paradoxes.

7. The letter \(h\) is chosen to correspond with the\(\mathfrak{H}\) that Curry 1942b uses todenote this property; likewise for \(u\) below.

8. Fitch considers the paradox in both its property-theoretic andset-theoretic versions (Fitch 1952: 89).

9. This is noted in Church’s contemporaneous review (Church1942).

10. Strictly speaking, in his presentation, the “names” arenumerals that denote numbers, not properties; properties are thengiven numerical codes.

11. Geach actually uses diagonalization based on a theory of syntax toachieve the required self-reference. He adds that “we mightinstead use the familiar Gödelian devices” for simulatingself-reference using diagonalization based on a theory of arithmetic.See entry onGödel’s incompleteness theorems.

12. On the relationship between Curry’s paradox andLöb’s theorem, see van Benthem(1978), who remarks on the “strange anti-parallel betweenGödel’s proof, where a well-known semantic paradox inspireda formal result in terms of provability rather than truth, andLöb’s proof, where a semantic paradox … was extractedfrom a formal result about provability” (1978: 59).

13. This is not to be confused with a result in proof theory known asCurry’s Lemma following Anderson &Belnap (1975: 136).

14.Zardini (2015), who calls Cont the rule of “absorption”,notes that given MP it entails there can be no Curry sentence for an“unacceptable sentence” \(\pi\).

15. It is also closely related to the method used in Löb’soriginal proof of Löb’s theorem (1955).

16. A third counterpart of the Curry-Paradox Lemma replaces Cont with arule which, in light of Peirce’s Law \(\vdash_{\mathcal{T}}((\alpha {\rightarrow} \beta ) {\rightarrow}\alpha)){\rightarrow}\alpha\), could be called Peirce’s Rule:

(PR)

If \(\alpha {\rightarrow}\beta \vdash_{\mathcal{T}} \alpha\) then \(\vdash_{\mathcal{T}} \alpha\).

The proof of the Lemma is then structurally analogous to that insection 3.1. Here, too, one could instead use IdL and Peirce’s Law. Cf.Bunder 1986, Rogerson& Restall 2004. Unlike theprinciples used so far, Peirce’s Law and Peirce’s Rulefail in intuitionistic logic.

17. However, once a response rejects enough standard logical principlesinvolving \({\rightarrow}\), it may be indeterminate whether it countsas a Curry-completeness or Curry-incompleteness response. That’sbecause the connective \({\rightarrow}\) appearing in the definitionof a Curry sentence (Definition 1) was stipulated to be aconditional.

18. The grounds given for denying that a target theory \(\mathcal{T}\) isCurry-complete are typically parallel to grounds for denying thatthere is a sentence \(\lambda\) such that it and its negation\(\lnot\lambda\) are intersubstitutable according to \(\mathcal{T}\).And the respective moves will have parallel philosophicalramifications. An exception may be the early response to Curry’sparadox by Fitch (1952, 1969). Fitchpresents several possible restrictions on the inference rules of anatural deduction system that are motivated specifically byCurry’s paradox. Since their effect is to ensure violations of(Property) and(Set), these are Curry-incompleteness responses. In particular,Fitch’s systems allow a derivation of \(\vdash_{\mathcal{T_P}} h\ \epsilon \ h {\rightarrow}\pi\). Yet, based on global features ofthat derivation, his “special restriction” and“restriction on nonrecurrence” would both block itsextension to a derivation of \(\vdash_{\mathcal{T_P}} h \ \epsilon \h\). For helpful discussion, see Anderson1975 and Rogerson 2007.

19. Counterparts of the Curry-Paradox Lemma can be formulated usingprinciples involving negation. For example, \(\lnot\alpha\vdash_{\mathcal{T}} \alpha {\rightarrow}\beta\) together with theclassicalreductio rule

yield Peirce’s Rule ofnote 16. (Of course, in a language whose only “conditional”\(\alpha {\rightarrow}\beta\) is the material conditional\(\lnot\alpha \lor \beta\), the corresponding version of Curry’sparadox will involve negation. In that setting, Curry’s paradoxposes a less distinctive challenge.)

20. Like Geach, they add that Curry’s paradox shows that “theproperties of implication … do us in on their own”. (By“implication”, they mean a conditional; seenote 23.)

21. Meyer et al. (1979) cite the relevant logics \(E\) and \(R\); thesame is true of the contraction-free relevant logic \(RWX\) (Slaney1989). Glutty responses to paradoxthatdo admit of Curry-complete theories include Priest2006, Beall2009, and Beall 2015.

22. See also Restall 1993. Geach citesMoh’s paper, but there is little reason to think he hadŁ\(_{3}\) in mind as a logic with which one “might… hope” to avoid the Liar paradox but which remainsvulnerable to Curry’s paradox. That is because Curry sentenceswith respect to the primitive conditional \({\rightarrow}\), whichfails to satisfy Cont, pose no problem for a theory based onŁ\(_{3}\). Gappy responses to paradox that admit ofnontrivial Curry-complete theories include Kripke1975, White1979 and Field 2008.

23. See also Curry & Feys 1958: 259. By“implication”, Prior means a conditional. Several yearsearlier, Quine had remarked that the use of “conditional”in place of “implication” is “encouragingly on theincrease” (1953: 451).

24. The fact that Curry’s paradox calls attention to this generalstructure could perhaps be what Curry et al.(1972) have in mind when they refer to Curry’s paradox as“the generalized Russell paradox”. However, their point islikely a different one, namely that Russell’s paradox becomes aspecial case of (the property- or set-theoretic form of)Curry’s paradox provided \(\lnot\alpha\) is defined as \(\alpha{\rightarrow}\bot\), as is possible in intuitionistic and classicallogic.

25. The same result obtains if the requirement that \(\odot\) be a Curryconnective is replaced with the requirement that it be a Curry\('\)connective, where this is defined by replacing P2 with

(P2′)

If \(\odot\alpha \vdash_{\mathcal{T}} \alpha\) then \(\vdash_{\mathcal{T}} \alpha\). (Seenote 16.)

For what are, in effect, quite a number of other ways of generalizingthe Curry-Paradox Lemma, see Bimbó2006 and Humberstone 2006.

26. In the case of Russell’s paradox in set theory, \(\mu\) can be\(\{x: x \notin x\} \in \{x: x \notin x\}\). In the case of the Liarparadox, \(\mu\) can be \(T\langle\lambda\rangle\) where \(\lambda\)is \(\lnot T\langle\lambda\rangle\).

27. See also Curry & Feys 1958: 259.They add: “If one insists on regarding [\(\odot\alpha\), definedas \(\alpha {\rightarrow}\pi\)] as a species of negation, then thatnegation is a minimal negation”. As noted below, this isstrictly speaking incorrect. The explanation is that Curry’sproof of the corresponding claim, in Curry1952, assumes that the conditional is characterized using an extension of thesequent calculus in Curry 1950:32–33, which includes the rule of structural exchange.This is what guarantees that \(\alpha \vdash (\alpha {\rightarrow}\pi){\rightarrow}\pi\), i.e., \(\alpha \vdash _{\mathcal{T}}\odot\odot\alpha\).

28. These include the relevant logics \(T\) and \(E\) of Anderson& Belnap 1975, and many of thelogics explored in Brady 2006.

29.More recent work includes Bacon 2015, Barrio et al. forthcoming, Cook2014, Field 2017, Mares & Paoli 2014, Meadows 2014, Murzi 2014,Murzi & Rossi forthcoming, Murzi & Shapiro 2015, Nicolai &Rossi 2017, Rosenblatt 2017, Tajer & Pailos 2017, Priest 2015,Shapiro 2013 & 2015, Wansing & Priest 2015, Weber 2014,Zardini 2013 & 2014.

30. The predicate form of v-Curry is discussed in detail by Beall &Murzi (2013); it is mentioned by Whittle (2004) and Shapiro(2011). The connective and predicate forms are equivalent provided\(Val(\langle\alpha\rangle, \langle\beta \rangle)\) and \(\alpha{\Rightarrow}\beta \) are intersubstitutable according to\(\mathcal{T}_{V}\). Accordingly, while the predicate form isasemantic paradox (in the sense that it concerns a featureof expressions that depends on their interpretation), it can beobtained by constructing a property- or set-theoretic Curry sentence\(\kappa\) intersubstitutable according to \(\mathcal{T}_{V}\) with\(\kappa {\Rightarrow}\pi\) and ensuring that the latter sentence isin turn intersubstitutable with \(Val(\langle\kappa\rangle,\langle\pi\rangle)\).

31. The status of rules such as VP and VD is a topic of controversy: see Field2017.

32.Indeed, Nicolai & Rossi(forthcoming) argue that v-Curry paradoxes motivate responsesaccording to which a third structural principle, namely thereflexivity principle \(\phi \vdash_{\mathcal{T}_{V}}\! \phi\)codified by Id, can fail in cases where \(\phi \) contains a predicateexpressing the relation \(\vdash_{\mathcal{T}_{V}}\).

33. The non-uniformityconsidered at the start ofsection 4 was a matter of giving different responses to ordinary Curry paradoxes arisingin different domains (such as property theory set theory). The non-uniformityconsidered insection 5.2 was a matter of different connectives failing to qualify asCurry connectives for different reasons.