“Curry’s paradox”, as the term is used by philosophers today, refers to a wide variety of paradoxes ofself-reference or circularity that trace their modern ancestry to Curry (1942b) and Löb (1955).^[1] The common characteristic of these so-called Curry paradoxes is theway they exploit a notion of implication, entailment, or consequence,either in the form of a connective or in the form of a predicate. Curry’s paradox arises in a number of different domains. LikeRussell’s paradox, it can take the form of a paradox ofset theory or the theory of properties. But it can also take the form of a semantic paradox, closely akin to theLiar paradox. Curry’s paradox differs from both Russell’s paradox andthe Liar paradox in that it doesn’t essentially involve thenotion of negation. Common truth-theoretic versions involve a sentencethat says of itself that if it is true then an arbitrarily chosenclaim is true, or—to use a more sinister instance—says ofitself that if it is true then every falsity is true. The paradox isthat the existence of such a sentence appears to imply the truth ofthe arbitrarily chosen claim, or—in the more sinisterinstance—of every falsity. In this entry, we show how thevarious Curry paradoxes can be constructed, examine the space ofavailable solutions, and explain some ways Curry’s paradox issignificant and poses distinctive challenges.

• 1. Introduction: Two Guises of the Paradox
  • 1.1 An Informal Argument
  • 1.2 A Constraint on Theories
  • 1.3 Overview
• 2. Constructing Curry Sentences
  • 2.1 Curry’s First Method, and Set-Theoretic Curry Sentences
  • 2.2 Curry’s Second Method, and Truth-Theoretic Curry Sentences
• 3. Deriving the Paradox
  • 3.1 The Curry-Paradox Lemma
  • 3.2 Alternative Premises
• 4. Responses to Curry’s Paradox
  • 4.1 Curry-Incompleteness Responses
  • 4.2 Curry-Completeness Responses
    • 4.2.1 Contraction-Free Responses
    • 4.2.2 Detachment-Free Responses
    • 4.2.3 Application to the Informal Argument
• 5. The Significance of Curry’s Paradox
  • 5.1 Dashing Hopes for Solutions to Negation Paradoxes
    • 5.1.1 Paraconsistent Solutions Frustrated
    • 5.1.2 Paracomplete Solutions Frustrated
  • 5.2 Pointing to a General Paradox Structure
• 6. Validity Curry
  • 6.1 Connective Form
  • 6.2 Predicate Form
  • 6.3 Significance
• Bibliography
  • Key Historical Sources
  • Other References
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction: Two Guises of the Paradox

1.1 An Informal Argument

Suppose that your friend tells you: “If what I’m sayingusing this very sentence is true, then time is infinite”. Itturns out that there is a short and seemingly compelling argument forthe following conclusion:

(P)

The mereexistence of your friend’s assertion entails (or has as aconsequence) that time is infinite.

Many hold that (P) is beyond belief (and, in that sense, paradoxical),even if time is indeed infinite. Or, if that isn’t bad enough,consider another version, this time involving a claim known to befalse. Let your friend say instead: “If what I’m sayingusing this very sentence is true, then all numbers are prime”.Now,mutatis mutandis, the same short and seeminglycompelling argument yields (Q):

(Q)

The mereexistence of your friend’s assertion entails (or has as aconsequence) that all numbers are prime.

Here is the argument for(P). Let \(k\) be the self-referential sentence your friend uttered,simplified somewhat so that it reads “If \(k\) is true then timeis infinite”. In view of what \(k\) says, we know this much:

(1)

Under the supposition that \(k\) is true, it is the case thatif \(k\)is true then time is infinite.

But, of course, we also have

(2)

Under the supposition that \(k\) is true, it is the case that \(k\)is true.

Under the supposition that \(k\) is true, we have thus derived aconditional together with its antecedent. Usingmodus ponenswithin the scope of the supposition, we now derive theconditional’s consequent under that same supposition:

(3)

Under the supposition that \(k\) is true, it is the case thattime is infinite.

The rule of conditional proof now entitles us to affirm a conditionalwith our supposition as antecedent:

(4)

If \(k\) is true then time is infinite.

But, since (4) just is \(k\) itself, we thus have

(5)

\(k\) is true.

Finally, putting (4) and (5) together bymodus ponens, weget

(6)

Time is infinite.

We seem to have established that time is infinite using no assumptionsbeyond the existence of the self-referential sentence \(k\), along withthe seemingly obvious principles about truth that took us to (1) and also from (4) to (5). And the same goes for(Q), since we could have used the same form of argument to reach the falseconclusion that all numbers are prime.

1.2 A Constraint on Theories

One challenge posed by Curry’s paradox is to pinpoint what goeswrong in the foregoing informal argument for(P),(Q) or the like. But starting with Curry’s initialpresentation in Curry 1942b (see the supplementary document onCurry on Curry’s Paradox), discussionof Curry’s paradox has usually had a different focus. It hasconcerned various formal systems —most often set theories ortheories of truth. In this setting what poses the paradox is a proofthat the system has a particular feature. Typically, the feature atissue istriviality. A theory is said to be trivial, orabsolutely inconsistent, when it affirms every claim that isexpressible in the language of the theory.^[2]

An argument establishing that a particular formal theory is trivialwill pose a problem if either of the following is the case: (i) wewish to use the formal theory in our inquiries, as we use set theorywhen doing mathematics, or (ii) we wish to use the formal theory inorder to model features of language or thought, in particular theclaims to which some speakers or thinkers are committed. Either way,the target theory’s triviality would show that it is inadequatefor its intended purpose. So this is a second challenge posed byCurry’s paradox.

To spell out the sense in which Curry’s paradox constrainstheories we need to say what aCurry sentence is. Informally,a Curry sentence is a sentence that is equivalent, by the lights ofsome theory, to a conditionalwith itself as antecedent. Forexample, one might think of the argument ofsection 1.1 as appealing to an informal theory of truth. Then the sentence“\(k\) is true” serves as a Curry sentence for thattheory. That is because, given what our informal theory tells us aboutwhat \(k\)’s truth involves, “\(k\) is true” shouldbe equivalent to “If \(k\) is true, then time is infinite”(since this conditional is \(k\) itself).

In what follows, the notation \(\vdash_{\mathcal{T}} \alpha\) is usedto say that theory \(\mathcal{T}\) contains sentence \(\alpha\), and\(\Gamma \vdash_{\mathcal{T}} \alpha\) is used to say that \(\alpha\)follows from the premises collected in \(\Gamma\) according to\(\mathcal{T}\) (i.e., according to \(\mathcal{T}\)’sconsequence relation \(\vdash_{\mathcal{T}}\)).^[3] Except insection 4.2.1, however, we will be concerned only with claims about what followsaccording to the theory from a single premise, i.e., claims expressed by sentences of form\(\gamma \vdash_{\mathcal{T}} \alpha\). (We rely on context to make clear where such a sentence is being used and where it is only being mentioned.)

Two sentences (in the languageof theory \(\mathcal{T}\)) will be calledintersubstitutableaccording to \(\mathcal{T}\) provided the truth of any claim of theform \(\Gamma \vdash_{\mathcal{T}} \alpha\) is unaffected bysubstitutions of one for the other within \(\alpha\) or within any ofthe sentences in \(\Gamma\). Finally, we assume that the languagecontains a connective \({\rightarrow}\) that serves, in some suitablesense, as a conditional. For purposes of the following definition, wedon’t place any specific requirements on the behavior of thisconditional. We can now define the notion of aCurry sentence fora sentence-theory pair.

The various versions of Curry’s paradox arise from the existenceof arguments in favor of the following very general claim. (Thesearguments, which rest on assumptions about the conditional\({\rightarrow}\), will be discussed in detail insection 3.)

An argument that appears to establish the Troubling Claim will countas paradoxical provided there is also compelling reason to believethat this claim is false. A counterexample to the Troubling Claimwould be any theory \(\mathcal{T}\) and sentence \(\pi\) such thatthere is a Curry sentence for \(\pi\) and \(\mathcal{T}\) but it isnot the case that \(\vdash_{\mathcal{T}} \pi\).

As noted above, Curry’s paradox is often understood as achallenge to the existence of nontrivial theories. Given the TroublingClaim, a theory will be trivial whenever a Curry sentence can beformulated forany sentence in the language of the theory.Indeed, triviality follows from a weaker condition, which thefollowing definition makes explicit.

While one instance of \(\pi'\) satisfying the condition (ii) would be\(\pi\) itself, another instance would be an “explosive”sentence \(\bot\) that is contained in a theory only ifeverysentence is contained in the theory.^[5]

The Troubling Claim now has an immediate consequence: a Curry-completetheory must contain every sentence in its language.

Again, any argument that appears to establish the Troubling Corollarywill count as paradoxical provided that there is compelling reason tobelieve that there are nontrivial theories (indeed true theories) that areCurry-complete.

1.3 Overview

For the remainder of this entry, Curry’s paradox will beunderstood as imposing a paradoxical constraint on theories, namelythe one stated by the above Troubling Corollary. Presenting a versionof Curry’s paradox, understood this way, involves doing twothings:

• arguing that \(\mathcal{T}\) is Curry-complete, for some apparentlynontrivial target theory \(\mathcal{T}\), and

• giving an argument for the Troubling Claim.^[6]

Sections2 and3 discuss these two tasks in that order. For now, the basic idea can beconveyed using the example of the self-referential sentence \(k\) thatreads “If \(k\) is true then time is infinite”. First,given our understanding of truth, we recognize that the sentence“\(k\) is true” is intersubstitutable with “If \(k\)is true, then time is infinite”. Second, the informal argumentofsection 1.1 derives a paradoxical conclusion from this equivalence. Readerschiefly interested in the logical principles involved in that argumentand related ones, and the options for resisting such arguments, maywish to turn tosection 3.

2. Constructing Curry Sentences

As it is standardly presented today, Curry’s paradox afflicts“naive” truth theories (those featuring a“transparent” truth predicate) and “naive” settheories (those featuring unrestricted set abstraction). This sectionwill explain how each kind of theory can give rise to Curry sentences.We start, however, with a version that concerns theories ofproperties, a version that more closely resemblesCurry’s formulation. (The supplementary documentCurry on Curry’s Paradox briefly characterizes the targets of Curry’s own versions ofthe paradox.)

A theory of properties featuresunrestricted propertyabstraction provided that for any condition statable in thelanguage of the theory, there exists a property that (according to thetheory) is exemplified by precisely the things that meet thiscondition. Consider a theory \(\mathcal{T_P}\) formulated in alanguage with a property abstraction device \([x: \phi x]\) and anexemplification relation \(\epsilon\). For example, if \(\phi(t)\)says that the object which the term \(t\) stands for is triangular,\(t \ \epsilon \ [x: \phi x]\) says that this object exemplifies theproperty of triangularity. Then, given unrestricted propertyabstraction, we should have the following principle.

In effect, Curry (1942b) sketches two“methods of constructing” Curry sentences using hiscounterpart of (Property). He says that the first is “based onthe Russell paradox”, while the second is “based on theEpimenides paradox”. Although both methods areproperty-theoretic, the first method yields a precursor ofset-theoretic versions of Curry’s paradox, while the secondyields a precursor of truth-theoretic versions.

2.1 Curry’s First Method, and Set-Theoretic Curry Sentences

The version of Russell’s paradox which Curry’sfirst method resembles is the one that concerns propertyexemplification. Its topic is the property of being such thatonefails to exemplify oneself. We obtain a property-theoretic Currysentence by considering instead the property of being such thatone exemplifies oneself only if time is infinite. Say that weintroduce the name \(h\) for that property, by stipulating \(h =_{def}[x: x \ \epsilon \ x {\rightarrow}\pi]\), where the sentence \(\pi\)says that time is infinite.^[7] Applying the principle(Property) to the sentence \(h \ \epsilon \ h\), we find:

In other words, \(h \ \epsilon \ h\) is a Curry sentence for \(\pi\)and \(\mathcal{T_P}\).

Curry’s first method subsequently gave rise to set-theoreticCurry sentences. A theory of sets featuresunrestricted setabstraction provided that for any condition statable in thelanguage of the theory, there exists a set that (according to thetheory) contains all and only the things that meet this condition. Let\(\mathcal{T_S}\) be our theory of sets, formulated in a language thatexpresses set abstraction using \(\{ x: \phi x\}\) and set membershipusing \(\in\). Then the counterpart of(Property) is

To obtain a set-theoretic Curry sentence, consider the set consistingof anything that is a member of itself only if time is infinite. Saythat we introduce the name \(c\) for that set, by stipulating \(c=_{def} \{ x: x \in x {\rightarrow}\pi \}\). Applying the principle(Set) to the sentence \(c \in c\), we find:

In other words, \(c \in c\) is a Curry sentence for \(\pi\) and\(\mathcal{T_S}\).

The set-theoretic version of Curry’s paradox was introduced inFitch 1952^[8] and is also presented in Moh 1954 andPrior 1955.

2.2 Curry’s Second Method, and Truth-Theoretic Curry Sentences

Despite his remark about the “Epimenides paradox”, a formof the Liar paradox, Curry’s second method is a variantof a related semantic paradox, Grelling’s paradox.^[9] In its original form, Grelling’s paradox considers a propertypossessed by many words, namely the property a word has when itfails to exemplify the property it stands for (Grelling& Nelson 1908). For example, theword “offensiveness” has that property: it fails toexemplify the property it stands for, since it isn’t offensive(see entry onparadoxes and contemporary logic). In effect, Curry considers instead the property a word has provideditexemplifies the property it stands for only if time isinfinite. Now suppose that our theory introduces a name \(u\) forthis property. Curry then shows how to construct a sentence that(speaking informally) says that the name \(u\) exemplifies theproperty it stands for. He shows that this sentence will serve as aCurry sentence for a theory of properties and the denotation of names.^[10]

Though this method of obtaining a Curry sentence is based on asemantic feature of expressions, it still relies on propertyabstraction. Nonetheless, it can be viewed as a precursor to a whollysemantic version. (Rather than consider the above-introducedproperty,one could consider thepredicate “applies to itself only if timeis infinite”.) Accordingly, as Geach(1955) and Löb (1955) werethe first to show, Curry sentences can be obtained using semanticprinciples alone, without any reliance on property abstraction. Theirroute corresponds to the informal argument, insection 1.1, involving the self-referential sentence \(k\) that reads “If\(k\) is true then time is infinite.”

For this purpose, let \(\mathcal{T_T}\) be a theory of truth, where\(T\) is the truth predicate. Assume the “transparency”principle

To obtain a Curry sentence using this principle, assume there is asentence \(\xi\) that is \(T\langle \xi \rangle {\rightarrow}\pi\).^[11] Then it follows immediately from (Truth) that

In other words, \(T\langle \xi \rangle\) is a Curry sentence for\(\pi\) and \(\mathcal{T_T}\).

Geach notes that the semantic paradox that results from a sentencelike \(T\langle \xi \rangle\) resembles “the Curry paradox inset theory”. Löb, who doesn’t mention Curry’swork, credits the paradox to a referee’s observation about theproof of what is now known as Löb’s theorem concerningprovability (see entry onGödel’s incompleteness theorems). The referee, now known to have been Leon Henkin (Halbach& Visser 2014: 257), suggestedthat the method Löb used in his proof “leads to anew derivation of paradoxes in natural language”, namely theinformal argument ofsection 1.1 above.^[12]

3. Deriving the Paradox

Suppose that we have used one of the above methods to show, for sometheory of truth, sets, or properties, that the theory isCurry-complete (in virtue, say, of containing a Curry sentence foreach sentence of the language, or for an explosive sentence). Toconclude that the theory in question is trivial, it now suffices togive an argument for the Troubling Claim. This is the claim that forevery theory \(\mathcal{T}\), if there is a Curry sentence for \(\pi\)and \(\mathcal{T}\), then \(\vdash_{\mathcal{T}} \pi\). Such anargument will make use of assumptions about the logical behavior ofthe conditional \({\rightarrow}\) mentioned inDefinition 1. Assuming the Troubling Claim must be resisted, this accordinglyplaces constraints on the behavior of this conditional.

3.1 The Curry-Paradox Lemma

To start, here is a very general limitative result, a close variant ofthe Lemma in Curry 1942b.^[13]

Here MP is a version ofmodus ponens, and Cont is a principleofcontraction: two occurrences of the sentence \(\alpha\)are “contracted” into one. (We will soon encounter relatedprinciples that are more commonly referred to as contraction.^[14]) TheCurry-Paradox Lemma entails that any Curry-complete theory mustviolate one or more of Id, MP or Cont on pain of triviality.

To prove the Lemma one shows that Id, MP and Cont, together with the“Curry-intersubstitutivity” of \(\kappa\) with \(\kappa{\rightarrow}\pi\), suffice to establish \(\vdash_{\mathcal{T}} \pi\).The following derivation resembles the informal argument ofsection 1.1. That argument also included a subargument for the principle Cont,which will be examined below.

Section 4 will discuss ways in which each of the two principles concerning\({\rightarrow}\) assumed in the Curry-Paradox Lemma might bejustified or rejected.

3.2 Alternative Premises

There are counterparts of the Curry-Paradox Lemma that invokealternative sets of logical principles (see, e.g., Rogerson& Restall 2004 and Bimbó2006). Probably the most commonversion replaces therules Id and Cont with correspondinglaws:

The derivation now goes as follows:

A second common counterpart of the Curry-Paradox Lemma is due to Meyer,Routley, and Dunn (1979).^[15] It uses two principles concerning conjunction: the law form ofmodus ponens and the idempotency of conjunction.

(Idem\(_{\wedge}\))

The sentences \(\alpha\) and \(\alpha \wedge \alpha\) are intersubstitutable according to \(T\)

This time the derivation goes as follows:

Formulating the Curry-Paradox Lemma using Cont, rather than ContL orMPL, will make it easier to call attention (in the next section) tosignificant differences within the class of responses that reject both of the latter principles.^[16]

4. Responses to Curry’s Paradox

Responses to Curry’s paradox can be divided into two classes,based on whether they accept the Troubling Corollary that allCurry-complete theories are trivial.

• Curry-incompleteness responses accept the TroublingCorollary. However, they deny that the target theories of properties,sets or truth are Curry-complete. Curry-incompleteness responses can,and usually do, embrace classical logic.

• Curry-completeness responses reject the Troubling Corollary;they insist that there can be nontrivial Curry-complete theories. Anysuch theory must violate one or more of the logical principles assumedin the Curry-Paradox Lemma. Since classical logic validates those principles, these responses invoke a non-classical logic.^[17]

There is also the option of advocating a Curry-incompleteness responseto Curry paradoxes arising in one domain, say set theory, whileadvocating a Curry-completeness response to Curry paradoxes arising inanother domain, say property theory (e.g.,Field 2008; Beall 2009).

4.1 Curry-Incompleteness Responses

Examples of prominent theories of truth that supplyCurry-incompleteness responses to Curry’s paradox include Tarski’s hierarchical theory, therevision theory of truth (Gupta & Belnap 1993) and thecontextualist approaches (Burge1979, Simmons 1993, and Glanzberg 2001, 2004). These theories all restrict the “naive” transparency principle(Truth). For an overview, see the entry on theLiar paradox. In thecontext of set theory Curry-incompleteness responses includeRussellian type theories and various theories that restrict the “naive” set abstraction principle(Set). See the entries onRussell’s paradox andalternative axiomatic set theories.

In general, the considerations relevant to evaluating mostCurry-incompleteness responses don’t appear to be specific toCurry’s paradox, but pertain equally to the Liar paradox (in thetruth-theoretic domain) and Russell’s paradox (in the set- and property-theoretic domains).^[18] For that reason the rest of this entry will focus on Curry-completeness responses, thoughsection 6.3 briefly returns to the distinction in the context of so-calledvalidity Curry paradoxes.

4.2 Curry-Completeness Responses

Curry-completeness responses to Curry’s paradox hold that thereare theories that are Curry-complete yet nontrivial; such a theorymust violate one or more of the logical principles assumed in theCurry-Paradox Lemma. Since the rule Id has generally been leftunquestioned (but see French 2016 and Nicolai& Rossi forthcoming), this has meant denying that theconditional \({\rightarrow}\) of a nontrivial Curry-complete theorysatisfies both MP and Cont. Accordingly, responses have fallen intotwo categories.

(I)

The most common strategy has been to accept that such a theory’s conditional obeys MP, but deny that it obeys Cont. Since Cont is a contraction principle, such responses can be calledcontraction-free. This strategy was first proposed by Moh (1954), who is cited approvingly by Geach (1955) and Prior (1955).

(II)

A second and much more recent strategy is to accept that such a theory’s conditional obeys Cont, but deny that it obeys MP (sometimes called the rule of “detachment”). Such responses can be calleddetachment-free. This strategy is advocated, in different ways, by Ripley (2013) and Beall (2015).

Each category of Curry-completeness responses can in turn besubdivided according to how it blocks purported derivations of Contand MP.

4.2.1 Contraction-Free Responses

The principle Cont that is rejected by contraction-free responsesfollows from two standard principles. These are single-premiseconditional proof and a slightly more general version ofmodusponens, involving at most one premise \(\gamma\):

(MP′)

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

(CP)

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

Contraction-free responses must thus reject one or the other of thesetwo principles for the conditional of a nontrivial Curry-completetheory. Accordingly, two subcategories of theorists in category (I)can be identified:

(Ia)

Astrongly contraction-free response denies that \({\rightarrow}\) obeys MP′ (e.g., Mares & Paoli 2014; Slaney 1990; Weir 2015; Zardini 2011).

(Ib)

Aweakly contraction-free response accepts that \({\rightarrow}\) obeys MP′, but denies that it obeys CP (e.g., Field 2008; Beall 2009; Nolan 2016).

The reason why responses in category (Ib) only count asweakly contraction-free is that, as steps 1–3 show,they accept the contraction principle according to which if \(\alpha \vdash_{\mathcal{T}} \alpha {\rightarrow}\beta\) then \(\alpha \vdash_{\mathcal{T}} \beta\).

Proponents of strongly contraction-free responses hold that MP′doesn’t properly express the relevant form ofmodusponens. They typically present their own form of that rule in a“substructural” framework, specifically one that lets usdistinguish between what follows from a premisetaken onceand what follows from the same premisetaken twice. (See the entry onsubstructural logics.) Accordingly, MP′ needs to be replaced by

(MP″)

If \(\gamma \vdash_{\mathcal{T}} \alpha {\rightarrow}\beta\) and \(\gamma \vdash_{\mathcal{T}} \alpha\) then \(\gamma, \gamma \vdash_{\mathcal{T}} \beta\)

and the rule of “structural contraction” needs to berejected:

(sCont)

If \(\Gamma, \gamma, \gamma \vdash_{\mathcal{T}} \beta\) then \(\Gamma, \gamma \vdash_{\mathcal{T}} \beta\)

It is because they reject structural contraction that stronglycontraction-free approaches can claim to preservemodus ponens despite rejecting MP′ (see Shapiro 2011,Zardini 2013, and Ripley 2015a).

Strongly contraction-free responses also need to block a derivation ofMP′ using a pair of principles involving conjunction:

(MP′\(_{\land}\))

If \(\gamma \vdash_{\mathcal{T}} \alpha {\rightarrow}\beta\) and \(\delta \vdash_{\mathcal{T}} \alpha\) then \(\gamma \wedge \delta \vdash_{\mathcal{T}} \beta\)

(Idem\(_{\wedge}\))

The sentences \(\alpha\) and \(\alpha \wedge \alpha\) are intersubstitutable according to \(T\)

Avoiding this derivation of MP′ requires denying that there is aconjunction \(\wedge\) that obeys both MP′\(_{\wedge}\) andIdem\(_{\wedge}\). According to many strongly contraction-freeresponses (e.g., Mares & Paoli 2014; Zardini 2011), one kind ofconjunction—the “multiplicative” kind, or“fusion”—obeys MP′\(_{\wedge}\) but notIdem\(_{\wedge}\), whereas another kind—the“additive” kind—obeys Idem\(_{\wedge}\) but not MP′\(_{\wedge}\) (see the entry onlinear logic, and Ripley 2015a). Ifthe substructural framework discussed above is used, the failure ofMP′\(_{\wedge}\) amounts to the fact that for additiveconjunction, \(\gamma, \delta \vdash_{\mathcal{T}} \beta\) is notequivalent to \(\gamma \wedge \delta \vdash_{\mathcal{T}}\beta\).

As for weakly contraction-free responses, the failure of CP hassometimes been motivated using “worlds” semantics of thesort that involve a distinction between logicallypossibleandimpossible worlds (e.g., Beall2009; Nolan 2016). To refute CP we need the truth of \(\alpha\vdash_\mathcal{T} \beta\) and the falsity of \(\vdash_\mathcal{T}\alpha{\rightarrow}\beta\). On the target “worlds”approaches \(\vdash_\mathcal{T}\) is defined as truth preservationover a proper subset of worlds (in a model), namely, the“possible worlds” of the model. Hence, for \(\alpha\vdash_\mathcal{T} \beta\) to be true is for there to be no possibleworld (in any model) at which \(\alpha\) is true and \(\beta\) untrue.In turn, to refute \(\vdash_\mathcal{T}\alpha{\rightarrow}\beta\) weneed a possible world at which \(\alpha{\rightarrow}\beta\) is untrue.How does that happen? Because connectives are defined in a way thattakes account ofall (types of) worlds in the model (possibleand, if there be any, impossible) there’s an option for\(\alpha{\rightarrow}\beta\) to be untrue at a possible world invirtue of \(\alpha\) being true and \(\beta\) being untrue at animpossible world. And that’s just what happens on thetarget approaches. (Exactly how one defines the truth-at-a-world andfalsity-at-a-world conditions for the arrow depends on the exact“worlds” approach at issue.)

4.2.2 Detachment-Free Responses

Detachment-free responses must block a straightforward derivation ofMP based on a principle of transitivity together with the converse ofsingle-premise conditional proof:

(Trans)

If \(\alpha \vdash_{\mathcal{T}} \beta\) and \(\vdash_{\mathcal{T}} \alpha\), then \(\vdash_{\mathcal{T}} \beta\)

(CCP)

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

There are two subcategories of theorists in category (II):

(IIa)

Astrongly detachment-free response denies that \({\rightarrow}\) obeys CCP (Goodship 1996; Beall 2015).

(IIb)

Aweakly detachment-free response accepts that \({\rightarrow}\) obeys CCP, but rejects Trans (Ripley 2013).

The reason why responses in category (IIb) are onlyweaklydetachment-free is that CCP, which these responses accept, can beregarded as a kind of detachment principle for the conditional.

One strategy for replying to the charge that detachment-free responsesare counterintuitive has been to appeal to a connection betweenconsequence and our acceptance and rejection of sentences. Accordingto this connection, whenever it is the case that \(\alpha\vdash_{\mathcal{T}} \beta\), this means (or at least implies) that itis incoherent by the lights of theory \(\mathcal{T}\) to accept\(\alpha\) while rejecting \(\beta\) (see Restall 2005). Now supposethat, by the lights of a theory \(\mathcal{T}\), it is incoherent toreject \(\alpha\) and it is also incoherent to accept \(\alpha\) whilerejecting \(\beta\). Then, Ripley (2013) argues, there need be nothingincoherent by the theory’s lights about rejecting \(\beta\), aslong as one doesn’t also accept \(\alpha\). There is thus roomto give up Trans and adopt a weakly detachment-free response toCurry’s paradox. Beall’s defense of the stronglydetachment-free approach rests on related considerations. He argues,in effect, that a principle weaker than CCP can play the relevant rolein constraining the combinations of acceptance and rejection ofsentences including \(\alpha\), \(\beta\), and \(\alpha{\rightarrow}\beta\).

4.2.3 Application to the Informal Argument

The approaches to Curry’s paradox just distinguished find fault with different inferences and sub-conclusions of the informal paradoxical argument insection 1.1. A strongly contraction-free response corresponds to blocking step (3)of that argument, since it rejects MP′. A weaklycontraction-free response instead blocks step (4), since it rejectsCP. Neither kind of detachment-free response will accept the reasoningin step (3). Since they accept Cont, detachment-free responses allowus to derive theconclusion of (4), whence weaklydetachment-free responses further allow us to derive the conclusion of(3) by CCP. However, both kinds of detachment-free response findfault with the final move by MP to (6).

5. The Significance of Curry’s Paradox

In this section, we explain some distinctive lessons that can belearned by considering Curry’s paradox. For discussion of thekinds of significance that versions of Curry’s paradox share with related paradoxes, see the entries onRussell’s paradox and theLiar paradox.

5.1 Dashing Hopes for Solutions to Negation Paradoxes

Starting with Church (1942), Moh(1954), Geach(1955), Löb (1955) and Prior(1955), discussion of Curry’sparadox has emphasized that it differs from Russell’s paradox,and the Liar paradox, in that it doesn’t “involv[e] negation essentially” (Anderson 1975: 128).^[19] One reason the negation-free status of Curry’s paradox mattersis that it renders the paradox resistant to some resolutions thatmight be adequate for such “negation paradoxes”.

Geach argues that Curry’s paradox poses a problem for anyproponents of naive truth theory or naive set theory who, faced withnegation paradoxes,

might … hope to avoid [these paradoxes] by using a logical systemin which ‘\(p\) if and only if not-\(p\)’ were a theoremfor some interpretations of ‘\(p\)’ without our being ableto infer thence any arbitrary statement…. (Geach1955: 71)

The problem, he says, is that Curry’s paradox “cannot beresolved merely by adopting a system that contains a queer sort ofnegation”. Rather, “if we want to retain the naive view oftruth, or the naive view of classes …, then we must modify theelementary rules of inference relating to ‘if’”(1955: 72). Geach’s view of thesignificance of Curry’s paradox is closely echoed by Meyer,Routley, and Dunn (1979: 127). Theyconclude that Curry’s paradox frustrates those who had“hoped that weakening classical negation principles” would resolve Russell’s paradox.^[20]

In short, the point is that there are non-classical logics with weaknegation principles that resolve Russell’s paradox and the Liar,yet remain vulnerable to Curry’s paradox. These are logics withthe following features:

(a)

They can serve as the basis for a nontrivial theory according to which some sentence is intersubstitutable with its own negation.

(b)

They can’t serve as the basis for a nontrivial theory that is Curry-complete.

While it is unclear which logics Geach may have had in mind, there areindeed non-classical logics that meet these two conditions. Theoriesbased on these logics accordingly remain vulnerable to Curry’sparadox.

5.1.1 Paraconsistent Solutions Frustrated

Meyer, Routley, and Dunn (1979) callattention to one class of logics that meet conditions (a) and (b). They are among theparaconsistent logics, which are logicsaccording to which a sentence together with its negation will not entail any arbitrary sentence. Paraconsistent logics can be used to obtaintheories that resolve Russell’s paradox, and the Liar, byembracing negation inconsistency without succumbing to triviality.

According to such a theory \(\mathcal{T}\), sentences \(\lambda\) and\(\lnot\lambda\) can be intersubstitutable, as long as both \(\vdash _{\mathcal{T}} \lambda\) and \(\vdash _{\mathcal{T}} \lnot \lambda\).Such theories are “glutty”, in the sense that they affirm some sentence together with its negation (see entry ondialetheism). Yet a number of prominent paraconsistent logics can’t serve asthe basis for Curry-complete theories on pain of triviality. Suchlogics are sometimes said to fail to be “Curry paraconsistent” (Slaney 1989).^[21]

5.1.2 Paracomplete Solutions Frustrated

Many of the non-classical logics that have been proposed to underwriteresponses to Russell’s paradox and the Liar paradox areparacomplete logics, logics that reject the law of excludedmiddle. These logics make possible “gappy” theories. Inparticular, where \(\lambda\) and \(\lnot\lambda\) areintersubstitutable according to such a theory \(\mathcal{T}\), it willfail to be the case that \(\vdash _{\mathcal{T}} \lambda \lor \lnot\lambda\). Some of these paracomplete logics likewise meet conditions(a) and (b).

One example is the logic Ł\(_{3}\) based on the three-valuedtruth-tables of Łukasiewicz (see, e.g.,Priest 2008). Since it meets condition (a), Ł\(_{3}\)offers a possible response to Russell’s paradox and theLiar—in particular, a gappy response. Yet consider theiterated conditional \(\alpha {\rightarrow}(\alpha{\rightarrow}\beta)\), which we abbreviate as \(\alpha \Rightarrow\beta\). Suppose that a Curry sentence for \(\pi\) and anŁ\(_{3}\)-based theory \(\mathcal{T}\) is redefined to be anysentence \(\kappa\) intersubstitutable with \(\kappa \Rightarrow\pi\). Then \(\mathcal{T}\) will meet all the conditions of theCurry-Paradox Lemma, as was first noted by Moh(1954). Hence, as long as there is a \(\kappa\) that isintersubstitutable with \(\kappa \Rightarrow \pi\) according to\(\mathcal{T}\), then \(\vdash _{\mathcal{T}} \pi\). Consequently Ł\(_{3}\) won’t underwrite a response to Curry’s paradox.^[22]

To summarize: Curry’s paradox stands in the way of someotherwise available avenues for resolving semantic paradoxes by meansof glutty or gappy theories. As a result, the need to evadeCurry’s paradox has played a significant role in the developmentof non-classical logics (e.g., Priest 2006;Field 2008).

5.2 Pointing to a General Paradox Structure

The negation-free status of Curry’s paradox matters for a secondreason. Prior makes the following important point:

We can … say not only that Curry’s paradox does notinvolve negation but that even Russell’s paradox presupposesonly those properties of negation which it shares with implication. (Prior 1955: 180)^[23]

What he has in mind is that Russell’s paradox and Curry’sparadox can be understood as resulting from thesame generalstructure, which can be instantiated either using negation or using a conditional.^[24]

The general structure can be made explicit by defining a type of unaryconnective that gives rise to Curry’s paradox, and showing howthis type is exemplified both by negation and by a unary connectivedefined in terms of a conditional.

The Generalized Curry-Paradox Lemma can now be instantiated in twodifferent ways, so as to yield either Curry’s paradox or anegation paradox:

• To obtain Curry’s paradox, let the unary connective \(\odot\) besuch that \(\odot\alpha\) is \(\alpha {\rightarrow}\pi\), and let\(\mu\) be a sentence intersubstitutable with \(\mu{\rightarrow}\pi\) according to \(\mathcal{T}\). Then P1 amounts tothe instance of MP used in our derivation of the Curry-Paradox Lemma,while P2 is nothing other than our rule Cont.

  \[\tag{MP}\textrm{If }\vdash_{\mathcal{T}} \alpha {\rightarrow}\beta\textrm{ and }\vdash_{\mathcal{T}} \alpha\textrm{ then }\vdash_{\mathcal{T}} \beta\]\[\tag{Cont}\textrm{If }\alpha \vdash_{\mathcal{T}} \alpha {\rightarrow}\beta\textrm{ then }\vdash_{\mathcal{T}} \alpha {\rightarrow}\beta\]

• To obtain a negation paradox, let \(\odot\alpha\) be \(\lnot\alpha\),and let \(\mu\) be a sentence intersubstitutable with \(\lnot\mu\)according to \(\mathcal{T}\).^[26] Then P1 amounts to an instance ofex contradictionequodlibet (or “explosion”), while P2 is areductio principle.

  \[\tag{ECQ}\textrm{If }\vdash_{\mathcal{T}} \alpha\textrm{ and }\vdash_{\mathcal{T}} \lnot\alpha\textrm{ then }\vdash_{\mathcal{T}} \beta\] \[\tag{Red}\textrm{If }\alpha \vdash_{\mathcal{T}} \lnot\alpha\textrm{ then }\vdash_{\mathcal{T}} \lnot\alpha\]

Prior’s point is that the features of negation that are relevantto Russell’s paradox or the Liar paradox areexhausted byits status as a Curry connective. This makes clear why theseparadoxes do not depend on features of negation, such as excludedmiddle or double negation elimination, that fail to hold innonclassical theories where negation remains a Curry connective (e.g., in intuitionistic theories, where ECQ and Red both hold).^[27]

Moreover, a Curry connective need not be very negation-like at all. It may fail to be even aminimal negation (see entry onnegation), since it need not obey the law of double introduction:

For example, suppose that \(\odot\alpha\) is \(\alpha{\rightarrow}\pi\). Then in order for \(\odot\) to obey DI, it wouldhave to be the case that \(\alpha \vdash _{\mathcal{T}} (\alpha{\rightarrow}\pi) {\rightarrow}\pi\). That principle is violated by anumber of non-classical theories for which \(\odot\), when defined this way,does qualify as a Curry connective.^[28]

To summarize: Curry’s paradox points to a general structureinstantiated by a wide range of paradoxes. This structuredoesn’t itself involve negation, but it is also displayed byparadoxes that (unlike Curry’s paradox) do essentially involvenegation, such as Russell’s paradox and the Liar paradox.

The issue of which paradoxes display a common structure becomesimportant in light of the “principle of uniform solution”influentially advocated by Priest (1994). According to this principle,paradoxes that belong to the “same kind” should receivethe “same kind of solution”. Suppose that we delimit onekind of paradox as follows:

Assuming one accepts the principle of uniform solution, the questionbecomes what counts as proposing a uniform solution to all generalizedCurry paradoxes. In particular, does it suffice to show, for everyinstance of the kind thus delimited, that what appears to be a Curryconnective in fact fails to be one? It would seem that this shouldindeed be enough. It’s unclear why uniformity shouldadditionally require that all seeming Curry connectives fail toqualify as suchin virtue of violating the samecondition. For instance, suppose that negation and our unaryconnective defined using \({\rightarrow}\) both appear to satisfy thegeneralized principle P2, in the former case because \({\lnot}\)appears to obey Red and in the latter case because \({\rightarrow}\)appears to obey Cont. Unless these two appearances share a commonsource (e.g., an implicit reliance on structural contraction, asclaimed by Zardini 2011), there need be nothing objectionablynon-uniform about taking one appearance at face value while dismissingthe other as deceptive. (For discussion of the philosophical issuehere, applied to a different class of paradoxes, see the exchange inSmith 2000 and Priest 2000.)

If that is right, the desideratum that generalized Curry paradoxes beresolved uniformly needn’t discriminate between the variouslogically revisionary solutions that have been pursued. These includethe following three options:

• One might hold that it is principle P1 alone that fails when\(\odot\alpha\) is instantiated as \(\lnot\alpha\) (to get a negationparadox), whereas it is P2 alone that fails when \(\odot\alpha\) isinstantiated as \(\alpha {\rightarrow}\pi\) (to get a Curry paradox).On this approach, ECQ and Cont fail, while Red and MP hold (Priest1994, 2006).

• One might hold that P2 alone fails for both instantiations of\(\odot\). On this approach, Red and Cont fail, while ECQ and MP hold(Field 2008; Zardini 2011).

• One might hold that P1 alone fails for both instantiations of\(\odot\). On this approach, ECQ and MP fail, while Red and Cont hold(Beall 2015; Ripley 2013).

Thus, for example, Priest’s own approach would count asresolving Curry’s paradox and the Liar paradox uniformlyquaexamples of generalized Curry paradox. This would be the casedespite the fact that Priest evaluates Liar sentences as both true andfalse, whereas he rejects the claim that Curry sentences are true.

In any event, Curry’s paradox raises challenges in connectionwith the issue of what type of uniformity should be required ofsolutions to various paradoxes (see also Zardini 2015). Priest himself calls attention to akind of paradox narrower than the generalized Curry paradoxes, a kindwhose instances include the negation paradoxes butexcludeCurry’s paradox. This kind is picked out by Priest’s“Inclosure Schema” (2002); see the entry onself-reference. One ongoing dispute is about whether there might be a version ofCurry’s paradox that counts as an “inclosureparadox”, though it resists Priest’s uniform dialetheicsolution to such paradoxes (see the exchange in Beall2014b, Weber etal. 2014, and Beall 2014a, aswell as Pleitz 2015).

6. Validity Curry

The last decade (as of the date of this version of this entry) haswitnessed a boom in attention to Curry paradoxes, and perhapsespecially to what have been calledvalidity Curry orv-Curry paradoxes (Whittle 2004; Shapiro 2011; Beall & Murzi 2013).^[29] V-Curry involves Curry sentences that specifically invoke atheory’s consequence or “validity” relation, byusing either a conditional or a predicate that purports to expresstheory \(\mathcal{T}\)’s relation \(\vdash_\mathcal{T}\) in thelanguage of \(\mathcal{T}\) itself.

6.1 Connective Form

For one form of v-Curry paradox, let the conditional mentioned in the definition of a Curry sentence (Definition 1) be aconsequence connective \({\Rightarrow}\). A sentencewith \({\Rightarrow}\) as its major operator is to be interpretedthus: “That \(p\) entails (according to \(\mathcal{T}\)) that\(q\)”. We now immediately obtain property-theoretic, set-theoretic ortruth-theoretic versions of Curry’s paradox, provided only that\({\Rightarrow}\) meets the conditions MP and Cont of theCurry-Paradox Lemma.

What makes this instance of the Curry-ParadoxLemma particularly troublesome is that it poses an obstacle to onecommon response to Curry’s paradox, namely theweakly contraction-free response discussed insection 4.2.1. That response depended on rejecting the rule CP of single-premise conditional proof, onedirection of the single-premise “deduction theorem”. Butthis is a rule that has seemed difficult to resist for a consequenceconnective (Shapiro 2011; Weber 2014; Zardini 2013). If \(\beta\) isa consequence of \(\alpha\) according to the consequence relation oftheory \(\mathcal{T}\), where this theory has \({\Rightarrow}\) asits own consequence connective, then \(\mathcal{T}\) must surelycontain the consequence claim \(\alpha{\Rightarrow}\beta\). Likewise, this variety of Curry paradox posesan obstacle fordetachment-free responses, which requirerejecting the rule MP. If a theory with its own consequenceconnective contains both \(\alpha\) and the consequence conditional\(\alpha {\Rightarrow}\beta\), then it must surely contain \(\beta\)as well. Or so, at least, it has seemed. Admittedly, the proponent ofaweakly detachment-free response will argue that MP for \({\Rightarrow}\) illicitly builds in transitivity (seesection 4.2.2). Still, what seems inescapable is the converse of CP, the rule CCP that is the otherdirection of the single-premise deduction theorem. If a theorycontains the consequence conditional \(\alpha {\Rightarrow}\beta\),then surely \(\beta\) follows from \(\alpha\) according to thetheory. That would still rule out astrongly detachment-freeresponse.

6.2 Predicate Form

A second form of v-Curry paradox arises for a theory \(\mathcal{T}_V\)whose subject-matter includes the single-premise consequence relation\(\vdash_{\mathcal{T}_{V}}\) that obtains, according to that verytheory, between sentences in its language.^[30] Let this relation be expressed by the predicate \(Val(x,y)\), andassume further that there is a sentence \(\chi\) that is either\(Val(\langle\chi\rangle, \langle\pi\rangle)\), or is at least intersubstitutable with the latter according to \(\mathcal{T}_V\). One form of v-Curryparadox employs two principles governing \(Val\), which we call“validity detachment” and “validity proof”following Beall & Murzi (2013).

Using these principles, we get the following quick argument for\(\vdash_{\mathcal{T}_{V}} \pi\).

As applied to thispredicate form of v-Curry, a weaklycontraction-free response would resist the “contraction”from step 2 to step 4 by rejecting the rule VP, and adetachment-free response would reject VD, even in the zero-premise form used at step 6. Again, though, bothVP and zero-premise VD have seemed inescapable in view of the intendedinterpretation of the predicate \(Val\) (Beall& Murzi 2013; Murzi 2014; Murzi & Shapiro 2015; Priest 2015;Zardini 2014).^[31] Finally, even if VD is rejected as illicitly involving transitivity, what seems inescapable is the converse of VP. If so, that would at least rule out astrongly detachment-free response.

An arguably more powerful version of v-Curry reasoning is presented byShapiro (2013) and Field(2017: 7). This reasoning can take eitherconnective or predicate form, but it doesn’t depend on CP or VP.Here we give the predicate form using \(Val\). As above, we firstderive that \(\chi \vdash_{\mathcal{T}_{V}} \pi\) using VD. In view of themeaning of \(Val\), the conclusion that \(\chi \vdash_{\mathcal{T}_{V}}\pi\) shows that \(Val(\langle\chi\rangle, \langle\pi\rangle)\) istrue, i.e., that \(\chi\) is true. But if \(\chi\) is true and \(\chi \vdash_{\mathcal{T}_{V}}\pi\), then it would seem\(\pi\) must also be true. Sinceweakly detachment-free(nontransitive) responses to v-Curry do allow the derivation of \(\chi\vdash_{\mathcal{T}_{V}} \pi\), this reasoning poses an objection tosuch responses as well.

6.3 Significance

If, in fact, v-Curry paradoxes aren’t amenable to weaklycontraction-free or strongly detachment-free responses, then (assuming the rule Id is retained) thespace of Curry-complete responses is restricted tostronglycontraction-free andweakly detachment-free responses.The former responses, as explained insection 4.2.1, are typically presented by reformulatingmodus ponens (or detachment for the validity predicate) in a substructural deduction system and rejecting the structural contraction rule sCont. Thelatter responses, as explained insection 4.2.2, reject the structural principle of transitivity. For this reason,v-Curry paradoxes have sometimes been taken to motivatesubstructural consequence relations (e.g.,Barrio et al. forthcoming; Beall & Murzi2013; Ripley 2015a; Shapiro 2011, 2015).^[32]

The lively and wide-ranging debate on v-Curry paradoxes has resultedin genuine progress in our understanding of Curry paradoxes. In theend, what has become clear is that while v-Curry paradoxes may invitedifferent resolutions from non-v-Curry paradoxes, they remain withinthe same mold as generalized Curry paradoxes. In particular, in thegeneral template ofsection 5.2 one may take \(\odot\) to express (either as a predicate or as aconnective) consequence in light of \(\vdash_\mathcal{T}\) itself.This is the heart of v-Curry. Inasmuch as there are (many) different(formal) consequence relations definable over our language (e.g.,logical consequence in virtue of logical vocabulary, epistemicconsequence in virtue of logical-plus-epistemic vocabulary, and so on)there are thereby many different v-Curry paradoxes that may arise.Still, the space of solutions to these paradoxes is the space ofsolutions to the generalized Curry paradoxes canvassed in thisentry.

There remain, however, at least two reasons v-Curry paradoxes meritseparate attention. First, as noted above, two categories ofCurry-complete solutions — the weakly contraction-free andstrongly detachment-free options — have appeared especiallyproblematic in the case of v-Curry paradoxes. Second, suppose that onetreats an ordinary Curry paradox (property-theoretic, set-theoretic orsemantic) in a Curry-complete fashion. There may still be reason totreat the corresponding (connective or predicate) v-Curry paradox in aCurry-incomplete fashion, perhaps in virtue of seeing a theory’sconsequence relation as essentially beyond capture by any connectiveor predicate in the language of the theory (see, e.g., Myhill 1975;Whittle 2004). Thus, a “non-uniform” solution to ordinaryCurry paradoxes and their v-Curry counterparts may — onceagain — be a motivatednon-uniformity.^[33]

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dialetheism |Gödel, Kurt: incompleteness theorems |lambda calculus, the |liar paradox |logic: algebraic propositional |logic: combinatory |logic: linear |logic: paraconsistent |logic: substructural |negation |paradoxes: and contemporary logic |Russell’s paradox |self-reference |set theory: alternative axiomatic theories

Acknowledgments

We are grateful to Julien Murzi, Lorenzo Rossi, and an anonymousreferee for detailed comments that led to clarifications andimprovements. We also wish to thank the participants of our 2016graduate seminar at UConn on this topic.