## 1. Introduction: Two Guises of the Paradox


### 1.1 An Informal Argument




Suppose that your friend tells you: "If what I'm saying
using this very sentence is true, then time is infinite". It
turns out that there is a short and seemingly compelling argument for
the following conclusion:



(P)
The mere
existence of your friend's assertion entails (or has as a
consequence) 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 be
false. Let your friend say instead: "If what I'm saying
using this very sentence is true, then all numbers are prime".
Now, *mutatis mutandis*, the same short and seemingly
compelling argument yields (Q):



(Q)
The mere
existence of your friend's assertion entails (or has as a
consequence) 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 time
is infinite". In view of what \(k\) says, we know this much:



(1)
Under the supposition that \(k\) is true, it is the case
 that *if* \(k\) *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 a
conditional together with its antecedent. Using *modus ponens*
within the scope of the supposition, we now derive the
conditional's consequent under that same supposition:



(3)
Under the supposition that \(k\) is true, it is the case
 that *time is infinite*.

The rule of conditional proof now entitles us to affirm a conditional
with 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 by *modus ponens*, we
get



(6)
Time is infinite.

We seem to have established that time is infinite using no assumptions
beyond the existence of the self-referential sentence \(k\), along with
the 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 false
conclusion that all numbers are prime.


### 1.2 A Constraint on Theories



One challenge posed by Curry's paradox is to pinpoint what goes
wrong in the foregoing informal argument for
 (P),
 (Q) or the like. But starting with Curry's initial
presentation in Curry 1942b (see the supplementary document on 
 Curry on Curry's Paradox), discussion
of Curry's paradox has usually had a different focus. It has
concerned various formal systems --most often set theories or
theories of truth. In this setting what poses the paradox is a proof
that the system has a particular feature. Typically, the feature at
issue is *triviality*. A theory is said to be trivial, or
*absolutely inconsistent*, when it affirms every claim that is
expressible in the language of the
 theory.[2]




An argument establishing that a particular formal theory is trivial
will pose a problem if either of the following is the case: (i) we
wish to use the formal theory in our inquiries, as we use set theory
when doing mathematics, or (ii) we wish to use the formal theory in
order to model features of language or thought, in particular the
claims to which some speakers or thinkers are committed. Either way,
the target theory's triviality would show that it is inadequate
for its intended purpose. So this is a second challenge posed by
Curry's paradox.



To spell out the sense in which Curry's paradox constrains
theories we need to say what a *Curry sentence* is. Informally,
a Curry sentence is a sentence that is equivalent, by the lights of
some theory, to a conditional *with itself as antecedent*. For
example, one might think of the argument of
 section 1.1
 as appealing to an informal theory of truth. Then the sentence
"\(k\) is true" serves as a Curry sentence for that
theory. That is because, given what our informal theory tells us about
what \(k\)'s truth involves, "\(k\) is true" should
be 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 used
to 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 \(\mathcal{T}\)'s
consequence relation
 \(\vdash\_{\mathcal{T}}\)).[3]
 Except in
 section 4.2.1,
 however, we will be concerned only with claims about what follows
according 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 language
of theory \(\mathcal{T}\)) will be called *intersubstitutable*
according to \(\mathcal{T}\) provided the truth of any claim of the
form \(\Gamma \vdash\_{\mathcal{T}} \alpha\) is unaffected by
substitutions of one for the other within \(\alpha\) or within any of
the sentences in \(\Gamma\). Finally, we assume that the language
contains a connective \({\rightarrow}\) that serves, in some suitable
sense, as a conditional. For purposes of the following definition, we
don't place any specific requirements on the behavior of this
conditional. We can now define the notion of a *Curry sentence for
a sentence-theory pair*.




**Definition 1** (Curry sentence) Let \(\pi\) be a
sentence of the language of \(\mathcal{T}\). A Curry sentence for
\(\pi\) and \(\mathcal{T}\) is any sentence \(\kappa\) such that
\(\kappa\) and \(\kappa {\rightarrow}\pi\) are intersubstitutable
according to
 \(\mathcal{T}\).[4]




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




**Troubling Claim** For every theory \(\mathcal{T}\), and
any sentence \(\pi\) in the language of \(\mathcal{T}\), if there is a
Curry sentence for \(\pi\) and \(\mathcal{T}\), then
\(\vdash\_{\mathcal{T}} \pi\).




An argument that appears to establish the Troubling Claim will count
as paradoxical provided there is also compelling reason to believe
that this claim is false. A counterexample to the Troubling Claim
would be any theory \(\mathcal{T}\) and sentence \(\pi\) such that
there is a Curry sentence for \(\pi\) and \(\mathcal{T}\) but it is
not the case that \(\vdash\_{\mathcal{T}} \pi\).



As noted above, Curry's paradox is often understood as a
challenge to the existence of nontrivial theories. Given the Troubling
Claim, a theory will be trivial whenever a Curry sentence can be
formulated for *any* sentence in the language of the theory.
Indeed, triviality follows from a weaker condition, which the
following definition makes explicit.




**Definition 2** (Curry-complete theory) A theory
\(\mathcal{T}\) is Curry-complete provided that for every sentence
\(\pi\) in the language of \(\mathcal{T}\), there is some \(\pi'\)
such that (i) there is a Curry sentence for \(\pi'\) and
\(\mathcal{T}\) and (ii) if \(\vdash\_{\mathcal{T}} \pi'\) then
\(\vdash\_{\mathcal{T}} \pi\).




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 if *every*
sentence is contained in the
 theory.[5]




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




**Troubling Corollary** Every Curry-complete theory is
trivial.




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


### 1.3 Overview



For the remainder of this entry, Curry's paradox will be
understood as imposing a paradoxical constraint on theories, namely
the one stated by the above Troubling Corollary. Presenting a version
of Curry's paradox, understood this way, involves doing two
things:


* arguing that \(\mathcal{T}\) is Curry-complete, for some apparently
nontrivial target theory \(\mathcal{T}\), and
* giving an argument for the Troubling
 Claim.[6]



Sections
 2
 and
 3
 discuss these two tasks in that order. For now, the basic idea can be
conveyed using the example of the self-referential sentence \(k\) that
reads "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 argument
of
 section 1.1
 derives a paradoxical conclusion from this equivalence. Readers
chiefly interested in the logical principles involved in that argument
and related ones, and the options for resisting such arguments, may
wish to turn to
 section 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" set
theories (those featuring unrestricted set abstraction). This section
will explain how each kind of theory can give rise to Curry sentences.
We start, however, with a version that concerns theories of
 properties, a version that more closely resembles
Curry's formulation. (The supplementary document
 Curry on Curry's Paradox
 briefly characterizes the targets of Curry's own versions of
the paradox.)



A theory of properties features *unrestricted property
abstraction* provided that for any condition statable in the
language of the theory, there exists a property that (according to the
theory) is exemplified by precisely the things that meet this
condition. Consider a theory \(\mathcal{T\_P}\) formulated in a
language with a property abstraction device \([x: \phi x]\) and an
exemplification 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 the
property of triangularity. Then, given unrestricted property
abstraction, we should have the following principle.




**(Property)** For every open sentence \(\phi\) with one
free variable, and every term \(t\), the sentences \(t \ \epsilon \
[x: \phi x]\) and \(\phi t\) are intersubstitutable according to
\(\mathcal{T\_P}\).




In effect, Curry (1942b) sketches two
"methods of constructing" Curry sentences using his
counterpart of (Property). He says that the first is "based on
the Russell paradox", while the second is "based on the
Epimenides paradox". Although both methods are
property-theoretic, the first method yields a precursor of
set-theoretic versions of Curry's paradox, while the second
yields 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's
first method resembles is the one that concerns property
exemplification. Its topic is the property of being such that *one
fails to exemplify oneself*. We obtain a property-theoretic Curry
sentence by considering instead the property of being such that
*one exemplifies oneself only if time is infinite.* Say that we
introduce 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:




\(h \ \epsilon \ h\) and \(h \ \epsilon \ h {\rightarrow}\pi\) are
intersubstitutable according to \(\mathcal{T\_P}\).




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-theoretic
Curry sentences. A theory of sets features *unrestricted set
abstraction* provided that for any condition statable in the
language of the theory, there exists a set that (according to the
theory) contains all and only the things that meet this condition. Let
\(\mathcal{T\_S}\) be our theory of sets, formulated in a language that
expresses set abstraction using \(\{ x: \phi x\}\) and set membership
using \(\in\). Then the counterpart of
 (Property)
 is




**(Set)** For every open sentence \(\phi\) with one free
variable, and every term \(t\), the sentences \(t \in \{ x: \phi x\}\)
and \(\phi t\) are intersubstitutable according to
\(\mathcal{T\_S}\).




To obtain a set-theoretic Curry sentence, consider the set consisting
of anything that is a member of itself only if time is infinite. Say
that 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:




\(c \in c\) and \(c \in c {\rightarrow}\pi\) are intersubstitutable
according to \(\mathcal{T\_S}\).




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 in
Fitch
 1952[8]
 and is also presented in Moh 1954 and
Prior 1955.


### 2.2 Curry's Second Method, and Truth-Theoretic Curry Sentences



Despite his remark about the "Epimenides paradox", a form
of the Liar paradox, Curry's second method is a variant
of a related semantic paradox, Grelling's
 paradox.[9]
 In its original form, Grelling's paradox considers a property
possessed by many words, namely the property a word has when it
*fails to exemplify the property it stands for* (Grelling
& Nelson 1908). For example, the
word "offensiveness" has that property: it fails to
exemplify the property it stands for, since it isn't offensive
(see entry on
 paradoxes and contemporary logic).
 In effect, Curry considers instead the property a word has provided
it *exemplifies the property it stands for only if time is
infinite*. Now suppose that our theory introduces a name \(u\) for
this property. Curry then shows how to construct a sentence that
(speaking informally) says that the name \(u\) exemplifies the
property it stands for. He shows that this sentence will serve as a
Curry sentence for a theory of properties and the denotation of
 names.[10]




Though this method of obtaining a Curry sentence is based on a
*semantic feature of expressions*, it still relies on property
abstraction. Nonetheless, it can be viewed as a precursor to a wholly
semantic version. (Rather than consider the above-introduced *property*,
one could consider the *predicate* "applies to itself only if time
is infinite".) Accordingly, as Geach
(1955) and Lob (1955) were
the first to show, Curry sentences can be obtained using semantic
principles alone, without any reliance on property abstraction. Their
route corresponds to the informal argument, in
 section 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




**(Truth)** For every sentence \(\alpha\), the sentences
\(T\langle \alpha \rangle\) and \(\alpha\) are intersubstitutable
according to \(\mathcal{T\_T}\).




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




\(T\langle \xi \rangle\) and \(T\langle \xi \rangle {\rightarrow}\pi\)
are intersubstitutable according to \(\mathcal{T\_T}\).




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 sentence
like \(T\langle \xi \rangle\) resembles "the Curry paradox in
set theory". Lob, who doesn't mention Curry's
work, credits the paradox to a referee's observation about the
proof of what is now known as Lob's theorem concerning
provability (see entry on
 Godel's incompleteness theorems).
 The referee, now known to have been Leon Henkin (Halbach
& Visser 2014: 257), suggested
that the method Lob used in his proof "leads to a
new derivation of paradoxes in natural language", namely the
informal argument of
 section 1.1
 above.[12] 


## 3. Deriving the Paradox



Suppose that we have used one of the above methods to show, for some
theory of truth, sets, or properties, that the theory is
Curry-complete (in virtue, say, of containing a Curry sentence for
each sentence of the language, or for an explosive sentence). To
conclude that the theory in question is trivial, it now suffices to
give an argument for the Troubling Claim. This is the claim that for
every theory \(\mathcal{T}\), if there is a Curry sentence for \(\pi\)
and \(\mathcal{T}\), then \(\vdash\_{\mathcal{T}} \pi\). Such an
argument will make use of assumptions about the logical behavior of
the conditional \({\rightarrow}\) mentioned in
 Definition 1.
 Assuming the Troubling Claim must be resisted, this accordingly
places 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 of
the Lemma in Curry
 1942b.[13]





**Curry-Paradox Lemma** Suppose that theory
\(\mathcal{T}\) and sentence \(\pi\) are such that (i) there is a
Curry sentence for \(\pi\) and \(\mathcal{T}\), (ii) all instances of
the identity rule (Id) \(\alpha \vdash\_{\mathcal{T}} \alpha\) hold,
and (iii) the conditional \({\rightarrow}\) satisfies both of the
following principles:

 

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

 

\[\tag{Cont} \textrm{If } \alpha \vdash\_{\mathcal{T}} \alpha
{\rightarrow}\beta \textrm{ then } \vdash\_{\mathcal{T}} \alpha
{\rightarrow}\beta \]


Then \(\vdash\_{\mathcal{T}} \pi\).




Here MP is a version of *modus ponens*, and Cont is a principle
of *contraction*: two occurrences of the sentence \(\alpha\)
are "contracted" into one. (We will soon encounter related
principles that are more commonly referred to as
 contraction.[14]) The
Curry-Paradox Lemma entails that any Curry-complete theory must
violate 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 of
 section 1.1.
 That argument also included a subargument for the principle Cont,
which will be examined below.

 

\[
\begin{array}{rll}
1 & \kappa \vdash\_{\mathcal{T}} \kappa & \textrm{Id}\\
2 & \kappa \vdash\_{\mathcal{T}} \kappa {\rightarrow}\pi & \textrm{1 Curry-intersubstitutivity}\\
3 & \vdash\_{\mathcal{T}} \kappa {\rightarrow}\pi & \textrm{2 Cont}\\
4 & \vdash\_{\mathcal{T}} \kappa & \textrm{3 Curry-intersubstitutivity}\\
5 & \vdash\_{\mathcal{T}} \pi & \textrm{3, 4 MP}
\end{array}
\]


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


### 3.2 Alternative Premises



There are counterparts of the Curry-Paradox Lemma that invoke
alternative sets of logical principles (see, e.g., Rogerson
& Restall 2004 and Bimbo
2006). Probably the most common
version replaces the *rules* Id and Cont with corresponding
*laws*:

 

\[\tag{IdL}
\vdash\_{\mathcal{T}} \alpha {\rightarrow}\alpha
\]

\[
\tag{ContL} 
\vdash\_{\mathcal{T}} (\alpha {\rightarrow}(\alpha {\rightarrow}\beta)) {\rightarrow}(\alpha {\rightarrow}\beta) 
\]


The derivation now goes as follows:

 

\[
\begin{array}{rll}
1 & \vdash\_{\mathcal{T}} \kappa {\rightarrow}\kappa &
\textrm{IdL }\\
2 & \vdash\_{\mathcal{T}} \kappa {\rightarrow}(\kappa {\rightarrow}\pi) &
\textrm{1 Curry-intersubstitutivity }\\
3 & \vdash\_{\mathcal{T}} (\kappa {\rightarrow}(\kappa {\rightarrow}\pi)) {\rightarrow}(\kappa {\rightarrow}\pi) &
\textrm{2 ContL }\\
4 & \vdash\_{\mathcal{T}} \kappa {\rightarrow}\pi &
\textrm{2, 3 MP }\\
5 & \vdash\_{\mathcal{T}} \kappa &
\textrm{4 Curry-intersubstitutivity }\\
6 & \vdash\_{\mathcal{T}} \pi &
\textrm{4, 5 MP }\\
\end{array}
\]


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 of
*modus ponens* and the idempotency of conjunction.



\[\tag{MPL}
\vdash\_{\mathcal{T}} ((\alpha {\rightarrow}\beta) \wedge \alpha) {\rightarrow}\beta
\]


(Idem\(\_{\wedge}\))
The sentences
 \(\alpha\) and \(\alpha \wedge \alpha\) are intersubstitutable
 according to \(T\)

This time the derivation goes as follows:

 

\[
\begin{array}{rll}
1 & \vdash\_{\mathcal{T}} ((\kappa {\rightarrow}\pi) \wedge \kappa) {\rightarrow}\pi &
 \textrm{MPL }\\
2 & \vdash\_{\mathcal{T}} (\kappa \wedge \kappa) {\rightarrow}\pi &
 \textrm{1 Curry-intersubstitutivity }\\
3 & \vdash\_{\mathcal{T}} \kappa {\rightarrow}\pi &
 \textrm{2 Idem\(\_{\wedge}\) }\\
4 & \vdash\_{\mathcal{T}} \kappa &
 \textrm{4 Curry-intersubstitutivity }\\
5 & \vdash\_{\mathcal{T}} \pi &
 \textrm{3, 4 MP }\\
\end{array}
\]


Formulating the Curry-Paradox Lemma using Cont, rather than ContL or
MPL, will make it easier to call attention (in the next section) to
significant 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 all
Curry-complete theories are trivial.


* *Curry-incompleteness responses* accept the Troubling
Corollary. 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. Any
such theory must violate one or more of the logical principles assumed
in 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 response
to Curry paradoxes arising in one domain, say set theory, while
advocating a Curry-completeness response to Curry paradoxes arising in
another domain, say property theory (e.g.,
Field 2008; Beall 2009).


### 4.1 Curry-Incompleteness Responses



Examples of prominent theories of truth that supply
Curry-incompleteness responses to Curry's paradox include
 Tarski's hierarchical theory, the 
 revision theory of truth
 (Gupta & Belnap 1993) and the
contextualist approaches (Burge
1979, Simmons 1993, and 
Glanzberg 2001, 2004). These theories all
 restrict the "naive" transparency principle
 (Truth).
 For an overview, see the entry on the 
 Liar paradox.
 In the
context of set theory Curry-incompleteness responses include
Russellian type theories and various theories that restrict the
 "naive" set abstraction principle
 (Set).
 See the entries on
 Russell's paradox
 and
 alternative axiomatic set theories.




In general, the considerations relevant to evaluating most
Curry-incompleteness responses don't appear to be specific to
Curry's paradox, but pertain equally to the Liar paradox (in the
truth-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, though
 section 6.3
 briefly returns to the distinction in the context of so-called
validity Curry paradoxes.


### 4.2 Curry-Completeness Responses




Curry-completeness responses to Curry's paradox hold that there
are theories that are Curry-complete yet nontrivial; such a theory
must violate one or more of the logical principles assumed in the
Curry-Paradox Lemma. Since the rule Id has generally been left
unquestioned (but see French 2016 and Nicolai
& Rossi forthcoming), this has meant denying that the
conditional \({\rightarrow}\) of a nontrivial Curry-complete theory
satisfies both MP and Cont. Accordingly, responses have fallen into
two 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
 called *contraction-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 called
 *detachment-free*. This strategy is advocated, in different
 ways, by Ripley (2013) and Beall (2015).



Each category of Curry-completeness responses can in turn be
subdivided according to how it blocks purported derivations of Cont
and MP.


#### 4.2.1 Contraction-Free Responses




The principle Cont that is rejected by contraction-free responses
follows from two standard principles. These are single-premise
conditional proof and a slightly more general version of *modus
ponens*, 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 
 \(\vdash\_{\mathcal{T}} \alpha {\rightarrow}\beta\)


\[
\begin{array}{rll}
1 & \alpha \vdash\_{\mathcal{T}} \alpha {\rightarrow}\beta &
 \\
2 & \alpha \vdash\_{\mathcal{T}} \alpha &
 \textrm{Id }\\
3 & \alpha \vdash\_{\mathcal{T}} \beta &
 \textrm{1, 2 MP' }\\
4 & \vdash\_{\mathcal{T}} \alpha {\rightarrow}\beta &
 \textrm{3 CP }\\
\end{array}
\]



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



(Ia)
A *strongly contraction-free* response denies that
 \({\rightarrow}\) obeys MP' (e.g., Mares & Paoli 2014;
 Slaney 1990; Weir 2015; Zardini 2011).
(Ib)
A *weakly 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 as
*weakly* 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 of *modus
ponens*. They typically present their own form of that rule in a
"substructural" framework, specifically one that lets us
distinguish between what follows from a premise *taken once*
and what follows from the same premise *taken twice*. (See
 the entry on
 substructural 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 be
rejected:



(sCont)
If \(\Gamma, \gamma, \gamma \vdash\_{\mathcal{T}} \beta\) then
 \(\Gamma, \gamma \vdash\_{\mathcal{T}} \beta\)



It is because they reject structural contraction that strongly
contraction-free approaches can claim to preserve
*modus ponens* despite rejecting MP' (see Shapiro 2011,
Zardini 2013, and Ripley 2015a).




Strongly contraction-free responses also need to block a derivation of
MP' 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 \(T\)


\[
\begin{array}{rll}
1 & \gamma \vdash\_{\mathcal{T}} \alpha {\rightarrow}\beta &
 \\
2 & \gamma \vdash\_{\mathcal{T}} \alpha &
 \\
3 & \gamma \wedge \gamma \vdash\_{\mathcal{T}} \beta &
 \textrm{1, 2 MP'\(\_{\wedge}\) }\\
4 & \gamma \vdash\_{\mathcal{T}} \beta &
 \textrm{3 Idem\(\_{\wedge}\) }\\
\end{array}
\]



Avoiding this derivation of MP' requires denying that there is a
conjunction \(\wedge\) that obeys both MP'\(\_{\wedge}\) and
Idem\(\_{\wedge}\). According to many strongly contraction-free
responses (e.g., Mares & Paoli 2014; Zardini 2011), one kind of
conjunction--the "multiplicative" kind, or
"fusion"--obeys MP'\(\_{\wedge}\) but not
Idem\(\_{\wedge}\), whereas another kind--the
"additive" kind--obeys Idem\(\_{\wedge}\) but not
 MP'\(\_{\wedge}\) (see the entry on
 linear logic,
 and Ripley 2015a). If
the substructural framework discussed above is used, the failure of
MP'\(\_{\wedge}\) amounts to the fact that for additive
conjunction, \(\gamma, \delta \vdash\_{\mathcal{T}} \beta\) is not
equivalent to \(\gamma \wedge \delta \vdash\_{\mathcal{T}}
\beta\).




As for weakly contraction-free responses, the failure of CP has
sometimes been motivated using "worlds" semantics of the
sort that involve a distinction between logically *possible*
and *impossible* worlds (e.g., Beall
2009; 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 preservation
over 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 possible
world (in any model) at which \(\alpha\) is true and \(\beta\) untrue.
In turn, to refute \(\vdash\_\mathcal{T}\alpha{\rightarrow}\beta\) we
need a possible world at which \(\alpha{\rightarrow}\beta\) is untrue.
How does that happen? Because connectives are defined in a way that
takes account of *all* (types of) worlds in the model (possible
and, if there be any, impossible) there's an option for
\(\alpha{\rightarrow}\beta\) to be untrue at a possible world in
virtue of \(\alpha\) being true and \(\beta\) being untrue at an
*impossible* world. And that's just what happens on the
target approaches. (Exactly how one defines the truth-at-a-world and
falsity-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 of
MP based on a principle of transitivity together with the converse of
single-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
 \(\alpha \vdash\_{\mathcal{T}} \beta\)


\[
\begin{array}{rll}
1 & \vdash\_{\mathcal{T}} \alpha {\rightarrow}\beta &
 \\
2 & \vdash\_{\mathcal{T}} \alpha &
 \\
3 & \alpha \vdash\_{\mathcal{T}} \beta &
 \textrm{1 CCP }\\
4 & \vdash\_{\mathcal{T}} \beta &
 \textrm{2, 3 Trans }\\
\end{array}
\]



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



(IIa)
A *strongly detachment-free* response denies that
 \({\rightarrow}\) obeys CCP (Goodship 1996; Beall 2015).
(IIb)
A *weakly detachment-free* response accepts that
 \({\rightarrow}\) obeys CCP, but rejects Trans (Ripley
 2013).



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




One strategy for replying to the charge that detachment-free responses
are counterintuitive has been to appeal to a connection between
consequence and our acceptance and rejection of sentences. According
to this connection, whenever it is the case that \(\alpha
\vdash\_{\mathcal{T}} \beta\), this means (or at least implies) that it
is incoherent by the lights of theory \(\mathcal{T}\) to accept
\(\alpha\) while rejecting \(\beta\) (see Restall 2005). Now suppose
that, by the lights of a theory \(\mathcal{T}\), it is incoherent to
reject \(\alpha\) and it is also incoherent to accept \(\alpha\) while
rejecting \(\beta\). Then, Ripley (2013) argues, there need be nothing
incoherent by the theory's lights about rejecting \(\beta\), as
long as one doesn't also accept \(\alpha\). There is thus room
to give up Trans and adopt a weakly detachment-free response to
Curry's paradox. Beall's defense of the strongly
detachment-free approach rests on related considerations. He argues,
in effect, that a principle weaker than CCP can play the relevant role
in constraining the combinations of acceptance and rejection of
sentences 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 in 
 section 1.1.
 A strongly contraction-free response corresponds to blocking step (3)
of that argument, since it rejects MP'. A weakly
contraction-free response instead blocks step (4), since it rejects
CP. Neither kind of detachment-free response will accept the reasoning
in step (3). Since they accept Cont, detachment-free responses allow
us to derive the *conclusion* of (4), whence weakly
detachment-free responses further allow us to derive the conclusion of
(3) by CCP. However, both kinds of detachment-free response find
fault 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 be
learned by considering Curry's paradox. For discussion of the
kinds of significance that versions of Curry's paradox share
 with related paradoxes, see the entries on
 Russell's paradox
 and the
 Liar paradox.


### 5.1 Dashing Hopes for Solutions to Negation Paradoxes




Starting with Church (1942), Moh
(1954), Geach
(1955), Lob (1955) and Prior
(1955), discussion of Curry's
paradox 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 matters
is that it renders the paradox resistant to some resolutions that
might be adequate for such "negation paradoxes".




Geach argues that Curry's paradox poses a problem for any
proponents of naive truth theory or naive set theory who, faced with
negation paradoxes,



> 
> 
> 
> might ... hope to avoid [these paradoxes] by using a logical system
> in which '\(p\) if and only if not-\(p\)' were a theorem
> for some interpretations of '\(p\)' without our being able
> to infer thence any arbitrary statement.... (Geach
> 1955: 71)
> 
> 
> 




The problem, he says, is that Curry's paradox "cannot be
resolved merely by adopting a system that contains a queer sort of
negation". Rather, "if we want to retain the naive view of
truth, or the naive view of classes ..., then we must modify the
elementary rules of inference relating to 'if'"
(1955: 72). Geach's view of the
significance of Curry's paradox is closely echoed by Meyer,
Routley, and Dunn (1979: 127). They
conclude 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 weak
negation principles that resolve Russell's paradox and the Liar,
yet remain vulnerable to Curry's paradox. These are logics with
the 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 are
indeed non-classical logics that meet these two conditions. Theories
based on these logics accordingly remain vulnerable to Curry's
paradox.


#### 5.1.1 Paraconsistent Solutions Frustrated




Meyer, Routley, and Dunn (1979) call
attention to one class of logics that meet conditions (a) and (b).
 They are among the 
 paraconsistent logics,
 which are logics
according to which a sentence together with its negation will 
not entail any arbitrary sentence. Paraconsistent logics can be used to obtain
theories that resolve Russell's paradox, and the Liar, by
embracing 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 on
 dialetheism).
 Yet a number of prominent paraconsistent logics can't serve as
the basis for Curry-complete theories on pain of triviality. Such
logics 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 underwrite
responses to Russell's paradox and the Liar paradox are
*paracomplete logics*, logics that reject the law of excluded
middle. These logics make possible "gappy" theories. In
particular, where \(\lambda\) and \(\lnot\lambda\) are
intersubstitutable according to such a theory \(\mathcal{T}\), it will
fail 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 L\(\_{3}\) based on the three-valued
truth-tables of Lukasiewicz (see, e.g.,
Priest 2008). Since it meets condition (a), L\(\_{3}\)
offers a possible response to Russell's paradox and the
Liar--in particular, a gappy response. Yet consider the
iterated conditional \(\alpha {\rightarrow}(\alpha
{\rightarrow}\beta)\), which we abbreviate as \(\alpha \Rightarrow
\beta\). Suppose that a Curry sentence for \(\pi\) and an
L\(\_{3}\)-based theory \(\mathcal{T}\) is redefined to be any
sentence \(\kappa\) intersubstitutable with \(\kappa \Rightarrow
\pi\). Then \(\mathcal{T}\) will meet all the conditions of the
Curry-Paradox Lemma, as was first noted by Moh
(1954). Hence, as long as there is a \(\kappa\) that is
intersubstitutable with \(\kappa \Rightarrow \pi\) according to
\(\mathcal{T}\), then \(\vdash \_{\mathcal{T}} \pi\). Consequently
 L\(\_{3}\) won't underwrite a response to Curry's
 paradox.[22]





To summarize: Curry's paradox stands in the way of some
otherwise available avenues for resolving semantic paradoxes by means
of glutty or gappy theories. As a result, the need to evade
Curry's paradox has played a significant role in the development
of 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 second
reason. Prior makes the following important point:



> 
> 
> 
> We can ... say not only that Curry's paradox does not
> involve negation but that even Russell's paradox presupposes
> only 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's
paradox can be understood as resulting from the *same general
structure*, which can be instantiated either using negation or
 using a
 conditional.[24]





The general structure can be made explicit by defining a type of unary
connective that gives rise to Curry's paradox, and showing how
this type is exemplified both by negation and by a unary connective
defined in terms of a conditional.




**Definition 3** (Curry connective)
 Let \(\pi\) be a sentence in the language of
theory \(\mathcal{T}\). The unary connective \(\odot\) is a Curry
connective for \(\pi\) and \(\mathcal{T}\) provided it satisfies two
principles:

 

\[\tag{P1}
\textrm{If} \vdash\_{\mathcal{T}} \alpha \textrm{ and } \vdash\_{\mathcal{T}} \odot\alpha \textrm{ then } \vdash\_{\mathcal{T}} \pi.
\]

 

\[\tag{P2}
\textrm{If } \alpha \vdash\_{\mathcal{T}} \odot\alpha \textrm{ then } \vdash\_{\mathcal{T}} \odot\alpha.
\]




**Generalized Curry-Paradox Lemma** Suppose that
\(\mathcal{T}\) is such that Id holds and that for some pair of
sentences \(\pi\) and \(\mu\), (i) \(\mu\) and \(\odot\mu\) are
intersubstitutable according to \(\mathcal{T}\) and (ii) \(\odot\) is
a Curry connective for \(\pi\) and \(\mathcal{T}\). In that case
 \(\vdash\_{\mathcal{T}}
 \pi\).[25]




*Proof*: 

 

\[
\begin{array}{rll}
1 & \mu \vdash\_{\mathcal{T}} \mu &
 \textrm{Id }\\
2 & \mu \vdash\_{\mathcal{T}} \odot\mu &
 \textrm{1 Curry-intersubstitutivity }\\
3 & \vdash\_{\mathcal{T}} \odot\mu &
 \textrm{2 P2 }\\
4 & \vdash\_{\mathcal{T}} \mu &
 \textrm{3 Curry-intersubstitutivity }\\
5 & \vdash\_{\mathcal{T}} \pi &
 \textrm{3, 4 P1 }\\
\end{array}
\]




The Generalized Curry-Paradox Lemma can now be instantiated in two
different ways, so as to yield either Curry's paradox or a
negation paradox:


* To obtain Curry's paradox, let the unary connective \(\odot\) be
such 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 to
the 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 of *ex contradictione
quodlibet* (or "explosion"), while P2 is a
*reductio* 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 relevant
to Russell's paradox or the Liar paradox are *exhausted by
its status as a Curry connective*. This makes clear why these
paradoxes do not depend on features of negation, such as excluded
middle or double negation elimination, that fail to hold in
nonclassical 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 a *minimal negation* (see entry on
 negation),
 since it need not obey the law of double introduction:



\[\tag{DI}
\alpha \vdash \_{\mathcal{T}} \odot\odot\alpha.
\]



For example, suppose that \(\odot\alpha\) is \(\alpha
{\rightarrow}\pi\). Then in order for \(\odot\) to obey DI, it would
have to be the case that \(\alpha \vdash \_{\mathcal{T}} (\alpha
{\rightarrow}\pi) {\rightarrow}\pi\). That principle is violated by a
number 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 structure
instantiated by a wide range of paradoxes. This structure
doesn't itself involve negation, but it is also displayed by
paradoxes that (unlike Curry's paradox) do essentially involve
negation, such as Russell's paradox and the Liar paradox.




The issue of which paradoxes display a common structure becomes
important 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 receive
the "same kind of solution". Suppose that we delimit one
kind of paradox as follows:




**Definition 4** (Generalized Curry paradox) We have a
generalized Curry paradox in any case where the assumptions stated in
the Generalized Curry-Paradox Lemma appear to hold.




Assuming one accepts the principle of uniform solution, the question
becomes what counts as proposing a uniform solution to all generalized
Curry paradoxes. In particular, does it suffice to show, for every
instance of the kind thus delimited, that what appears to be a Curry
connective in fact fails to be one? It would seem that this should
indeed be enough. It's unclear why uniformity should
additionally require that all seeming Curry connectives fail to
qualify as such *in virtue of violating the same
condition*. For instance, suppose that negation and our unary
connective defined using \({\rightarrow}\) both appear to satisfy the
generalized 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 common
source (e.g., an implicit reliance on structural contraction, as
claimed by Zardini 2011), there need be nothing objectionably
non-uniform about taking one appearance at face value while dismissing
the other as deceptive. (For discussion of the philosophical issue
here, applied to a different class of paradoxes, see the exchange in
Smith 2000 and Priest 2000.)



If that is right, the desideratum that generalized Curry paradoxes be
resolved uniformly needn't discriminate between the various
logically revisionary solutions that have been pursued. These include
the 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 negation
paradox), whereas it is P2 alone that fails when \(\odot\alpha\) is
instantiated as \(\alpha {\rightarrow}\pi\) (to get a Curry paradox).
On this approach, ECQ and Cont fail, while Red and MP hold (Priest
1994, 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 as
resolving Curry's paradox and the Liar paradox uniformly *qua
examples of generalized Curry paradox*. This would be the case
despite the fact that Priest evaluates Liar sentences as both true and
false, whereas he rejects the claim that Curry sentences are true.



In any event, Curry's paradox raises challenges in connection
with the issue of what type of uniformity should be required of
solutions to various paradoxes (see also Zardini 2015). Priest himself calls attention to a
kind of paradox narrower than the generalized Curry paradoxes, a kind
whose instances include the negation paradoxes but *exclude*
Curry's paradox. This kind is picked out by Priest's
"Inclosure Schema" (2002);
 see the entry on
 self-reference.
 One ongoing dispute is about whether there might be a version of
Curry's paradox that counts as an "inclosure
paradox", though it resists Priest's uniform dialetheic
solution to such paradoxes (see the exchange in Beall
2014b, Weber et
al. 2014, and Beall 2014a, as
well as Pleitz 2015).


## 6. Validity Curry




The last decade (as of the date of this version of this entry) has
witnessed a boom in attention to Curry paradoxes, and perhaps
especially to what have been called *validity Curry* or
*v-Curry* paradoxes (Whittle 2004;
 Shapiro 2011; Beall & Murzi
 2013).[29]
 V-Curry involves Curry sentences that specifically invoke a
theory's consequence or "validity" relation, by
using either a conditional or a predicate that purports to express
theory \(\mathcal{T}\)'s relation \(\vdash\_\mathcal{T}\) in the
language 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 a *consequence connective* \({\Rightarrow}\). A sentence
with \({\Rightarrow}\) as its major operator is to be interpreted
thus: "That \(p\) entails (according to \(\mathcal{T}\)) that
\(q\)". 
We now immediately obtain property-theoretic, set-theoretic or
truth-theoretic versions of Curry's paradox, provided only that
\({\Rightarrow}\) meets the conditions MP and Cont of the
Curry-Paradox Lemma. 




What makes this instance of the Curry-Paradox
Lemma particularly troublesome is that it poses an obstacle to one
common response to Curry's paradox, namely the *weakly
 contraction-free* response discussed in
 section 4.2.1. That response depended on
 rejecting the rule CP of single-premise conditional proof, one
direction of the single-premise "deduction theorem". But
this is a rule that has seemed difficult to resist for a consequence
connective (Shapiro 2011; Weber 2014; Zardini 2013). If \(\beta\) is
a consequence of \(\alpha\) according to the consequence relation of
theory \(\mathcal{T}\), where this theory has \({\Rightarrow}\) as
its own consequence connective, then \(\mathcal{T}\) must surely
contain the consequence claim \(\alpha
{\Rightarrow}\beta\). Likewise, this variety of Curry paradox poses
an obstacle for *detachment-free* responses, which require
rejecting the rule MP. If a theory with its own consequence
connective 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 of
a *weakly* detachment-free response will argue that MP for
 \({\Rightarrow}\) illicitly builds in transitivity
 (see section 4.2.2). Still, what seems
 inescapable is the converse of CP, the rule CCP that is the other
direction of the single-premise deduction theorem. If a theory
contains the consequence conditional \(\alpha {\Rightarrow}\beta\),
then surely \(\beta\) follows from \(\alpha\) according to the
theory. That would still rule out a *strongly* detachment-free
response.


### 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 very
theory, between sentences in its
 language.[30]
 Let this relation be expressed by the predicate \(Val(x,y)\), and
assume 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-Curry
paradox employs two principles governing \(Val\), which we call
"validity detachment" and "validity proof"
following Beall & Murzi (2013).



\[\tag{VD}
\textrm{If }\gamma \vdash\_{\mathcal{T}\_{V}} Val(\langle\alpha\rangle, \langle\beta\rangle)\textrm{ and }\gamma \vdash\_{\mathcal{T}\_{V}} \alpha\textrm{ then }\gamma \vdash\_{\mathcal{T}\_{V}} \beta
\]

 

\[\tag{VP}
\textrm{If }\alpha \vdash\_{\mathcal{T}\_{V}} \beta\textrm{ then }\vdash\_{\mathcal{T}\_{V}} Val(\langle\alpha\rangle, \langle\beta\rangle)
\]


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

 

\[
\begin{array}{rll}
1 & \chi \vdash\_{\mathcal{T}\_{V}} \chi &
 \textrm{Id }\\
2 & \chi \vdash\_{\mathcal{T}\_{V}} Val(\langle\chi\rangle, \langle\pi\rangle) &
 \textrm{2 Curry-intersubstitutivity }\\
3 & \chi \vdash\_{\mathcal{T}\_{V}} \pi &
 \textrm{1, 2 VD }\\
4 & \vdash\_{\mathcal{T}\_{V}} Val(\langle\chi\rangle, \langle\pi\rangle) &
 \textrm{3 VP }\\
5 & \vdash\_{\mathcal{T}\_{V}} \chi &
 \textrm{4 Curry-intersubstitutivity }\\
6 & \vdash\_{\mathcal{T}\_{V}} \pi &
 \textrm{4, 5 VD }\\
\end{array}
\]


As applied to this *predicate form* of v-Curry, a weakly
contraction-free response would resist the "contraction"
from step 2 to step 4 by rejecting the rule VP, and a
detachment-free response would reject VD, even in the zero-premise form used at step 6. Again, though, both
VP and zero-premise VD have seemed inescapable in view of the intended
interpretation 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 a *strongly* detachment-free response.



An arguably more powerful version of v-Curry reasoning is presented by
Shapiro (2013) and Field
(2017: 7). This reasoning can take either
connective or predicate form, but it doesn't depend on CP or VP.
Here we give the predicate form using \(Val\). As above, we first
derive that \(\chi \vdash\_{\mathcal{T}\_{V}} \pi\) using VD. In view of the
meaning of \(Val\), the conclusion that \(\chi \vdash\_{\mathcal{T}\_{V}}
\pi\) shows that \(Val(\langle\chi\rangle, \langle\pi\rangle)\) is
true, 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. Since *weakly* detachment-free
(nontransitive) responses to v-Curry do allow the derivation of \(\chi
\vdash\_{\mathcal{T}\_{V}} \pi\), this reasoning poses an objection to
such responses as well.


### 6.3 Significance



If, in fact, v-Curry paradoxes aren't amenable to weakly
contraction-free or strongly detachment-free responses, then (assuming the rule Id is retained) the
space of Curry-complete responses is restricted to *strongly
contraction-free* and *weakly detachment-free* responses.
The former responses, as explained in
 section 4.2.1,
 are typically presented by reformulating *modus ponens* (or detachment for the validity predicate) in a substructural deduction system and rejecting the structural contraction rule sCont. The
latter responses, as explained in
 section 4.2.2,
 reject the structural principle of transitivity. For this reason,
v-Curry paradoxes have sometimes been taken to motivate
*substructural* consequence relations (e.g.,
Barrio et al. forthcoming; Beall & Murzi
2013; Ripley 2015a; Shapiro 2011, 2015).[32]



The lively and wide-ranging debate on v-Curry paradoxes has resulted
in genuine progress in our understanding of Curry paradoxes. In the
end, what has become clear is that while v-Curry paradoxes may invite
different resolutions from non-v-Curry paradoxes, they remain within
the same mold as generalized Curry paradoxes. In particular, in the
general template of
 section 5.2
 one may take \(\odot\) to express (either as a predicate or as a
connective) 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, epistemic
consequence 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 of
solutions to the generalized Curry paradoxes canvassed in this
entry.



There remain, however, at least two reasons v-Curry paradoxes merit
separate attention. First, as noted above, two categories of
Curry-complete solutions -- the weakly contraction-free and
strongly detachment-free options -- have appeared especially
problematic in the case of v-Curry paradoxes. Second, suppose that one
treats an ordinary Curry paradox (property-theoretic, set-theoretic or
semantic) in a Curry-complete fashion. There may still be reason to
treat the corresponding (connective or predicate) v-Curry paradox in a
Curry-incomplete fashion, perhaps in virtue of seeing a theory's
consequence relation as essentially beyond capture by any connective
or predicate in the language of the theory (see, e.g., Myhill 1975;
Whittle 2004). Thus, a "non-uniform" solution to ordinary
Curry paradoxes and their v-Curry counterparts may -- once
again -- be a motivated
non-uniformity.[33]