Curry's paradox is aparadox in which an arbitrary claimF is proved from the mere existence of a sentenceC that says of itself "IfC, thenF". The paradox requires only a few apparently-innocuous logical deduction rules. SinceF is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in variouslogics, including certain forms ofset theory,lambda calculus, andcombinatory logic.

The paradox is named after the logicianHaskell Curry, who wrote about it in 1942.^[1] It has also been calledLöb's paradox afterMartin Hugo Löb,^[2] due to its relationship toLöb's theorem.

Claims of the form "ifA, thenB" are calledconditional claims. Curry's paradox uses a particular kind of self-referential conditional sentence, as demonstrated in this example:

Even thoughGermany does not borderChina, the example sentence certainly is a natural-language sentence, and so the truth of that sentence can be analyzed. The paradox follows from this analysis. The analysis consists of two steps. First, common natural-language proof techniques can be used to prove that the example sentence is true[steps 1–4 below]. Second, the truth of the sentence can be used to prove that Germany borders China[steps 5–6]:

1. The sentence reads "If this sentence is true, then Germany borders China"  [repeat definition to get step numbering compatible tothe formal proof]
2. If the sentence is true, then it is true.  [obvious, i.e., atautology]
3. If the sentence is true, then: if the sentence is true, then Germany borders China.  [replace "it is true" by the sentence's definition]
4. If the sentence is true, then Germany borders China.  [contract repeated condition]
5. But 4. is what the sentence says, so it is indeed true.
6. The sentence is true[by 5.], and[by 4.]: if it is true, then Germany borders China.
   So, Germany borders China.  [modus ponens]

Because Germany does not border China, this suggests that there has been an error in one of the proof steps. The claim "Germany borders China" could be replaced by any other claim, and the sentence would still be provable. Thus every sentence appears to be provable. Because the proof uses only well-accepted methods of deduction, and because none of these methods appears to be incorrect, this situation is paradoxical.^[3]

The standard method for provingconditional sentences (sentences of the form "ifA, thenB") is called "conditional proof". In this method, in order to prove "ifA, thenB", firstA is assumed and then with that assumptionB is shown to be true.

To produce Curry's paradox, as described in the two steps above, apply this method to the sentence "if this sentence is true, then Germany borders China". HereA, "this sentence is true", refers to the overall sentence, whileB is "Germany borders China". So, assumingA is the same as assuming "IfA, thenB". Therefore, in assumingA, we have assumed bothA and "IfA, thenB". Therefore,B is true, bymodus ponens, and we have proven "If this sentence is true, then 'Germany borders China' is true." in the usual way, by assuming the hypothesis and deriving the conclusion.

Now, because we have proved "If this sentence is true, then 'Germany borders China' is true", then we can again apply modus ponens, because we know that the claim "this sentence is true" is correct. In this way, we can deduce that Germany borders China.

The example in the previous section used unformalized, natural-language reasoning. Curry's paradox also occurs in some varieties offormal logic. In this context, it shows that if we assume there is a formal sentence (X →Y), whereX itself is equivalent to (X →Y), then we can proveY with a formal proof. One example of such a formal proof is as follows. For an explanation of the logic notation used in this section, refer to thelist of logic symbols.

1. X := (X →Y)
   
   assumption, the starting point, equivalent to "If this sentence is true, thenY"
2. X →X
   
   law of identity
3. X → (X →Y)
   
   substitute right side of 2, sinceX is equivalent toX →Y by 1
4. X →Y
   
   from 3 bycontraction
5. X
   
   substitute 4, by 1
6. Y
   
   from 5 and 4 bymodus ponens

An alternative proof is viaPeirce's law. IfX =X →Y, then (X →Y) →X. This together with Peirce's law ((X →Y) →X) →X andmodus ponens impliesX and subsequentlyY (as in above proof).

The above derivation shows that, ifY is an unprovable statement in a formal system, then there is no statementX in that system such thatX is equivalent to the implication (X →Y). In other words, step 1 of the previous proof fails. By contrast, the previous section shows that in natural (unformalized) language, for every natural language statementY there is a natural language statementZ such thatZ is equivalent to (Z →Y) in natural language. Namely,Z is "If this sentence is true thenY".

Even if the underlying mathematical logic does not admit any self-referential sentences, certain forms of naive set theory are still vulnerable to Curry's paradox. In set theories that allowunrestricted comprehension, we can prove any logical statementY by examining the set$${\displaystyle X\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left\{x\mid (x\in x)\to Y\right\}.}$$One then shows easily that the statement${\displaystyle X\in X}$ is equivalent to${\displaystyle (X\in X)\to Y}$. From this,${\displaystyle Y}$ may be deduced, similarly to the proofs shown above. ("${\displaystyle X\in X}$" stands for "this sentence".)

Therefore, in a consistent set theory, the set${\displaystyle \left\{x\mid (x\in x)\to Y\right\}}$ does not exist for falseY. This can be seen as a variant onRussell's paradox, but is not identical. Some proposals for set theory have attempted to deal with Russell's paradox not by restricting the rule of comprehension, but by restricting the rules of logic so that it tolerates the contradictory nature of the set of all sets that are not members of themselves. The existence of proofs like the one above shows that such a task is not so simple, because at least one of the deduction rules used in the proof above must be omitted or restricted.

Curry's paradox may be expressed in untypedlambda calculus, enriched byimplicational propositional calculus. To cope with the lambda calculus's syntactic restrictions,${\displaystyle m}$ shall denote the implication function taking two parameters, that is, the lambda term${\displaystyle ((mA)B)}$ shall be equivalent to the usualinfix notation${\displaystyle A\to B}$.

An arbitrary formula${\displaystyle Z}$ can be proved by defining a lambda function${\displaystyle N:=\lambda p.((mp)Z)}$, and${\displaystyle X:=(\mathsf{\text{Y}}N)}$, where${\displaystyle \mathsf{\text{Y}}}$ denotes Curry'sfixed-point combinator. Then${\displaystyle X=(NX)=((mX)Z)}$ by definition of${\displaystyle \mathsf{\text{Y}}}$ and${\displaystyle N}$, hence theabove sentential logic proof can be duplicated in the calculus:^[4][5]

Insimply typed lambda calculus, fixed-point combinators cannot be typed and hence are not admitted.

Curry's paradox may also be expressed incombinatory logic, which has equivalent expressive power tolambda calculus. Any lambda expression may be translated into combinatory logic, so a translation of the implementation of Curry's paradox in lambda calculus would suffice.

The above term${\displaystyle X}$ translates to${\displaystyle (rr)}$ in combinatory logic, where$${\displaystyle r=\mathsf{\text{S}}(\mathsf{\text{S}}(\mathsf{\text{K}}m)(\mathsf{\text{S}}\mathsf{\text{I}}\mathsf{\text{I}}))(\mathsf{\text{K}}Z);}$$hence^[6]$${\displaystyle (rr)=((m(rr))Z).}$$

Curry's paradox can be formulated in any language supporting basic logic operations that also allows a self-recursive function to be constructed as an expression. Two mechanisms that support the construction of the paradox areself-reference (the ability to refer to "this sentence" from within a sentence) andunrestricted comprehension in naive set theory. Natural languages nearly always contain many features that could be used to construct the paradox, as do many other languages. Usually, the addition of metaprogramming capabilities to a language will add the features needed. Mathematical logic generally does not allow explicit reference to its own sentences; however, the heart ofGödel's incompleteness theorems is the observation that a different form of self-reference can be added—seeGödel number.

The rules used in the construction of the proof are therule of assumption for conditional proof, the rule ofcontraction, andmodus ponens. These are included in most common logical systems, such as first-order logic.

In the 1930s, Curry's paradox and the relatedKleene–Rosser paradox, from which Curry's paradox was developed,^[7][1] played a major role in showing that various formal logic systems allowing self-recursive expressions areinconsistent.

The axiom of unrestricted comprehension is not supported bymodern set theory, and Curry's paradox is thus avoided.

• Fixed-point combinator
• Girard's paradox
• Liar paradox
• List of paradoxes
• Richard's paradox
• Zermelo–Fraenkel set theory

• Beall, J. C."Curry's paradox". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.ISSN 1095-5054.OCLC 429049174.
• Cantini, Andrea."Paradoxes and Contemporary Logic". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.ISSN 1095-5054.OCLC 429049174.
• Penguins Rule the Universe: A Proof that Penguins Rule the Universe, a brief and entertaining discussion of Curry's paradox.

• Mathematical paradoxes
• Mathematical logic
• Paradoxes of naive set theory
• Self-referential paradoxes

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