Curry's Paradox - Formal Logic

Formal Logic

The example in the previous section used unformalized, natural-language reasoning. Curry's paradox also occurs in formal logic. In this context, it shows that if we assume there is a formal sentence (X → Y), where X itself is equivalent to (X → Y), then we can prove Y with a formal proof. One example of such a formal proof is as follows.

1. X → X

rule of assumption, also called restatement of premise or of hypothesis

2. X → (X → Y)

substitute right side of 1, since X is equivalent to X → Y by assumption

3. X → Y

from 2 by contraction

4. X

substitute 3, since X = X → Y

5. Y

from 4 and 3 by modus ponens

Therefore, if Y is an unprovable statement in a formal system, there is no statement X in that system such that X is equivalent to the implication (X → Y). By contrast, the previous section shows that in natural (unformalized) language, for every natural language statement Y there is a natural language statement Z such that Z is equivalent to (Z → Y) in natural language. Namely, Z is "If this sentence is true then Y".

In specific cases where the classification of Y is already known, few steps are needed to reveal the contradiction. For example, when Y is "Germany borders China," it is known that Y is false.

1. X = X → Y

assumption

2. X = X → false

substitute known value of Y

3. X = ¬X ∨ false

implication

4. X = ¬X

identity