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

Read more about this topic:  Curry's Paradox

Famous quotes containing the words formal and/or logic:

    That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through formal instruction.
    Fred Rogers (20th century)

    It is the logic of our times,
    No subject for immortal verse—
    That we who lived by honest dreams
    Defend the bad against the worse.
    Cecil Day Lewis (1904–1972)