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Haskell Curry first postulated what is now known as Curry’s paradox in a paper in 1941. (Footnote: Haskell B Curry, The paradox of Kleene and Rosser Transactions of the American Mathematical Society 50.3 (1941): 454-516.) An example of what is claimed to be a natural language instance of Curry’s paradox (also known as Lob’s paradox) is (see Wikipedia: Curry’s paradox):

(A) If this sentence is true, then Germany borders China.

The paradox is supposedly that if the sentence A is true then it must be true that Germany borders China. The supposed paradox relies on one accepting that the sentence A is “true”.

But the problem with the sentence A is, of course, that there is no clear definition of what is meant by the occurrence of “this sentence” within A. If:

“this sentence”

is meant to be precisely the same as the symbol sequence:

If this sentence is true, then Germany borders China.

then we have by substitution:

If “If this sentence is true, then Germany borders China.” is true, then Germany borders China

and again:

If “If “If this sentence is true, then Germany borders China.” is true, then Germany borders China.” is true, then Germany borders China.

and so on, and so on, … an infinite recursion, and “this sentence” is never defined.

So where is the paradox? There isn’t one, unless you accept that (A) is “true”. But why would anyone accept that a sentence that has no clear definition is “true”?

## Formal logic

There are claims that the natural language “paradox” above can be derived from formal logic. By replacing “Germany borders China”, by Y, where Y can be any proposition, we have:

If this sentence is true, then Y.

There have been various claims that the above sentence can be derived from formal logic, see for example Wikipedia: Curry’s paradox, and Stanford Encyclopedia of Philosophy: Curry’s Paradox.

However, these claims all rely on the use of inconsistent systems, so there is no justification for claiming that there is a real paradox.

### Assumption of logical equivalence

The other approach is to assert that in certain formal systems there can be a logical equivalence of truth values of certain terms. So there is an assertion that in certain formal systems, there can exist a proposition X, such that, for all Y, X → Y is true if and only if X is true.

But if that is the case, then it is easily shown that this leads to a contradiction:

Assume that there can exist a proposition X such that, for all Y:

X ≡ X → Y

is true (where A ≡ B indicates logical equivalence). This is equivalent to:

X ≡ ¬X ∨ Y

So, if X is true, then so is:

¬X ∨ Y

which gives us that:

X ≡ Y

But since Y can be any proposition, then Y can be ¬X, which would give that:

X ≡ ¬X

On the other hand, if X is not true, then we have that ¬X is true, and so we have that

¬X ≡ ¬(¬X ∨ Y)

In formal logic, ¬(¬X ∨ Y) is the same as:

X ∧ ¬Y

so that we have:

¬X ≡ X ∧ ¬Y

which is a contradiction. (Footnote: Also, since Y can be any proposition, then Y can be X, which would give that:
¬X ≡ X

The alleged “paradox”, as in the previous section, relies on there being a formal system which is both

(i) consistent and

(ii) can include propositions X where for all Y, X ≡ X → Y (where ≡ denotes logical equivalence).

Clearly, because of the contradiction indicated above, there can be no such formal system, and so there is no paradox.

### Assumption of equality

In the natural language sentence

If this sentence is true, then Y.

if “this sentence”

could mean precisely the same as the symbol sequence:

If this sentence is true, then Y.

then, ipso facto, the two expressions “this sentence” and If this sentence is true, then Y. must then also be logically equivalent as in the above section, and so the analysis above must hold for that case also.

### Conclusion

The term paradox is usually used to refer to a situation where there is an apparent contradiction which is surprising. But Curry’s paradox, like many other so-called paradoxes, the Curry paradox isn’t really a paradox at all, because it’s not at all surprising. It’s simply a case of an inconsistent system producing a contradiction - we know that inconsistent systems produce contradictions, so there isn’t any surprise at the occurrence of a contradiction.

Footnotes:

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## NEWS

### Lebesgue Measure

There is now a new page on a contradiction in Lebesgue measure theory.

### Illogical Assumptions

There is now a new page Halbach and Zhang’s Yablo without Gödel which analyzes the illogical assumptions used by Halbach and Zhang.

### Easy Footnotes

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

### O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

### Previous Blog Posts

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

– a page with general links:

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

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