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# Curry’s paradox (Lob’s paradox)

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

which is a contradiction.

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

which is a contradiction.)

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.

Curry’s so-called paradox isn’t a paradox at all.

#### Other paradoxes

Footnotes:

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