Claim. It is not the case that all logically equivalent designators are co-designative.

Proof. Assume (to reach a contradiction) that all logically equivalent designators are co-designative.

Consider the following definite description,

ψ1: ‘the least natural number that is neither designated by an English definite description consisting of exactly twenty-seven words nor designated by “the king of Sweden”’

There are at most n^27 twenty-seven word definite descriptions (where n is the total number of English words). Each definite description designates at most one natural number. Hence, all the 27-word definite descriptions of English can designate at most n^27 natural numbers. But there are infinitely many natural numbers, so there are some numbers (infinitely many in fact) that cannot be designated by an English definite description consisting of exactly twenty-seven words. Thus, there is a least natural number that is neither designated by an English definite description consisting of exactly twenty-seven words nor designated by ‘the king of Sweden’.

Exactly one number satisfies the matrix of the proper definite description ψ1, therefore ψ1 designates that number.

Consider the following definite description, which is logically equivalent to ψ1,

ψ2: ‘the least natural number that is not designated by an English definite description consisting of exactly twenty-seven words and is not designated by “the king of Sweden”’

By assumption, since ψ2 is logically equivalent to ψ1, ψ2 designates the same number as ψ1. It follows that, this number, i.e. the least natural number that is not designated by an English definite description consisting of exactly twenty-seven words (and not designated by ‘the king of Sweden’) is also designated by ψ2, which is an English definite description consisting of exactly twenty-seven words. Contradiction.

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