Many thanks for your comments, Lou and Bruno. I read and pondered,
and finally concluded that the paths taken by each of you exceed
my competencies. I subsequently sent your comments to Professor
Johnstone—-I trust this is acceptable—asking him if he would care to
respond with a brief sketch of the reasoning undergirding his critique,
which remains anchored in Gödel’s theorem, not in the writings of others
about Gödel’s theorem. Herewith his reply:

Since no one commented on the reasoning supporting the conclusions reached in the two cited articles, let me attempt to sketch the crux of the case presented.

The Liar Paradox contains an important lesson about meaning. A statement that says of itself that it is false, gives rise to a paradox: if true, it must be false, and if false, it must be true. Something has to be amiss here. In fact, what is wrong is the statement in question is not a statement at all; it is a pseudo-statement, something that looks like a statement but is incomplete or vacuous. Like the pseudo-statement that merely says of itself that it is true, it says nothing. Since such self-referential truth-evaluations say nothing, they are neither true nor false. Indeed, the predicates ‘true’ and ‘false’ can only be meaningfully applied to what is already a meaningful whole, one that already says something.

The so-called Strengthened Liar Paradox features a pseudo-statement that says of itself that it is neither true nor false. It is paradoxical in that it apparently says something that is true while saying that what it says it is not true. However, the paradox dissolves when one realizes that it says something that is apparently true only because it is neither true nor false. However, if it is neither true nor false, it is consequently not a statement, and hence it says nothing. Since it says nothing, it cannot say something that is true. The reason why it appears to say something true is that one and the same string of words may be used to make either of two declarations, one a pseudo-statement, the other a true statement, depending on how the words refer.

Consider the following example. Suppose we give the name ‘Joe’ to what I am saying, and what I am saying is that Joe is neither true nor false. When I say it, it is a pseudo-statement that is neither true nor false; when you say it, it is a statement that is true. The sentence leads a double life, as it were, in that it may be used to make two different statements depending on who says it. A similar situation can also arise with a Liar sentence: if the liar says that what he says is false, then he is saying nothing; if I say that what he says is false, then I am making a false statement about his pseudo-statement.

This may look like a silly peculiarity of spoken language, one best ignored in formal logic, but it is ultimately what is wrong with the Gödel sentence that plays a key role in Gödel’s Incompleteness Theorem. That sentence is a string of symbols deemed well-formed according to the formation rules of the system used by Gödel, but which, on the intended interpretation of the system, is ambiguous: the sentence has two different interpretations, a self-referential truth-evaluation that is neither true nor false or a true statement about that self-referential statement. In such a system, Gödel’s conclusion holds. However, it is a mistake to conclude that no possible formalization of Arithmetic can be complete. In a formal system that distinguishes between the two possible readings of the Gödel sentence (an operation that would considerably complicate the system), such would no longer be the case.

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