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.
********
Cheers,
Maxine
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