[FOM] The liar "revenge"?
nweaver at math.wustl.edu
Tue Jul 21 00:20:17 EDT 2015
Charles Parsons wrote:
> It's another matter whether mathematicians should be interested in
> these problems. They have given rise to some nice mathematical logic,
> but I would not want to argue that (at least after the clarifications in
> the twenty years after Russell's paradox and then Tarski's work on truth
> and definability) that the Liar and related problems are a problem for
> the foundations of mathematics.
Are you satisfied with Tarski's characterization of what constitutes
a truth predicate? I mean his view that this should be understood
relative to a metasystem and his criterion that the metasystem
should prove every instance of the T-scheme. This characterization
suffers from the defect that general laws such as "for every sentence A,
the sentence `A implies A' is true" can fail, in the sense that a
metasystem which proves every instance of the T-scheme for some object
language could also consistently prove the negation of such a law for
that language. So it fails to capture basic features of our intuitive
conception of truth.
Or did you mean Tarski's (and Vaught's) construction of a truth predicate
for any interpreted language? The problem I have with this construction
is that it only applies to languages with set models. It does not apply,
for instance, to any language in which the word "set" is interpreted to
range over all sets --- more pointedly, it does not apply to any language
in which the construction itself can be formulated.
These seem to me to be basic problems with Tarski's work on truth.
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