Re: The fallacy of strengthened liar's paradox.

On Jan 2, 5:53 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough) wrote:
LauLuna says.

I believe that the EXTENSION of the extensible concepts are
spread, so to say, along a hierarchy of logical levels; however, the
concepts themselves remain the same, because their INTENSIONAL
contents remain the same across the hierarchy of logical levels.

That's an intuitively appealing approach, but I don't know of
a way to make it rigorous.

Perhaps Tarski overlooked the distinction between concept extension
and concept intension. In order to be capable of entertaining thoughts
like 'the extension of the truth predicate must be conceived of as
distributed into several logical levels', we must be able to retain
the unitary intensional content of the concept of truth. Otherwise,
the theory becomes itself unspeakable.

Well, there *is* a unifying notion of truth given by
Tarski's truth schema

    True(#Phi) <-> Phi

The only difference between the various levels of truth
is simply what formulas the truth predicate is allowed
to range over.

Please, let me focus on this. If there is a unifying notion of truth
that remains the same across all levels, languages or interpretations
where it is applied, then there is a single concept of truth.

But this unifying concept has different extensions at different
logical levels, for the extension of a concept at a logical level is
just what it can 'range over' at that level, or, in other words, what
it can be used to refer to at that level.

So the concept of truth is extensible in some sense that is to be made
rigorous elsewhere.

I hope you agree with this. But we've arrived at the topic of possible
domains of reference (or universes of discourse); the realms our
expressions can range over are our domains of reference. To avoid
paradox we need to reject the possibility of a direct or indirect self-
reference implying a self-attribution of truth/falsity: whatever is
able to attribute truth/falsity cannot lie within its own domain of
reference.

So, it is the 'attempt' at self-attribution of a truth value what must
be wrong in sentences like the Liar and the Truth-teller:

(1) this sentence is false

(2) this sentence is true

This contrasts with:

(3) this sentence has exactly ten letters

and the like.

We cannot reject now (1) and (2) by resorting to equivocation since we
have a unitary notion of truth we can always understand; we can just
say that (1) and (2) fail in their 'attempted' self-attribution.

But why do they do so while (3) obviously succeeds in univocally (and
falsely) speaking of itself?

Here is where the distinction sentence/proposition plays a role.
Sentences are not true/false by themselves because they have no
semantical dimension by themselves, only the information content they
convey according to some interpretation can be such. Then if we read
(1) and (2) litterally they are false, but this is not quite fair for
we can transform them like follows:

(1*) the proposition this sentence expresses is false

(2*) the proposition this sentence expresses is true

We must concede that (1*) and (2*) are ultimately referring to
propositons, not sentences, no matter how you reword them, because
their predicates are the kind of predicates that can only be
attributed to propositions. So the difference between (3) and (1*)-
(2*) is that the proposition expressed by (3) refers to the sentence
that expresses it but not to itself, while (1*) and (3*), in order to
acquire any meaning in English, would have to express strictly self-
referential propositions.

I guess that, if you persist in attributing true/falsity to sentences,
you'll be forced to hesitate between these positions:

1. There is no unique concept of truth (you need this to get out of
the Liar)
2. There is a unifying concept of truth (you need this to express 1.)

The drawback is, of course, that we can reproduce the Liar in that
same language in what you express 1.

Best regards
.

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