Re: The fallacy of strengthened liar's paradox.

LauLuna says...

On Dec 31 2007, 3:22=A0pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:

Basically what I think you are proposing is a hierarchy of
types, where each type can be associated with a recursive
ordinal. The collection of all recursive ordinals is not
a recursive set. So there is no algorithm to determine
whether an ordinal is recursive or not.

Yes, it could be a way of putting it. I'm suggesting that the natural
language can reach beyond the recursive, since it can in principle
build new expressions upon (the name of) any algorithm.

That doesn't mean that natural language can reach beyond the
recursive. Given any recursive ordinal alpha, the operation
alpha --> alpha+1 is perfectly recursive. Given any r.e.
collection O of recursive ordinals, there is a recursive
operation that constructs from O a new recursive ordinal
larger than any element of O.

Anyway, it seems the Liar sentence can be denied a univocal
propositional content for two main different reasons:

1) Because its subject fails to denote anything (the 'attempted' self-
reference is impossible).

There are two parts to the Liar sentence, the subject and
the predicate:

Part 1: "This sentence"
Part 2: "is not true"

There is no problem with part 1, at least of we say that
a sentence need not be semantically meaningful to be a
sentence (it only needs to be syntactically correct).
So the problem is with part 2.

But the second part is as syntactically correct as the first.

What I meant to say was that noun phrase "This sentence" is
*semantically* meaningful. It has a referent, namely the sentence
"This sentence is not true". In contrast, the predicate "is true"
is not semantically meaningful in all cases. As Tarski pointed
out, there is no consistent meaning to a truth predicate.

Actually, I don't think we can split the sentence into two parts for
an analysis. They interact. The problem arises from the conjunction
because it is the predicate what shows that 'this sentence' is
actually attempting to refer to a proposition and not just to a
sentence.

No, "This sentence" refers to a sentence, and not a proposition.
I'm assuming that "is true" is a predicate on *sentences*, not
propositions.

A formal Russellian approach does not suffice here, that's true. But
try to analyze where the contradiction comes from.

It seems to come from assumption that "truth" is a property.
The liar paradox can be interpreted as showing that truth *isn't*
a property, although truth for a particular language can have
a truth predicate that works for that language.

The inconsistency stems precisely from the possibility of PA plus T to
express a sentence pronouncing itself not true. More concretely, the
so enriched PA would have a sentence L such that

L is true <-> ~T('L') <-> L is not true

where 'L' is the code of L.

It would happen that L expressed no proposition and had no truth
value, and at the same time that L expressed an arithmetical
proposition with a definite truth value.

Hence the contradiction. So, it can be accounted for in terms of the
impossibility of self-reference.

I don't know what you mean by that. It's impossible to *avoid*
self-reference (in a language as powerful as PA). If
you banish direct self-reference, then you will still have
indirect self-reference (via coding). You can't banish
all self-reference. But you *can* banish truth predicates.

I can concede that the concept of truth is in some sense extensible,
like the concept of set, of proposition, etc. But not that it does not
exist.

Truth for a particular *language* exists. You just can't have
a single notion of truth that works for all languages.

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.

--
Daryl McCullough
Ithaca, NY

.

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