Re: The fallacy of strengthened liar's paradox.
- From: LauLuna <laureanoluna@xxxxxxxx>
- Date: Tue, 1 Jan 2008 17:43:17 -0800 (PST)
On Dec 31 2007, 3:22 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
LauLuna says...
On Dec 29, 5:14=A0pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
LauLuna says...
Assume there is a collection C of syntactic marks or types (expressing
levels, interpretations or whatever is required for disambiguation) by
which natural language can be so extended that any expression gets
logically disambiguated. C should be a part of a grammar able to
disambiguate all expressions; as such C must be nameable. Now consider
LC =A0'this sentence is meaningless in all levels/interpretations in C'
That shows that the collection C cannot be expressed in natural language.
It doesn't show that C does not exist.
How could a set of symbols belonging to a grammar (i.e. an algorithm
generating formulas) be unnameable in natural language?
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.
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.
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.
2) Because the natural language truth predicate is ambiguous.
I'd say the first is a rather Russellian approach while the second is
a Tarskian-Quinean one.
As I see it, the difficulties the first approach appears to encounter
(namely, the apparent existence of successful self-references) get
solved by the distinction between sentences and propositions, and the
qualified rejection of self-reference, which states that no
proposition refers to itself but some propositions do refer to the
sentences that express them.
In contrast, the second approach seems to me to be unbearably
counterintuitive.
The "Russellian" approach by itself doesn't work unless you
*also* adopt the Tarskian position that there is no truth
predicate. Peano Arithmetic has no sentences that refer to
themselves, but if you introduce a truth predicate
T(x) with the interpretation that
T(x) <-> x is the code for a true sentence
then the system becomes inconsistent. So it seems to
me that the Tarskian approach is necessary, whether it
is counterintuitive or not.
A formal Russellian approach does not suffice here, that's true. But
try to anlye where the contradiction comes from.
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 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. 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.
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.
Regards
.
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