Re: Solution to Liar Paradox

On Mar 9, 6:24 am, Barb Knox <s...@xxxxxxxxx> wrote:
In article <MPG.205a7d2bbf56a2b1989...@xxxxxxxxxxxx>,
David Marcus <DavidMar...@xxxxxxxxxxxxxx> wrote:

LauLuna wrote:
[snip]
All
that Lan Wen proves is that if there is a sentence able to say of
itself that it is false, then we have a contradiction; so there is no
such sentence. I dare say we knew this before.

He does more than that: He gives a syntactic characterization of what
sentence-like things can have truth values. And, he shows that the Liar
is just one of a huge class of similar paradoxical sets of English
sentences. The latter is important because it makes it clear that the
paradox is simply a manifestation in English of the general mathematical
principle that systems of equations need not have solutions. This should
help us emotionally stop assuming that all sets of apparently sensible
English sentences must have truth assignments that satisfy them.

Having an allowable truth assignment does not by itself solve the Liar
and related paradoxes. For example, consider Curry's paradox; the
following sentence DOES have a consistent truth assignment for all its
parts, but is still a paradox:
If this whole sentence is true then Santa Clause exists.

Let P be the sentence so we have (where , means equivalence):

P , P => SCE

If SCE=FALSE then it's

P , P => FALSE
P , ~P v FALSE
P , ~( P ^ ~FALSE )
P , ~(P ^ TRUE)
P , ~(P)
P , ~P

i.e. "This is false."

P is not a paradox. It merely states that SCE. If SCE=FALSE you can
call it a "Paradox" (by the CBL definition it is) since it proves
FALSE. But then there is no solution and it is merely the Liar. So P
is a generalization of the Liar with a variable that produces the Liar
when the variable is FALSE.

C-B

It is consistent for that sentence to be true; it's not consistent for
it to be false (since (F -> anything) is in fact always true).
Therefore by pure logic we have shown that the sentence is true and thus
Santa Clause exists.

The remaining problem
is: why is the Liar not such sentence in spite of all appearances?

Because the translation into English hides the fact. The English
sentence has the appearance of something that should have a truth value.
But, appearances can be deceiving, as the careful mathematical analysis
shows. The boolean equation that corresponds to the English sentence
clearly doesn't have a solution.

--
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| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
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Relevant Pages

  • Practical Application of the "Liars Paradox"
    ... The Liar Paradox. ... "Truth" for English sentences is not ...
    (sci.math)
  • Practical Application of the "Liars Paradox"
    ... The Liar Paradox. ... "Truth" for English sentences is not ...
    (sci.logic)
  • Re: Solution to Liar Paradox
    ... It is like Russell's paradox: ... why is the Liar not such sentence in spite of all appearances? ... self-reference, negation, substitution and consistency. ... is not a sentence because then English sentences are not complete, ...
    (sci.logic)
  • Re: Post Axiom Syndrome
    ... Consider the Liar paradox. ... The Liar is not so bad, but with further primary objectives in an ... ZF is inconsistent, because the set-theoretical universe is infinite, ...
    (sci.logic)
  • Re: My investigations into Godels Incompleteness Theorem
    ... the Liar, is not literally self-referential ... understood or explained in the literature. ... When CBL is used to formalize the Liar Paradox, ... particular sets and systems of representation we formalize these ...
    (sci.logic)
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