Re: The fallacy of strengthened liar's paradox.

Newberry says...

On Jan 2, 9:40=A0am, stevendaryl3...@xxxxxxxxx (Daryl McCullough) wrote:

42. No sentence that is syntactically identical to sentence number 42 is
true.

56: Sentence number 42 is meaningless.

Where is the contradiction?

The same reasoning that leads to the conclusion that sentence 42
is meaningless allows you to conclude that *any* sentence syntactically
identical to sentence 42 is meaningless (at least in a context in
which "sentence 42" refers to the above sentence).

Let C = the set of all sentences that are syntactically identical
to sentence number 42. Each sentence in C is either true, false,
or meaningless. So either (1) at least one sentence in C is true, or
(2) at least one sentence in C is false, or (3) all sentences in C
are meaningless. Let's consider these possibilities.

Case (1) Assume that some sentence, call it "P", in C is true.
By definition of C, P must be syntactically of the form
"No sentence that is syntactically identical to sentence
number 42 is true." If P is meaningful, then it is equivalent
to the following sentence:

"No sentence in C is true."

But P *is* a sentence in C. That's a contradiction.
So Case (1) is impossible.

Case (2) Assume that some sentence, call it "P", in C is false.
By definition of C, P must be syntactically of the form
"No sentence that is syntactically identical to sentence
number 42 is true." If P is false, then its negation is true.
The negation of P is equivalent to the following sentence:

"Some sentence that is syntactically identical to sentence 42
is true."

which is equivalent to the following sentence:

"Some sentence in C is true."

But we already agreed that that is impossible (in case (1)).
So Case (2) is impossible, also.

Case (3) Assume that every sentence in C is meaningless.

From this it follows that:

"No sentence in C is true."

From this, together with the definition of C, the following
sentence follows:

"No sentence that is syntactically identical to sentence 42
is true."

But that sentence *itself* is syntactically identical to
sentence 42. So that sentence itself must be meaningless!
So in particular, it can't be *true*. But it follows
from the assumption that every sentence in C is meaningless.
So that assumption cannot be true. So Case 3 is impossible.

That's the paradox: We can prove that either Case (1),
Case (2), or Case (3) must hold. But we also can show
that none of these cases is possible. That's a contradiction.

--
Daryl McCullough
Ithaca, NY

.

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