Re: A missing definition in "Gödel's Proof" by Nagel & Newman (open letter)

On Thu, 01 Nov 2007 16:37:19 -0000, translogi <wilemien@xxxxxxxxxxxxxx>
wrote:


Is a truth-table for that statement that shows it to be true in all
combinations of T and F for p and q considered to be a valid
proof of its truth?

NO. Sorry I need to use capitals here.

Something is a proof if there is a derivation of the theorem from
the axioms and inference rules. A proof is very strict thing.

Right.

"As part of his predicate calculus, Frege developed a strict definition
of a 'proof'. In essence, he defined a proof to be any finite sequence
of well-formed statements such that each statement in the sequence
either is an axiom or follows from previous members by a valid rule of
inference. A proof of the statement B from the premises A1,...,An is any
finite sequence of statements (with B the final statement in the
sequence) such that each member of the sequence: (a) is one of the
premises A1,...,An, or (b) is an axiom, or (c) follows from previous
members of the sequence by a rule of inference. This is essentially the
definition of a proof that logicians still use today." (Ed Zalta)

Though the problem is that many authors refer to meta-proofs as "proofs"
too...

In the present context (this thread) it might help to avoid confusion if
we call the proofs you have in mind here: /formal proofs/.


Truth tables do not prove anything.

Right. If "proof" refers to a formal proof here. But of course a truth
table (meta-)proves/shows that the formula in question is a _tautology_.


But you can say (if you really like them) they show it

And if you are very philosophical you can follow Wittgenstein
"Every tautology itself shows that it is a tautology." 6,127

Right.

6.1 The propositions of logic are tautologies.

6.113
It is the characteristic mark of logical propositions that
one can perceive in the symbol alone that they are true
[i.e. "logically true" --G.F.]

6.126
Whether a proposition belongs to logic can be calculated by
calculating the logical properties of the _symbol_.

And this we do when we prove a logical proposition. For without
troubling ourselves about a sense and a meaning, we form the
logical propositions out of others by mere _symbolic rules_.

We prove a logical proposition by creating it out of other logical
propositions by applying in succession certain operations, which
again generate tautologies out of the first. [...]

Naturally this way of showing that its propositions are tautologies
is quite unessential to logic. Because the propositions, from which
the proof starts, must show without proof that they are tautologies.

6.1262
Proof in logic is only a mechanical expedient to facilitate
the recognition of tautology, where it is complicated.

6.127 [...]
Every tautology itself shows that it is a tautology.


F.

P.S.
Of course, W's claims are only true for /propositional logic/.

--

E-mail: info<at>simple-line<dot>de
.

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