'When Are Relations Neither True Nor False? [Logic]

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From: Marshall on 3 Mar 2010 03:27

On Mar 2, 10:19 pm, Newberry <newberr...(a)gmail.com> wrote:
> On Mar 2, 10:15 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
> > I just don't see any reason to give up a perfectly straightforward and
> > sensible argument (like that above) in order to fit your intuitions
> > that, if (3) is true, then (2) is neither true nor false. Perhaps
> > t-relevance logic is useful, but it does not seem to be satisfactory
> > for mathematical reasoning. In mathematics, the argument I outlined
> > above is a perfectly good argument for (3).
>
> > Seems to me that you've got quite a job convincing mathematicians
> > otherwise.
>
> I encourage you to look at the big picture and not just pick at one
> particular aspect. And do not worry about mathematicians. You can make
> your own judgement, can you not?

Big picture is, you're proposing making logic more complicated
and less effective, and nothing you've proposed has any
benefit to offset the costs.

Marshall

From: Jesse F. Hughes on 3 Mar 2010 06:38

Newberry <newberryxy(a)gmail.com> writes:

> On Mar 2, 10:15 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> I just don't see any reason to give up a perfectly straightforward and
>> sensible argument (like that above) in order to fit your intuitions
>> that, if (3) is true, then (2) is neither true nor false. Perhaps
>> t-relevance logic is useful, but it does not seem to be satisfactory
>> for mathematical reasoning. In mathematics, the argument I outlined
>> above is a perfectly good argument for (3).
>>
>> Seems to me that you've got quite a job convincing mathematicians
>> otherwise.
>
> I encourage you to look at the big picture and not just pick at one
> particular aspect. And do not worry about mathematicians. You can make
> your own judgement, can you not?

I've made my judgment. There's absolutely nothing wrong or fishy with
the following argument.

If there are counterexamples to FLT, then there are counterexamples
with prime exponent.

There are not counterexamples with prime exponent.

Therefore, there are no counterexamples to FLT.

The idea that the second premise is not true unless the conclusion is
false is simply nonsense to me.

By the way, it is not that I reject your aims just because of this one
example. For years now, I've told you that I haven't any clue why you
think that a vacuously true universal statement is not true. This
notion simply holds no intuitive plausibility for me, but it is the
single motivation for your attempts[1] at reforming logic.

Footnotes:
[1] Attempts that have, thus far, not introduced a single deductive
rule for predicate logic.

--
"[In the movie, Tom Green] delivers a child, severs the umbilicus with
his teeth and then swings the baby over his head before tenderly
handing it to the stunned, blood-spattered mother[...] This was, I
have to say, a bit much." -- New York Times movie reviewer A. O. Scott

From: Aatu Koskensilta on 3 Mar 2010 07:13

Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> So of course I found it difficult when a technical judgment was
> rendered on the ground of vague and subjective opinion.

If a brief glance doesn't convince you that G�del's proof isn't a formal
proof you need of course take a closer look.

> I could easily cite one or two "crazy" informalities in Schoenfield's
> book but that wouldn't make the proofs in his book no less formal than
> the proofs in Godel's paper!

Right, most proofs we find in Shoenfield are no more formal in any
technical sense than G�del's was. Take for instance the proof of Lemma 2
in Section 2.4:

We use induction on the length of u. Write u as v v1 ... vk, where v is
a symbol of index k and v1 ... vk are designators. If the occurrence of
a symbol in question is the initial v, then it begins u. Otherwise, the
occurrence is in some vi, and hence, by induction hypothesis, begins an
occurrence of a designator in vi. Hence it begins an occurrence of a
designator in u.

Taking a good long look at this, we find it is by no stretch of the
imagination a sequence of formulas in a formal language in which every
formula is either an axiom (of some unspecified formal system) or
follows from earlier formulas by a formally defined rule of inference.

> "Any any proof in mathematics" is way too vague to really understand
> what you said here.

Can you think of an example of a proof in mathematics that can't be
formalized in a suitable formal system?

> We all know the criteria for determining whether or not his proof is
> formal: if there's a formal system where what he asserted is a theorem,
> then his proof is a formal proof.

No one but you has proposed this bizarre criterion. According to your
criterion pretty much any and every piece of reasoning, such as

I like boiled broccoli. Therefore, as it says in Melvyn Pike's _Titus
Groans_, Clarice came up to her sister's side and they both looked at
him. It follows, by Shelah's possible cofinality theory and the book of
Job, that sometimes snails try to eat people (with very little
success). From this it is immediate that for no set A is there a set of
everything not contained in A.

is a formal proof.

> If you say, for example, his theorem can be proven in the formal PRA
> then there's a formal proof for what he asserted in the paper, hence
> at the same time saying there's no formal proof for what he asserted
> is contradictory.

No one has claimed there's no formal proof of the incompleteness
theorem. Indeed, we use the fact that (one half of) the first
incompleteness theorem is formally provable in theories meeting certain
conditions in the usual proofs of the second incompleteness theorem.

> Godel's proof isn't a formal proof for a _different reason_, despite
> that the formula (~Bew_c[(17 Gen r)] /\ ~Bew_c[Neg(17 Gen r)]) is a
> theorem in a "proving"/"encoding" formal system, such as PRA.

I'm afraid I can make nothing of these odd proclamations.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Aatu Koskensilta on 3 Mar 2010 07:16

Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes:

> I like boiled broccoli. Therefore, as it says in Melvyn Pike's _Titus
> Groans_, Clarice came up to her sister's side and they both looked at
> him. It follows, by Shelah's possible cofinality theory and the book of
> Job, that sometimes snails try to eat people (with very little
> success). From this it is immediate that for no set A is there a set of
> everything not contained in A.

The relevant reference is of course _Titus Groan_ and not _Titus Groans_
as I erroneously wrote above.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Aatu Koskensilta on 3 Mar 2010 08:15

Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> Aatu Koskensilta wrote:
>
>> That the proof doesn't involve e.g. any model theoretic
>> considerations we ascertain simply by inspecting the proof and the
>> relevant definitions.
>
> Really?

Yes. Your somewhat odd reasoning about "0=0" has no apparent relevance.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus