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

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From: Nam Nguyen on 3 Mar 2010 23:13

Aatu Koskensilta wrote:
> 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, *you're a debater who doesn't seem to have the in-good-faith spirit*
in debating technical matter. Here is my full "Really?" paragraph:

>> Really? How about:
>>
>> (1) "0=0 is true" is true
>> (2) "0=0 is true"
>> (3) "0=0"
>> (4) 0=0
>>
>> Isn't it true that there's convention that in certain proper context
>> (1) asserts the same truth as (4)? [And (4) is just a formula!]

Your virtually snipping all of what I said isn't really an issue: it's
that there's a sincere technical question *you didn't bother to respond*,
then all you had was asserting my reason was "odd" and had "no apparent
relevance".

Apparently, debating to you means you simply ignore what people ask and
then say whatever you feel pleased.

What a pathetic way of arguing technical matter!

From: Aatu Koskensilta on 4 Mar 2010 00:13

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

> Apparently, debating to you means you simply ignore what people ask and
> then say whatever you feel pleased.

Well, what relevance do you take your somewhat odd reasoning about "0=0"
to have to the observation that G�del's proof doesn't involve any model
theoretic considerations?

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

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

From: Newberry on 4 Mar 2010 00:23

On Mar 3, 9:37 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > This is my main motivation:
>
> > Gödel's sentence has the same form as (3.1):
>
> > ~(Ex)(Ey)(Pxy & Qy) (4.1)
>
> > Pxy means that x is the proof of y, where x, y are Gödel numbers of
> > wffs or sequences of wffs. Q has been constructed such that only one y
> > = m satisfies it, and m is the Gödel number of (4.1).
> > Assume that Gödel's sentence (4.1) is not derivable, i.e. that
>
> > ~(Ex)Pxm (4.2)
>
> > is true. Then (4.1) is ~(T v F). Thus if Gödel's sentence is not
> > derivable it is neither true nor false.
>
> So, you want to deny that Goedel's theorem is true.

We better get this straigh first. No. I do not want to deny that
Goedel's theorem is true.

> The best way to
> do that is to deny that mathematical logic is the right logic.
> Instead, we will suppose that sentences which are vacuously true are
> in fact neither true nor false.
>
> Of course, your attempt is really *weird* at best. If 4.2 is true
> then 4.1 is neither true nor false, but 4.1 is used in the derivation
> of 4.2.

What do you mean by this? How is 4.1 used in the derivation of 4.2?

> So, if 4.2 is not true, then 4.1 may well be true after all!

It may in the sense that one of its presuppositions is true. But then
our system would be inconsistent. Not sure what your point is.

>
> The situation is indeed very similar to my example. The fact is that
> I find my example not controversial in the least (nor your example,
> for that matter).
>
> 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.

We will get to your example in a minute. But just out of curiosity, is
this how the proof actually goes?

>
> I sincerely doubt that you find this bit of reasoning dubious. I
> honestly cannot see anything the least bit controversial with it.
> Rather, it seems to me, you want to deny Goedel's theorem is true

No.

(for
> whatever reason) and so you want to deny that this bit of simple
> argument is correct.
>
> --
> "Yup, you guessed it. If worse comes to worse, I *will* turn to the
> Army to help me with mathematicians. And then mathematicians don't
> think the NSA or CIA can save your asses, as generals LIKE me."
> -- James Harris's latest foray into mathematical logic.- Hide quoted text -
>
> - Show quoted text -

From: Aatu Koskensilta on 4 Mar 2010 00:30

MoeBlee <jazzmobe(a)hotmail.com> writes:

> P.S. And note that now you're arguing the OPPOSITE of your earlier
> challenge against the claim that Godel's proof can be formalized.

My (very tentative) impression is that Nam didn't intend to argue this
opposite challenge. Rather, he is, rather bafflingly, arguing those who
assert that G�del's proof wasn't a formal proof but can be formalized
are contradicting themselves. This is a sound line of thought on the
bizarre and pointless definition of formal proof he gave earlier.

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

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

From: Jesse F. Hughes on 4 Mar 2010 09:24

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

> On Mar 3, 9:37 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> Newberry <newberr...(a)gmail.com> writes:
>> > This is my main motivation:
>>
>> > Gödel's sentence has the same form as (3.1):
>>
>> > ~(Ex)(Ey)(Pxy & Qy) (4.1)
>>
>> > Pxy means that x is the proof of y, where x, y are Gödel numbers of
>> > wffs or sequences of wffs. Q has been constructed such that only one y
>> > = m satisfies it, and m is the Gödel number of (4.1).
>> > Assume that Gödel's sentence (4.1) is not derivable, i.e. that
>>
>> > ~(Ex)Pxm (4.2)
>>
>> > is true. Then (4.1) is ~(T v F). Thus if Gödel's sentence is not
>> > derivable it is neither true nor false.
>>
>> So, you want to deny that Goedel's theorem is true.
>
> We better get this straigh first. No. I do not want to deny that
> Goedel's theorem is true.

Well, I'm sorry if I misrepresented your opinion, but you *just*
suggested that if (4.2) is true, then (4.1) is neither true nor false
and hence is not true.

Maybe what you mean is this: Goedel's theorem is a theorem of PA in
classical logic, but you are interested in a different logic. In this
different logic, the analog to Goedel's theorem is neither true nor
false. Is that a correct assessment? (Of course, there is no real
alternative logic yet, but let us ignore that fact for now.)

>> The best way to
>> do that is to deny that mathematical logic is the right logic.
>> Instead, we will suppose that sentences which are vacuously true are
>> in fact neither true nor false.
>>
>> Of course, your attempt is really *weird* at best. If 4.2 is true
>> then 4.1 is neither true nor false, but 4.1 is used in the derivation
>> of 4.2.
>
> What do you mean by this? How is 4.1 used in the derivation of 4.2?

Ignore these comments. I got things backwards here.

[...]

>> The situation is indeed very similar to my example. The fact is that
>> I find my example not controversial in the least (nor your example,
>> for that matter).
>>
>> 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.
>
> We will get to your example in a minute. But just out of curiosity, is
> this how the proof actually goes?

I don't know. As far as I recall, the first premise was well-known
prior to Wiles's proof. I don't know whether Wiles's proof amounts to
a proof of the second premise.

I'm pretty sure that there are examples of this kind of reasoning in
ordinary mathematics, whether the proof of FLT takes this form or
not. It has been a *long* time, however, since I did any real
mathematics myself, so I'm sorry that no examples come to mind.

A similar bit of reasoning that seems utterly uncontroversial to me is
this. Suppose I have a proof of

~(Ex)(Px & Qx) (1)

and also a proof of

~(Ex)(Px & ~Qx). (2)

Then I may conclude

~(Ex)Px. (3)

This seems perfectly sensible to me. Unfortunately, according to your
statements on presuppositions, if I were to conclude (3), then (1) and
(2) are not true (since they are neither true nor false) and hence, I
assume, cannot be used in a proof of (3). Oops!

Again, I can't give you any examples of this form of reasoning from
ordinary mathematics. Perhaps someone else can.

One last example: suppose I have two counties, B and C. In county B,
there are no Republicans. In C, no one voted for Obama. Then

In counties B and C, there are no Republicans who voted for Obama. (a)

is true. Unfortunately, the two consequences

In B, there are no Republicans who voted for Obama. (b)

and

In C, there are no Republicans who voted for Obama. (c)

are neither true nor false (according to you), even though (in
classical logic, at least) the statement (a) is equivalent to the
conjunction (b) & (c). I just don't see why I should think that (a)
is true, but (b) is neither true nor false.

Finally, it seems that your notions are very sensitive to what we take
to be atomic predicates. In a language in which "is large and round"
is a predicate R and "is square" a predicate S,

~(Ex)(Rx & Sx)

is true. In a language in which "is large" is a predicate L and "is a
round square" is a predicate T, the equivalent statement

~(Ex)(Lx & Tx)

is neither true nor false, even though it means the same thing. This
situation strikes me as a problem.

--
Jesse F. Hughes
Did you lay down in heaven? Did you wake up in hell?
I bet you never guessed that it would be so hard to tell.
-- The Flatlanders, /Judgment Day/