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

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From: Newberry on 4 Mar 2010 12:17

On Mar 4, 6:24 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(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.)

Let us clarify this once and for all. Here is Goedel's original
theorem.

For every omega-consistent recursive class k of FORMULAS there are
recursive CLASS SIGNS r such that neither v Gen r nor Neg(v Gen r)
belong to Flg(k).

This says that every such system is SYNTACTICALLY incomplete. Note
that that the above formulation does not say anything truth. This was
proved for Principia Mathematica, as far as I know it applies to PA,
and I believe that in fact it does generalize to ANY system capable of
arithmetic.

I am not denying any of this. Note that the system I am proposing is
syntactically incomplete.

The following formulation due to Keene is indeed false:

Any effectively generated theory capable of expressing elementary
arithmetic cannot be both consistent and complete. In particular, for
any consistent, effectively generated formal theory that proves
certain basic arithmetic truths, there is an arithmetical statement
that is true, [1] but not provable in the theory (Kleene 1967, p.
250).

The theorem (not formally stated by Goedel) that there are true but
unprovable formulae, the Second Theorem and Tarski's theorem
generalize to a vast array of systems but NOT to every system capable
of arithmetic. Also if you want to claim that there are TRUE but
unprovable formulae I do not know if you can do it by purely syntactic
means as some people are claiming in this thread.

I have quoted the following before:

"The proof by Goedel and Tarski that a language cannot contain its own
semantics applied only to languages without truth gaps."

Outline of a Theory of Truth
Saul Kripke
The Journal of Philosophy, Vol. 72, No. 19, Seventy-Second Annual
Meeting American Philosophical Association, Eastern Division. (Nov. 6,
1975), p. 714.

Note that the system I am proposing does have truth gaps.

>
> >> 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/- Hide quoted text -
>
> - Show quoted text -

From: David Bernier on 4 Mar 2010 12:23

Jesse F. Hughes wrote:
> 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:
>>>> G�del'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 G�del numbers of
>>>> wffs or sequences of wffs. Q has been constructed such that only one y
>>>> = m satisfies it, and m is the G�del number of (4.1).
>>>> Assume that G�del'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 G�del'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.

I read:
[ Newberry:]
" G�del's sentence has the same form as (3.1):"

I've been wondering if there's a typo. there and if it ought to
be numbered in the quote above (4.1) and not (3.1).

> 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.)

David Bernier

From: Aatu Koskensilta on 4 Mar 2010 12:30

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

> Let us clarify this once and for all. Here is Goedel's original
> theorem.

Quoting G�del's original statement is nothing but pointless
obscurantism.

> Also if you want to claim that there are TRUE but unprovable formulae
> I do not know if you can do it by purely syntactic means as some
> people are claiming in this thread.

Who are these people? If we state a theorem using the notion of
arithmetical truth obviously we must introduce and make use of this
concept also in the proof.

But let's consider the following variant of the incompleteness theorem
in which we certainly don't invoke in any essential way the notion of
arithmetical truth:

For any formal theory in which statements of the form "the Diophantine
equation D(x1, ..., xn) = 0 has no solutions" can be expressed, either
the theory proves "the Diophantine equation D(x1, ..., xn) = 0 has no
solutions" for some Diophantine equation D(x1, ..., xn) = 0 that has
solutions, or there are infinitely many Diophantine equations D(x1, ...,
xn) = 0 which have no solutions but for which "the Diophantine equation
D(x1, ..., xn) = 0 has no solutions" is not provable.

How would you apply your logical ideas, about statements neither true
nor false, about meaninglessness, about presuppositions, here?

--
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 12:39

David Bernier <david250(a)videotron.ca> writes:

> Jesse F. Hughes wrote:
>> 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.
>
> I read:
> [ Newberry:]
> " Gödel's sentence has the same form as (3.1):"
>
> I've been wondering if there's a typo. there and if it ought to
> be numbered in the quote above (4.1) and not (3.1).

I assumed it was a typo and he meant (4.1).

--
"If you have a really big idea, you can get a measure of how big it is
by how much people resist the obvious. From what I've seen, I have a
REALLY, REALLY, *REALLY*, BIG DISCOVERY!!!"
--James Harris: If I'm not important, how come people ignore me?

From: MoeBlee on 4 Mar 2010 12:42

On Mar 3, 10:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:

> Aatu, *you're a debater who doesn't seem to have the in-good-faith spirit*
> in debating technical matter.

It's revealing that you choose to take your conversation with Aatu as
a "debate".

MoeBlee