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

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From: Newberry on 2 Apr 2010 00:35

On Apr 1, 3:09 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote:
> Newberry says...
>
> >Let me ask you something. Let
>
> >~(Ex)(Ey)(Pxy & Qy) (1)
>
> >be Goedel's sentence. [Pxy: x is a proof of y, Qy is such that only
> >one y=m satisfies it & m = #(1)]. You are absolutely convinced that PA
> >is consistent i.e. (1) is true. That is, the search for x and y will
> >never terminate. How do you know that the search for x and y will
> >never terminate?
>
> Suppose you find two natural numbers m and n such that,
> P(n,m) & Q(m). Then you can easily prove (Ex) (Ey) (Pxy & Qy).
> You can also "decode" n to get a proof of ~(Ex)(Ey)(Pxy & Qy).
> So you would, in that case, have a proof of a contradiction.
>
> If your axioms are consistent, the the above case cannot happen.
> To say that "the above case cannot happen" is to say that there
> are no natural numbers m and n such that P(n,m) & Q(m). Which
> is formalized as ~(Ex)(Ey)(Pxy & Qy).
>
> So the assumption that your system is consistent directly leads
> to conclusion (1).

What I was getting at is how we know that the system is consistent.
And if we do know it then we also know that certain seraches will
never terminate. Can we apply this knowledge to Diophantine equations?

From: Newberry on 2 Apr 2010 00:37

On Mar 31, 5:12 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
> >Why are you stating this so categorically? Look at this excerpt from
> >Gaifman:
>
> >QUOTE
> >Line 1: The sentence on line 1 is not true.
> >Line 2: The sentence on line 1 is not true.
>
> >The standard evaluation rule for a sentence of the form "The sentence
> >written in/on ... is true" is roughly this:
> >(*) Go to ... and evaluate the sentence written there. If that
> >sentence is true, so is "The sentence written in ... is true" , else
> >the latter is false.
>
> In general, any *procedure* used to evaluate the truth of sentences
> in a self-referential language is incomplete, in the sense that there
> are true sentences that are not evaluated as true by the procedure.
>
> This is easy to see: Let P be some procedure to evaluate the truth
> of sentences. Then consider the sentence
>
> "When procedure P is applied to this sentence, the result is not true"

And what procedure would it be? It cannot be Gaifman's procedure
because the sentence above does not have the form "The sentence
written in/on ... is true"
>
> Contrary to your claims about sentences that fail to express a possible
> state of the world, the above sentence makes a perfectly definite claim:
> That a certain procedure applied to a certain sentence does not produce
> a certain result. If the procedure is actually applied to that sentence,
> and run to completion, then we can just check to see what the result is.
>
> What this shows is that for any sound procedure that purports to evaluate
> the truth of sentences, there is a true (in the sense of corresponding to
> the facts) sentence that the procedure fails to evaluate as true. So *every*
> procedure for evaluating truth of sentences is incomplete. A procedure
> can be complete for a certain *class* of sentences, for example, those
> expressible in a limited language, but there will always be an extended
> language that the procedure fails on.
>
> --
> Daryl McCullough
> Ithaca, NY

From: Daryl McCullough on 2 Apr 2010 06:19

Newberry says...
>
>On Mar 31, 5:12=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:

>> In general, any *procedure* used to evaluate the truth of sentences
>> in a self-referential language is incomplete, in the sense that there
>> are true sentences that are not evaluated as true by the procedure.
>>
>> This is easy to see: Let P be some procedure to evaluate the truth
>> of sentences. Then consider the sentence
>>
>> "When procedure P is applied to this sentence, the result is not true"
>
>And what procedure would it be?

Well, for example, the search for a proof for the statement. Or
Gaifman's procedure.

>It cannot be Gaifman's procedure because the sentence above does
>not have the form "The sentence written in/on ... is true"

Then, as I said, it's a true sentence that Gaifman's procedure does
not return true for.

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 2 Apr 2010 06:23

Newberry says...

>What I was getting at is how we know that the system is consistent.

In the case of PA, it's because we know that everything it says
about the natural numbers is true.

Basically, the axioms of PA consist of:
1. Axioms that recursively define plus and times in terms of successor.
2. Axioms that say that zero is the smallest natural natural number,
and that successor is 1-1.
3. The induction axioms, which basically say that every natural
number is obtained from zero by repeatedly applying the successor
function.

--
Daryl McCullough
Ithaca, NY

From: Jesse F. Hughes on 2 Apr 2010 06:16

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

> I do not know how it will turn out. I forgot who proved that the
> square root of 2 was irrational and what his proof looked like. Maybe
> your version is something concocted by the modern mathematicians who
> take classical logic for granted.

It was due to nameless Pythagorean. It was a geometric proof, rather
than the more familiar algebraic proof, but I don't know the details.

> Maybe it will turn invalid, maybe valid with some modifications or
> added assumptions. Mind you the Greeks did not have the concept that
> the vacuous sentences were true. The traditional syllogism
> presupposes that the subject class is non- empty.

I'll betcha that the mathematical proofs of, say, Euclid, do not
follow the logical restrictions of Aristotle's categorical logic.
But, again, I'm no expert on this by any stretch of the imagination.

I would've thought that you had certain aims for your logic. I
would've thought that, for instance, you would want that, if a set of
sentences T entails P in your logic, then that same set of sentences
entails P in classical logic. Roughly, that classical logic makes
more things true, but doesn't make different things true. If so, of
course, you'd have to drop the claim (recently made) that

~(Ex)(Px & Qx) -> (Ex)Px.

Similarly, of course, I would expect that if your logic proves P, then
so does classical logic.

Right now, I'm not sure whether you've considered questions like
this. If not, you prob'ly oughta.

--
Jesse F. Hughes

"When you try to kiss a girl, it's hard not to get spit on the girl."
-- Quincy P. Hughes, age 3 (almost 4)