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

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From: Jesse F. Hughes on 25 Mar 2010 13:39

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

>> Evidently, the empty set is not closed under the successor operation
>> (according to you), since
>>
>> (Ay)( y in {} -> s(y) in {} )
>>
>> is neither true nor false. So my question is this: since the empty
>> set is not closed under the successor operation, would you say that
>>
>> (Ax)( (Ay)(y in x -> s(y) in x) -> (x is infinite) )
>>
>> is a true statement?
>
> I think it is true.
>
>> (I hope not.)
>
> Why?

Well, because that sort of reasoning will lead to trouble, I believe.

Let me ask you a question. In general, is the statement

(Ax)(Px -> Qx) -> (Ex)Px

true?

If so, we're going to have some problems.

The following is a theorem of classical arithmetic and (I assume) also
in your system.

(An)((Ea,b)( a^2/b^2 = n ) -> (Ea,b)( a^2/b^2 = n and gcd(a,b) = 1 )).

Note that this statement is not meaningless, since indeed
(En)(Ea,b)(a^2/b^2 = 1) is true.

From this, it follows that

(Ea,b)( a^2/b^2 = 2 ) -> (Ea,b)( a^2/b^2 = 2 and gcd(a,b) = 1 ),

right? Simple instantiation of 2 for n. Is that move legal? (I
assume that at this point, we have no proof that 2 is irrational,
since this is part of such a proof.)

Hence,

(Aa,b)( a^2/b^2 = 2 -> gcd(a,b) != 1 ) -> ~(Ea,b)( a^2/b^2 = 2 ).

Now, if (Ax)(Px -> Qx) -> (Ex)Px is true in general, then we have both

(Aa,b)( a^2/b^2 = 2 -> gcd(a,b) != 1 ) -> ~(Ea,b)( a^2/b^2 = 2 ) and
(Aa,b)( a^2/b^2 = 2 -> gcd(a,b) != 1 ) -> (Ea,b)( a^2/b^2 = 2 ).

Unfortunately, we also can prove

(Aa,b)( a^2/b^2 = 2 -> gcd(a,b) != 1 ).

Thus, we conclude both ~(Ea,b)( a^2/b^2 = 2 ) and
(Ea,b)( a^2/b^2 = 2 ). Which is bad.

Again, where did I go wrong?

--
One these mornings gonna wake | Ain't nobody's doggone business how
up crazy, | my baby treats me,
Gonna grab my gun, kill my baby. | Nobody's business but mine.
Nobody's business but mine. | -- Mississippi John Hurt

From: Jesse F. Hughes on 25 Mar 2010 13:40

Newberry <newberryxy(a)gmail.com> writes:
> On Mar 24, 9:34 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> Paradoxical in what sense?
>
> Does not everybody know what the paradox of material implication is?

I'm just a simple housewife (with somewhat hairy legs). Why not tell
me what you mean by "the paradox of material implication" and why it's
a paradox?

--
Jesse F. Hughes
"The future is a fascinating thing, and so is history. And you people
are a fascinating part of history, for those in the future."
-- James S. Harris is fascinating, too

From: Daryl McCullough on 25 Mar 2010 14:00

Jesse F. Hughes says...
>
>Newberry <newberryxy(a)gmail.com> writes:
>> On Mar 24, 9:34=A0am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>>
>>> Paradoxical in what sense?
>>
>> Does not everybody know what the paradox of material implication is?
>
>I'm just a simple housewife (with somewhat hairy legs). Why not tell
>me what you mean by "the paradox of material implication" and why it's
>a paradox?

Wikipedia has a list of theorems of classical logic that it calls
"paradoxes of material implication":

http://en.wikipedia.org/wiki/Paradoxes_of_material_implication

There's nothing paradoxical about any of them, other than the
fact that they may be counter-intuitive to someone who is a
complete newbie to formal logic.

Ultimately, the people on this newsgroup who object to standard
mathematics are really objecting to the idea that there can be
such a thing as a counter-intuitive result. The ultimate logic
would be one in which it is impossible to prove any result that
you couldn't already guess was true.

--
Daryl McCullough
Ithaca, NY

From: MoeBlee on 25 Mar 2010 17:48

On Mar 25, 1:00 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:

> Wikipedia has a list of theorems of classical logic that it calls
> "paradoxes of material implication":
>
> http://en.wikipedia.org/wiki/Paradoxes_of_material_implication
>
> There's nothing paradoxical about any of them

The discussion there about (P&Q) -> R |- (P -> R) v (Q -> R)

is at least somewhat interesting.

MoeBlee

From: Jesse F. Hughes on 25 Mar 2010 18:05

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

> On Mar 25, 1:00 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> wrote:
>
>> Wikipedia has a list of theorems of classical logic that it calls
>> "paradoxes of material implication":
>>
>> http://en.wikipedia.org/wiki/Paradoxes_of_material_implication
>>
>> There's nothing paradoxical about any of them
>
> The discussion there about (P&Q) -> R |- (P -> R) v (Q -> R)
>
> is at least somewhat interesting.

Yes, but their example ("If I close switch A and switch B, the light
will go on. Therefore, it is either true that if I close switch A the
light will go on, or that if I close switch B the light will go on.")
is poorly chosen, since P, Q and R stand for propositions, while "I
close switch A (or B)" is an action. (I'm not sure what type of
sentence "The light will go on," is -- it's not an action, in the
sense of dynamic logic, but rather it describes a change in the
world.)

Their example is better understood in dynamic logic rather than
propositional logic.

No matter, the fix is easy -- indeed, I'll go fix it now.

--
Meaningless movies
on the screen behind the band that's blowing Waterboys,
throwing shapes "My Love is My Rock
Half of the music is on tape in the Weary Land"