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

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From: Daryl McCullough on 26 Mar 2010 06:40

Newberry says...

>Are you saying in a roundabout manner that in classical logic (P & ~P)
>-> Q is a tautology? Well we know that. But what does it have to do
>with anything?

I'm saying that none of the "paradoxes of material implication" are
paradoxical.

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 26 Mar 2010 06:43

Newberry says...
>
>On Mar 25, 3:37=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:
>> Newberry says...
>>
>> >In my logic the Liar paradox can be expressed as follows.
>>
>> > =A0 ~(Ex)(Ey)(Pxy & Qy) =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0=
> =A0 =A0(L)
>>
>> >where Pxy means that x is a proof of y, Q is satisfied by only one y =3D
>> >m, and m is Goedel number of (L).
>>
>> That's not the Liar sentence.
>
>The sentence seemingly says about itself that it is not provable.

That's a Godel sentence, not the Liar sentence.

>SInce Tarski's theorem does not apply we can equate provability with
>truth.

Why does that follow? You can't possibly do that. What is provable
depends on what axioms you have. Different axioms leads to different
notions of provable.

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 26 Mar 2010 06:45

Newberry says...
>
>On Mar 25, 3:49=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:
>> Newberry says...
>>
>> >Tarski's theorem does not apply to formal systems with gaps. I think
>> >it is preferable.
>>
>> If you the way you express Tarski's theorem is like this, then truth
>> gaps don't change anything:
>>
>> There is no formula T(x) such that if x is a Godel code of a true
>> sentence, then T(x) is true, and otherwise, ~T(x) is true.
>>
>> Anyway, *why* is it preferable to have a formal system for which Tarki's
>> theorem does not apply? Preferable for what purpose?
>
>If truth is expressible then truth can be equivalent to provabilty.

That doesn't follow.

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 26 Mar 2010 06:49

Newberry says...
>
>On Mar 25, 3:38=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:
>> Newberry says...
>>
>> >If you take the position that there are truth value gaps then the Liar
>> >papradox is solvable in English.
>>
>> What does it mean to be "solvable" and why do you want it to be solvable?
>
>It mean that there is a plausible explanation why there is no
>inconsistency. I do not like inconsistencies.

The Liar sentence is not *expressible* in any standard mathematical
theory (PA or ZFC). So you don't have to do anything to keep the Liar
from spoiling the consistency of those languages.

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 26 Mar 2010 06:54

Nam Nguyen says...

>Daryl McCullough wrote:
>
>>
>> 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.
>
>That's a grossly erroneous over-generalization. It's many of those who
>defend standard mathematics who erroneously object to the counter-intuitive
>_but real nature_ of mathematics: relativity of truth and provability.

Huh? Standard mathematics perfectly well takes into account the
"relativity of truth". Truth is relative to an interpretation. So
your objection makes no sense.

--
Daryl McCullough
Ithaca, NY