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

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From: Jesse F. Hughes on 27 Mar 2010 00:17

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

> On Mar 26, 3:49 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> wrote:
>> 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.
>
> Why you think you have to tell me that I do not know. If you lool a
> few lines above you will see that I was talking about the Liar paraox
> in the natural language.
>

But what you say is a non-sequitur. Perhaps, if the liar paradox is
neither true nor false in English, then it is not a paradox. But this
observation has nothing at all to do with your primary aim: that is,
to deny that vacuously true universal statements are true.

--
Jesse F. Hughes

"You people are the diminishment of a world."
-- James S. Harris, to mathematicians.

From: Jesse F. Hughes on 27 Mar 2010 00:22

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

> On Mar 26, 3:53 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> Newberry <newberr...(a)gmail.com> writes:
>> > On Mar 25, 10:26 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> >> Newberry <newberr...(a)gmail.com> writes:
>> >> > On Mar 24, 3:32 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> >> >> Newberry <newberr...(a)gmail.com> writes:
>> >> >> > Plus
>>
>> >> >> > (x)((x = x + 1) -> (x = x + 2))
>>
>> >> >> > does not look particularly meaningful to me.
>>
>> >> >> I don't believe you.
>>
>> >> > Trust me.
>>
>> >> >> You know what it means. It's perfectly clear
>> >> >> what it means. It means that whenever x = x + 1, then x = x + 2.[1]
>>
>> >> > The sentence "if it rains then some roads are wet" describes a
>> >> > possible state of affairs. I can picture to myself what it means. I
>> >> > can even picture "if it rains then no roads are wet." It is still
>> >> > conceivable although very unlikely. "If it rains and does not rain
>> >> > then the roads are wet" does not describe any possible state of
>> >> > affairs. I cannot picture to myself what it expresses.
>>
>> >> Is the statement "Honesty is a virtue" meaningful? What do you
>> >> picture when you think about that statement?
>>
>> > It can certainly be analyzed into something imaginable.
>>
>> Well, have at it!
>>
>> >> As usual, your claim that meaning involves picturing various states of
>> >> affairs is silliness. I can understand various theorems about, say,
>> >> infinite dimensional spaces. I daresay that I know those theorems are
>> >> meaningful, even though I cannot picture a space with more than three
>> >> dimensions.
>>
>> > This argument is indeed silly. These theorems are about Certesian
>> > products R x R x R x R ... If you understand numbers, real numbers
>> > and cartesian products then you of course understand statements about
>> > sets of n-tuples of real numbers. If the product has less than 4
>> > dimensions then it can also be understood as staments about the
>> > physical space.
>>
>> So? You said that I have to be able to picture it.
>
> You can picture 2 + 2 = 4. For example the union of a set of two red
> apples with a set of two green apples is a set of four apples. The
> number 2 is the set of all sets of cardinality 2. The number 4 is the
> set of all sets of cardinality 4. From the natural numbers you
> construct rational numbers and real numbers and Cartesian producst of
> real numbers ...

Well, yes and no. I can construct them in a certain sense, but I
surely can't picture them in any reasonable sense. I'm a simple
housewife, and I find myself utterly incapable, for instance, of
picturing the difference between a regular polygon with 999 sides and
a regular polygon with 1000 sides. I'm just that limited.

And yet, you want me to picture infinite dimensional spaces before I
can assent that theorems regarding those spaces are true.

Some unanswered points are left below.

>> > You are saying what the world would look like if x = x + 1. No such
>> > word is possible so it is not possible to say or even to imagine what
>> > such a world would look like.
>>
>> No, I'm not saying what the world would look like if x = x + 1. I'm
>> merely pointing out a single consequence of that equation. Indeed,
>> this consequence is *true* in those structures in which x = x + 1.
>> (As Nam pointed out, such structures do exist, you know.)
>>
>>
>>
>> >> >> [1] In fact, this statement seems obviously true! Suppose
>> >> >> x = x + 1. Then we may substitute x + 1 for x in the right hand side
>> >> >> of the equation x = x + 1, thus:
>>
>> >> >> x = x + 1
>> >> >> = (x + 1) + 1
>> >> >> = x + 2.
>>
>> >> >> I see nothing the least bit fishy about this reasoning.
--
Jesse F. Hughes
"Conviction of fraud can mean jail time. It can mean social censor. It
can mean big headlines where mathematicians take "perp walks" before a
jeering public." -- JSH on the "censor" that awaits mathematicians.

From: Newberry on 27 Mar 2010 00:34

On Mar 26, 3:56 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > On Mar 25, 3:05 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> >> MoeBlee <jazzm...(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.)
>
> > Do you think that propositions cannot be about actions?
>
> Sure, they can, but "I close the switch" is not a proposition. "I am
> closing the switch" or "I have closed the switch" are propositions.

I am not following.
>
> --
> "Maya Nahib is not a Checotah Indian! [...] Maya Nahib is an Englishman!"
> "Are you telling us that a civilized white man could kill and ravish
> and destroy with all the brutality of a savage?"
> -- Adventures by Morse radio program (1944)- Hide quoted text -
>
> - Show quoted text -

From: Newberry on 27 Mar 2010 00:53

On Mar 26, 4:24 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> 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 means that there is a plausible explanation why there is no
> >inconsistency. I do not like inconsistencies.
>
> I don't see how truth gaps help in the Liar paradox.
>
> Suppose you have a truth predicate T(x) and you have a sentence L
> (with Godel number #L) of the form
>
> forall x, A(x) -> ~T(x)
>
> and you have a theorem
>
> forall x, A(x) <-> x=#L
>
> Then L cannot be true, and cannot be false. So L falls into a "truth gap"..
> But then what about the sentence
>
> "L is not true"
>
> which is formalized by
>
> ~T(#L)
>
> Is that true, or does that have a truth gap, as well? We just
> agreed that L was not true, so if T(x) is a truth predicate,
> that should be formalized by ~T(#L). From that, surely it follows
> that
>
> "forall x, if x=#L, then ~T(x)"
>
> Since x=#L <-> A(x), then surely it follows that
>
> "forall x, A(x) -> ~T(x)"
>
> So L follows from the claim that L falls in the truth gap. Truth
> gaps *don't* help with the Liar paradox.

If you do not mind I will leave the Goedel numbers out as they are not
applicable to the natural language. It wil also streamaline the
discussion. Then let

L: ~T(L)

If v(L) = ~(T v F) then there is no contradiction. L is not true. The
argument usually goes "but that is what L says." But L does not say
anything.

From: Daryl McCullough on 27 Mar 2010 07:56

Newberry says...

>L: ~T(L)
>
>If v(L) = ~(T v F) then there is no contradiction. L is not true.

But *if* T is a truth predicate, then "L is not true" is formalized
by the statement ~T(L).

>The argument usually goes "but that is what L says." But L does not
>say anything.

It says "L is not true".

So your proposed resolution is complete nonsense.

--
Daryl McCullough
Ithaca, NY