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Next: Meaning, Presuppositions, Truth-relevance, Godel's Theorem andthe Liar Paradox
From: Newberry on 12 Aug 2010 00:25
On Aug 11, 11:03 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
> >But anyway the point is that IF
> >~(Ex)Pxm
> >then
> >~(Ex)[Pxm & ((x = x &) Qm)]
> >is vacuous.
>
> Why? In the case we are talking about,
> Qm is assumed to be true. In that case,
> Pxm & x=x & Qm
> means the same thing as
> Pxm. The conjunction of any statement S
> with any true statement produces a new
> statement that is equivalent to S.

Not in truth-relevant logic. I have already told you many times that
we were not using classical logic.

>
> >I do not know what you are trying to argue here.
>
> I'm arguing that there is no sensible notion of
> "vacuous" according to which ~(Ex)Pxm & ((x = x &) Qm)
> is vacuous.
>
>
>
>
>
> >By "vacuous" I mean that the subject class is empty.
>
> >> >In this sense the sentence is vacuous.
>
> >> >> If there actually *is*
> >> >> a proof of the Godel sentence, then that statement
> >> >> would be provably *false*. So that statement *could*
> >> >> be false, so it certainly is not vacuously true.
>
> >> >If ~(Ex)Pxm then it certainly is "vacuously true".
>
> >> So you are saying that if it is true, then it is vacuously
> >> true. What is the point of adding the adjective "vacuously"
> >> in this case? It contributes nothing. Just say "It is true".
>
> >By "it" I meant
> >~(Ex)[Pxm & ((x = x &) Qm)]
>
> which is equivalent to
> ~(Ex) Pxm
>
> >> Your vacuously true sentences has the main property that
> >> people would want for true sentences, namely that you can't
> >> derive a false statement from a collection of vacuously
> >> true statements.
>
> >In the logic of presuppositions vacuous sentences are NOT true.
>
> I know. My point is that you are making a *meaningless* distinction
> between vacuous statements and true statements. Vacuous statements
> share with true statements the desirable property that they can
> never be used to derive a *false* statement. Furthermore, a pair
> of vacuous statements can be used to derive a *non-vacuous* statement.
> For example:
>
> "All counterexamples to GC are multiples of 3"
>
> "All counterexamples of GC are of the form 2*p where p is prime"
>
> These two vacuous statements allow us to conclude:
>
> "There are no counterexamples to GC".
>
> So vacuous statements can be useful in deriving nonvacuous statements.

Whether this will be allowed or not in the logic I am proposing is
irrelevant. If a sound derivation system exists it will derive all the
true, non-vacuous sentences.

> You haven't given a coherent reason to care about the distinction between
> vacuous and nonvacuous statements.

I gave you two:
a) There is no way to claim that "All John's children are asleep" is
true if John has no children. The sentence does not correspond to the
actual state of affairs - there are no John's children anywhere who
are asleep.
b) We get a semantically complete arithmetic

>
> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
>
> - Show quoted text -

From: Newberry on 12 Aug 2010 09:26
On Aug 12, 5:46 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
>
>
>
>
>
>
> >On Aug 11, 11:03=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> >wrote:
> >> Newberry says...
>
> >> >But anyway the point is that IF
> >> >~(Ex)Pxm
> >> >then
> >> >~(Ex)[Pxm & ((x =3D x &) Qm)]
> >> >is vacuous.
>
> >> Why? In the case we are talking about,
> >> Qm is assumed to be true. In that case,
> >> Pxm & x=x & Qm
> >> means the same thing as
> >> Pxm. The conjunction of any statement S
> >> with any true statement produces a new
> >> statement that is equivalent to S.
>
> >Not in truth-relevant logic.
>
> I'm pointing out how incredibly stupid that notion of
> "truth relevant logic" is.
>
> >> For example:
>
> >> "All counterexamples to GC are multiples of 3"
>
> >> "All counterexamples of GC are of the form 2*p where p is prime"
>
> >> These two vacuous statements allow us to conclude:
>
> >> "There are no counterexamples to GC".
>
> >> So vacuous statements can be useful in deriving nonvacuous statements.
>
> >Whether this will be allowed or not in the logic I am proposing is
> >irrelevant.
>
> No, it's not irrelevant. You're proposing some rules for something
> that looks vaguely like logic. Should it really count as logic, or
> is it just goofing around with symbols? I think that to show that
> it counts as a kind of logic, it should be possible to derive something
> interesting with it.
>
> >> You haven't given a coherent reason to care about the distinction between
> >> vacuous and nonvacuous statements.
>
> >I gave you two:
> >a) There is no way to claim that "All John's children are asleep" is
> >true if John has no children.
>
> Sure there is. It's the same as claim that there is no person who is both
> awake and one of John's children. There is no problem with understanding
> what that claim means. Yes, if someone *says* "All John's children
> are asleep", then we assume that he is familiar with John's children
> (how else would he know whether they are asleep or not), and so would
> know whether he has any children at all. If John happened to have no
> children, then the speaker would have said "John has no children"
> rather than bringing up sleeping.
>
> The real thing that is going on here is an analysis of the possible
> intentions of the speaker. When someone tells us something, we try
> to figure out what purpose they are trying to accomplish, and we
> use our analysis of purpose to interpret what it is they are saying.
> We assume that the speaker is *not* telling us something that is
> tautologically true (because what would be the point in telling
> us something that we already know, or can easily figure out?) We
> assume that if someone leaves out some piece of information, it's
> either because they don't know it, or they don't want us to know
> it, or they thought that it was irrelevant to our purposes.
>
> None of this applies to *mathematical* statements. In the case
> of statements about arithmetic, it isn't that anyone *told* us
> those statements, and had some agenda for telling them. Instead,
> we are trying to *figure* out which statements are true and which
> are not. The communicative intent that is important in understanding
> natural language exchanges between humans is *not* important in
> mathematical proof. It's extremely weird to try to impose the
> same rules of "presupposition" to the case of mathematics.
>
> This endeavor seems to be starting with something kind of interesting,
> which is the notion of presuppositions in natural language, and applying
> it in a completely bizarre way.
>
> >b) We get a semantically complete arithmetic
>
> Why do you believe that?

Goedel's sentence is not true because it is vacuous, and we do not
regard vacuous sentences as true.

>
> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
>
> - Show quoted text -

From: James Burns on 12 Aug 2010 10:34
Daryl McCullough wrote:
> Newberry says...

>>a) There is no way to claim that "All John's children are asleep" is
>>true if John has no children.
>
>
> Sure there is. It's the same as claim that there is no person who is both
> awake and one of John's children. There is no problem with understanding
> what that claim means. Yes, if someone *says* "All John's children
> are asleep", then we assume that he is familiar with John's children
> (how else would he know whether they are asleep or not), and so would
> know whether he has any children at all. If John happened to have no
> children, then the speaker would have said "John has no children"
> rather than bringing up sleeping.
>
> The real thing that is going on here is an analysis of the possible
> intentions of the speaker. When someone tells us something, we try
> to figure out what purpose they are trying to accomplish, and we
> use our analysis of purpose to interpret what it is they are saying.
> We assume that the speaker is *not* telling us something that is
> tautologically true (because what would be the point in telling
> us something that we already know, or can easily figure out?) We
> assume that if someone leaves out some piece of information, it's
> either because they don't know it, or they don't want us to know
> it, or they thought that it was irrelevant to our purposes.

What you describe reminds me of conversational implicature.

Jim Burns

http://plato.stanford.edu/entries/implicature/
< H. P. Grice (1913�1988) was the first to systematically study
< cases in which what a speaker means differs from what the sentence
< used by the speaker means. Consider the following dialogue.
<
< 1. Alan: Are you going to Paul's party?
< Barb: I have to work.
<
< If this was a typical exchange, Barb meant that she is not going
< to Paul's party. But the sentence she uttered does not mean that
< she is not going to Paul's party. Hence Barb did not say that she
< is not going, she implied it. Grice introduced the technical terms
< implicate and implicature for the case in which what the speaker
< said is distinct from what the speaker thereby meant (implied, or
< suggested).[1] Thus Barb implicated that she is not going; that
< she is not going was her implicature. Implicating is what Searle
< (1975: 265�6) called an indirect speech act. Barb performed one
< speech act (meaning that she is not going) by performing another
< (saying that she has to work).

From: Newberry on 12 Aug 2010 23:40
On Aug 12, 8:48 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
> >Goedel's sentence is not true because it is vacuous, and we do not
> >regard vacuous sentences as true.
>
> On the contrary! We certainly do.
>
> Look, *EVERY* theorem of pure first order logic is, in a sense,
> vacuously true. For any other first-order theory, a sentence S
> is a theorem if there is a finite conjunction A1 & A2 & ... & An
> of axioms such that
>
> A1 & A2 & ... & An -> S
>
> is vacuously true. In a sense, then, logical deduction amounts to
> showing that certain sentences are vacuously true, given certain
> assumptions.
>
> Your goal of banishing the vacuously true sentences amounts to
> banishing the use of logic.

I do not think this is correct. For example

P v ~P

is a theorem of truth-relevant logic.

P v ~P v Q

is not. I suggest reading sectio 2.2.

>
> --
> Daryl McCullough
> Ithaca, NY

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Prev: Statistics for Business Decision making and Analysis Stine 2011 test bank and solution manual is available for purchase at affordable prices. Contact me at alltestbanks11[at]gmail.com to buy it today.
Next: Meaning, Presuppositions, Truth-relevance, Godel's Theorem andthe Liar Paradox