Meaning, Presuppositions, Truth-relevance, Godel's Theorem and the Liar Paradox
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From: Newberry on 10 Aug 2010 09:27 On Aug 10, 3:55 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >But it does not. Just because > > >~(Ex)(Px#G) > > >is true it does not follow that > > >~(Ex)(Ey)(Pxy & Qy) (G) > > >is true. > > In normal (nonstupid) semantics, it does follow. You can certainly > make up whatever semantics you like for first order logic, but what > is the point here? > > If there is only one number, #G, for which Qy holds, At least you are beginning to get the point. > then > Ey (Pxy & Qy) means the same thing as Px#G. Why in the *world* > would you want that not to be the case? Two reasons a) ~(Ex)(Ey)(Pxy & Qy) is a hierarchy of vacuous sentences. Some of us think those are really not true. b) We get a semantically consistent system. You seem to be going > out of your way to block the usefulness of first-order logic > for reasoning. How so? > -- > Daryl McCullough > Ithaca, NY
From: MoeBlee on 10 Aug 2010 11:56 On Aug 8, 8:01 pm, Don Stockbauer <donstockba...(a)hotmail.com> wrote: > Does any of this put food on the table? It gives me something interesting to read while I eat the food on the table. MoeBlee
From: Newberry on 11 Aug 2010 00:23 On Aug 10, 8:56 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Aug 8, 8:01 pm, Don Stockbauer <donstockba...(a)hotmail.com> wrote: > > > Does any of this put food on the table? > > It gives me something interesting to read while I eat the food on the > table. > > MoeBlee You can also read it when disposing of the processed food.
From: Newberry on 11 Aug 2010 00:39 On Aug 10, 8:20 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Aug 10, 3:55=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> If there is only one number, #G, for which Qy holds, > > >At least you are beginning to get the point. > > >> then > >> Ey (Pxy & Qy) means the same thing as Px#G. Why in the *world* > >> would you want that not to be the case? > > >Two reasons > >a) ~(Ex)(Ey)(Pxy & Qy) is a hierarchy of vacuous sentences. > > In what sense is it vacuous? Again, I have to simplify for the sake of brevity.Let us pick y = #G = m. We obtain ~(Ex)(Pxm & Qm) According to Strawson's logic of presupposition Pxm must be non-empty if the above is to be T v F. We happen to know that ~(Ex)Pxm 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". > > Maybe you mean that *if* it is true, then it is > vacuously true. That's stretching the meaning of > "vacuously true" beyond the breaking point. > > >Some of us think those are really not true. > > (1) It's not vacuous, and (2) vacuously true sentences > are *true*. I have told you many times that my paper was not about classical logic. In Strawson's logic of presuppositions sentences with empty subject class are considered ~(T v F). > > >b) We get a semantically consistent system. > >> You seem to be going out of your way to block the usefulness > >> of first-order logic for reasoning. > > >How so? > > Let me go through an example. Suppose I don't know whether > Goldbach's conjecture is true, or not. But I can prove the > following two statements: > > (1) "If x is a counterexample to GC, then x is a multiple of 3" > (2) "If x is a counterexample to GC, then x/2 is a prime number" > > In ordinary logic, (1) and (2) imply the conclusion: > "There are no counterexamples to GC". > > Assuming GC is true, then (1) and (2) are vacuous. > But those two vacuous statements imply the nonvacuous > statement, Goldbach's conjecture. If you want to say that > vacuous statements are not true, then presumably you are > blocked from proving (1) or (2), because neither of those > "lemmas" are true, in your semantics. > > Worse, since we don't know whether Goldbach's conjecture is > true, or not, we don't know whether (1) and (2) are vacuous > or not. > > So your semantics gets in the way of doing ordinary mathematical > proofs. You are making the assumption that x exists, so such proof is not necessarily ruled out. Anyway, as long as a sound derivation system exists it will prove all the true formulae except those that classical logic considers "vacuously true". And BTW, the old Greeks did not have the concept of empty subject class. Aristotle always assumes that the subject class is non-empty. > > -- > Daryl McCullough > Ithaca, NY
From: Newberry on 11 Aug 2010 09:14
On Aug 11, 4:50 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Aug 10, 8:20=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> >Two reasons > >> >a) ~(Ex)(Ey)(Pxy & Qy) is a hierarchy of vacuous sentences. > > >> In what sense is it vacuous? > > >Again, I have to simplify for the sake of brevity.Let us pick y = #G = m. > >We obtain > > >~(Ex)(Pxm & Qm) > > >According to Strawson's logic of presupposition Pxm must be non-empty > >if the above is to be T v F. > > I think that either you are misinterpreting Strawson, or that > Strawson's theory is complete nonsense. The statement > "~(Ex) (Pxm & Qm)" does not have "(Ex) Pxm" as a > presupposition. Why in the world would anyone think that? > > I understand why someone might consider > > ~(Ex)(Pxm & Qm) > > a little strangely written. Since Qm does not depend on x, > there is no reason to leave it inside the quantifier. You > could rewrite the formula in the form: > > ~(Qm & (Ex)Pxm) > > which is logically equivalent. We can certainly introduce > some kind of "normal form" for first-order logic, in which > we simplify formulas as much as possible, and rewrite them > so that formulas bound by quantifiers must all mention the > variable quantified. That's a tidying up rule, which has no > impact on the *meaning* of the formula. Or at least, it > *shouldn't* have any impact. ~(Ex)[Pxm & ((x = x &) Qm)] > > >We happen to know that > > >~(Ex)Pxm > > No, you don't happen to know that. What you know is that > *if* the theory is consistent, then ~(Ex) Pxm. We have no > way, in general, to know whether a theory is consistent > or not. So we have no way of knowing whether the sentence > > ~(Ex)(Pxm & Qm) > > is "vacuous" in your sense. A proof is necessary to reach > that conclusion. But if you need to *prove* that a sentence > is vacuously true, then it clearly is *not* vacuously true, > in any meaningful notion of the word "vacuous". If it is > difficult (but not impossible) to establish the truth of > something, then it means to me that it is not vacuous. A lot of people are absolutely convinced that e.g. PA is consistent. But anyway the point is that IF ~(Ex)Pxm then ~(Ex)[Pxm & ((x = x &) Qm)] is vacuous. I do not know what you are trying to argue here. 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)] > > 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. > > -- > Daryl McCullough > Ithaca, NY |