Meaning, Presuppositions, Truth-relevance, G�del's Theorem and the Liar Paradox
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From: Daryl McCullough on 11 Aug 2010 07:50 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. >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. >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". 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. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 11 Aug 2010 14:03 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. >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. You haven't given a coherent reason to care about the distinction between vacuous and nonvacuous statements. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 12 Aug 2010 08:46 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? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 12 Aug 2010 11:25 James Burns says... >Daryl McCullough wrote: >> 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/ > > 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). Thanks for the link. Yeah, I think that's really what's going on with the examples that Newberry brings up about "presuppositions" and "relevance". It is more fruitful to think of it in terms of conversational implicature than in terms of a new semantics for first-order logic. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 12 Aug 2010 11:48
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. -- Daryl McCullough Ithaca, NY |