Meaning, Presuppositions, Truth-relevance, Godel's Theorem and the Liar Paradox
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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 |