Meaning, Presuppositions, Truth-relevance, Godel's Theorem and the Liar Paradox
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From: Daryl McCullough on 9 Aug 2010 12:53 Newberry says... >http://www.scribd.com/doc/35519023/Meaning-Presuppositions-Truth-relevance-Godel-s-Theorem-and-the-Liar-Paradox I took a look at the paper. I understand the idea about presuppositions. The example from the paper: "All John's children are asleep". Rather than rendering it as Ax jc(x) -> as(x) (where jc(x) means x is one of John's children, and as(x) means x is asleep), Strawson wants to render it as a pair <(Ex jc(x)), (Ax jc(x) -> as(x))> where the first formula is the presupposition--it's what must be true in order to even evaluate the truth of the claim. If the presupposition is false, then the sentence is neither true nor false. That's perfectly sensible as a way of translating natural language into logic. Natural language claims don't neatly translate into first order logic (particularly because of the use of referring expressions that may be non-denoting). But this business of presuppositions has *nothing* to do with the Godel sentence. The Godel sentence is *already* first-order logic. It has no presuppositions (other than possibly the existence of natural numbers). The complaint about the translation of "All John's children are asleep" as "Ax jc(x) -> as(x)" is that it throws away information, namely the information about presuppositions. You *can't* recover the presuppositions from the first-order logic formula alone, because statements with different presuppositions give rise to the same first-order logic formula. Instead of "All John's children are asleep", one could instead say "There is no person who is both awake and a child of John", which has no presuppositions. But it translates to a first-order logic statement that is classically equivalent. First-order logic doesn't *have* presuppositions, which is the reason why it is inadequate for representing natural language statements that *do* have presuppositions. What you're doing is something quite weird, which is to insert presuppositions into first order logic, where there are no presuppositions and no *need* for presuppositions. I guess there are two kinds of philosophy: the kind that takes murky concepts and attempts to make them clear, and the kind that takes clear concepts and attempts to make them murky. Identifying the presuppositions in natural language is an example of the first, inserting presuppositions into first-order arithmetic statements is an example of the second. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 9 Aug 2010 15:47 Daryl McCullough says... > >Newberry says... > >>http://www.scribd.com/doc/35519023/Meaning-Presuppositions-Truth-relevance-Godel-s-Theorem-and-the-Liar-Paradox > >I took a look at the paper. I understand the idea about presuppositions. >The example from the paper: "All John's children are asleep". >Rather than rendering it as > >Ax jc(x) -> as(x) > >(where jc(x) means x is one of John's children, and as(x) means >x is asleep), Strawson wants to render it as a pair > ><(Ex jc(x)), (Ax jc(x) -> as(x))> > >where the first formula is the presupposition--it's what must be true >in order to even evaluate the truth of the claim. If the presupposition >is false, then the sentence is neither true nor false. I actually think that this business about presuppositions is of dubious use even in such a case. You would certainly feel misled if someone told you "All John's children are asleep" and it was really the case that John *had* no children. But the reasoning behind why you would feel misled has to do, not with the semantics of the statement, but rather with the assumptions about why someone would tell you such a thing. People don't typically go around saying anything that happens to be true. They usually say something because it is relevant to something else people are interested in. If John doesn't have any children, then someone is not likely to make claims about their being asleep (unless they are being intentionally misleading). In my opinion, it is clearer to try to separate these two aspects of communication: the semantics (what is the statement saying, and what would it mean for it to be true or false) and the pragmatics (what is the speaker's intentions behind making the statement). -- Daryl McCullough Ithaca, NY
From: Newberry on 9 Aug 2010 23:42 On Aug 9, 7:59 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > In article <3f8cee92-6dbb-4f44-beb7-0878d02b9...(a)p11g2000prf.googlegroups..com>, > Newberry says... > > > > >On Aug 9, 6:33=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> Newberry says... > > >> >The question was what is Goedel sentence. The formula I exhibited is > >> >Goedel's formula in many kinds of logic including PA. > > >> If it is sufficiently similar to the Godel formula for PA, then it > >> is nonsensical to say that it is neither true nor false. > > >Would you care to define "sufficiently similar" and show how your > >conclusion follows? > > The main ideas behind Godel's proof is > 1. Invent a coding for formulas so that every formula is associated > with a natural number (or an element of whatever the domain of the > theory is about) > 2. Define a formula Pr(x) such that Pr(x) holds of a natural number > x if and only if x is the code of a provable formula of whatever theory > we are talking about. > 3. Construct a sentence G such that G <-> ~Pr(#G) is a theorem, > where #G means the code for G. > > 1-3 is what I consider the essential features of what it means > for G to be a "Godel sentence". There a few details that can be > tweaked---for instance, 3 presupposes that there are constant > terms (e.g. numerals) for each element of the domain. That's > not essential; instead, we can have a formula Q(x) such > that > > G <-> Ax (Q(x) -> ~Pr(x)) > > and such that Q(x) holds if and only if x is the code for G. > > Anyway, in terms of 1-3, it is nonsensical to say that G is > neither true nor false. G is a specific formula. If that formula > is provable, then Pr(#G) holds (by definition, Pr(#G) holds > if G is provable). But G is the negation of that formula. So > G is the negation of a true sentence, and so is a false sentence. > > So if you say that G is not false, then it follows that G is > not provable, and from that it follows that ~Pr(#G) is true, > and from that, it follows that G is true. But it does not. Just because ~(Ex)(Px#G) is true it does not follow that ~(Ex)(Ey)(Pxy & Qy) (G) is true.
From: Newberry on 9 Aug 2010 23:54 On Aug 9, 9:53 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >http://www.scribd.com/doc/35519023/Meaning-Presuppositions-Truth-rele... > > I took a look at the paper. I understand the idea about presuppositions. > The example from the paper: "All John's children are asleep". > Rather than rendering it as > > Ax jc(x) -> as(x) > > (where jc(x) means x is one of John's children, and as(x) means > x is asleep), Strawson wants to render it as a pair > > <(Ex jc(x)), (Ax jc(x) -> as(x))> > > where the first formula is the presupposition--it's what must be true > in order to even evaluate the truth of the claim. If the presupposition > is false, then the sentence is neither true nor false. > > That's perfectly sensible as a way of translating natural language > into logic. Natural language claims don't neatly translate into > first order logic (particularly because of the use of referring > expressions that may be non-denoting). > > But this business of presuppositions has *nothing* to do with > the Godel sentence. The Godel sentence is *already* first-order > logic. It has no presuppositions (other than possibly the existence > of natural numbers). > > The complaint about the translation of "All John's children are asleep" > as "Ax jc(x) -> as(x)" is that it throws away information, namely the > information about presuppositions. You *can't* recover the presuppositions > from the first-order logic formula alone, because statements with > different presuppositions give rise to the same first-order logic > formula. Instead of "All John's children are asleep", one could > instead say "There is no person who is both awake and a child of > John", which has no presuppositions. But it translates to a > first-order logic statement that is classically equivalent. > > First-order logic doesn't *have* presuppositions, which is > the reason why it is inadequate for representing natural language > statements that *do* have presuppositions. What you're doing is > something quite weird, which is to insert presuppositions into > first order logic, where there are no presuppositions and no > *need* for presuppositions. > > I guess there are two kinds of philosophy: the kind that takes > murky concepts and attempts to make them clear, and the kind that > takes clear concepts and attempts to make them murky. Identifying > the presuppositions in natural language is an example of the first, > inserting presuppositions into first-order arithmetic statements > is an example of the second. Everybody knows that Goedel's sentence is in FOL and that FOL has no presuppositions. If you were under the impression that I was saying otherwise then I need to clarify that I was not. What I am saying is that a) you can do arithmetic in the logic of presuppositions b) you get a more well behaved system
From: Jesse F. Hughes on 9 Aug 2010 23:51
stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > I took a look at the paper. I understand the idea about presuppositions. > The example from the paper: "All John's children are asleep". > Rather than rendering it as > > Ax jc(x) -> as(x) > > (where jc(x) means x is one of John's children, and as(x) means > x is asleep), Strawson wants to render it as a pair > > <(Ex jc(x)), (Ax jc(x) -> as(x))> > > where the first formula is the presupposition--it's what must be true > in order to even evaluate the truth of the claim. If the presupposition > is false, then the sentence is neither true nor false. > > That's perfectly sensible as a way of translating natural language > into logic. Natural language claims don't neatly translate into > first order logic (particularly because of the use of referring > expressions that may be non-denoting). > > But this business of presuppositions has *nothing* to do with > the Godel sentence. The Godel sentence is *already* first-order > logic. It has no presuppositions (other than possibly the existence > of natural numbers). > > The complaint about the translation of "All John's children are asleep" > as "Ax jc(x) -> as(x)" is that it throws away information, namely the > information about presuppositions. You *can't* recover the presuppositions > from the first-order logic formula alone, because statements with > different presuppositions give rise to the same first-order logic > formula. Instead of "All John's children are asleep", one could > instead say "There is no person who is both awake and a child of > John", which has no presuppositions. But it translates to a > first-order logic statement that is classically equivalent. > > First-order logic doesn't *have* presuppositions, which is > the reason why it is inadequate for representing natural language > statements that *do* have presuppositions. What you're doing is > something quite weird, which is to insert presuppositions into > first order logic, where there are no presuppositions and no > *need* for presuppositions. I must say that this is by far the most insightful criticism of Newberry's discussion of Goedel's theorem thus far. -- Jesse F. Hughes "I am the next legend--living, breathing and solving mega problems in the here and now." -- James S. Harris |