'When Are Relations Neither True Nor False?
in [Math]
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From: MoeBlee on 26 Feb 2010 15:45 On Feb 25, 8:55 pm, Newberry <newberr...(a)gmail.com> wrote: > On Feb 25, 10:49 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: > > Newberry says... > > > >On Feb 25, 8:25=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > > >wrote: > > >> We prove an implication: If PA is consistent, then G. We don't > > >> have a proof (within PA) that PA is consistent, but we believe > > >> it's true. So G follows from the assumption that PA is consistent. > > > >And we believe that PA is consistent because all its axioms are > > >manifestly true? Is > > > > "All unicorns have two horns" > > > >manifestly true? > > > Which axiom of PA is analogous to "All unicorns have two horns"? > > No axiom. But PA uses the same logic. You think that if the axioms are > manifestly true then what the logic itself produces does not have to > be? So your original challenge was incorrectly stated. Indeed, it's not the axioms you object to but the logical system, which I and others have been pointing out. > And how about > > (x)((x # x) -> ( 1 = 2)) ? > > Is it manifestly true? The point has been made that we recognize that you object to the ordinary truth table for '->' . You don't advance the conversation by asking over and over about it. MoeBlee
From: MoeBlee on 26 Feb 2010 15:46 On Feb 25, 8:58 pm, Newberry <newberr...(a)gmail.com> wrote: > On Feb 25, 9:45 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Feb 25, 11:22 am, Newberry <newberr...(a)gmail.com> wrote: > > > > And we believe that PA is consistent because all its axioms are > > > manifestly true? > > > They're clearly true to a lot of people. They seem clearly true to me. > > Which axiom of PA doesn't strike you as true? > > > Meanwhile, in a theory such as Z, we prove that the axioms of PA are > > true. Though, I don't argue that that, in itself, has much > > epistemological force. > > > > "All unicorns have two horns" > > > > manifestly true? > > > Given ordinary predicate logic, if the sentence is taken as > > > Ax((x is a horse & x has a horn) -> x has two horns) > > > then it is clearly true in any model in which there are no horses with > > horns. > > "true in any model in which there are no horses with horns" is the > same as "manifestly true"? Okay, then my point is made in even greater force. MoeBlee
From: Newberry on 26 Feb 2010 23:53 On Feb 26, 12:46 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Feb 25, 8:58 pm, Newberry <newberr...(a)gmail.com> wrote: > > > > > > > On Feb 25, 9:45 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Feb 25, 11:22 am, Newberry <newberr...(a)gmail.com> wrote: > > > > > And we believe that PA is consistent because all its axioms are > > > > manifestly true? > > > > They're clearly true to a lot of people. They seem clearly true to me.. > > > Which axiom of PA doesn't strike you as true? > > > > Meanwhile, in a theory such as Z, we prove that the axioms of PA are > > > true. Though, I don't argue that that, in itself, has much > > > epistemological force. > > > > > "All unicorns have two horns" > > > > > manifestly true? > > > > Given ordinary predicate logic, if the sentence is taken as > > > > Ax((x is a horse & x has a horn) -> x has two horns) > > > > then it is clearly true in any model in which there are no horses with > > > horns. > > > "true in any model in which there are no horses with horns" is the > > same as "manifestly true"? > > Okay, then my point is made in even greater force. > If there are no unicorns in your model, what does the model tell you about unicorns?
From: Newberry on 27 Feb 2010 00:21 On Feb 26, 7:38 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Feb 26, 3:51 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> In any case, the proof of (*) is relatively simple and uses no > >> second-order stuff. The sentence (*) is plainly meaningful and I > >> daresay that *you know what it means*. > > > I know exactly what you want to say: ~T[(Ex)((x is a counterexample to > > GC) & (x < 27)). You are just sloppy in expressing it. > > No, it means what I wrote just below. There are no counterexamples to > GC that are less than 27. That's a plain English sentence. Let's > ignore formalisms for now. > > I just want to double check the following. Are you honestly telling > me that you're not sure whether the sentence > > There is no counterexample to GC that is less than 27. (*) > > is meaningful or not? That sentence means the same thing as (is > equivalent to) > > Any counterexample to GC is larger than or equal to 27. > > As far as you can tell, these two English sentences may be completely > meaningless, but the sentence > > It is not true that there is a counterexample to GC less than 27. (**) > > is meaningful. Perhaps as well, though you have not said so, the > sentence > > It is not false that there is no counterexample to GC less than 27. > > is meaningful. > > Moreover, you think that when I utter the possibly meaningless (*), I > really *mean* to say (**), but I'm merely sloppy. If I thought hard > about it, I'd realize that (**) is what I meant -- since, although I > didn't notice it, I can't be sure that (*) means anything at all. > > If this is honestly your opinion, I guess I have nothing to say. I > know what all these sentences mean. I know what they mean regardless > of whether GC is true or not. It really is that simple: if I know > what something means, then it's not meaningless. If I also know that > it's true, then it's silly to claim that it's neither true nor false > (but its negation is not true). People noticed long time ago (beginning with Strawson himself) that some sentences with empty subjects feel false and some of them just feel ..., well, empty. There is a lot of literature trying to explain the different intuitions. I recommed this highly interesting paper http://semanticsarchive.net/Archive/zAzNjllO/von.fintel.kof.pdf I do not know if von Fintel's theory is directly applicable tp your example. I am pointing out that some sentences with empty subjects feel false and some do not, and the different intuitions are difficult to explain. My view is that the negation of the former should be expressed as ~T(A) and the negation of the latter as ~A. You may feel that your example is clearly true because the natural language does not distinguish the two cases very well. And it may be the reason why the foundations of mathematics are such a mess. And since one variant is neither true nor false anyway then why bother using ~T(). We can simply abbreviate it by using negation. For what I know ordinary mathematics may well go on by using "there are no counterexamples to GC less than 27." But if you want to do any work in the foundations of mathematics you need higher precision. > (The following .sig was randomly chosen, but perfectly suitable -- > except I don't regard Newberry as a philosopher.) > -- > "How can people [philosophers] talk like that? Acting as if they're > /glad/ they don't know things! Finding out more and more things they > don't know! It's like children proudly coming to show you a full > potty!" -- Terry Pratchett, /Small Gods/
From: Newberry on 27 Feb 2010 00:27
On Feb 26, 9:45 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Newberry wrote: > > > On Feb 25, 8:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > > wrote: > > > Newberry says... > > > > >[...] Frege also believed that the > > > >axioms in his system were manifestly true. > > > > Well, he was mistaken. The belief that a system is consistent > > > can be mistaken. > > > Which of his axioms was not manifestly true? > > You state[1] that a function, too, can act as the indeterminate > element. This I formerly believed, but now this view seems > doubtful to me because of the following contradiction. Let w > be the predicate: to be a predicate that cannot be predicated > of itself. Can w be predicated of itself? From each answer > the opposite follows. > > [1] I'm quoting Russell from van Heijenoort, the "you" is Frege. At the > point "[1]" Russell refers to a passage in Begriffsschrift which is also > in van Heijenoort in English translation. Yoy just made my point. He formerly believed it until he found a contradiction. |