Torkel Franzen on truth
in [Logic]
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From: abo on 11 Nov 2007 16:29 On Nov 11, 10:23 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > Why is "good reason" subjective? What is "good reason" to you may not be "good reason" to someone else. For instance, a hard-core theist would hold that there is not "good reason" to discuss the existence of God. > And as for what Aunt Bessie has to do > with the question whether we have good reason to doubt, e.g., the > truth of PA, I'm completely stumped! Well, Aunt Bessie has good reason to go shopping every Thursday; other people don't. Yet you ask (I presume with a straight face) why "good reason" is subjective or what Aunt Bessie has to do with it.
From: Peter_Smith on 11 Nov 2007 17:06 On 11 Nov, 21:44, Peter_Smith <ps...(a)cam.ac.uk> wrote: > Bad argument. The fact that A has a good reason to do castle, and B > has a good reason to not to do castle doesn't make either reason > "subjective". A and B's situation in the game may be different, and it > could -- for all that has been said -- be an objective matter that > someone in A's position has a good reason to castle and someone in B's > situation has a good reason not to castle. Mutatis mutandis for > Bessie. Apologies -- that's very careless editing! "to do castle" should read, of course "to castle".
From: Newberry on 11 Nov 2007 18:54 On Nov 11, 11:31 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 11 Nov, 17:46, Newberry <newberr...(a)gmail.com> wrote: > > > > > > > On Nov 11, 12:29 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > On 11 Nov, 03:05, Newberry <newberr...(a)gmail.com> wrote: > > > > > On Nov 9, 4:17 am, aatu.koskensi...(a)xortec.fi wrote:> On 9 Oct, 10:20, Peter_Smith wrote: > > > > > > > Read "pretty good reason" to mean "at least pretty good reason, maybe > > > > > > conclusive reason". As it happens I think there are conclusive reasons > > > > > > to believe PA consistent. > > > > > > Yes, PA is obviously consistent. > > > > > OK, how do we reconcile it with this? > > > > > >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55 > > > > There is no conflict at all between what I said (something TF held > > > too), and that latter quote. To hold that PA is clearly consistent is > > > quite compatible with holding that, with some arbitrarily thrown- > > > together extension of Q, we won't in the general case know whether it > > > is consistent, and hence won't know whether its canonical Gödel > > > sentence is true. > > > So let's confine ourselves to PA for now. We can prove that it is > > consistent, that is we have proven G. How did we manage to do that > > without running into a contradiction? We did not simply add Con(T) as > > another axiom, we proved it. I suppose we proved it in some metatheory > > M. How do we know that M is consistent? > > Depends what M is: it if is a suitable set theory, by getting your > head around the idea of the structure of the iterative hierarchy. Let's assume then that M is a suitable set theory and that it is consistent. How do we prove that M is consistent? In a metatheory M_2? How do we know that M_2 is consistent? Which hierarchy did you have in mind, PA, M, M_2, M_3? If I get my head around this hierarchy M-omega does it mean that I am using a meta-meta-theory N? > (I know this might sound odd coming from someone whose day-job is as a > philosopher, but frankly, I do find "how we know?" questions are as > entirely boring applied to maths as applied to claims about medium- > sized dry goods. Scepticism either way is just uninteresting.)- I have not heard this one yet. I do not even quite understand what you are trying to say here. So let's just stick to the subjectmatter.
From: Aatu Koskensilta on 12 Nov 2007 06:37 On 2007-11-08, in sci.logic, george wrote: > On Nov 8, 1:01 am, Newberry <newberr...(a)gmail.com> wrote: >> I am not sure that I understand what Franzen is saying. > > Don't panic; neither did he. Given that the quoted passage is perfectly clear it seems you're suggesting Franz�n managed to write something eminently comprehensible without himself understanding any of it. This is a curious suggestion -- perhaps you have something more sensible in mind? -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 12 Nov 2007 06:39
On 2007-11-09, in sci.logic, abo wrote: > Conclusive! Obvious! Who could doubt what one learned as a young boy > in Sunday school? I don't know. What does one learn about Peano arithmetic as a young boy in Sunday school? -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |