On Sun, Jun 10, 2012 at 10:11 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:
> > On 09 Jun 2012, at 21:53, Abram Demski wrote: > > Bruno, Wei, > > I've been reading the book "saving truth from paradox" on and off, and it > has convinced me of the importance of the "inside view" way of doing > foundations research as opposed to the "outside view". > > At first, I simply understood Field to be referring to the language vs > meta-language distinction. He criticises other researchers for taking the > "outside view" of the system they are describing, meaning that they are > describing the theory from a meta-language which must necessarily exist > outside the theory. > > > Since Gödel we know that for "rich" theory we can embed the metatheory in > the theory. That is what Gödel's provability predicate does, and what > Kleene predicate does for embedding the reasoning on the Turing machines, > and the phi_i, in terms of number relations. > > Arithmetic contains its own interpreter(s). > > > > Hm, well, such is not the case for Truth. According to Tarski, we are forbidden from embedding the metalanguage in the language. This follows from simple, intuitive assumptions, showing that the Liar paradox will spring up in any 'reasonable' theory of truth. Kripke showed how to weaken our notion of truth to the point where the truth predicate could be within the language, but his theory does not allow is to say everything which seems natural to assert about truth, so many more theories have been created after. Every theory seems to suffer from the "revenge problem": in order to define a notion of truth which can fit into the language, a more complicated semantics for that truth predicate must be described. The Strengthened Liar Paradox is then describable in that semantics, if we try to fit it within the same language. So, we are again forced to create a meta-language outside of our language to describe its semantics. (But who describes the semantics of the meta-language?) So, the cases for syntactic meta-theories and semantic meta-theories diverge widely. > > > I thought that his complaint was frivolous; of course you need to describe > a theory of truth via a meta-language. That is part of the structure of the > problem. Yes, it makes the entire theory dubious; but without a concrete > alternative, the only reply to this is "such is life!". So I was confused > when he refused to take other logicians literally (accepting the logic > which they put forward as the logic which they put forward), and instead > claimed that their logic corresponded to the 1-higher theory (the > metalanguage in which they describe their theory). > > At some point, though, the technique "clicked" for me, and I understood > that he was saying something very different. For example, the outside view > of Kripke's theory of truth says that truth is a 'partial' notion, with an > extension and an anti-extension, but also a 'gap' between the two where it > is undefined. (It is a "gap theory".) > > > I am not sure I understand well. > > I hope the previous explanation helped. Field claims to get around the revenge problem by not really providing a meta-language to give a semantics to the truth predicate. He does provide something similar, but it is really a semantics for a restricted domain, to give an intuition for the working of the theory. (He argues that Kripke's semantics must be viewed in this way, too.) > > > On the inside view, however, it does not make this kind of commitment; it > does not claim there is a gap. What the theory says about itself makes no > commitment about the status of the (would-be) gap sentences; they could > well be both true and false. The "outside view" will insist on giving a > semantic status to these, but this is pathological; we cannot develop a > theory of truth in this way (we know that it leads to paradox). > > Instead, we need to take the inside view seriously, and develop theories > from that perspective. > > This generally means taking the truth predicate as basic, and looking for > deduction rules about it which capture what we want, rather than trying to > define its semantics in a set-theoretic or otherwise external way. > > I don't feel that I have an excellent grasp of this technique, though. So, > I'm looking for feedback. Do you have any thoughts or advice here? > > > Better! A theory. Not mine, but the one by the "rich" universal machine > itself (that I call Löbian). Basically a machine is Löbian if it is > universal (in Church Turing sense) and can prove (in a technical weak > sense) that she is universal. Basically it is a universal system + an > induction axiom (or axiom scheme). Examples are Peano Arithmetic, ZF, etc. > Yes, I should finally buy a book on this. :) > The machine's inside view is already unameable by the machine, it is a > "time" creator, (in some semantics), a kind of intuitionist knower. Yes, it > is important to take its view too. > > All löbian machines are able to distinguish two forms of self-reference: a > third person one, and a first person one. And other modalities, notably > those needed to extract physics from arithmetic (as UDA enforced). > > The computationalist hypothesis suggest using computer science and > mathematical logic for dealing with the complex aspects of relative > self-reference, in apparent simple ideal case. I think. > > Bruno > > > > > > Wei, > > Concerning your "undefinability of induction" paradox... > > In this view, the answer is more or less "there can be no truth predicate > which acts like that"... truth is an "open" notion, much like ordinals are > an open notion. > > To some extent, this is an acceptance of the fact that if an alien showed > up claiming to have a box which determined the truth or falsehood of any > statement, we should ascribe this 0 probability; or rather, we won't fully > understand the statement (there is no way to say such a thing; the idea is > incoherent). We can ascribe some probability to much weaker statements > concerning the connection between the output of the box and the truth of > statements, however. In particular, probability can be ascribed to any > partial notion of truth which can be discussed. > > This feels like "accepting the problem statement as a statement of the > solution". The problem is that there is no notion of semantics for which > allows a system to refer to all its own semantic values. The 'solution' is > to say that semantics simply "isn't like that" (there is no 'completion' of > the semantics). If we state these formally, the problem and the solution > are the same statement; it seems like we've made no progress! Again, any > comments on this approach are appreciated. > > Best, > > -- > Abram Demski > http://lo-tho.blogspot.com/ > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to email@example.com. > To unsubscribe from this group, send email to > everything-list+unsubscr...@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to firstname.lastname@example.org. > To unsubscribe from this group, send email to > everything-list+unsubscr...@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. > -- Abram Demski http://lo-tho.blogspot.com/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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