On 11 Jun 2012, at 20:08, Abram Demski wrote:

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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" wayof doing foundations research as opposed to the "outside view".At first, I simply understood Field to be referring to the languagevs meta-language distinction. He criticises other researchers fortaking the "outside view" of the system they are describing,meaning that they are describing the theory from a meta-languagewhich must necessarily exist outside the theory.Since Gödel we know that for "rich" theory we can embed themetatheory in the theory. That is what Gödel's provability predicatedoes, and what Kleene predicate does for embedding the reasoning onthe 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, weare forbidden from embedding the metalanguage in the language. Thisfollows from simple, intuitive assumptions, showing that the Liarparadox will spring up in any 'reasonable' theory of truth. Kripkeshowed how to weaken our notion of truth to the point where thetruth predicate could be within the language, but his theory doesnot allow is to say everything which seems natural to assert abouttruth, so many more theories have been created after. Every theoryseems to suffer from the "revenge problem": in order to define anotion of truth which can fit into the language, a more complicatedsemantics for that truth predicate must be described. TheStrengthened Liar Paradox is then describable in that semantics, ifwe try to fit it within the same language. So, we are again forcedto create a meta-language outside of our language to describe itssemantics. (But who describes the semantics of the meta-language?)So, the cases for syntactic meta-theories and semantic meta-theoriesdiverge widely.I thought that his complaint was frivolous; of course you need todescribe a theory of truth via a meta-language. That is part of thestructure of the problem. Yes, it makes the entire theory dubious;but without a concrete alternative, the only reply to this is "suchis life!". So I was confused when he refused to take otherlogicians literally (accepting the logic which they put forward asthe logic which they put forward), and instead claimed that theirlogic corresponded to the 1-higher theory (the metalanguage inwhich they describe their theory).At some point, though, the technique "clicked" for me, and Iunderstood that he was saying something very different. Forexample, the outside view of Kripke's theory of truth says thattruth 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 aroundthe revenge problem by not really providing a meta-language to givea semantics to the truth predicate. He does provide somethingsimilar, but it is really a semantics for a restricted domain, togive an intuition for the working of the theory. (He argues thatKripke's semantics must be viewed in this way, too.)

`Field wrote a book "science without number" which I found not really`

`convincing, except for Newtonian gravitation, but not physical`

`sciences or the theological sciences in general (rather trivially once`

`you assume the comp hypothesis). You might elaborate on his argument.`

`I am problem driven, and I think comp entails big change in`

`fundamental science that we have to take into account. I am not sure`

`it makes sense to interpret Kripke semantics literally, as opposed to`

`arithmetic and most of computer science. The right ([]*) modal`

`hypostases have no Kripke semantics, for example.`

`For arithmetic, I use Tarski theory of truth. So "ExP(x)" is true if`

`it exists a natural number n so that it is the case that P(n). That is`

`enough to describe the behavior of machines (but not their mind!).`

`Then I use, implicitly thanks to Solovay theorem, what simple`

`arithmetical machine (relations) can prove and not prove about them,`

`and how they can interpret their relation with some other universal`

`numbers. I let the machine develop her own semantics, and facilitate`

`my task by studying only the correct one (by definition).`

`Some Kripke semantics occur more or less naturally, but not all`

`arithmetical modalities have a Kripke semantics. Note that for G* you`

`can develop a semantics is term of sequences of Kripke models. Then`

`"p" is satisfied by such a sequence if p is eventually satisfy in the`

`models in that models sequence.`

Bruno

On the inside view, however, it does not make this kind ofcommitment; it does not claim there is a gap. What the theory saysabout itself makes no commitment about the status of the (would-be)gap sentences; they could well be both true and false. The "outsideview" will insist on giving a semantic status to these, but this ispathological; we cannot develop a theory of truth in this way (weknow that it leads to paradox).Instead, we need to take the inside view seriously, and developtheories from that perspective.This generally means taking the truth predicate as basic, andlooking for deduction rules about it which capture what we want,rather than trying to define its semantics in a set-theoretic orotherwise 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 oradvice here?Better! A theory. Not mine, but the one by the "rich" universalmachine itself (that I call Löbian). Basically a machine is Löbianif it is universal (in Church Turing sense) and can prove (in atechnical weak sense) that she is universal. Basically it is auniversal system + an induction axiom (or axiom scheme). Examplesare 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 isa "time" creator, (in some semantics), a kind of intuitionistknower. 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 othermodalities, notably those needed to extract physics from arithmetic(as UDA enforced).The computationalist hypothesis suggest using computer science andmathematical logic for dealing with the complex aspects of relativeself-reference, in apparent simple ideal case. I think.BrunoWei, Concerning your "undefinability of induction" paradox...In this view, the answer is more or less "there can be no truthpredicate which acts like that"... truth is an "open" notion, muchlike ordinals are an open notion.To some extent, this is an acceptance of the fact that if an alienshowed up claiming to have a box which determined the truth orfalsehood of any statement, we should ascribe this 0 probability;or rather, we won't fully understand the statement (there is no wayto say such a thing; the idea is incoherent). We can ascribe someprobability to much weaker statements concerning the connectionbetween the output of the box and the truth of statements, however.In particular, probability can be ascribed to any partial notion oftruth which can be discussed.This feels like "accepting the problem statement as a statement ofthe solution". The problem is that there is no notion of semanticsfor which allows a system to refer to all its own semantic values.The 'solution' is to say that semantics simply "isn't likethat" (there is no 'completion' of the semantics). If we statethese formally, the problem and the solution are the samestatement; it seems like we've made no progress! Again, anycomments on this approach are appreciated.Best, -- Abram Demski http://lo-tho.blogspot.com/ --You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-l...@googlegroups.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 GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.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.-- Abram Demski http://lo-tho.blogspot.com/ --You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.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.

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