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).
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.
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.
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/
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