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