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. 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".) 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?


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.


Abram Demski

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