[FOM] Proof "from the book"

[Jeffrey Ketland]

Aatu Koskensilta:

>We need to know very little about truth in order to
>convince ourselves
>that if we accept a theory T as true, then we should accept reflection
>principles for it.

If we accept "S is true", then yes, since the reflection scheme RFN(S) is
derivable from "S is true", using the Tarskian inductive definition. But
suppose we merely accept the theory S? (I.e., the axioms of S enter our
belief box.) How do we pass from acceptance of S to acceptance of the global
reflection principle "S is true"? I would argue that we do need to know
something about the structure of the semantical theory for the language of
S. The construction S |-> Tr(S) formalizes this, and when S satisfies
certain constraints, then Tr(S) proves "All theorems of S are true".

We appear to have to move to a higher level of abstraction (i.e., introduce
semantical notions not definable in the language of S) in order to "extract"
reflective consequences of a theory S. We must think not only of the objects
of S, but also of S's semantical relation to these objects. In other words,
we reflect on S itself, as a semantical representation of its intended
domain.

>One could argue that Tr(S), while not interpretable in S, is implicit in
>the acceptance
>of S.

But the Tarskian inductive truth definition really does lie outside what is
expressible in S, by Tarski's Theorem. Better to say that acceptance of the
reflection scheme RFN(S) is implicit in the acceptance of S, *because* we
are already committed, in the background as it were, to the Tarskian
inductive truth definition as the correct way of understanding sentences of
S as representations of its domain.

>>The truth
>>axioms of Tr(S) are not implicit anywhere in S. Rather, the arithmetic
>>conseqences of Tr(S) are implicit in S.
>>
>Well, implicit in the acceptance of S, I would say.

This reminds me of Lewis Carroll's famous dialogue, "What the Tortoise said
to Achilles", concerning the justification of MP. Is the proposition B
"implicit" in the assumptions A and A->B? Or, rather, is it *acceptance* of
B that implicit in acceptance of A and acceptance of A->B?
Or is it perhaps that our truth theory says that,

      If A is true and A->B is true, then B is true

For individual sentences, reflection (i.e., semantic ascent---the passage
from A to "A is true") can be shown to be conservative. For whole theories,
reflection goes non-conservative. In the case of acceptance of theories, we
have at the semantical level,

      If all theorems of S are true, then all instances of RFN(S) are true.

Despite the crucial fact that, by Goedel's results, RFN(S) is not provable
in S (if S is consistent).
My point is that the reflective consequences of a theory can only be
justified by appealing to a higher-order notion of truth, which lies outside
what is expressible in the theory itself.

Another curious aspect of this non-conservation of reflection, as applied to
theories, is that there are consistent theories whose reflective closure is
inconsistent. For example, PA + {~G} is consistent, but Tr(PA + {~G}) is
inconsistent. For Tr(PA + {~G}) implies "All theorems of PA + {~G} are
true", and thus "All theorems of PA are true", and thus Con(PA) and thus G.
But Tr(PA + {~G}) implies ~G.
We might call such theories "reflectively inconsistent".

--- Jeff
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jeffrey Ketland
School of Philosophy, Psychology and Language Sciences
University of Edinburgh, David Hume Tower
George Square, Edinburgh  EH8 9JX, United Kingdom
jeffrey.ketland at ed.ac.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~