[FOM] Feasible consistency---Truth Transfer Policy
V.Sazonov@csc.liv.ac.uk
V.Sazonov at csc.liv.ac.uk
Sat Oct 28 13:50:43 EDT 2006
Quoting Mirco Mannucci <mmannucc at cs.gmu.edu> Tue, 24 Oct 2006:
Dear Mirco,
> LOGICAL RULES ARE CREDIBILITY VALUES TRANSFER OPERATORS,
> from the premisses to the consequences.
>
>
> Now, I think this transfer should be made explicit by what I like to
> call a TRUTH TRANSFER POLICY
> (TTP). A TTP prescribes which credibility for a specific rule (say
> ^-introduction in
> Natural Deduction) one should assign to the consequents, knowing the
> credibility of
> the antecedents.
I have doubts in such an approach which, as you want, does not restrict
logic, but only introduces a credibility values *via TTP*.
You know, the following is formally (in fact, feasibly) consistent:
M(0), forall x (M(x) => M(x+1)) and not M(100).
(A kind of solution of Heap Paradox.) Here M (meaning intuitively
'medium number') is a formally definable predicate in an appropriately
formalised theory of feasible numbers in a suitably restricted logic.
In fact closure under successor also means that it is impossible to
show which exactly is the borderline x where M(x) & not M(x+1) holds.
(There is some other unusual effect. The details, which are quite
rigorous, are omitted.)
Which should be TTP for Modus Ponens which would guarantee such a
consistency in terms of credibility in the case of classical logic
without restrictions?
As I wrote in a previous posting, credibility values can be introduced,
but in a different way (via complexity of cut elimination or, more
properly speaking, complexity of elimination of abbreviations for
terms) which does not seem to be definable in terms of TTP.
Best wishes,
Vladimir Sazonov
P.S. I do not think that the subject 'paraconsistency' was
appropriate for this discussion.
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