Re: looking for a predicate hierarchy

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:o68o7slu8pj7$.1cpgy64c622lt$.dlg@xxxxxxxxxx:


Formulas with your implication potentially cannot handle
contradiction, that's what. The whole point of having 'T' as a
designated truth value is to allow models for expressions like (F /\
^F).

You forgot that everything is in the inference rules.

Not necessarily. The consequence relation { |- ) can be defined either
semantically (e.g. with truth tables for the connectives) or syntactically
(axioms and inference rules).

For a logic to be able to handle contradictions like (F /\ ~ F) for
example, the consequence relation F,~F|- P should fail for some P (the
non-explosion principle) otherwise the logic becomes trivial by the same
semantic argument as for the classical logic. With your '~' connective it's
just the case (trivialization) whilst it is not with the '^' connective
defined as 0->1, 1->0, _|_->_|_, T->T.

Now, are you claiming that you can produce a system of axioms and
inference rules that would show that the 4-valued logic with your "~" is
not trivial ?

Yes, a
contradiction cannot be constructed from 0 and 1 using /\ (AND), or
any conventional logic operators. This was a *desired* property, that
1 /\ 0 = 0, 1 V 0 = 1, after all. That alone does not make it trivial,
because the contradiction and uncertainty can still be produced. For
example by operations like consensus(+) and gullibility(*):

1 + 0 = _|_ 1 * 0 = T

??? What has it got to do with the price of fish ? What make the 4-valued
logic trivial is your "~" because as was said before, (F /\ ~F) has an
empty model (there is no valuation v(F) such that v(F /\ ~F) would be in
the designated truth set {t, T}) so any arbitrary P vacuously is a semantic
consequence of {F,~F}.



for further information see:

http://www.dmitry-kazakov.de/ada/fuzzy.htm#fuzzy_proposition


We'll tackle the fuzzy stuff after we've done with the simpler four-valued
case first ;)

.