Re: looking for a predicate hierarchy

On Sat, 23 Dec 2006 19:38:48 +0100 (CET), V.J. Kumar wrote:

"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.

Necessarily. The properties of a logical system are determined by the
inference rules.

Your argumentation seems to be based on an assumption that

(A /\ not A) => B

were somehow equivalent to

{ A, not A } |= B

but that's wrong.

Firstly the former by no means implies the latter.

Secondly the former is not trivially true:

A (A /\ not A) => B
------------------------------
0 1
1 1
_|_ 1 if B=1 or B=_|_, otherwise T
T 1 if B=1 or B=T, otherwise _|_

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,

This is not considered as a contradiction. Contradiction out of certain
evidences {0,1} is not constructed with either V or /\.

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,

Note, that ~ is not used in place of negation. This is again a jumpy
assumption. For negation "not" is still used. The idea behind this x=>y was
to define it in terms a set inclusion over subsets {0,1}^2. The background
is to make it compatible with a possibility measure, for further extension
into intuitionistic fuzzy logic and interpretation of x=>y as a conditional
y|x.

Merry Christmas,

--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de
.

Relevant Pages

  • Re: looking for a predicate hierarchy
    ... semantically (e.g. with truth tables for the connectives) or syntactically ... just the case (trivialization) whilst it is not with the '^' connective ... because the contradiction and uncertainty can still be produced. ... the designated truth set ) so any arbitrary P vacuously is a semantic ...
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  • Re: If (P & ~P) -> Q is not derivable then Goedels formula is not derivable
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  • Re: looking for a predicate hierarchy
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