Re: looking for a predicate hierarchy

On Tue, 26 Dec 2006 12:41:14 +0100 (CET), V.J. Kumar wrote:

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:jl3iup5zix87.d4njwfc8n3h1.dlg@xxxxxxxxxx:

On Mon, 25 Dec 2006 15:48:59 +0100 (CET), V.J. Kumar wrote:

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:dlxa9cude3z$.1eaoremttucpv$.dlg@xxxxxxxxxx:

Aha, but ~ is not allowed for use [*]. ~A is not a formula.

Yes, it's a reasonable requirement to forbid using '~' directly.
After all, you can just define your '=>' with a truth table. Alas,
it does not save the logic from trivialization. Just substitute
(A=>A) /\ ^(A=> A) for F/\ not F in the Lewis proof, where F was an
arbitrary formula and its negation, not just a variable. '^' is of
course the standard negation -- you cannot forbid that, can you ?

No, I can, reasoning about reasoning were suspicious anyway. So => is
not a connective.

If => is not a connective, then you lose ability to say for example "if it
rains we stay at home; it rains, so we stay at home". So without the
most basic rule of inference, how can you call your system a logic at all
?

There are two issues here, I don't know which one you referred, so I would
try give my opinion on both:

Issue I. Modus ponens

Modus ponens does not hold. But weaker forms of do. Like:

1. (1=A and (A=>B)) => B

i.e. it certainly rains so we stay at home.

2. (A and 1=(A=>B)) => B

i.e. we certainly stay at home when rains, it rains so we stay at home.

3. (A and (A=>B)) => (A and B)

if it rains and when it rains we stay at home then it rains and we stay at
home. (This one makes sense when things gets fuzzy, i.e. A were somewhere
between true and false.)

Issue II. Reasoning about logical consequence

I.e. whether the symbol => should be allowed in formulae. I don't
immediately see why it should. Empirically, the knowledge base is shaped as
a set of rules

IF P(A1,...An) THEN Q(B1,...Bm)

where P and Q are constructed out of terms bound by V, /\, not. You wanted
to consider rules as facts by allowing => in P and Q? Then I find it not
surprising that doing so we might get in trouble if we would attempt to
interpret => as |= and reverse. Just a guess.

BTW, I also saw somewhere a third implication, let's denote it #>. To
compare all three:

=> [ ~xVy ]
T 0 1 _|_
--------------------
T 1 _|_ 1 _|_
0 1 1 1 1
1 T 0 1 _|_
_|_ T T 1 1

Yhis trivializes your logic.

Actually I never cared to prove otherwise. You know, if I were a
mathematician at a university with much free time at my disposal... Anyway,
you pushed me (in so rare holidays (:-)) to make a quick check:

http://en.wikipedia.org/wiki/Principle_of_explosion

For => neither disjunctive syllogism nor reductio ad absurdum nor
contraposition hold. So neither of three proofs of explosion given there
work. Your turn. (:-))

#> [ like => when T<->0 and _|_<->1 for the premise ]
T 0 1 _|_
--------------------
T 1 1 1 1
0 1 _|_ 1 _|_
1 T T 1 1
_|_ T 0 1 _|_

This also trivializes your logic: (A->A) /\ ^(A->A has an empty model.

(_|_#>_|_) /\ not((_|_#>_|_) =_|_, you need T instead, which would give
[undesired] T#>A.

-> [ not xVy, like => when T<->_|_ for the premise ]
T 0 1 _|_
--------------------
T T T 1 1
0 1 1 1 1
1 T 0 1 _|_
_|_ 1 _|_ 1 _|_

For this, both the inference rule and the deduction theorem do not hold,
it's the same as if you did not define any implication.

I thought you were a champion of not x V y, (:-)) OK, I don't like it
either. Some people tried to save it (i.e. actually the rule (A => B) =
((not A) V B)) by tuning V and /\. But I don't see obvious reasons why this
rule should be preserved.

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