Re: looking for a predicate hierarchy

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:sl1nhiggnva5$.yggtbi64xjik.dlg@xxxxxxxxxx:

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

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.

Yes, I am aware of that trick wrt to MP. It does not strike me as very
convincing because you lose ability to reason purely in syntactic terms,
you need to appeal to the current truth value and make sure it is
actually true and not T. With weakened MP, you do not have a pure
deductive system that would rely only on a system of axioms and inference
rule(s) any more, but some hybrid. Whether the hybrid is ugly or not,
is a matter of taste. At least, it makes one question the very idea of
a contradiction handling logic.


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.

No, my point is that you cannot say IF...THEN in your language at all,
just because you don't have this IF..THEN which is merely a synonym for
=> (or vice versa). Sure, you can instead appeal to semantic
consequence relation |= but you should admit that there is no purely
syntactic deduction system in your logic any more by which you might have
been able to reason without resorting to truth tables or other semantic
means.


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 (:-))

Sorry about that ;)

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. (:-))


The disjunctive syllogism will hold for the formula (A=>A). Just
substitute (A->A) for the letter and follow the reasoning. Also, it's
much simpler to follow the semantic argument (the formula having an
empty model). When you do that, it'll become immediately obvious that
any formula with an empy model (such formula whose truth table contains
only 0 or _|_) will trivialize the logic we are discussing. Unless, of
course one resorts to hacks like appealing to truth tables to make sure
that for example MP can be used.



#> [ 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.

Actually, it's just the opposite: if I could get T for a formula and
its negation, the explosion would no have happened. With '#>', it's
impossible:

FOr the semantic conseqence relation |=

(A#>A), ^(A#>)|= X

The left side has an empty model because it evaluates either to 0 or _|_,
so any formula X on the right side is satisfied !

One can also claim that the disjunctive syllogism is alive simply because
0, and _|_ do not matter so Lewis' proof applies: (A#>) and ^(A#>) can
be seen as opposites.



-> [ 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, (:-))

As long as the logic can handle contradiction and does not explode, I do
not care how the implication is defined.

For a possible definition, see Avron's articles, e.g. "Value in four
values" and others. He defines 'a implies b' to be 1 for 0 and _|_ and b
otherwise. The definition avoids trivialization and preserves modus
ponens although it has some other drawbacks such as the deduction system
being too weak.

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.


.

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