Re: looking for a predicate hierarchy

On Tue, 26 Dec 2006 19:38:29 +0100 (CET), V.J. Kumar wrote:

"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.
[...]
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.

Right, but we have to loose that anyway. To me four-valued logic is just a
necessary step to a fuzzy one. There you will never have a chance to save
it. When it rains to 0.9 degree and does not to 0.6, then what?

I think that this step was already made by going four-valued. That ~ and =>
cannot be obtained is a necessary consequence of. Intuitionistically, one
cannot say anything about whether it does not rain, when it rains. It could
be 1 or _|_. And "it really rains" is a quark in both, which cannot be
extracted. This splits modus ponens into a pair:

IF A THEN B (what we do when it rains)
IF not A THEN B (what we do when it does not)

Because B alone were again unusable in further inferences we need another
two:

IF A THEN not B (what we don't do when it rains)
IF not A THEN not B (what we don't do when it does not rain)

A set-theoretic inclusion interpretation of the above were as follows. Let
A be a subset of some universe of events corresponding to rain. Let B be a
subset of the universe corresponding us sitting at home. Then the relation
of A to B is described by

B in A (B intersects A, a set-measure s of, i.e. a conditional s(B|A))
B in ~A (B intersects complement of A, conditional s(B|~A))
B outside A (complement of B intersects A, conditional s(~B|A)
B outside ~A (complements intersect, conditional s(~B|~A))

Individual events and sets of, like A and B cannot be directly observed. A
reasoning happens under some observation condition E which is again a
subset of the universe. The goal of inference is to obtain {s(B|E),
s(~B|E)} from available observations of A, i.e. from {s(A|E), s(~A|E),
s(A|~E), s(~A|~E)}. The logical symbols become the interpretation:

A | { s(A|E), s(~A|E)}
----+-------------------------
T | {0, 0}
0 | {0, 1}
1 | {1, 0}
_|_| {1, 1}

It is not all the information about A, we could have. s(A|~E) and s(~A|~E)
are missing. This is why we cannot introduce a formal implication which
would be based on just A (observed when E). We lack a complement
observation of A when ~E.

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.

But

((A or B) and not A) => B

is not universally true (for example in A=T, B=0). And we cannot use -> and
=> interchangeable. One should stick to one of them.

Also, it's
much simpler to follow the semantic argument (the formula having an
empty model).

As a programmer I prefer direct tests! Papers on paraconsistent logic make
me ill... (:-))

#> [ 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.

But it is T for which T#>A is universally true. The first row of the table.

-> [ 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.

Is this what you mean?

a> | T 0 1 _|_
---+-----------------
T | 0 0 _|__|_
0 | 1 1 1 1
1 | 0 0 _|__|_
_|_| 0 0 _|__|_

So 1=>1 were _|_. That's too strange.

The definition avoids trivialization and preserves modus
ponens although it has some other drawbacks such as the deduction system
being too weak.

And modus ponens in this form

(A and (A a> B)) a> B

does not hold in 1 a> 1:

(1 and (1 a> 1)) a> 1 = (1 and _|_) a> 1 = _|_ a> 1 = _|_

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

Relevant Pages

  • Re: looking for a predicate hierarchy
    ... "if it rains we stay at home; it rains, ... convincing because you lose ability to reason purely in syntactic terms, ... deductive system that would rely only on a system of axioms and inference ... syntactic deduction system in your logic any more by which you might have ...
    (comp.object)
  • Re: OT - A Modest Proposal
    ... I once sat on an airplane next to a seemingly intelligent woman who uttered the following words to me: "What happens to the sun when it rains?". ... It was clear that this woman had so little grasp of the universe of which she was a living, unique part that she felt that rain on Earth somehow affected the sun. ... And, unfortunately, there are so many ignorant people talking nonsense that they are have a real detrimental effect on the lives of everybody. ...
    (rec.outdoors.rv-travel)