In three different posts, Brent Meeker wrote :
> I'm not sure that logic in the formal sense can be right or wrong;
> it's a set of conventions about
> language and inference. About the only standard I've seen by which a
> logic or mathematical system
> could be called "wrong" is it if it is inconsistent, i.e. the axioms
> and rules of inference allow
> everything to be a theorem.
I disagree. The main lesson provided by the works of of Tarski and
Godel has shown us how far truth and consistency are different.
By the second incompleteness theorem: (with PA = Peano Arithmetic
Theory)
PA + "PA is consistent" is both consistent and correct
PA + "PA is not consistent" is consistent, but hardly correct!
I will come back on this. But if you recall that Consistent(p) = ~B~p,
then remember that all the followings are not equivalent from the (1
and 3) point of views of the machines: Bp, Bp & p, Bp & ~B~p, Bp & ~B~p
& p.
(if you prefer: p is provable, p is provable *and* p is true, p is
provable and p is consistent, p is provable and p is consistent and p
is true.
> I don't understand "assumptions about logic and math"? We don't need
> to make assumptions about them
> because they are rules we made up to keep us from reaching
> self-contradictions when making long
> complex inferences.
Logician are interested in correctness, and relative correctness. The
whole of model (not modal!) theory concerns those matter.
> They are rules about propositions and inferences. The propositions
> may be
> about an observation like "a species that used this kind of reasoning
> survived more frequently than
> those who used that kind." I might need logic to make further
> inferences, but I don't need
> assumptions about logic to understand it.
I agree if you talk of some minimal informal logic, like children seems
to develop in their early years. (cf Piaget, for examples). Now
concerning the many logics, it is different. There is a continuum of
logics ... each having apparently some domain of application. Fields
like "Categorical Logic" provides tools for many logics.
Linear logic take into account resources. For example, the following is
classically, intuitionisticaly and quantum logically valid:
If i have one dollar I can buy a box of cigarets
If I have one dollar I can buy a box of matches
Thus If I have one dollar I can buy a box of cigarets and I can buy a
box of matches.
ALL logics, when studied mathematically, are studied in the frame of
classical mathematics.
You will never find a treatise on Fuzzy logic with a theorem like "It
is 0,743 true that a fuzzy set A can be represented by a function from
A to the real line".
(ok a case could be made for intuitionist logic, due to the existence
of an intuitionist conception of math).
> Remember Cooper is talking about reasoning, reaching decisions, and
> taking actions - not just making
> truth preserving inferences from axioms. Classical logic applies to
> declarative, timeless sentences
> - a pretty narrow domain.
... called Platonia. Narrow?
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Tue Jul 11 2006 - 12:07:38 PDT