On 27 Sep 2011, at 13:49, Stephen P. King wrote:
On 9/26/2011 7:56 PM, Jason Resch wrote:
<snip>
For well-defined propositions regarding the numbers I think the
values are confined to true or false.
Jason
--
[SPK]
Not in general, unless one is only going to allow only Boolean
logics to exist. There have been proven to exist logics that have
truth values that range over any set of numbers, not just {0,1}.
Recall the requirement for a mathematical structure to exist: Self-
consistency.
Consistency is a notion applied usually to theories, or (chatty)
machines, not to mathematical structures.
A theory is consistent if it does not prove some proposition and its
negation. A machine is consistent if it does not assert a proposition
and its negation.
In first order logic we have Gödel-Henkin completeness theorem which
shows that a theory is consistent if and only if there is a
mathematical structure (called model) satisfying (in a sense which can
be made precise) the proposition proved in the theory.
Also, it is true that classical (Boolean) logic are not the only
logic. There are infinitely many logics, below and above classical
propositional logic. But this cannot be used to criticize the use of
classical logic in some domain.
All treatises on any non classical logic used classical (or much more
rarely intuitionistic) logic at the meta-level. You will not find a
book on fuzzy logic having fuzzy theorems, for example. Non classical
logics have multiple use, which are not related with the kind of ontic
truth we are looking for when searching a TOE.
Usually non classical logic have epistemic or pragmatic classical
interpretations, or even classical formulation, like the classical
modal logic S4 which can emulate intuitionistic logic, or the
Brouwersche modal logic B, which can emulate weak quantum logic. This
corresponds to the fact that intuitionist logic might modelize
constructive provability, and quantum logic modelizes observability,
and not the usual notion of classical truth (as used almost everywhere
in mathematics).
To invoke the existence of non classical logic to throw a doubt about
the universal truth of elementary statements in well defined domain,
like arithmetic, would lead to complete relativism, given that you can
always build some ad hoc logic/theory proving the negation of any
statement, and this would make the notion of truth problematic. The
contrary is true. A non classical logic is eventually accepted when we
can find an interpretation of it in the classical framework.
A non standard truth set, like the collection of open subsets of a
topological space, provided a classical sense for intuitionist logic,
like a lattice of linear subspaces can provide a classical
interpretation of quantum logic (indeed quantum logic is born from
such structures). It might be that nature observables obeys quantum
logic, but quantum physicists talk and reason in classical logic, and
use classical mathematical tools to describe the non classical
behavior of matter.
Bruno
http://iridia.ulb.ac.be/~marchal/
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