Title: Re: Fw: ' possible' reply to Bruno Marchal
At 13:26 +0200 23/08/2002, Lennart Nilsson wrote:

Terry Savage (TS) wrote:

First, we need to distinguish between the uses of 'possibility' and 'necessity' within the propositional (sentential) calculus (PC), and then within quantification theory (QT). Within PC we also have to note the difference between the purely syntactic formulas (without interpretation) and those same formulas imbedded in a semantic metalanguage. Taken solely syntactially, introducing the Modal operators produces an extension to PC, the motivation for which is obscure. If, instead of calling the modal symbols (diamond and box) 'possibility' and 'necessity', we called them 'blark' and 'blog', nobody would take the excersize seriously, except to note that one can add arbitrary connectives (or functions) to one formalism and produce an extended one. Calling these arbitrary symbols 'necessity' and 'possibility' seems to me to be a ruse to sneak obscure philosophical concepts into a simple excersize for rewriting marks on paper.


BM:
Right. That's why I use just "square" and "diamond" when
I teach modal Logic. I use "necessity" and "possibility"
only when I motivate the system S5.


AXIOMS:   <axioms of classical propositional logic>
          [](p -> q) -> ([]p -> []q)
          []p -> p
          []p -> [][]p
          <>p -> []<>p

RULES:    p p->q       p
         --------,    ---
            q         []p

Let us define a world by a (labelized) function from the set
of propositional letters {p, q, r, ...} in {O, 1}, and let us
say p is true in world W if W(p) = 1, and let us close those
world for classical propositional logic. Then, "defining" []p
by p is true in all world, and <>p by p is true in at least
one world, it should be obvious that we get a model for S5.
This is Leibniz semantics (= Kripke semantics where all worlds
are accessible from any world).


TS:
If we try to extend Modal Logic into QT, serious difficulties show up immediately. A striaghtforward interpretation results in the failure of Leibniz's Law of Identity. The Kripke solution for this is to postulate a model which ranges over multiple *possible* worlds. David Lewis, taking this to an extreme, maintains that these possible worlds somehow exist in parallel with our own world. I am reminded of Quine's despair at this kind of double-talk, which led him to give away the term 'exist'. It doesn't help me to understand the expression 'For some x, it's possible that Fx', by referring me to a set of possible worlds. The best advice re:Modal QT, comes from Jennifer Davoren  'One can also study predicate modal (or temporal) logics, extending predicate or first-order logics, but unless one has particularly good reasons, my advice is don't go there.' Lecture notes, ANU Summer School 2000, 'Non-classical Logic...'.


BM:
OK. Quantifying in modal context is difficult. According to
some author such quantification introduces essentialism.
I think this is not a defect but I agree this should be handle
with care. Now, relatively to my work, let us mention that:

-The quantification of the logic G and G* is semantically cristal
clear (all formulas admit non ambiguous arithmetical interpretation).
So the "essence" are mathematically well defined in term of
relations between numbers.
-In the short version of my thesis I have suppress any use of the
quantified logic.


TS:
John McCarthy has attacked the problem in another way, summarized by the title of one of his short notes: 'Modality, SI!, Modal logic, NO!'.


BM:
With all my respect for the inventor of LISP, I think this is a joke.
"Modal logic" is just an extending collection of mathematical tools for
clarifying the notions of modality. McCarthy predicative approach,
well known in modal logic, can be proved impossible for all the
first-person notion tackled in my thesis. (This is linked with Tarski
theorem saying that arithmetical truth cannot be predicatively defined in
arithmetic, or better the Kaplan Montague Benacerraf extension
of that result for the notion of knowledge). Look for "Benacerraf"
in the everything-list archive. http://www.escribe.com/science/theory/


-Bruno

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