[FOM] Modal logics of contingency

[Joao Marcos]

Joao Marcos wrote:
> >
> > This much to fix some intended meaning for the above operators.
> > Now, start from a modal logic which has only C (and D) as
> > primitive operators, alongside with the classical ones.  Call
> > any such a logic a *modal logic of contingency*.  Question:
> >
> > -o- Were such logics already investigated?  References?

Åke Persson wrote:
>
> Finally a question: What can a logic of 'p is contingent' and 'p is
> non-contingent' be used for?

Dear Åke:

Since I made my question I was kindly informed, off-list, of the
existence of the paper:

Humberstone, I.L.
The logic of non-contingency.
Notre Dame Journal of Formal Logic 36, No.2, 214-229 (1995).

   This is a wonderfully written paper with everything I needed:
history, motivations and results.  I guess you have an equally good
chance of finding an answer to your disquietudes there.

   My own interest on the matter was not directly related to
contingency, but to other closely related modal properties, not given
simply by boxes or diamonds.  This is it.  In modal logics of
provability à la Loeb-Boolos, *consistency* of a sentence is usually
equated to the non-provability of its negation (that is, to its
"possibility").  This is quite strong of an interpretation.  Another
possible interpretation takes a formula to be consistent if its holding
good implies its being provable.  *This* is quite close to what a
non-contingency operator does, and is the track I intended to explore.

Åke Persson also wrote:
>
>     "Intuitionism and constructivism has rejected reductio ad
>     absurdum as an unsure method of proof. Above is shown that
>     from ~M~p we can not conclude  Lp. We need to show ~M
>     (~p v ?p), i.e. both ~M~p ("it's not possible that not-p") and
>     ~M?p ("it's not possible that p is unknown") to conclude Lp.
>     Intuitionists are right in their critisism but hits the wrong
property
>     of logic. Double negation reduction is a most basically
>     foundation of all logic and will still remain untouched. There are
>     still no problems to reduce ~~p to p, to toggle beween a
>     meaning and its contrary meaning for every negation. Also for
>     every negation of truth we still can toggle between the contraries
>     Tp and Fp. As reductio ad absurdum proof takes place in
>     modal contexts we need to handle them by modal logic but for
>     that we need a modal logic strict constructive derived, like the
>     logic refered above."

   This is a completely different matter.  I can kind of understand that
one might be willing to retain double negation elimination as a valid
rule, together with its converse, inside some intuitionistic-like logic.
This is in fact feasible, if you can throw some convenient rules away.
But, on the other hand, even very basic versions of reductio ad absurdum
are sufficient to prove excluded middle, inside very weak logics.  I
wonder thus how is it that you plan to save reductio and still be
sufficiently "intuitionist"?

      JM

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