# [FOM] origins of completeness in modal logic

Allen Hazen allenph at unimelb.edu.au
Wed Mar 4 01:09:59 EST 2009

```John P. Burgess, about a decade back, published an interesting article in
the "Notre Dame Journal of Formal Logic" with a title like "The Correct
Modal Logic" or maybe the "True Modal Logic": a bit of conceptual analysis
to identify an intuitive sense of necessity, then some starts at
axiomatically characterizing the logic correct for that interpretation
(narrows it down, but doesn't get to a unique logic).  Aim is described by
comparison to what Kreisel did for classical logic in "Informal Rigor and
Completeness Proofs."  Recommended.

Earlier, around 1980, Nicholas D. Goodman published an article, "The Knowing
Mathematician" in (I think) the philosophy journal "Synthese", arguing that
S4 is the correct logic for a square interpreted as "it is informally
mathematically provable that": this led to a bit of a work by others: S.
Shapiro, ed., "Intensional Mathematics" (North-Holland, "Studies in Logic,"
beginning of the 1980s) contains a number of related papers.

Bull formulated an intuitionistic analogue of S5 (i.e. an S5-like modal
extension of intuitionistic propositional calculus) with the property that
its theorems are the "translations" (in the obvious, familiar, sense) of
valid formulas of monadic predicate logic: S5 is that if you like classical
m. predicate logic, Bull's if you prefer intuitionistic.  This drew a
cautiously approving comment from Heyting, which can be taken as (weak)
evidence that he at least thought this was along the right line...

Sorry not to give more precise bibliographical data: I'm in a rush.  Remind
me and I will try to be more precise(and to find Heyting's remark on Bull).

Allen Hazen
Philosophy (PASI)
University of Melbourne

On 3/3/09 5:07 AM, "Max Weiss" <30f0fn at gmail.com> wrote:

> For first-order logic, there is some intuitive notion of (classical)
> validity that is sufficiently stable to allow axiomatization and even
> the proof of a completeness theorem before the relevant semantical
> notions were completely formalized.  However, for modal logic this
> seems not clearly to be the case: the question of an arbitrary modal
> formula "is it valid?" tout court, seems simply ill-posed.
>

```