Semantics for System GHP, après Mates

By J. L. Speranza

I am seriously discussing, elsewhere, what interface we can build between Carnap and Grice.

You read secondary bibliography on Grice and you are surprised: no mention of Mates!

(I mean, superficial secondary biblio on Grice tends to be either too superficial, too secondary, superficial, secondary, or too focused!)

I once started a mini-fiche system: authors cited by Grice, authors citing Grice. Mates HAD to be there. It's author cited by Grice! (Along with Aristotle!)

Benson Mates is a genius. He was prof. emeritus of UC/Berkeley and brought many a joy to Grice's days there. He was _logic_ as understood at Berkeley, and unlike many other formal logicians of his day -- this was 1967 -- he always had a good ear for a "philosophy of ordinary language" as Grice's claim to fame was as being.

So, I was pretty moved when I read Grice's humble tribute to Quine, and see that Grice cares to mention Mates, "by word of mouth" and "via his _Elementary Logic_, a book I was familiar with. Of the zillion books on symbolic logic that students are aware of, Grice had to choose the one _I_ was familiar with, and that put Mates in my priority of things. Mates was a many-talented person and has research on stoic logic too.

Now, the semantics for System Q (for Grice) and G (for Myro and me) has a semantics apres Mates, then. This is 'informal' alla Hilbert and Carnap, in that the strings that make for the 'statements' in this sub-part of the system are sort of too metalogic.

In the long run, it amounts to things like

(x)Px

everything is a pirot.

You have to go to the set "P" and check with its extension: it has one member: "a".

So, yes "a is a Pirot".

Since "a" is the only member of "Pirot" ("Pirot" is a one-member class) it is 'yielded' as per a theorem, that

it is true that (x)Px

i.e "(x)Px" is true.

Since this is only for one interpretation I (where "P" is such a class) one cannot expect that it will be true regardless.

Now, for

(x)Px --> Px

Every pirot is a pirot

-- the scenario is different. For we don't need to _check_, but yet we rather, because --> may yield paradoxes if "P" is null.

Or

(x)(Px v - Px)

Everything is either a pirot or it isn't.

This yields okay in System Q and Sytem G-HP, because it is bivalent logic, and the rule of inference and truth-table for "-", (elimination and introduction), yield no intuitionistic clashes. But here the gist is in the interpretation of ' - ', which _is_ truth-functional, rather than on the interpretation of the predicate calculus qua predicate-calculus.

Etc.