[CL] Re: Some comments on future directions

[John F. Sowa]

Enrico,

That's true:

EF> But you DON'T have specialistic semantics builtin in CL.

You need a context mechanism, in which the modalities
are axiomatized at the metalevel.

EF> Either (a) you axiomatise the specialistic structures
 > in the logic at hand (think JUST to all the zillion
 > different kripke structures in modal logic: linear,
 > branching, interval, bounded, unbounded, discrete,
 > rational, dense, etc *time*; same for space; all the
 > proposed structures for space-time; all the variants
 > for beliefs, knowledge, intentions, etc etc), or
 > (b) these semantic structures are builtin in your logic,
 > possibly even as an additional dimension to your
 > favourite logic (e.g., CL). I am advocating (b), e.g.,
 > to have them builtin in the semantics.

I agree.  My proposal is (a), but to dump Kripke semantics.

Kripke had a brilliant idea in 1963, from which we all
learned a lot.  But Michael Dunn had an even better idea
in 1973, which made Kripke semantics obsolete.  Unfortunately,
very few logicians knew about it, Dunn did nothing to
publicize it, and his innovation was ignored.  I am one
of the few people who took Dunn's work seriously.  For
a brief summary of Dunn's semantics and its equivalence
to K's semantics, see Section 2 of

    http://www.jfsowa.com/pubs/laws.htm
    Laws, Facts, and Contexts

For the reason why K's semantics is obsolete, see Dana Scott's
article of 1970, in which he called the limitation to one
modality "one of the biggest mistakes of all in modal logic":

DS> The only way to have any philosophically significant results
 > in deontic or epistemic logic is to combine these operators with:
 > Tense operators (otherwise how can you formulate principles of
 > change?); the logical operators (otherwise how can you compare
 > the relative with the absolute?); the operators like historical
 > or physical necessity (otherwise how can you relate the agent
 > to his environment?); and so on and so on. (p. 143)

I realize that there are people who have designed Kripke models
for multiple modalities, but those constructions are a reductio
ad absurdum.  Instead of K's triples (set of worlds, accessibility
relation, evaluation function), people have proposed octuples
for BDI models (Beliefs, Desires, and Intentions).  If you include
all the verbs in any natural language, your tuples would get
very long -- you would need an incredible number of independent
accessibility relations thrown into your models.  That makes it
impossible to have a semantic representation for anything but
a toy subset of any NL.

EF> If, on the other hand, you want to axiomatise the structures
 > in your logic, you have to *prove* that the axiomatisation
 > is sound and complete wrt the original specific algebraic
 > specification, and that this axiomatisation is possible at all.

No.  You are thinking in terms of a different logic for every
modality.  What I am proposing is just one logic, in which
you add one or more axioms for each mode or intentional verb.
That is no more difficult than writing a program specification
in which you add one or more axioms for each feature.  For any
particular sentence or paragraph that contains one or more
intentional verbs, you add just those axioms that are needed
to define the verbs that actually occur in that text.  See
Sections 6 and 7 of the paper on Laws, Facts, and Contexts.

I admit that if you throw in too many axioms, the whole
thing might collapse into a contradiction.  But so what?
That's not a contradiction in your logic, but only in
the statements you made in that logic.  That happens
all the time.   It's known as a bug.

John