[FOM] Possibility, necessity, and realism.

dmehkeri@yahoo.ca dmehkeri at yahoo.ca
Sun Apr 12 18:29:35 EDT 2009

I am wondering whether a realistic interpretation of an intuitionistic modality pair as "possible" and "necessary" has gotten much attention. I am thinking of these propositions that are classically equivalent but constructively inequivalent:

(O1) []p \/ []~p
(O2) <>p -> []p
(O3) ~[]p -> []~p
(O4) <>~p -> ~<>p

(S1) <>p /\ <>~p
(S2) ~[]~~p /\ ~[]~p
(S3) ~[]p /\ ~[]~p

Using just normal intuitionistic modal axioms, O1 through O4 are progressively weaker statements. Adding classical logic, we also have O4 -> O1. Similarly for S1 through S3. 

O1 expresses that p is either necessarily true or necessarily false. I am going to call this "determinate". O2 expresses that if p is possible, then p is necessary, which seems to express that p is "categorical". S1 asserts that p and its negation are both possible, and this sounds like "contingent". To close under negation, I have included O3, O4, S2, and S3 :  ~O1 = ~O2 = ~O3 = S3; ~O4 = S2; ~S1 = ~S2 = O4; ~S3 = O3. S3 could be said "indeterminate", O3 "not indeterminate", and O4 "not contingent". I don't know a nice way to pronounce S2.

Of course it would be inconsistent to assert that a single statement is categorical and deny that it is determinate. But for an open sentence p(n) with one free variable, I could consistently assert that for all n, p(n) is categorical, while denying that for all n, p(n) is determinate. 

Should asserting the "categoricity" of all arithmetic statements -- that they are absolutely independent of all contingency -- be enough to qualify someone as a realist, at least with respect to the integers? And yet it doesn't seem to be enough to deduce the law of the excluded middle.



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