Adequacy Results for Some Priorean Modal Propositional Logics

Standard possible world semantics for propositional modal lan-guages ignore truth-value gaps. However, simple considerations suggest that it should not be so. In Section 1, I identify what I take to be a correct truth-clause for necessity under the assumption that some possible worlds are incomplete (i.e., “at” which some propositions lack a truthvalue). In Section 2, I build a world semantics, the semantics of TV-models, for standard modal propositional languages, which agrees with the truth-clause for necessity previously identi-fied. Sections 3–5 are devoted to systematic concerns. In particular, in Section 4, Prior’s system Q (propositional version) is given a TV-models semantics and proved adequate (i.e., sound and complete) with respect to it. 1 Incomplete worlds and modality Let a proposition be any statement that is actually true or false,1 and let us say that possible world w is (i) complete with respect to proposition p if and only if p is true or false at w, and (ii) complete (tout court) if and only if it is complete with respect to all propositions. Then by definition the actual world is complete. And a classical assumption in possible worlds semantics for propositional modal logics is that every possible world is complete. There are serious reasons to reject that assumption. Consider, for instance, the proposition ‘Socrates is mortal’, and assume (i) that there are possible worlds where Socrates does not exist, and (ii) that for ‘Socrates is mortal’ to have a truth-value at a world, Socrates must exist therein: two defendable assumptions, which jointly entail that there are worlds where ‘Socrates is mortal’ has no truth-value. Once it is granted that some propositions have no truth-value at some worlds, it is still not decided if and how these truth-value gaps are transmitted to more complex propositions. In this paper, we shall adopt the principle of contamination, according to which if a proposition has no truth-value at a given world, then every proposition containing the first thereby has no truth-value at that world. 1 Published in Notre Dame Journal of Formal Logic 40, issue 20, 236-249, 1999 which should be used for any reference to this work