Towards a New Logic of Indicative Conditionals

In this paper I will propose a refinement of the semantics of hypervaluations (Mura 2009), one in which a hypervaluation is built up on the basis of a set of valuations, instead of a single valuation. I shall define validity with respect to all the subsets of valuations. Focusing our attention on the set of valid sentences, it may easily shown that the rule substitution is restored and we may use valid schemas to represent classes of valid sentences sharing the same logical form. However, the resulting semantical theory TH turns out to be throughout a modal three-valued theory (modal symbols being definable in terms of the non modal connectives) and a fragment of it may be considered as a three-valued
version of S5 system. Moreover, TH may be embedded in S5, in the sense that for every formula ϕ of
TH there is a corresponding formula ϕ' of S5 such that ϕ' is S5-valid iff ϕ is TH-valid.
The fundamental property of this system is that it allows the definition of a purely semantical relation of logical consequence which is coextensive to Adams’ p-entailment with respect to simple conditional sentences, without being defined in probabilistic terms. However, probability may be well be defined on the lattice of hypervaluated tri-events, and it may be proved that Adam’s p-entailment, once extended to all tri-events, coincides with our notion of logical consequence as defined in semantical terms.