Possible worlds and the contingency of logic

Abstract

In modal semantics, when speaking of possible worlds, there seems to be the tacit assumption that logical reasoning will stay constant throughout. That is to say that a logical reasoning valid at one world is valid in all worlds, hence necessary. But what happens then if we decide to consider possible worlds semantics where different worlds may respond to different logics? What then becomes necessary? In this thesis, we expand the possible world semantics for modal logics by not assuming one ‘type’ of possible worlds in a model, but by considering that different possible worlds might reason under different logics. We focus ourselves on a setting where we combine classical and intuitionistic worlds.

We use ⊢, to denote pure propositional intuitionistic reasoning even if the language contains □. In that sense, formulas of the form □ A behave as propositional variables as far as ⊢, is concerned. Likewise we consider the ⊢ relation for classical reasoning. We define so-called mixed models which are tuples ⟨W, R, {lw}w∈W , {Tw}w∈W ⟩, where lw ∈ {i, c} and Tw a set of modal formulas such that 1. ⊥ ∈/ Tw 2. Tw ⊢lw φ ⇒ φ ∈ Tw 3. □φ ∈ Tw ⇐⇒ ∀v(wRv ⇒ Tv ⊢lv φ) 4. ¬□φ ∈ Tw ⇐⇒ ∃u(wRu ∧ Tu ⊢lu ¬φ) We prove soundness of the intuitionistic normal modal logic iK+ (bem) wrt mixed models, where bem is short for ‘Box Excluded Middle’ and denotes the axiom □A ∨ ¬□A. The logic iK has well-studied birelational semantics with an R relation for the □ and ≤ for intuitionistic implication (Bozic and Dosen 1984). We prove soundness and completeness for iK + (bem) with respect to these birelational semantics together with the birelational model frame condition. w ≤ v ⇒ ∀z(wRz ⇒ vRz). We conclude completeness for iK + (bem) wrt mixed models. These results pave the way for new semantic constructions of Kripke models, raising intriguing mathematical and philosophical questions. It invites us to consider the implementation of more logics, possibly non-comparable, in this construction.