(Star-based) Three-valued Kripke-style semantics for pseudo- and weak-Boolean logics

This article investigates Kripke-style semantics for two sorts of logics: pseudo-Boolean (pB) and weak-Boolean (wB) logics. As examples of the first, we introduce G ₃ and .G ₃ is the three-valued Dummett-Gödel logic; is the modal logic S5 but with its orthonegation replaced by a pB (or Heyting) negation. Examples of wB logic are is G ₃ with a wB negation in place of its pB negation; is S5 with a wB negation replacing its orthonegation. For each system, we provide a three-valued Kripke-style semantics with and without star operation (which is like the star operation of the Routley-Meyer semantics of relevance logics). We prove soundness and completeness theorems in each case. Note that wB logics may be equivalent to logics with Baaz's projection Δ. We finally introduce the G ₃ and the both with Δ and show that they are equivalent to and respectively.