(Star-Based) three-valued Kripke-style semantics for pseudo- and weak-Boolean logics
E. Yang · Logic Journal of IGPL · 2011
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 G3 and S53pB.G3 is the three-valued Dummett–Gödel logic; S53pB is the modal logic S5 but with its orthonegation replaced by a pB (or Heyting) negation. Examples of wB logic are G3wB and S53wB.G3wB is G3 with a wB negation in place of its pB negation; S53wB 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 G3 and the S53pB both with Δ and show that they are equivalent to G3wB and S53wB, respectively.