Disentangling FDE-Based Paraconsistent Modal Logics

On Definability of Connectives and Modal Logics over FDE

“…Almost all logics considered in this section were studied in [10], and although their presentation here will be self-contained, we refer the interested reader to that paper for further information, including presentations of some axiomatizations and tableau calculi.…”

“…Semantics for BK 2 + 2 − , BK FS bl , and MBL The study of modal bilattice logic, MBL, was motivated in [12] by obtaining a modal extension of the four-valued logic FDE characterized by possible worlds models with a four-valued accessibility relation between possible worlds. In [10] it was shown that MBL can be faithfully embedded into a logic characterized by the class of models of the fusion BK 2 × BK 2 of two copies of BK 2 , enriched with the binary connectives ⊗ and ⊕ and the constants ⊤, b, and n from bilattice logic. Moreover, it was shown that BK 2 × BK 2 and the Fischer Servi-style modal logic BK FS are weakly definitionally equivalent.…”

“…, A n ) and the L-formula A that is guaranteed to exist by Theorem 2.6 share their verification and their falsification conditions, the two formulas are provably strongly equivalent in every logic under consideration in which strong implication, ⇒, is either primitive or definable. In [9,10] it was shown that provable strong equivalence is a congruence relation in BK and in BK 2 , so that ⋆(A 1 , . .…”

“…First we recall the notion of definitional equivalence of logics from [5], which was also considered as basic in [13] and [10].…”

“…In Item 3 of the original definition from [5], conditions A ⇔ ρθ(A) ∈ L 1 and B ⇔ θρ(B) ∈ L 2 are replaced by conditions (A, ρθ(A)) ∈Ω(L 1 ) and respectively (B, θρ(B)) ∈ Ω(L 2 ), whereΩ(L i ) denotes the so called Tarski's congruence over logic L i , i.e., the largest congruence on the algebra of formulas with the universe Form(L i ) that respects all theories over L i (see [2,4] for details). However, in many logics L with strong negation, in particular in all logics considered in this article, its Tarski's congruence is determined by the condition A ⇔ B ∈ L (see [10] for details).…”