A Kripke Semantics for Hajek's BL

Abstract

We provide a generalisation of Kripke semantics for Petr Hajek's Basic Logic and prove soundness and completeness of the same with respect to our semantics. We find this semantics easily specialises to the linearly ordered Kripke frames for Godel-Dummett logic which BL properly contains. Our soundness, deduction theorem and completeness arguments further strengthen this analogy. This paper extends the insights of [10] from GBL ew f to the case of BL.Proposition 2. The natural deduction system BL (Figure 1) has the same derivable formulas as the Hilbert-style system BL H of [7]:As in the proof of Proposition 1.5 in [10]. Left-to-right follows by induction on the structure of the natural deduction proof once one shows each instance of a natural deduction rule translates to a theorem of BL H . We amend the provability ordering from [10] for the present system:Since BL H results from GBL ew f by adding (A14), all other cases are as in [10], except (A14) which simply says that the provability relation is linearly ordered. For the right to left direction of the 'iff', this follows by induction on the BL H derivation of ψ 1 → ... → ψ n → φ once we show that each of the axioms of BL H are theorems of BL. The only case left to verify then is (A14), and this is an axiom in the natural deduction calculus BL hence always provable in that calculus.Note 3. Gödel-Dummett logic GD results from BL by adjoining the structural rule of contraction. BL and MV-algebrasWe situate the algebraic semantics characterising BL in terms of the somewhat larger theory of residuated lattices.