A Kripke Semantics for Hajek's BL
Andrew Lewis-Smith (University of Sheffield)
TL;DR
This paper develops a Kripke semantics for Hajek's Basic Logic (BL), demonstrating its soundness and completeness, and showing how it specializes to Godel-Dummett logic, thereby extending previous work on intuitionistic Lukasiewicz logic.
Contribution
It introduces a new Kripke semantics for BL, establishing soundness and completeness, and connects BL to Godel-Dummett logic through specialized frames.
Findings
Semantics is sound and complete for BL
Specializes to Godel-Dummett logic frames
Extends previous semantics for Lukasiewicz logic
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 our previous paper, "A Kripke Semantics for Intuitionistic Lukasiewicz logic," to the case of Hajeks' BL.
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