Hypersequent and Labelled Calculi for Intermediate Logics PDF
Preview Hypersequent and Labelled Calculi for Intermediate Logics
Hypersequent and Labelled Calculi for Intermediate Logics Lara Spendier (Joint work with A. Ciabattoni and P. Maffezioli) [email protected] ViennaUniversityofTechnology September 19, 2013 Cut-free calculi Cut-free calculi Cut-free calculi Cut-free calculi Intermediate logics: Two approaches Intermediate logics ... Logics between intuitionistic and classical logic Semantic approach Syntactic approach imposing on intuitionistic extending intuitionistic logic Kripke frames additional with Hilbert axioms conditions on the (transitive and reflexive) accessibility Hypersequent calculi relation (cid:54) Labelled calculi (Propositional) Intermediate logics Example Jankov (De Morgan) logic: IL + Frame condition ∀x∀y∀z((x (cid:54)y ∧x (cid:54)z)→∃w(y (cid:54)w ∧z (cid:54)w)) “Equivalent” axiom ¬α∨¬¬α G¨odel logic: IL + Frame condition ∀x∀y∀z((x (cid:54) y ∧x (cid:54) z) → (y (cid:54) z ∨z (cid:54) y)) “Equivalent” axiom (α ⊃ β)∨(β ⊃ α) Bd : IL + 2 Frame condition ∀x∀y∀z((x (cid:54) y ∧y (cid:54) z) → (y (cid:54) x ∨z (cid:54) y)) “Equivalent” axiom α∨(α ⊃ (β∨¬β)) Intermediate logics: Two approaches frame conditions Intuitionistic Logic Hilbert axioms Intermediate Logics Intermediate logics: Two approaches Cut-free labelled calculi frame conditions Intuitionistic Logic Hilbert axioms Intermediate Logics Intermediate logics: Two approaches structural rules Cut-free labelled calculi transform frame conditions Intuitionistic Logic Hilbert axioms Intermediate Logics
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