Basic proof theory
Displaying modal logic
Cut-elimination for a logic with definitions and induction
Theoretical Computer Science - Special issue on proof-search in type-theoretic languages
A Schütte-Tait Style Cut-Elimination Proof for First-Order Gödel Logic
TABLEAUX '02 Proceedings of the International Conference on Automated Reasoning with Analytic Tableaux and Related Methods
A Note on Linear Kripke Models
Journal of Logic and Computation
How to universally close the existential rule
LPAR'10 Proceedings of the 17th international conference on Logic for programming, artificial intelligence, and reasoning
A local system for intuitionistic logic
LPAR'06 Proceedings of the 13th international conference on Logic for Programming, Artificial Intelligence, and Reasoning
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Gödel-Dummett logic is an extension of first-order intuitionistic logic with the linearity axiom (A ⊃ B)∨(B ⊃ A), and the so-called "quantifier shift" axiom ∀x(A ∨ B(x)) ⊃ A ∨ ∀xB(x). Semantically, it can be characterised as a logic for linear Kripke frames with constant domains. Gödel-Dummett logic has a natural formalisation in hypersequent calculus. However, if one drops the quantifier shift axiom, which corresponds to the constant domain property, then the resulting logic has to date no known hypersequent formalisation. We consider an extension of hypersequent calculus in which eigenvariables in the hypersequents form an explicit part of the structures of the hypersequents. This extra structure allows one to formulate quantifier rules which are more refined. We give a formalisation of Gödel-Dummett logic without the assumption of constant domain in this extended hypersequent calculus. We prove cut elimination for this hypersequent system, and show that it is sound and complete with respect to its Hilbert axiomatic system.