A gentzen system for conditional logic

Abstract

Conditional logic is the deductive system 〈 $$\mathcal{L}$$ , ⊨〉 where $$\mathcal{L}$$ is the set of propositional connectives {∧, ∨,′} and ⊨ is the structural finitary consequence relation on the absolutely free algebra $$F_{m_\mathcal{L} } $$ that preserves degrees of truth over the structure of truth values 〈C, ≤〉. HereC is the non-commutative regular extension of the 2-element Boolean algebra to 3 truth values {t, u, f}, andf<u<t. In this paper we give a Gentzen type axiomatization for conditional logic.