Abstract

Given a logical system with disjunction (V) and negation (¬), we call implication, the derived operation defined by p ⇒ q ≡ (¬p) V q. Conditional logic is the 3-valued logic associated with short-circuit evaluation, and up to anti-isomorphism, is the unique non-commutative regular extension of Boolean logic to 3 truth values. The two anti-isomorphic conditional logics (left to right and right to left evaluation) yield quite different implications. We study the 3-element algebras associated with each of these operations, and for each of them a complete set of laws and a recursive formula for the free spectrum is obtained.

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