We discuss an algebraic approach to propositional logic with side effects. To this end, we use Hoare’s conditional [1985], which is a ternary connective comparable to if-then-else. Starting from McCarthy’s notion of sequential evaluation [1963] we discuss a number of valuation congruences and we introduce Hoare-McCarthy algebras as the structures that characterize these congruences.
 We propose a combination of Kleene’s three-valued logic and ACP process algebra via the guarded commandconstruct. We present an operational semantics in SOS-style, and a completeness result. © 1998 Elsevier Science B.V. All rights reserved. ABSTRACT In this paper, we survey 3-valued logics and their complete axiomatizations, one of which may be new. We then propose a 4-valued, functionally complete logic that incorporates these 3-valued systems and provide notations for interesting operators and subsystems.