A uniform quantificational logic for algebraic notions ofcontext

Abstract

A quantificational framework of formal reasoning is proposed, which emphasises the pattern of entering and exiting context. Contexts are modelled by an algebraic structure which reflects the order and manner in which context is entered into and exited from. The equations of the algebra partitions context terms into equivalence classes. A formal semantics is defined, containing models that map equivalence classes of certain context terms to sets of first order structures. The corresponding Hilbert system incorporates the algebraic equations as axioms asserted in context. In this way a uniform logic for arbitrary algebras of context is obtained. Soundness and completeness are proved. In semigroups of contexts, where combination of contexts is associative, finite ground algebraic equations correspond to contingent equivalence between certain logical formulas. Systems for sets and multisets of contexts are obtained by presenting their respective algebras as associativity plus finite ground equations. Some contextual reasoning systems in the literature are inherently associative, and we present those as special cases.