Contextual Deduction Theorems

Studia Logica 99 (1-3):279-319 (2011)
Abstract
Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case
Keywords Deductive system  deduction-detachment theorem  protoalgebraic  algebraizable  quasivariety  EDPC  distributive  modular  substructural logics
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Josep Maria Font (2013). The Simplest Protoalgebraic Logic. Mathematical Logic Quarterly 59 (6):435-451.
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