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Information frames, implication systems and modalities. (English) Zbl 0920.03027
The paper studies substructural logics with modal operators. Their semantics is based on ‘information frames’ (or ‘quantale frames’). Such a frame is a structure $$(Q,\circ, 1, \sqsubseteq)$$, where $$Q$$ is a non-empty set (of ‘pieces of information’), $$\sqsubseteq$$ is a complete lattice ordering of $$Q$$, $$\circ$$ is a semigroup operation on $$Q$$ with the unit $$1$$ which is distributive over arbitrary sums in $$(Q,\sqsubseteq)$$. Some extra conditions on information frames (commutativity, contraction, expansion, monotonicity) can be used in semantics for relevance logics and linear logic. For interpreting the modal operators $$\square, \diamondsuit$$, a binary relation $$R$$ (with some special properties) is incorporated into an information frame. Proof systems are defined as labelled deductive systems containing the basic system LKE. The main completeness theorem states that $$\vdash _{\text{LKE+C}}A$$ iff $$A$$ is a logical consequence of $$1$$ in every C-model, where C is some specific set of axioms. A more detailed version of the paper is M. D’Agostino, D. M. Gabbay, and A. Russo [Stud. Logica 59, 65-102 (1997; Zbl 0888.03011)].
##### MSC:
 03B45 Modal logic (including the logic of norms) 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 03B20 Subsystems of classical logic (including intuitionistic logic) 06F05 Ordered semigroups and monoids
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