An inferentially many-valued two-dimensional notion of entailment.(English)Zbl 1423.03068

Summary: Starting from the notions of $$q$$-entailment and $$p$$-entailment, a two-dimensional notion of entailment is developed with respect to certain generalized $$q$$-matrices referred to as $$\mathsf B$$-matrices. After showing that every purely monotonic single-conclusion consequence relation is characterized by a class of $$\mathsf B$$-matrices with respect to $$q$$-entailment as well as with respect to $$p$$-entailment, it is observed that, as a result, every such consequence relation has an inferentially four-valued characterization. Next, the canonical form of $$\mathsf B$$-entailment, a two-dimensional multiple-conclusion notion of entailment based on $$\mathsf B$$-matrices, is introduced, providing a uniform framework for studying several different notions of entailment based on designation, antidesignation, and their complements. Moreover, the two-dimensional concept of a $$\mathsf B$$-consequence relation is defined, and an abstract characterization of such relations by classes of $$\mathsf B$$-matrices is obtained. Finally, a contribution to the study of inferential many-valuedness is made by generalizing Suszko’s Thesis and the corresponding reduction to show that any $$\mathsf B$$-consequence relation is, in general, inferentially four-valued.

MSC:

 03B50 Many-valued logic 03B22 Abstract deductive systems
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