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.


03B50 Many-valued logic
03B22 Abstract deductive systems
Full Text: DOI


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