# An Inferentially Many-Valued Two-Dimensional Notion of Entailment

## DOI:

https://doi.org/10.18778/0138-0680.46.3.4.05## Keywords:

Inferential many-valuedness, two-dimensional entailment, B-matrices, B-consequence relations, monotonic consequence relations, q-entailment, p-entailment, Suszko Reduction## Abstract

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 B-matrices. After showing that every purely monotonic singleconclusion consequence relation is characterized by a class of 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 B-entailment, a two-dimensional multiple-conclusion notion of entailment based on 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 B-consequence relation is defined, and an abstract characterization of such relations by classes of 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 B-consequence relation is, in general, inferentially four-valued.

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*Bulletin of the Section of Logic*,

*46*(3/4), 233–262. https://doi.org/10.18778/0138-0680.46.3.4.05

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