Geodesic
An Inferentially Many-Valued Two-Dimensional Notion of Entailment
Bulletin of the Section of Logic, Tome 46 (2017) no. 3-4.

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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.
Keywords: Inferential many-valuedness, two-dimensional entailment, B-matrices, B-consequence relations, monotonic consequence relations, q-entailment, p-entailment, Suszko Reduction 
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     author = {Blasio, Carolina and Marcos, Jo\~ao and Wansing, Heinrich},
     title = {An {Inferentially} {Many-Valued} {Two-Dimensional} {Notion} of {Entailment}},
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Blasio, Carolina; Marcos, João; Wansing, Heinrich. An Inferentially Many-Valued Two-Dimensional Notion of Entailment. Bulletin of the Section of Logic, Tome 46 (2017) no. 3-4. http://geodesic.mathdoc.fr/item/BSL_2017_46_3-4_a1/