An Inferentially Many-Valued Two-Dimensional Notion of Entailment

Carolina Blasio; João Marcos; Heinrich Wansing

Bulletin of the Section of Logic (2017)

  • Volume: 46, Issue: 3/4
  • ISSN: 0138-0680

Abstract

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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.

How to cite

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Carolina Blasio, João Marcos, and Heinrich Wansing. "An Inferentially Many-Valued Two-Dimensional Notion of Entailment." Bulletin of the Section of Logic 46.3/4 (2017): null. <http://eudml.org/doc/295524>.

@article{CarolinaBlasio2017,
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.},
author = {Carolina Blasio, João Marcos, Heinrich Wansing},
journal = {Bulletin of the Section of Logic},
keywords = {Inferential many-valuedness; two-dimensional entailment; B-matrices; B-consequence relations; monotonic consequence relations; q-entailment; p-entailment; Suszko Reduction},
language = {eng},
number = {3/4},
pages = {null},
title = {An Inferentially Many-Valued Two-Dimensional Notion of Entailment},
url = {http://eudml.org/doc/295524},
volume = {46},
year = {2017},
}

TY - JOUR
AU - Carolina Blasio
AU - João Marcos
AU - Heinrich Wansing
TI - An Inferentially Many-Valued Two-Dimensional Notion of Entailment
JO - Bulletin of the Section of Logic
PY - 2017
VL - 46
IS - 3/4
SP - null
AB - 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.
LA - eng
KW - Inferential many-valuedness; two-dimensional entailment; B-matrices; B-consequence relations; monotonic consequence relations; q-entailment; p-entailment; Suszko Reduction
UR - http://eudml.org/doc/295524
ER -

References

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