An Inferentially Many-Valued Two-Dimensional Notion of Entailment

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.An Inferentially Many-Valued Two-Dimensional Notion of Entailment 235 of a p-matrix [7,8]. A q-matrix (quasi matrix) is a structure A, D + , D − , where A is an algebra similar to a propositional language L, and where D + and D − are subsets of A, and D + ∩ D − = ∅. Usually, D + is referred to as the set of designated values and D − as the set of antidesignated values. A p-matrix (plausibility matrix) is a structure A, D + , D * , where A is an algebra similar to a propositional language L and D + ⊆ D * ⊆ A. The set D * is usually referred to as the set of plausible, non-antidesignated values.We adopt a compact notation that avoids superscripts and the barnotation for set-theoretic complementation, introducing the symbols Y, Y , N, and N to denote, respectively, the sets of designated, non-designated (V \ Y), antidesignated, and non-antidesignated (V \N) values. With a cognitive twist, they might be taken as representing acceptance, non-acceptance, rejection and non-rejection. 1

Suszko’s Problem: Mixed Consequence and Compositionality

“…The following corollary puts upper bounds on Suszko rank for monotonic consequence relations with various syntactic properties. This is the exact analog of Corollary 1 in French and Ripley (2018), and of statements (M1) and (M2) in Blasio et al (2018):…”

“…Our work was developed independently of the work ofBlasio et al (2018), and only referred to the work ofFrench and Ripley (2018) at the time we submitted the first version of this paper. We are grateful to J. Marcos for bringing his joint paper with C. Blasio and H. Wansing to our notice.…”

From many-valued consequence to many-valued connectives

“…The fact that at most 4 values are needed to semantically characterize a monotonic consequence relation was also established by Blasio et al (2017) and by French and Ripley (2018), but with no heed paid to truth-functionality. The addition from Theorem 3.21 was that when the consequence relation admits only regular connectives, regularity ensures that the reduction to four values is moreover truth-functional, that is compositional on truth values.…”

What is a logical theory? On theories containing assertions and denials

“…Unravelling the definitions, we may say that (3) This follows from Prop. 3.5, using the fact that C2 ⊲ C2 = C2, proved in item (2).…”

“…As soon as one entertains the possibility of a notion of compatibility that allows either for gappy or for glutty reasoning, the corresponding notion of theory and the associated space of theories will have to be suitably adapted. A general framework allowing for non-Tarskian notions of consequence that are characterized by non-canonical valuations referring to more than two logical values is set up in [2]. Extending our present foray into the land of theories towards covering manydimensional notions of entailment looks like a natural plan of attack for the future.…”