Beyond Three Inferential Values

Abstract

The actual introduction of a non-reflexive and non-idempotent q-consequence gave birth to the concept of logical three-valuedness based on the idea of noncomplementary categories of rejection and acceptance. A q-consequence may not have bivalent description, the property claimed by Suszko’s Thesis on logical two-valuedness, (ST), of structural logics, i.e. structural consequence operations. Recall that (ST) shifts logical values over the set of matrix values and it refers to the division of matrix universe into two subsets of designated and undesignated elements using their characteristic functions as logical valuations, cf. [4] The extension of the idea operates with three-valued function, with the third value ascribed to those elements of the matrix which are neither rejected nor accepted. Accordingly, the logical three-valuedness departs naturally from the division of the matrix universe into three subsets and the (ST) counterpart says that any inference based on a structural q-consequence may have a bivalent or a three-valued description.

After a short presentation of the three-valued inferential framework, we discuss a solution for further exploration of the idea leading to logical n-valuedness for n > 3. Apparently, the first step in that direction is easy and it consists of a division of the matrix universe into more than three subsets. The next move, i.e. a definition of a matrix consequence-like relation being neither a consequence nor a q-consequence, seems extremely difficult. Therefore, here we consider only finite linear matrices with one-argument functions “labelling” respective matrix subsets. By means of these functions it is possible to represent a q-consequence as a “partial” Tarski’s consequence and, ultimately, to define a logically more-valued consequence-like relation. We believe, that the present partial proposal deserves an attention by itself but also that it may lead to a general approach to logically many-valued inference.