Inferential Intensionality

The paper is a study of properties of quasi-consequence operation which is a key notion of the so-called inferential approach in the theory of sentential calculi established in [5].The principal motivation behind the quasi-consequence, q-consequence for short, stems from the mathematical practice which treats some auxiliary assumptions as mere hypotheses rather than axioms and their further occurrence in place of conclusions may be justified or not. The main semantic feature of the q-consequence reflecting the idea is that its rules lead from the non-rejected assumptions to the accepted conclusions. First, we focus on the syntactic features of the framework and present the q-consequence as related to the notion of proof. Such a presentation uncovers the reasons for which the adjective ”inferential” is used to characterize the approach and , possibly, the term ”inference operation” replaces ”q-consequence”. It also shows that the inferential approach is a generalisation of the Tarski setting and, therefore, it may potentially absorb several concepts from the theory of sentential calculi, cf. [10]. However, as some concrete applications show, see e.g. [4], the new approach opens perspectives for further exploration. The main part of the paper is devoted to some notions absent in Tarski approach. We show that for a given q-consequence operation W instead of one W -equivalence established by the properties of W we may consider two congruence relations. For one of them the current name is preserved and for the other the term ”W -equality” is adopted. While the two relations coincide for any W which is a consequence operation, for an arbitrary W the inferential equality and the inferential equivalence may differ. Further to this we introduce the concepts of inferential extensionality and intensionality for q-consequence operations and connectives. Some general results obtained in Section 3 sufficiently confirm the importance of these notions. To complete a view, in Section 4 we apply the new intensionality-extensionality distinction to inferential extensions of a version of the Lukasiewicz four valued modal logic.