Truth-Value Constants in Multi-Valued Logics
Nissim Francez, Michael Kaminski · Outstanding contributions to logic · 2024
In some presentations of classical and intuitionistic logics, the objectlanguage is assumed to contain (two) truth-value constants : ⊤ (verum) and ⊥ (falsum), that are, respectively, true and false under every bivalent valuation. We are interested to define and study analogical constants ‡ , 1 ≤ i ≤ n , that in an arbitrary multi-valued logic over truth-values V = { v 1, . . . , vn } have the truth-value vi under every (multi-valued) valuation. As is well known, the absence or presence of such constants has a significant deductive impact on the logics studied. We define such constants proof-theoretically via their associated I / E -rules in a natural-deduction proof system. In particular, we propose a generalization of the notions of contradiction and explosiveness of a logic to the context of multi-valued logics.