Non-monotonic Weak Kleene Logics

AGUSTINA BORZI; MARTINA ZIRATTU

Ciudad de México

Workshop; 15th Latin American Workshop on Logic, Languages, Algorithms and New Methods of Reasoning (LANMR); 2023

Facultad de Ingeniería y Facultad de Ciencias, UNAM

When working on a multiple-premise and multiple-conclusion setting, in most logical systems the comma on the premise side matches the behavior of the object language conjunction, whilst on the conclusion side it matches that of the disjunction. However, this is not the case for Weak Kleene logics, which have a three-valued semantics in which the non-classical truth-value behaves in an “infectious” way, i.e. as an absorbing or zero element for all the operations, and the third value can be designated (resulting in the paraconsistent logic PWK) or not (resulting in the paracomplete logic WK3). In this systems, the metalinguistic comma alluded earlier doesn’t behave as the expected binary associative connective featured in the language. This suggests that there are two distinct ways of conjoining (disjoining) formulae: one can either use conjunction (disjunction) or the comma to gather premises (conclusions) together in PWK (WK3), resulting in a different set of validities in each case. Our main objective here is to restore the equivalence of the metalinguistic comma to that of the operations available in the language. To do so, we present what we call "uniform Weak Kleene logics", which turn out to be non-monotonic, and give a sequent calculi presentation for all the systems introduced here.