Some remarks on the logic of vagueness
Ayda I. Arruda, Elias Humberto Alves · 1979
Our objective here is to offer a contribution to the study of the logic of vagueness. We restrict ourselves to “vagueness related to negation”. This kind of vagueness happens when the law of excluded middle or the law of contradiction is not valid. We also consider only cases in which these two laws are not equivalent (when these laws are not valid but equivalent, we obtain systems related to dialectical logic like those suggested in [3]). Consequently, we have four cases of vagueness related to negation: 1) The law of excluded middle is valid but not the law of contradiction; 2) The law of contradiction is valid but not the law of excluded middle; 3) One of these laws is valid but not both; 4) Both these laws are not valid. For each case we present a logical system that formalizes the situation. We develop here only the propositional calculi; the corresponding first-order predicate calculi are easy to construct. The logical system for the first case is the system C1 of da Costa [2]. In a certain sense, the logical systems for the first two cases are “duals”. The semantical study of the systems presented here will be done in [1].