On Vidal's trivalent explanations for defective conditional in mathematics

ABSTRACT The paper deals with a problem posed by Mathieu Vidal to provide a formal representation for defective conditional in mathematics Vidal, M. [(2014). The defective conditional in mathematics. Journal of Applied Non-Classical Logics, 24(1–2), 169–179]. The key feature of defective conditional is that its truth-value is indeterminate if its antecedent is false. In particular, we are interested in two explanations given by Vidal with the use of trivalent logics. By analysing a simple argument from plane geometry, where defective conditional is in use, he gives two trivalent formal explanations for it. For both explanations, Vidal rigorously shows that (most well-known) trivalent logics cannot adequately represent defective conditional. Preserving Vidal's criteria of defective conditional ad max, we indicate some arguable points in his explanations and present an alternative explanation containing the original conjunction and disjunction in order to show that there are trivalent logics that might be an adequate formal explanation for defective conditional.