Conditionals, probability, and non-triviality
\textit{David Lewis}, in his ``Probabilities of conditionals and conditional probability'' [Philosophical Review 85, 279-315 (1976)], showed that, on plausible assumptions about probability functions, Adams's project of setting the probability of conditionals equal to the conditional probability trivialises the probability functions [\textit{E. W. Adams}, The logic of conditionals (1975; Zbl 0324.02002)]. Here, the authors prove: (a) That the implication fragment of intuitionist logic is the weakest logic for which the probability of a conditional is the conditional probability; (b) That adding negation (even with very weak assumptions) is sufficient to trivialise the resulting system. The system of \textit{B. C. van Fraassen} [``Probabilities of conditionals'', in: Foundations of probability theory, statistical inference, and statistical theories in science, Vol. I, Proc. Int. Colloq., London/Can. 1973, 261-300 (1976; Zbl 0339.02025)]\ succumbs to triviality of Adams's thesis is added to it; (c) There are various logics with \(\&\), \(\vee\), and \(F\), stronger than minimal logic but weaker than classical logic, which have a probabilistic semantics.
implication fragment of intuitionist logic
scientific article; zbMATH DE number 819339