François Lepage PARTIAL PROBABILITY FUNCTIONS AND INTUITIONISTIC LOGIC

We first present a sound and complete system of intuitionistic logic augmented with Nelson’s strong negation and interpreted in Kripke’s models’ structure. Then, we show that intuitionistic logic can naturally be interpreted in a modal trivalent logic (propositions are true, false or undefined). Secondly, we introduce a partial probability interpretation, which is inspired by Popper’s conditional probability functions and are characterized by the fact that conditions are not propositions but rather sets of propositions. Finally, we define the notion of probabilistic validity and we show using Kripke’s models that the intuitionistic system is sound and complete. 1. Motivation and background One starting point of this paper is partial logic. In [5] and [6], Lapierre and I showed that partial logic can be interpreted as a three-valued logic, the third value being the undefined value. Following Thijsse [13], we used Saturated deductively closed consistent sets (SDCCS) for the completeness proof. A set is SDCCS if it is deductively closed and if it contains (A ∨ B) only if it contains either A or B. If we restrict ourselves to partial propositional logic, the following system is proved to be sound and complete for Kleene strong 3-valued logic. ∗This work was supported by the Social Sciences and Humanities Research Council of Canada. Thanks to Grzegorz Malinowski who found a flaw in the first version of this paper.