Matteo Pascucci

This article concerns the treatment of propositional quantification in a framework of labelled natural deduction for modal logic developed by Basin, Matthews and Viganò. We provide a detailed analysis of a basic calculus that can be used for a proof-theoretic rendering of minimal normal multimodal systems with quantification over stable domains of propositions. Furthermore, we consider variations of the basic calculus obtained via relational theories and domain theories allowing for quantification over possibly unstable domains of propositions. The main result of the article is that fragments of the labelled calculi not exploiting reductio ad absurdum enjoy the Church–Rosser property and the strong normalization property; such result is obtained by combining Girard’s method of reducibility candidates and labelled languages of lambda calculus codifying the structure of modal proofs.

The deontic status of tautologies and contradictions is one of the major puzzles for authors of early works on deontic logic. It is well-known that von Wright addresses this problem by adopting a Principle of Deontic Contingency, which says that tautologies are not necessarily obligatory and contradictions are not necessarily forbidden. A more radical solution is proposed by Anderson within a reductionist approach to deontic logic and consists in restricting the range of application of deontic modalities to contingent propositions. Anderson’s solution has not received much attention in the literature, despite reflecting a typical feature of ordinary deontic reasoning, where non-contingent propositions are rarely, if ever, taken into account. In the present article we explore some of its formal consequences, providing a taxonomy of the properties of the Andersonian operators of obligation and permission for contingent propositions, O′ and P′, in the class of normal alethic systems.

This article concerns the metatheory of a class of modal logics whose language includes propositional constants of various kinds. The main novelties are the use of general frames with specific restrictions and the definition of the strict range of a formula. Many examples from the literature are treated within the framework provided and some traditional model-theoretic issues such as preservation results concerning the validity of formulas and definability results concerning frame properties are addressed.

This work concerns the use of operators for past and future contingency in Priorean temporal logic. We will develop a system named C_t, whose language includes a propositional constant and prove that (I) C_t is complete with respect to a certain class of general frames and (II) the usual operators for past and future necessity are definable in such system. Furthermore, we will introduce the extension C_t(lin) that can be interpreted on linear and transitive general frames. The theoretical result of the current work is that contingency can be treated as a primitive notion in reasoning about temporal modalities.