Bivalence and Future Contingency

Peer reviewe

Studies on modal logics of time and space

This dissertation presents original results in Temporal Logic and Spatial Logic. Part I concerns Branching-Time Logic. Since Prior 1967, two main semantics for Branching-Time Logic have been devised: Peircean and Ockhamist semantics. Zanardo 1998 proposed a general semantics, called Indistinguishability semantics, of which Peircean and Ockhamist semantics are limit cases. We provide a finite axiomatization of the Indistinguishability logic of upward endless bundled trees using a non-standard inference rule, and prove that this logic is strongly complete. In Part II, we study the temporal logic given by the tense operators F for future and P for past together with the derivative operator , interpreted on the real numbers. We prove that this logic is neither strongly nor Kripke complete, it is PSPACE-complete, and it is finitely axiomatizable. In Part III, we study the spatial logic given by the derivative operator and the graded modalities {n | n in N}. We prove that this language, call it L, is as expressive as the first-order language Lt of Flum and Ziegler 1980 when interpreted on T3 topological spaces. Then, we give a general definition of modal operator: essentially, a modal operator will be defined by a formula of Lt with at most one free variable. If a modal operator is defined by a formula predicating only over points, then it is called point-sort operator. We prove that L, even if enriched with all point-sort operators, however enriched with finitely many modal operators predicating also on open sets, cannot express Lt on T2 spaces. Finally, we axiomatize the logic of any class between all T1 and all T3 spaces and prove that it is PSPACE-complete.Open Acces

Ockhamist Propositional Dynamic Logic: a natural link between PDL and CTL

International audienceWe present a new logic called Ockhamist Propositional Dynamic Logic, OPDL, which provides a natural link between PDL and CTL*. We show that both PDL and CTL* can be polynomially embedded into OPDL in a rather simple and direct way. More generally, the semantics on which OPDL is based provides a unifying framework for making the dynamic logic family and the temporal logic family converge in a single logical framework. Decidability of the satisfiability problem for OPDL is studied in the paper

Deontic Modality in Rationality and Reasoning

Deontic Modality in Rationality and Reasoning Lay Summary Alessandra Marra The present dissertation investigates certain facets of the logical structure of oughts – where “ought” is used as a noun, roughly meaning obligation. I do so by following two lines of inquiry. The first part of the thesis places oughts in the context of practical rationality. The second part of the thesis concerns the inference rules governing arguments about oughts, and specifically the inference rule of Reasoning by Cases. These two lines of inquiry, together, aim to expound upon oughts in rationality and reasoning. The methodology used in this dissertation is the one of philosophical logic, in which logical, qualitative models are developed to support and foster conceptual analysis. The dissertation consists of four main chapters. The first two chapters are devoted to the role of oughts in practical rationality. I focus on the so-called Enkratic principle of rationality, which – in its most general formulation – requires that if an agent believes sincerely and with conviction that she ought to do X, then she intends to X. I develop a logical framework to investigate the (static and dynamic) relation between those oughts believed by the agent and her intentions. It is shown that, under certain minimal assumptions, the Enkratic principle of rationality is a principle of limited validity. The following two chapters of the dissertation constitute a study of the classical inference rule of Reasoning by Cases, which – in its simplest form – moves from the premises “A or B”, “if A then C” and “if B then C” to the conclusion “C”. Recent literature has called the validity of Reasoning by Cases into question, with the most influential counterexample being the so-called Miners’ Puzzle – an instance of Reasoning by Cases where “C” involves oughts. I provide a unifying explanation of why the Miners’ Puzzle emerges. It is shown that, within specific boundaries, Reasoning by Cases is a valid inference rule

Temporal STIT logic and its application to normative reasoning

International audienceI present a variant of STIT with time, called T-STIT (Temporal STIT), interpreted in standard Kripke semantics. On the syntactic level, T-STIT is nothing but the extension of atemporal individual STIT by: (i) the future tense and past tense operators, and (ii) the operator of group agency for the grand coalition (the coalition of all agents). A sound and complete axiomatisation for T-STIT is given. Moreover, it is shown that T-STIT supports reasoning about interesting normative concepts such as the concepts of achievement obligation and commitment

Labeled natural deduction for temporal logics

Nonostante la notevole rilevanza delle logiche temporali in molti campi dell'informatica, la loro analisi teorica non è certo da ritenersi conclusa. In particolare, molti sono i punti ancora aperti nell'ambito della teoria della dimostrazione, specialmente se consideriamo le logiche temporali di tipo branching. Il principale contributo di questa tesi consiste nella presentazione di un approccio modulare per la definizione di sistemi di deduzione naturale etichettata per un'ampia gamma di logiche temporali. Viene innanzitutto proposto un sistema per la logica temporale minimale di Prior; si mostra quindi come estenderlo in maniera modulare allo scopo di trattare logiche più complesse, quali ad esempio LTL. Viene infine proposta un'estensione al caso delle logiche branching, concentrando l'attenzione sulle logiche con semantica di tipo Ockhamist e bundled. Per i sistemi proposti, viene condotta una dettagliata analisi dal punto di vista della teoria della dimostrazione. In particolare, nel caso delle logiche del tempo discreto, per le quali si richiedono regole che modellino un principio di induzione, viene definita una procedura di normalizzazione ispirata da quelle dei sistemi per l'Aritmetica di Heyting. Come conseguenza, si ottiene una dimostrazione puramente sintattica della consistenza dei sistemi.Despite the great relevance of temporal logics in many applications of computer science, their theoretical analysis is far from being concluded. In particular, we still lack a satisfactory proof theory for temporal logics and this is especially true in the case of branching-time logics. The main contribution of this thesis consists in presenting a modular approach to the definition of labeled (natural) deduction systems for a large class of temporal logics. We start by proposing a system for the basic Priorean tense logic and show how to modularly enrich it in order to deal with more complex logics, like LTL. We also consider the extension to the branching case, focusing on the Ockhamist branching-time logics with a bundled semantics. A detailed proof-theoretical analysis of the systems is performed. In particular, in the case of discrete-time logics, for which rules modeling an induction principle are required, we define a procedure of normalization inspired to those of systems for Heyting Arithmetic. As a consequence of normalization, we obtain a purely syntactical proof of the consistency of the systems