Inference Rules in some temporal multi-epistemic propositional logics

Multi-modal logics are among the best tools developed so far to analyse human reasoning and agents’ interactions. Recently multi-modal logics have found several applications in Artificial Intelligence (AI) and Computer Science (CS) in the attempt to formalise reasoning about the behavior of programs. Modal logics deal with sentences that are qualified by modalities. A modality is any word that could be added to a statement p to modify its mode of truth. Temporal logics are obtained by joining tense operators to the classical propositional calculus, giving rise to a language very effective to describe the flow of time. Epistemic logics are suitable to formalize reasoning about agents possessing a certain knowledge. Combinations of temporal and epistemic logics are particularly effective in describing the interaction of agents through the flow of time. Although not yet fully investigated, this approach has found many fruitful applications. These are concerned with the development of systems modelling reasoning about knowledge and space, reasoning under uncertainty, multi-agent reasoning et c. Despite their power, multi modal languages cannot handle a changing environment. But this is exactly what is required in the case of human reasoning, computation and multi-agent environment. For this purpose, inference rules are a core instrument. So far, the research in this field has investigated many modal and superintuitionistic logics. However, for the case of multi-modal logics, not much is known concerning admissible inference rules. In our research we extend the investigation to some multi-modal propositional logics which combine tense and knowledge modalities. As far as we are concerned, these systems have never been investigated before. In particular we start by defining our systems semantically; further we prove such systems to enjoy the effective finite model property and to be decidable with respect to their admissible inference rules. We turn then our attention to the syntactical side and we provide sound and complete axiomatic systems. We conclude our dissertation by introducing the reader to the piece of research we are currently working on. Our original results can be found in [9, 4, 11] (see Appendix A). They have also been presented by the author at some international conferences and schools (see [8, 10, 5, 7, 6] and refer to Appendix B for more details). Our project concerns philosophy, mathematics, AI and CS. Modern applications of logic in CS and AI often require languages able to represent knowledge about dynamic systems. Multi-modal logics serve these applications in a very efficient way, and we would absorb and develop some of these techniques to represent logical consequences in artificial intelligence and computation

Sequence Semantics for Modelling Reason-based Preferences

We study how the non-classical n-ary operator circle times, originally intended to capture the concept of reparative obligation, can be used in the context of social choice theory to model preferences. A novel possible-world model-theoretic semantics, called sequence semantics, was proposed for the operator. In this paper, we propose a sound and complete axiomatisation of a minimal modal logic for the operator, and we extend it with axioms suitable to model social choice consistency principles such as extension consistency and contraction consistency. We provide completeness results for such extensions

Non-normal modal logics, quantification, and deontic dilemmas. A study in multi-relational semantics

This dissertation is devoted to the study of non-normal (modal) systems for deontic logics, both on the propositional level, and on the first order one. In particular we developed our study the Multi-relational setting that generalises standard Kripke Semantics. We present new completeness results concerning the semantic setting of several systems which are able to handle normative dilemmas and conflicts. Although primarily driven by issues related to the legal and moral field, these results are also relevant for the more theoretical field of Modal Logic itself, as we propose a syntactical, and semantic study of intermediate systems between the classical propositional calculus CPC and the minimal normal modal logic K

Sequence Semantics for Norms and Obligations

This paper presents a new version of the sequence semantics presented at DEON 2014. This new version allows us for a capturing the distinction between logic of obligations and logic of norms. Several axiom schemata are discussed, while soundness and completeness results are proved

Admissible inference rules in the linear logic of knowledge and time LTK

The paper investigates admissible inference rules for the multi-modal logic LTK, which describes a combination of linear time and knowledge. This logic is semantically defined as the set of all LTK-valid formulae, where LTK-frames are multi-modal Kripke-frames combining a linear and discrete representation of the flow of time with special S5-like modalities, defined at each time cluster and representing knowledge. We start by revising the effective finite model property in this particular case, while the central part of the paper is devoted to constructing special n-characterising models for LTK. Such structeres allow us to find an algorithm determining admissible inference rules in LTK; the main result of this work is that LTK is decidable with respect to inference rules. \ua9 Copyright 2006 Oxford University Press

Quantification in Some Non-normal Modal Logics

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal systems by Stolpe (Logic Journal of the IGPL11(5), 557\u2013575, 2003), but unfortunately that work used only models and not frames

Combining time and knowledge, semantic approach

The paper investigates a semantic approach for combining knowledge and time. We introduce a multi-modal logic LF containing modalities for knowledge and time in a semantic way, as the set of all F-valid formulae for a class of special frames F. The main result of our paper is the theorem stating that LTK is decidable and giving a resolving algorithm. The result is proven by using standard tools: filtration, bulldozing and contracting p-morphisms

An axiomatisation for the multi-modal logic of knowledge and linear time LTK

The paper aims at providing the multi-modal propositional logic LTK with a sound and complete axiomatisation. This logic combines temporal and epistemic operators and focuses on m odeling the behaviour of a set of agents operating in a system on the background of a temporal framework. Time is represented as linear and discrete, whereas knowledge is modeled as an S5-like modality. A further modal operator intended to represent environment knowledge is added to the system in order to achieve the expressive power sufficient to describe the piece of information available to the agents at each moment in the flow of time

Semantics for modelling reason-based preferences

In [13] the authors developed a logical system based on the definition of a new non-classical connective 97 originally capturing the notion of reparative obligation. The operator 97 and the system were proved to be appropriate for rather handling well-known contrary-to-duty paradoxes. Later on, a suitable model-theoretic possible-world semantics has been developed [4,5]. In this paper we show how a version of this semantics can be used to develop a sound and complete logic of preference and offer a suitable possible-world semantics. The semantics is a sequence-based non-normal one extending and generalising semantics for classical modal logics

Legal responsibility for the acts of others: A logical analysis

This paper offers a logical analysis of two cases where legal responsibility may emerge for the acts of others: (a) reflex responsibility, and (b) responsibility in the negotiorum gestio doctrine. The current contribution works within a fresh multi-modal system where the new operators are introduced for denoting intentions and actions in the interest of other agents, and the objectively ideal sets of actions for agents