On rich modal logics

I thank to my advisor, João Marcos, for the intellectual support and patience that devoted me along graduate years. With his friendship, his ability to see problems of the better point of view and his love in to make Logic, he became a great inspiration for me. I thank to my committee members: Claudia Nalon, Elaine Pimentel and Benjamin Bedregal. These make a rigorous lecture of my work and give me valuable suggestions to make it better. I am grateful to the Post-Graduate Program in Systems and Computation that accepted me as student and provided to me the propitious environment to develop my research. I thank also to the CAPES for a 21 months fellowship. Thanks to my research group, LoLITA (Logic, Language, Information, Theory and Applications). In this group I have the opportunity to make some friends. Someone of them I knew in my early classes, they are: Sanderson, Haniel and Carol Blasio. Others I knew during the course, among them I’d like to cite: Patrick, Claudio, Flaulles and Ronildo. I thank to Severino Linhares and Maria Linhares who gently hosted me at your home in my first months in Natal. This couple jointly with my colleagues of student flat Fernado, Donátila and Aline are my nuclear family in Natal. I thank my fiancée Luclécia for her precious a ective support and to understand my absence at home during my master. I thank also my parents Manoel and Zenilda, my siblings Alexandre, Paulo and Paula.Without their confidence and encouragement I wouldn’t achieve success in this journey. If you want the hits, be prepared for the misses Carl Yastrzemski

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.

Proof theory of quantified modal logics

We introduce labelled sequent calculi for indexed modal logics. We prove that the structural rules of weakening and contraction are height-preserving admissible, that all rules are invertible, and that cut is admissible. Then we prove that each calculus introduced is sound and complete with respect to the appropriate class of transition frames.

A Finite Model Property for Gödel Modal Logics

A new semantics with the finite model property is provided and used to establish decidability for Gödel modal logics based on (crisp or fuzzy) Kripke frames combined locally with Gödel logic. A similar methodology is also used to establish decidability, and indeed co-NP-completeness for a Gödel S5 logic that coincides with the one-variable fragment of first-order Gödel logic.

A Hennessy-Milner Property for Many-Valued Modal Logics

A Hennessy-Milner property, relating modal equivalence and bisimulations, is defined for many-valued modal logics that combine a local semantics based on a complete MTL-chain (a linearly ordered commutative integral residuated lattice) with crisp Kripke frames. A necessary and sufficient algebraic condition is then provided for the class of image-finite models of these logics to admit the Hennessy-Milner property. Complete characterizations are obtained in the case of many-valued modal logics based on BL-chains (divisible MTL-chains) that are finite or have universe [0,1], including crisp Lukasiewicz, Gödel, and product modal logics.

Logics for belief functions on MV-algebras

In this paper we present a generalization of belief functions over fuzzy events. In particular we focus on belief functions defined in the algebraic framework of finite MV-algebras of fuzzy sets. We introduce a fuzzy modal logic to formalize reasoning with belief functions on many-valued events. We prove, among other results, that several different notions of belief functions can be characterized in a quite uniform way, just by slightly modifying the complete axiomatization of one of the modal logics involved in the definition of our formalism. © 2012 Elsevier Inc. All rights reserved.

A syntactic realization theorem for justification logics

Justification logics are refinements of modal logics where modalities are replaced by justification terms. They are connected to modal logics via so-called realization theorems. We present a syntactic proof of a single realization theorem that uniformly connects all the normal modal logics formed from the axioms \$mathsfd\$, \$mathsft\$, \$mathsfb\$, \$mathsf4\$, and \$mathsf5\$ with their justification counterparts. The proof employs cut-free nested sequent systems together with Fitting's realization merging technique. We further strengthen the realization theorem for \$mathsfKB5\$ and \$mathsfS5\$ by showing that the positive introspection operator is superfluous.

Decidability for some justification logics with negative introspection

Justification logics are modal logics that include justifications for the agent's knowledge. So far, there are no decidability results available for justification logics with negative introspection. In this paper, we develop a novel model construction for such logics and show that justification logics with negative introspection are decidable for finite constant specifications.

Modal interpolation via nested sequents

The main method of proving the Craig Interpolation Property (CIP) constructively uses cut-free sequent proof systems. Until now, however, no such method has been known for proving the CIP using more general sequent-like proof formalisms, such as hypersequents, nested sequents, and labelled sequents. In this paper, we start closing this gap by presenting an algorithm for proving the CIP for modal logics by induction on a nested-sequent derivation. This algorithm is applied to all the logics of the so-called modal cube.

A preference-based semantics for CTD reasoning

In [8] the authors developed a logical system based on the definition of a new non-classical connective ⊗ capturing the notion of reparative obligation. The system proved to be appropriate for handling well-known contrary-to-duty paradoxes but no model-theoretic semantics was presented. In this paper we fill the gap and define a suitable possible-world semantics for the system for which we can prove soundness and completeness. The semantics is a preference-based non-normal one extending and generalizing semantics for classical modal logics.

Group Beliefs : Studies on the Nature and Logic of Collective Doxastic Attitudes

This thesis studies the nature and logic of collective doxastic attitudes, or what is referred to in ordinary language as "group beliefs". Beliefs and other intentional attitudes are attributed to groups and collections of people, and such attributions are used to explain and predict the actions of groups. The thesis develops an understanding of group beliefs as voluntarily adopted views or acceptances rather than as ordinary beliefs. Such an understanding can provide new answers to questions concerning collective knowledge and justification of group beliefs, and it allows developing modal logics with collective doxastic and epistemic notions. The thesis consists of six articles. The first three articles are philosophical studies concerned with the nature of group beliefs. The last three articles are logical studies that aim at developing proof-theoretical calculi for reasoning about collective doxastic attitudes.

By indirections find directions out : thinkable worlds in Abbott and Vonnegut

This thesis is concerned with the interaction between literature and abstract thought. More specifically, it studies the epistemological charge of the literary, the type of knowledge that is carried by elements proper to fictional narratives into different disciplines. By concentrating on two different theoretical methods, the creation of thought experiments and the framing of possible worlds, methods which were elaborated and are still used today in spheres as varied as modal logics, analytic philosophy and physics, and by following their reinsertion within literary theory, the research develops the theory that both thought experiments and possible worlds are in fact short narrative stories that inform knowledge through literary means. By using two novels, Abbott’s Flatland and Vonnegut’s The Sirens of Titan, that describe extra-dimensional existence in radically different ways, respectively as a phenomenologically unknowable space and as an outward perspective on time, it becomes clear that literature is constitutive of the way in which worlds, fictive, real or otherwise, are constructed and understood. Thus dimensions, established through extensional analogies as either experimental knowledge or modal possibility for a given world, generate new directions for thought, which can then take part in the inductive/deductive process of scientia. By contrasting the dimensions of narrative with the way that dimensions were historically constituted, the research also establishes that the literary opens up an infinite potential of abstract space-time domains, defined by their specific rules and limits, and that these different experimental folds are themselves partaking in a dimensional process responsible for new forms of understanding. Over against science fiction literary theories of speculation that posit an equation between the fictive and the real, this thesis examines the complex structure of many overlapping possibilities that can organise themselves around larger compossible wholes, thus offering a theory of reading that is both non-mimetic and non-causal. It consequently examines the a dynamic process whereby literature is always reconceived through possibilities actualised by reading while never defining how the reader will ultimately understand the overarching structure. In this context, the thesis argues that a causal story can be construed out of any one interaction with a given narrative—underscoring, for example, the divinatory strength of a particular vision of the future—even as this narrative represents only a fraction of the potential knowledge of any particular literary text. Ultimately, the study concludes by tracing out how novel comprehensions of the literary, framed by the material conditions of their own space and time, endlessly renew themselves through multiple interactions, generating analogies and speculations that facilitate the creation of new knowledge.

Lower complexity bounds in justification logic

Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pure justification logics is typically lower than that of the corresponding modal logics. Moreover, the so-called reflected fragments, which still contain complete information about the respective justification logics, are known to be in~NP for a wide range of justification logics, pure and hybrid alike. This paper shows that, under reasonable additional restrictions, these reflected fragments are NP-complete, thereby proving a matching lower bound. The proof method is then extended to provide a uniform proof that the corresponding full pure justification logics are $\Pi^p_2$-hard, reproving and generalizing an earlier result by Milnikel.