Clausal Resolution for Modal Logics of Confluence

We present a clausal resolution-based method for normal multimodal logics of confluence, whose Kripke semantics are based on frames characterised by appropriate instances of the Church-Rosser property. Here we restrict attention to eight families of such logics. We show how the inference rules related to the normal logics of confluence can be systematically obtained from the parametrised axioms that characterise such systems. We discuss soundness, completeness, and termination of the method. In particular, completeness can be modularly proved by showing that the conclusions of each newly added inference rule ensures that the corresponding conditions on frames hold. Some examples are given in order to illustrate the use of the method.Comment: 15 pages, 1 figure. Preprint of the paper accepted to IJCAR 201

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Philosophical logics - a survey and a bibliography

Intensional logics attract the attention of researchers from differing academic backgrounds and various scientific interests. My aim is to sketch the philosophical background of alethic, doxastic, and deontic logics, their formal and metaphysical presumptions and their various problems and paradoxes, without attempting formal rigor. A bibliography, concise on philosophical writings, is meant to allow the reader\u27s access to the maze of literature in the field

From 2-sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

We extend to natural deduction the approach of Linear Nested Sequents and 2-sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction---only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalisation, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (following Scott) for formulating sound deduction rules

Standpoint Logic: A Logic for Handling Semantic Variability, with Applications to Forestry Information

It is widely accepted that most natural language expressions do not have precise universally agreed definitions that fix their meanings. Except in the case of certain technical terminology, humans use terms in a variety of ways that are adapted to different contexts and perspectives. Hence, even when conversation participants share the same vocabulary and agree on fundamental taxonomic relationships (such as subsumption and mutual exclusivity), their view on the specific meaning of terms may differ significantly. Moreover, even individuals themselves may not hold permanent points of view, but rather adopt different semantics depending on the particular features of the situation and what they wish to communicate. In this thesis, we analyse logical and representational aspects of the semantic variability of natural language terms. In particular, we aim to provide a formal language adequate for reasoning in settings where different agents may adopt particular standpoints or perspectives, thereby narrowing the semantic variability of the vague language predicates in different ways. For that purpose, we present standpoint logic, a framework for interpreting languages in the presence of semantic variability. We build on supervaluationist accounts of vagueness, which explain linguistic indeterminacy in terms of a collection of possible interpretations of the terms of the language (precisifications). This is extended by adding the notion of standpoint, which intuitively corresponds to a particular point of view on how to interpret vague terminology, and may be taken by a person or institution in a relevant context. A standpoint is modelled by sets of precisifications compatible with that point of view and does not need to be fully precise. In this way, standpoint logic allows one to articulate finely grained and structured stipulations of the varieties of interpretation that can be given to a vague concept or a set of related concepts and also provides means to express relationships between different systems of interpretation. After the specification of precisifications and standpoints and the consideration of the relevant notions of truth and validity, a multi-modal logic language for describing standpoints is presented. The language includes a modal operator for each standpoint, such that \standb{s}\phi means that a proposition $\phi$ is unequivocally true according to the standpoint $s$ --- i.e.\ $\phi$ is true at all precisifications compatible with $s$. We provide the logic with a Kripke semantics and examine the characteristics of its intended models. Furthermore, we prove the soundness, completeness and decidability of standpoint logic with an underlying propositional language, and show that the satisfiability problem is NP-complete. We subsequently illustrate how this language can be used to represent logical properties and connections between alternative partial models of a domain and different accounts of the semantics of terms. As proof of concept, we explore the application of our formal framework to the domain of forestry, and in particular, we focus on the semantic variability of `forest'. In this scenario, the problematic arising of the assignation of different meanings has been repeatedly reported in the literature, and it is especially relevant in the context of the unprecedented scale of publicly available geographic data, where information and databases, even when ostensibly linked to ontologies, may present substantial semantic variation, which obstructs interoperability and confounds knowledge exchange

Belief Semantics of Authorization Logic

Authorization logics have been used in the theory of computer security to reason about access control decisions. In this work, a formal belief semantics for authorization logics is given. The belief semantics is proved to subsume a standard Kripke semantics. The belief semantics yields a direct representation of principals' beliefs, without resorting to the technical machinery used in Kripke semantics. A proof system is given for the logic; that system is proved sound with respect to the belief and Kripke semantics. The soundness proof for the belief semantics, and for a variant of the Kripke semantics, is mechanized in Coq

One-Variable Fragments of First-Order Many-Valued Logics

In this thesis we study one-variable fragments of first-order logics. Such a one-variable fragment consists of those first-order formulas that contain only unary predicates and a single variable. These fragments can be viewed from a modal perspective by replacing the universal and existential quantifier with a box and diamond modality, respectively, and the unary predicates with corresponding propositional variables. Under this correspondence, the one-variable fragment of first-order classical logic famously corresponds to the modal logic S5. This thesis explores some such correspondences between first-order and modal logics. Firstly, we study first-order intuitionistic logics based on linear intuitionistic Kripke frames. We show that their one-variable fragments correspond to particular modal Gödel logics, defined over many-valued S5-Kripke frames. For a large class of these logics, we prove the validity problem to be decidable, even co-NP-complete. Secondly, we investigate the one-variable fragment of first-order Abelian logic, i.e., the first-order logic based on the ordered additive group of the reals. We provide two completeness results with respect to Hilbert-style axiomatizations: one for the one-variable fragment, and one for the one-variable fragment that does not contain any lattice connectives. Both these fragments are proved to be decidable. Finally, we launch a much broader algebraic investigation into one-variable fragments. We turn to the setting of first-order substructural logics (with the rule of exchange). Inspired by work on, among others, monadic Boolean algebras and monadic Heyting algebras, we define monadic commutative pointed residuated lattices as a first (algebraic) investigation into one-variable fragments of this large class of first-order logics. We prove a number of properties for these newly defined algebras, including a characterization in terms of relatively complete subalgebras as well as a characterization of their congruences