What is modal logic

Invited Lectures.- Tableau Algorithms for Description Logics.- Modality and Databases.- Local Symmetries in Propositional Logic.- Comparison.- Design and Results of TANCS-2000 Non-classical (Modal) Systems Comparison.- Consistency Testing: The RACE Experience.- Benchmark Analysis with FaCT.- MSPASS: Modal Reasoning by Translation and First-Order Resolution.- TANCS-2000 Results for DLP.- Evaluating *SAT on TANCS 2000 Benchmarks.- Research Papers.- A Labelled Tableau Calculus for Nonmonotonic (Cumulative) Consequence Relations.- A Tableau System for Godel-Dummett Logic Based on a Hypersequent Calculus.- An Analytic Calculus for Quantified Propositional Godel Logic.- A Tableau Method for Inconsistency-Adaptive Logics.- A Tableau Calculus for Integrating First-Order and Elementary Set Theory Reasoning.- Hypertableau and Path-Hypertableau Calculi for some Families of Intermediate Logics.- Variants of First-Order Modal Logics.- Complexity of Simple Dependent Bimodal Logics.- Properties of Embeddings from Int to S4.- Term-Modal Logics.- A Subset-Matching Size-Bounded Cache for Satisfiability in Modal Logics.- Dual Intuitionistic Logic Revisited.- Model Sets in a Nonconstructive Logic of Partial Terms with Definite Descriptions.- Search Space Compression in Connection Tableau Calculi Using Disjunctive Constraints.- Matrix-Based Inductive Theorem Proving.- Monotonic Preorders for Free Variable Tableaux.- The Mosaic Method for Temporal Logics.- Sequent-Like Tableau Systems with the Analytic Superformula Property for the Modal Logics KB, KDB, K5, KD5.- A Tableau Calculus for Equilibrium Entailment.- Towards Tableau-Based Decision Procedures for Non-Well-Founded Fragments of Set Theory.- Tableau Calculus for Only Knowing and Knowing At Most.- A Tableau-Like Representation Framework for Efficient Proof Reconstruction.- The Semantic Tableaux Version of the Second Incompleteness Theorem Extends Almost to Robinson's Arithmetic Q.- System Descriptions.- Redundancy-Free Lemmatization in the Automated Model-Elimination Theorem Prover AI-SETHEO.- E-SETHEO: An Automated3 Theorem Prover.

Automated Reasoning with Analytic Tableaux and Related Methods: International Conference, TABLEAUX 2000 St Andrews, Scotland, UK, July 3-7, 2000 Proceedings

. In this paper we prove that the collection of all weakly distributive lattice ordered groups is a radical class and that it fails to be a torsion class.

Lattice ordered groups

Day 5: Applications and Extensions The precise topics for the course will depend in part on the interests and background of the students attending the lectures. This document contains an introduction to the course including some pointers to relevant literature. The focus is on the basics of neighborhood semantics with only pointers to literature relating to the more advanced topics. The website for the course is staff.science.uva.nl/∼epacuit/nbhd esslli.html On this website you will find the lecture slides (and possibly course notes) updated daily. Two other sources will also be useful:

Neighborhood Semantics for Modal Logic

Stone coalgebras

In the present paper, we introduce a multi-type calculus for the logic of measurable Kleene algebras, for which we prove soundness, completeness, conservativity, cut elimination and subformula property. Our proposal imports ideas and techniques developed in formal linguistics around the notion of structural control.

Kleene algebras, adjunction and structural control

We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. Second, we provide embeddings of the calculus in the theorem prover Isabelle for formalising proofs about D.EAK. As a case study we show that the solution of the muddy children puzzle is derivable for any number of muddy children. Third, there is a set of meta-tools, that allows us to adapt the tool for a wide variety of user defined calculi.

Tool support for reasoning in display calculi

Sahlqvist-style correspondence results remain a perennial theme and an active topic of research within modal logic. Recently, there has been interest in extending the classical results in this area to the modal mu-calculus [7]. For instance, in [8] van Benthem, Bezhanishvili and Hodkinson define a class of Sahlqvist formulas for the modal mu-calculus, all of which have frame correspondents in first-order logic with least fixed points (FO+LFP). The analysis pursued in that paper is model theoretic in the tradition of the well known Sahlqvist-van Benthem algorithm. Parallel to the model theoretic approach to this type of result, there exists an algebraic-algorithmic approach (see e.g., [3, 4, 5]) which derives correspondence (and canonicity) results by means of ‘calculi of correspondence’ consisting of simple derivation rules which depend for their soundness on the order theoretic properties of the operations interpreting the logical connectives in the algebraic semantics. As indicated in Part 1 [2], these rules are divided into approximation and adjunction rules, together with the Ackermann rules used to eliminate propositional variables. As also indicated in [2], the major challenge to extending the algebraic-algorithmic approach to the modal mu-calculus is obtaining sound approximation and adjunction rules for formulas involving fixed point binders: in the binder-free setting each connective can be treated in isolation on its own order-theoretic merits; in the presence of binders the the analysis must take into account the order-theoretic behaviour of whole formulas. As a result the derivation rules become significantly more involved.

/pdf/algorithmic-correspondence-for-intuitionistic-modal-mu-53hr551rij.pdf

Algorithmic correspondence for intuitionistic modal mu-calculus, Part 2

Dynamic Logics (DLs) form a large family of nonclassical logics, and perhaps the one enjoying the widest range of applications. Indeed, they are designed to formalize change caused by actions of diverse nature: updates on the memory state of a computer, displacements of moving robots in an environment, measurements in models of quantum physics, belief revisions, knowledge updates, etc. In each of these areas, DL-formulas express properties of the model encoding the present state of affairs, as well as the preand post-conditions of a given action. Actions are semantically represented as transformations of one model into another, encoding the state of affairs after the action has taken place. DL-languages are expansions of classical (static) logic with dynamic operators, parametrized with actions; dynamic operators are modalities interpreted in terms of the transformation of models corresponding to their action-parameters. However, when dynamic logics feature both dynamic and ‘static’ modalities, as in the case of the Dynamic Epistemic Logics (DEL), they typically lose many desirable properties, such as the closure under uniform substitution, which facilitate a smooth algebraic and proof-theoretic treatment. This and other difficulties make their algebraic and proof-theoretic theory presently underdeveloped, and the few existing proposals seem quite ad hoc: in particular, no proposed calculus in the literature enjoys the minimal conditions enabling the specification of the meaning of a logical symbol in the sense of proof-theoretic semantics. We developed a family of display-style, cut-free sequent calculi for dynamic epistemic logics on both an intuitionistic and a classical base [7]. Like the standard display calculi, these calculi are modular: just by modifying the structural rules according to Dosen’s principle [12], these calculi are generalizable both to different Dynamic Logics (Epistemic, Deontic, etc.) and to different propositional bases (Linear, Relevant, etc.). Moreover, the rules they feature agree with the standard relational semantics for dynamic epistemic logics, while a ‘natural’ coalgebraic semantics was introduced in [7]. However, these calculi still feature a peculiar set of contextual rules, i.e. structural and operational rules involving a context (understood as the precondition for the applicability of an action), which make them non-standard as display calculi. To fix this shortcoming, in [8] we propose a new class of more general sequent calculi (which we call dynamic calculi) that overcome the need of contextual rules and enjoy other benefits, including a natural proof-theoretic semantics. In this talk we present two concrete examples of dynamic calculi for one specific dynamic epistemic logic [2], both in its classical and in its intuitionistic version, where the Intuitionistic logic for Epistemic Action and Knowledge was intoduced via duality in [10] defining dynamic algebraic semantics on intuitionistic base. These calculi can be understood as generalizations of the display calculi [4], since they feature: 1. constituents of several types (e.g. of type AG for ‘agents’ or ACT for ‘actions’), along with propositional constituents, which in their turn can be of different types (e.g. intuitionistic propositions iPROP, or classical propositions cPROP, etc.); N. Galatos, A. Kurz, C. Tsinakis (eds.), TACL 2013 (EPiC Series, vol. 25), pp. 85–87 85 Sequent calculus for dynamic epistemic logic Greco, Kurz and Palmigiano 2. non-contextual1 structural connectives (structural conjunction and disjunction, implication and disimplication — also called exclusion —, and their neutral elements) by which structures are built, where a structure can be homogeneous (e.g. α ∧ β is of type ACT ×ACT ) or heterogeneous (e.g. α ∧ A is of type ACT × PROP).2 3. structural introduction rules, which constitues a pure structural calculus because the structural and operational level are now completely separated (e.g. (X ` Y)∧ (W ` Z) / X∧W ` Y∧ Z); 4. unary translation rules, to pass from structural to operational level: the flattening rule, making it possible to transform homogeneous structures into operational formulas (e.g. A ∧ B ` X / A∧B ` X, where A∧B is a propositional formula of type PROP×PROP→ PROP), and the currying rule, making it possible to transform heterogeneous structures into ‘parametrized’ operational formulas (e.g. α ∧ A ` X / 〈α〉A ` X, where 〈α〉A is a propositional modal formula of type ACT ×PROP→ PROP). These calculi are interesting from both a methodological and an applicative viewpoint. Methodologically, the dynamic calculi achieve a unified framework simultaneously accounting for different logical behaviors: modalities (e.g. exponentials or temporal operators) and quantifiers can be seen in terms of combinations of heterogeneous components by means of operational rules at the merging-level; the behavior of a single modal operator (e.g. monotonicity) or the interaction between modalities (e.g. dynamic and epistemic operators) can be captured by means of structural rules at the merging-level. We will also discuss the categorial semantics, which seems the most ‘appropriate’ for this kind of calculi. From the point of view of the applications, dynamic calculi provide a way of generating specific display calculi for a wide array of logics — which includes but is not limited to dynamic (epistemic) logics — simply by deriving the appropriate rules; this allows to obtain meta-theorems such as cut elimination [8] uniformly for each specific calculus as a byproduct of the general theory of dynamic calculi. The dynamic calculi enjoy the strong version of Wansing’s fundamental properties of segregation, symmetricity and explicitness [12]. Therefore, they share all the benefits that come from the display calculi, but are more general: for instance they capture also dynamic poly-modal logics on substructural base.

/pdf/dynamic-sequent-calculus-for-the-logic-of-epistemic-actions-3vv5q6wss6.pdf

Dynamic sequent calculus for the logic of Epistemic Actions and Knowledge

We introduce the logic LRC, designed to describe and reason about agents' abilities and capabilities in using resources. The proposed framework bridges two - up to now - mutually independent strands of literature: the one on logics of abilities and capabilities, developed within the theory of agency, and the one on logics of resources, motivated by program semantics. The logic LRC is suitable to describe and reason about key aspects of social behaviour in organizations. We prove a number of properties enjoyed by LRC (soundness, completeness, canonicity, disjunction property) and its associated analytic calculus (conservativity, cut elimination and subformula property). These results lay at the intersection of the algebraic theory of unified correspondence and the theory of multi-type calculi in structural proof theory. Case studies are discussed which showcase several ways in which this framework can be extended and enriched while retaining its basic properties, so as to model an array of issues, both practically and theoretically relevant, spanning from planning problems to the logical foundations of the theory of organizations.

The Logic of Resources and Capabilities

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Alessandra Palmigiano

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