One Variable Relevant Logics are S5ish

A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory (display calculus). It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework.

The Tarskian classical relevant logic \(\mathsf {TR}\) arises from Tarski’s work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables, presented by Tarski and Givant their book, A Formalization of Set Theory without Variables, and summarized in the first nine sections. \(\mathsf {TR}\) is closely related to the well-known logic \(\mathsf {KR}\). Every formula of relevance logic has a corresponding sentence in Tarski’s extended first-order logic of binary relations with operators on the relation symbols. A formula is in \(\mathsf {TR}\) (by definition), or in \(\mathsf {KR}\) (by a theorem), if and only if its corresponding sentence can be proved in first-order logic, using at most four variables, from the assumptions that all binary relations are dense and, for \(\mathsf {TR}\), commute under composition, or, for \(\mathsf {KR}\), are symmetric. The vocabulary of \(\mathsf {TR}\) is the same as the classical relevant logic \(\mathsf {CR^*}\) proposed by Meyer and Routley but \(\mathsf {TR}\) properly contains \(\mathsf {CR^*}\). The frames characteristic for \(\mathsf {TR}\) are the ones that are characteristic for \(\mathsf {CR^*}\) and satisfy an extra frame condition. There are formulas in \(\mathsf {TR}\) (but not in \(\mathsf {CR^*}\)) that correspond to this frame condition and provide a counterexample to a theorem of T. Kowalski. The frames characteristic for \(\mathsf {TR}\), or \(\mathsf {KR}\), are the ones whose complex algebras are integral dense relation algebras that are commutative, or symmetric, respectively. For both classes, the number of isomorphism types grows like the number of isomorphism types of ternary relations. Asymptotic formulas are obtained for both classes. Similar results apply to a hierarchy of logics defined by the number of variables used in the first-order proofs of their corresponding sentences.

Boolean contact algebras constitute a suitable algebraic theory for qualitative spatial reasoning. They are Boolean algebras with an additional contact relation grasping the topological aspect of spatial entities. In this paper we present a logic that combines propositional logic, relevance logic, and modal logic to reason about Boolean contact algebras. This is done in two steps. First, we use the relevance logic operators to obtain a logic suitable for Boolean algebras. Then we add modal operators that are based on the contact relation. In both cases we present axioms that are equivalent to requirement that every frame for the logic is indeed a Boolean algebra resp. Boolean contact algebra. We also provide a natural deduction system for this logic by defining introduction and eliminations rules for each logical operator. The system is shown to be sound. Furthermore, we sketch an implementation of the natural deduction system in the functional programming language and interactive theorem prover Coq.

The technique of modeling propositional logics by means of finite algebras is at least as old as many-valued logic and has provided much information about such non-classical systems as relevant logics, linear logic, and connexive logic. The problem of enumerating these algebras is one of satisfying finite constraints and so solutions to it generalise widely. Generate and Test methods suffer from a prohibitive exponential complexity, which has generally inhibited investigation except in trivial cases.

We introduce a general notion of semi-implication which generalizes both the implications used in classical, intuitionistic, and many other logics, as well as those used in relevance logics. It is mainly based on the relevant deduction property (RDP)—a weak form of the classical-intuitionistic deduction theorem which has motivated the design of the intensional fragments of the relevance logic R. However, CL ↔, the pure equivalential fragment of classical logic, also enjoys the RDP with respect to ↔. We show that in the language of → this is the only exception. This observation leads to an adequate definition of semi-implication, according to which a finitary logic L has a semi-implication → iff L has a strongly sound and complete Hilbert-type system which is an extension by axiom schemas of HR → (the standard Hilbert-type system for the implicational fragment of R). We also show that in the presence of a conjunction, or a disjunction, or an implication, a connective → of a logic L is a semi-implication iff it is an implication (i.e. it satisfies the classical-intuitionistic deduction theorem), and the same is true if L is induced by a matrix which has a single designated value or a single non-designated value.

This paper explores formal verification in the area of database technology, in particular how to reason about queries and updates in a database system. The formalism is sufficiently general to be applicable to relational and graph databases. We first define a domain-specific language consisting of nested query and update primitives, and give its operational semantics. Queries are in full first-order logic. The problem we try to solve is whether a database satisfying a given pre-condition will satisfy a given post-condition after execution of a given sequence of queries and updates. We propose a weakest-precondition calculus and prove its correctness. We finally examine a restriction of our framework that produces formulas in the guarded fragment of predicate logic and thus leads to a decidable proof problem.

First-order temporal logic, the extension of first-order logic with operators dealing with time, is a powerful and expressive formalism with many potential applications. This expressive logic can be viewed as a framework in which to investigate problems specified in other logics. The monodic fragment of first-order temporal logic is a useful fragment that possesses good computational properties such as completeness and sometimes even decidability. Temporal logics of knowledge are useful for dealing with situations where the knowledge of agents in a system is involved. In this paper we present a translation from temporal logics of knowledge into the monodic fragment of first-order temporal logic. We can then use a theorem prover for monodic first-order temporal logic to prove properties of the translated formulas. This allows problems specified in temporal logics of knowledge to be verified automatically without needing a specialized theorem prover for temporal logics of knowledge. We present the translation, its correctness, and examples of its use.

In this article we investigate the family of independence-friendly (IF) logics in the equality-free setting, concentrating on questions related to expressive power. Various natural equality-free fragments of logics in this family translate into existential second-order logic with prenex quantification of function symbols only and with the first-order parts of formulae equality-free. We study this fragment of existential second-order logic. Our principal technical result is that over finite models with a vocabulary consisting of unary relation symbols only, this fragment of second-order logic is weaker in expressive power than first-order logic (with equality). Results about the fragment could turn out useful for example in the study of independence-friendly modal logics. In addition to proving results of a technical nature, we address issues related to a perspective from which IF logic is regarded as a specification framework for games, and also discuss the general significance of understanding fragments of second-order logic in investigations related to non-classical logics.

We present a fuzzy (or quantitative) version of the van Benthem theorem, which characterizes propositional modal logic as the bisimulation-invariant fragment of first-order logic. Specifically, we consider a first-order fuzzy predicate logic along with its modal fragment, and show that the fuzzy first-order formulas that are non-expansive w.r.t. the natural notion of bisimulation distance are exactly those that can be approximated by fuzzy modal formulas.

This article aims to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. It was presented at the 36th International Conference on Logic Programming (ICLP).

This paper presents a neighborhood semantics for logics of entailment. It begins with a minimal system Min that expresses the most fundamental assumptions about the entailment relation, and continues by examining various extensions that reflect further assumptions that might be made about entailment. This leads first to the logic B that is the basic relevant logic, and then to more powerful systems. All of these logics are proved to be sound and strongly complete. With B the neighborhood semantics meets the Routley–Meyer relational semantics for relevant logic; these connections are examined. The minimal and basic entailment logics are shown to have the finite model property, and hence to be decidable.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

First-order logic augments the expressive power of propositional logic as it links the logical assertions to properties of objects of some non-empty universe: the domain of discourse. This is achieved by allowing the propositional symbols to take arguments that range over elements of the domain of discourse. These are now called predicate symbols and are interpreted as relations on the domain. Elements of the domain of discourse are denoted by terms built up from variables, constants, and functions applied to other terms. First-order logic also expands the lexicon of propositional logic with the quantifiers “for all” and “there exists” that are interpreted consistently with their natural language meaning.This chapter is devoted to classical first-order logic. Our presentation is similar to the one conducted for propositional logic. We first define the syntax of first-order logic, followed by its semantics. Next we define a proof system for it and present the fundamental theoretical results of soundness and completeness. We also discuss the decision problems related to this logic. The remaining sections of the chapter cover variations and extensions of first-order logic, as well as first-order theories.

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

An Equivalence between Dependencies in Nested Databases and a Fragment of Propositional Logic