Logics

Recently, historians have discussed the relevance of the nineteenth-century mathematical discipline of projective geometry for early modern classical logic in relation to possible solutions to semantic problems facing it. In this paper, I consider Hegel’s Science of Logic as an attempt to provide a projective geometrical alternative to the implicit Euclidean underpinnings of Aristotle’s syllogistic logic. While this proceeds via Hegel’s acceptance of the role of the three means of Pythagorean music theory in Plato’s cosmology, the relevance of this can be separated from any fanciful “music of the spheres” approach by the fact that common mathematical structures underpin both music theory and projective geometry, as suggested in the name of projective geometry’s principal invariant, the “harmonic cross-ratio”. Here, I demonstrate this common structure in terms of the phenomenon of “inverse foreshortening”. As with recent suggestions concerning the relevance of projective geometry for logic, Hegel’s modifications of Aristotle respond to semantic problems of his logic. Full article

We revise our former definition of graph operations and correspondingly adapt the construction of graph term algebras. As a first contribution to a prospective research field, Universal Graph Algebra, we generalize some basic concepts and results from algebras to graph algebras. To tackle this generalization task, we revise and reformulate traditional set-theoretic definitions, constructions and proofs in Universal Algebra by means of more category-theoretic concepts and constructions. In particular, we generalize the concept of generated subalgebra and prove that all monomorphic homomorphisms between graph algebras are regular. Derived graph operations are the other main topic. After an in-depth analysis of terms as representations of derived operations in traditional algebras, we identify three basic mechanisms to construct new graph operations out of given ones: parallel composition, instantiation, and sequential composition. As a counterpart of terms, we introduce graph operation expressions with a structure as close as possible to the structure of terms. We show that the three mechanisms allow us to construct, for any graph operation expression, a corresponding derived graph operation in any graph algebra. Full article

We show that intuitionistic propositional logic is Carnap categorical: the only interpretation of the connectives consistent with the intuitionistic consequence relation is the standard interpretation. This holds with respect to the most well-known semantics relative to which intuitionistic logic is sound and complete; among them Kripke semantics, Beth semantics, Dragalin semantics, topological semantics, and algebraic semantics. These facts turn out to be consequences of an observation about interpretations in Heyting algebras. Full article

This paper proposes a bilateral analysis of connexivity, presenting a bilateral natural deduction system for a weak connexive logic. The proposed logic deviates from other connexive logics and other bilateral logics in the following respects: (1) The logic induces a difference in meaning between inner and outer occurrences of negation in the connexive axioms. (2) The logic allows incoherence—assertion and denial of the same formula—while still being non-trivial. Full article

In this paper we explain the different meanings of the word “logic” and the circumstances in which it makes sense to use its singular or plural form. We discuss the multiplicity of logical systems and the possibility of developing a unifying theory about them, not itself a logical system. We undertake some comparisons with other sciences, such as biology, physics, mathematics, and linguistics. We conclude by delineating the origin, scope, and future of the journal Logics. Full article

We build and study dynamic versions of epistemic logic. We study languages parameterized by an action signature that allows one to express epistemic actions such as (truthful) public announcements, completely private announcements to groups of agents, and more. The language $\mathcal{L}$(Σ) is modeled on dynamic logic. Its sentence-building operations include modalities for the execution of programs, and for knowledge and common knowledge. Its program-building operations include action execution, composition, repetition, and choice. We consider two fragments of $\mathcal{L}(\mathbf{\Sigma })$. In ${\mathcal{L}}_1(\mathbf{\Sigma })$, we drop action repetition; in ${\mathcal{L}}_0(\mathbf{\Sigma })$, we also drop common knowledge. We present the syntax and semantics of these languages and sound proof systems for the validities in them. We prove the strong completeness of a logical system for ${\mathcal{L}}_0(\mathbf{\Sigma })$ and the weak completeness of one for ${\mathcal{L}}_1(\mathbf{\Sigma })$. We show the finite model property and, hence, decidability of ${\mathcal{L}}_1(\mathbf{\Sigma })$. We translate ${\mathcal{L}}_1(\mathbf{\Sigma })$ into PDL, obtaining a second proof of decidability. We prove results on expressive power, comparing ${\mathcal{L}}_1(\mathbf{\Sigma })$ with modal logic together with transitive closure operators. We prove that a logical language with operators for private announcements is more expressive than one for public announcements. Full article

The extension of the (ordinary) institution theory of Goguen and Burstall, known as the theory of stratified institutions, is a general axiomatic approach to model theories where the satisfaction is parameterized by states of models. Stratified institutions cover a uniformly wide range of applications from various Kripke semantics to various automata theories and even model theories with partial signature morphisms. In this paper, we introduce two natural concepts of logical interpolation at the abstract level of stratified institutions and we provide some sufficient technical conditions in order to establish a causality relationship between them. In essence, these conditions amount to the existence of nominals structures, which are considered fully and abstractly. Full article

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation. Finally, we discuss ways of extending these representations to lattices with a conditional or implication operation. Full article

This paper is an overview of some recent and ongoing developments of formal logical systems designed for reasoning about systems of rational agents who act in pursuit of their individual and collective goals, explicitly specified in the language as arguments of the strategic operators, in a socially interactive context of collective objectives and attitudes which guide and constrain the agents’ behavior. Full article