Logics doi: 10.3390/logics4020005

Authors: Logan Swanson

The class of tier-based strictly 2-local (TSL2) languages has been shown to be useful in modeling patterns across different linguistic domains. This paper discusses the learnability of the intersection closure of the TSL2 languages, multi-TSL2 (MTSL2). I present two learning algorithms, one that learns a relevant subclass of MTSL in polynomial time, and one that learns MTSL proper but requires potentially exponential time. Both algorithms generalize across tree-based and string-based data representations. I show that each algorithm correctly learns its target class from a limited sample of positive data, and discuss the tradeoffs between the two. The success of these algorithms delivers a key learning result for subregular linguistics, and demonstrates the utility of subregular language classes in developing a unified learning theory that spans different linguistic domains.

Logics doi: 10.3390/logics4020004

Authors: Odysseus Makridis

We undertake a thorough examination of George Boole’s claim that, as he discovered by means of his algebra, the law of idempotence is “more fundamental” than the law of non-contradiction (The Laws of Thought, Chapter III, Proposition IV). There is a paucity of sources investigating this subject (with a notable exception being (Béziau 2018)). We query Boole’s claim; we examine if and how we can make sense of it; we identify the notable Aristotelian precedent of philosophical reflections on relative fundamentality of logical principles; and we inquire as to what philosophical view of logic is consistent with Boole’s way of thinking about logical principles. Boole’s thinking is apparently burdened by a metaphysically laden view of logic. We argue in detail that it is a radically different way of thinking about logic—a formalist view that regards logic as manipulation of symbolic resources, congenial to logical positivism—which allows us to make some tentative sense of claims about relative fundamentality of logical laws, insofar as we can define such a notion in a meaningful way. However, on the other hand, entanglements in metaphysically laden phantasmagorias fail to support (or perhaps even fail to make sense of) Boole’s claim. In order to substantiate the metalogical and philosophical–logical claims, we advance and construct formal derivations within different Boolean languages with a view to showing how idempotence is primary in some formal systems, but it is derivable (from non-contradiction) in other systems. Hence, Boole’s claim, as we can make sense of it (as relative derivability), is language-dependent, and we argue that this is consistent with a certain philosophical view of what logic is.

Logics doi: 10.3390/logics4010003

Authors: Jiayi Zhou

This article investigates two distinct classes of intensifiers in Mandarin Chinese, ordinary intensifiers (Class I) and subjective intensifiers (Class II). Intensifiers from Class II, such as zhēn “really”, cannot be used in the following cases: (i) interrogatives, (ii) sentences containing epistemic modals, (iii) sentences with negation or as conditional antecedents, (iv) sentences containing attitude predicates, and (v) contexts lacking firsthand experience. This paper argues that evidentiality, as a conceptual framework concerning the source of knowledge, can account for these phenomena related to Class II intensifiers. Specifically, in this study, evidentiality constraints on both the subject and the manner of the information source. The subject must be the speaker, and the information must be acquired as firsthand experience.

Logics doi: 10.3390/logics4010002

Authors: Rui Li

Kripke semantics for intuitionistic predicate logic IQC is often viewed as a forcing relation between posets and formulas. In this paper, we further introduce Cohen forcing into semantics. In particular, we use generic filters to interpret the double-negation translations from classical first-order logic to the intuitionistic version. It explains how our method interprets classical theories into constructive ones. In addition, our approach is generalized to Friedman’s A-translation. Consequently, we propose an optimal A-translation that extends the class of theorems that are conserved from a classical theory to its intuitionistic counterpart.

Logics doi: 10.3390/logics4010001

Authors: Jiahong Wang

Golog Tibetan grammaticalizes both evidentiality and egophoricity, but the two categories interact in a constrained way: evidential marking neutralizes the binary egophoric versus non-egophoric contrast. This paper develops LEE (Logic of Evidentiality and Egophoricity), a multi-modal logic that formalizes this interaction. LEE employs operators □EGO, □SENS, and □INF for egophoric, sensory-evidential, and inferential-evidential markers, respectively. The blocking effect is captured by axioms □σφ → (□EGOφ ↔ ◇EGOφ) for σ ∈ {SENS, INF}. This paper establishes soundness, completeness, and decidability for LEE. Three empirical puzzles receive unified explanation: (i) blocking of egophoric vs. non-egophoric contrasts under evidential marking, (ii) semantic bleaching of egophoric morphology in evidential contexts, and (iii) the unidirectional nature of the evidential–egophoric interaction.

Logics doi: 10.3390/logics3040017

Authors: Ling Sun

Traditional degree semantics approaches have aimed to pin down the inherent class of adjectives. This paper presents a novel dynamic perspective, where the classification of an adjective is dynamic and syntactically dependent. Using measurement theory and fuzzy set analysis, the proposed framework defines dynamic patterns of adjective classes with a set of axioms and integrates these patterns with syntactic structures to explain the flexibility and constraints observed in adjectival expressions. Employing Mandarin data, the paper illustrates how different syntactic constructions select specific adjective classes, thereby affecting their distribution and interpretation. This approach not only accommodates cross-linguistic variations but also provides a more comprehensive understanding of the semantics of adjectives.

Logics doi: 10.3390/logics3040016

Authors: Emil Eva Rosina Franci Mangraviti

We offer a framework that captures both context-dependency and vagueness of predicate meanings—illustrated by the politically relevant case of woman—as an interaction of lexical meaning and Question under Discussion (‘QUD’). We extend existing approaches to non-maximality to superficially polysemous predicates like woman and show that this is conceptually coherent and insightful for a linguistic analysis of political debates about gender invitation policies: while there are (i) clear, semantically true, and (ii) strictly unacceptable cases of x is a woman, there are also (iii) merely pragmatically acceptable cases (‘like a woman with respect to the QUD’) as well as (iv) truly vague ones. We argue that this four-way division is the least complex model that captures current gender discourses in a harm-reducing, trans-inclusive way. This offers a new perspective on the semantics–pragmatics interface of predicate meanings.

Logics doi: 10.3390/logics3040015

Authors: Marcelo Esteban Coniglio Ana Claudia Golzio Kaique Matias de Andrade Roberto

José Morgado introduced in 1962 a novel notion of hyperlattices, which he called reticuloides. In his master’s thesis submitted in 1971 (under the supervision of Newton da Costa), Antonio M. Sette introduced a new class of implicative hyperlattices (here called SIHLs) based on Morgado’s hyperlattices. He also extended SIHLs by adding a unary hyperoperator, thus defining a class of hyperalgebras (denoted SHCω) corresponding to da Costa algebras for Cω, thereby providing suitable semantics for the logic Cω. In this paper, after providing a (hyper)lattice-theoretic characterization of Sette’s implicative hyperlattices and proving some basic results on SIHLs, we introduce a class of swap structures—special hyperalgebras over the signature of Cω that arise naturally from implicative lattices. We prove that these swap structures are indeed SHCω. Finally, we demonstrate that the class SHCω, as well as the aforementioned swap structures, characterizes the logic Cω.

Logics doi: 10.3390/logics3040014

Authors: Niccolò Rossi

If one believes that 2+2=4, then one also believes that either 2+2=4 or 971 is a cousin prime number. This follows from doxastic logics based on standard Kripke relational semantics, which validate disjunction introduction for belief. However, this principle does not hold in topic-sensitive semantics. An agent who lacks the concept of a ‘cousin prime number’ may be unable to entertain, and thus unable to believe, any proposition involving that concept. I argue that while disjunction introduction may fail for belief—and for other epistemic states that presuppose belief—it does hold for certain states that do not require belief. In this paper, I focus on the notion of commitment to the truth. Drawing on the concept of logical grounding, I propose formal semantics that preserve the requirement of topic-grasping, but weaken it in a way that allows for a more standard treatment of disjunction.

Logics doi: 10.3390/logics3040013

Authors: Mark A. Winstanley

Rationality has long been considered the quintessence of humankind. However, psychological experiments revealing reliable divergences in performances on reasoning tasks from normative principles of reasoning have cast serious doubt on the venerable dogma that human beings are rational animals. According to the standard picture, reasoning in accordance with principles based on rules of logic, probability theory, etc., is rational. The standard picture provides the backdrop for both the rationality and irrationality thesis, and, by virtue of the competence-performance distinction, diametrically opposed interpretations of reasoning experiments are possible. However, the standard picture rests on shaky foundations. Jean Piaget developed a psychological theory of reasoning, in which logic and mathematics are continuous with psychology but nevertheless autonomous sources of knowledge. Accordingly, logic, probability theory, etc., are not extra-human norms, and reasoners have the ability to reason in accordance with them. In this paper, I set out Piaget’s theory of rationality, using intra- and interpropositional reasoning as illustrations, and argue that Piaget’s theory of rationality is compatible with the standard picture but actually undermines it by denying that norms of reasoning based on logic are psychologically relevant for rationality. In particular, rather than logic being the normative benchmark, I argue that rationality according to Piaget has a psychological foundation, namely the reversibility of the operations of thought constituting cognitive structures.

Logics doi: 10.3390/logics3040012

Authors: Claire Ott

Analogies are an important part of human cognition for learning and discovering new concepts. There are many different approaches to defining analogies and how new ones can be found or constructed. We propose a novel approach in the tradition of structure mapping using colored multigraphs to represent domains. We define a category of colored multigraphs in order to utilize some Category Theory (CT) concepts. CT is a powerful tool for describing and working with structure-preserving maps. There are many useful applications for this theory in cognitive science, and we want to introduce one such application to a broader audience. CT and the concepts used in this paper are introduced and explained. We show how the category theoretical concepts product and pullback can be used with the category of colored multigraphs to find possible analogies between domains using different requirements. The dual notion of a pullback, the pushout, is then used as conceptual blending to generate a new domain.

Logics doi: 10.3390/logics3030011

Authors: Stipe Marić

Inquisitive modal logic InqML⊞ is a natural generalization of basic modal logic, with ⊞ as a primitive modal operator. In this paper, we study the bisimulation quotients in the logic InqML⊞. For a given inquisitive modal model M=(W,Σ,V), we first show that the bisimilarity relation is an equivalence relation on W and that there is the largest bisimulation on M. We then define the bisimulation quotient and prove that a model is connected to its bisimulation quotient by a surjective bounded morphism. Finally, we prove that two models are globally bisimilar if and only if their bisimulation quotients are isomorphic.

Logics doi: 10.3390/logics3030010

Authors: Chrysafis Hartonas

We present an extension and generalization of Sahlqvist–van Benthem correspondence to the case of distribution-free modal logic, with, or without negation and/or implication connectives. We follow a reductionist strategy, reducing the correspondence problem at hand to the same problem, but for a suitable system of sorted modal logic (the modal companion of the distribution-free system). The reduction, via a fully abstract translation, builds on the duality between normal lattice expansions and sorted residuated frames with relations (a generalization of classical Kripke frames with relations). The approach is scalable and it can be generalized to other systems, with or without distribution, such as distributive modal logic, or substructural logics with, or without additional modal operators.

Logics doi: 10.3390/logics3030009

Authors: Giacomo Bergami Oliver Robert Fox

This work contextualizes the possibility of deriving a unifying artificial intelligence framework by walking in the footsteps of General, Explainable, and Verified Artificial Intelligence (GEVAI): by considering explainability not only at the level of the results produced by a specification but also considering the explicability of the inference process as well as the one related to the data processing step, we can not only ensure human explainability of the process leading to the ultimate results but also mitigate and minimize machine faults leading to incorrect results. This, on the other hand, requires the adoption of automated verification processes beyond system fine-tuning, which are essentially relevant in a more interconnected world. The challenges related to full automation of a data processing pipeline, mostly requiring human-in-the-loop approaches, forces us to tackle the framework from a different perspective: while proposing a preliminary implementation of GEVAI mainly used as an AI test-bed having different state-of-the-art AI algorithms interconnected, we propose two other data processing pipelines, LaSSI and EMeriTAte+DF, being a specific instantiation of GEVAI for solving specific problems (Natural Language Processing, and Multivariate Time Series Classifications). Preliminary results from our ongoing work strengthen the position of the proposed framework by showcasing it as a viable path to improve current state-of-the-art AI algorithms.

Logics doi: 10.3390/logics3030008

Authors: Uwe Wolter

Logics of Statements in Context have been proposed as a general framework to describe and relate, in a uniform and unifying way, a broad spectrum of logics and specification formalisms, which also comprise “open formulas”. In particular, it has been shown that we can define arbitrary first-order “open formulas” in arbitrary categories. At present, there are two deficiencies. In the general case, only signatures with predicate symbols are considered and institutions of statements in context are only defined for single signatures. In this paper, we elaborate the special case of traditional many-sorted first-order logic. We show that any many-sorted first-order signature Σ with predicates and (!) operation symbols gives rise to an institution FLΣ of Σ-statements in context and that any signature morphism results in a comorphism between the corresponding institutions. We prove that we obtain a functor FL:Sig→coIns from the category of signatures into the category of institutions and comorphisms. We construct a corresponding Grothendieck institution FL♯ and prove that FL♯ is, indeed, an extension of the traditional institution of first-order logic, which only comprises “closed formulas”. We also investigate substitutions in detail and discuss (elementary) diagrams in the sense of traditional first-order logic.

Logics doi: 10.3390/logics3030007

Authors: Mansooreh Kimiagari

I aim to examine Val Plumwood’s feminist account of logic, as presented by Plumwood, using the frameworks developed by Elliott and McKaughan, and Intemann. Plumwood argues that relevance logic is the appropriate logical system for feminist reasoning. I intend to assess whether this constitutes a legitimate incorporation of values into logic. To this end, I evaluate the aims of Plumwood as a case study. Additionally, I trace the values embedded in my chosen case to determine whether feminist values advance the epistemic and social objectives of the research.

Logics doi: 10.3390/logics3030006

Authors: Alexandru Baltag Johan van Benthem

We study knowable informational dependence between empirical questions, modeled as continuous functional dependence between variables in a topological setting. We also investigate epistemic independence in topological terms and show that it is compatible with functional (but non-continuous) dependence. We then proceed to study a stronger notion of knowability based on uniformly continuous dependence. On the technical logical side, we determine the complete logics of languages that combine general functional dependence, continuous dependence, and uniformly continuous dependence.

Logics doi: 10.3390/logics3020005

Authors: Ali Baheri Peng Wei

We present Stratified Metric Temporal Logic (SMTL), a novel formalism for specifying and verifying the properties of complex cyber–physical systems that exhibit behaviors across multiple temporal and abstraction scales. SMTL extends existing temporal logics by incorporating a stratification operator, enabling the association of temporal properties with specific abstraction levels. This allows for the natural expression of multi-scale requirements while maintaining formal reasoning about inter-level relationships. We formalize the syntax and semantics of SMTL, proving that it strictly subsumes metric temporal logic (MTL) and offers enhanced expressiveness by capturing properties unattainable in existing logics. Numerical simulations comparing agents operating under MTL and SMTL specifications show that SMTL enhances agent coordination and safety, reducing collision rates without substantial computational overhead or compromising path efficiency. These findings highlight SMTL’s potential as a valuable tool for designing and verifying complex multi-agent systems operating across diverse temporal and abstraction scales.

Logics doi: 10.3390/logics3020004

Authors: Mikhail Moshkov

In this paper, we study three algorithmic problems involving computation trees: the optimization, solvability, and satisfiability problems. The solvability problem is concerned with recognizing computation trees that solve problems. The satisfiability problem is concerned with recognizing sentences that are true in at least one structure from a given set of structures. We study how the decidability of the optimization problem depends on the decidability of the solvability and satisfiability problems. We also consider various examples with both decidable and undecidable solvability and satisfiability problems.

Logics doi: 10.3390/logics3020003

Authors: Chrysafis Hartonas

This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, as a recent article by Bezhanishvili, de Groot, Dmitrieva and Morachini, who studied a distribution-free version of Dunn’s positive modal logic (PML). Unlike PML, we consider logics that may drop distribution and that are equipped with both an implication connective and modal operators. We adopt a uniform relational semantics approach, relying on recent results on representation and duality for normal lattice expansions. We prove canonicity and completeness in the relational semantics of the minimal distribution-free normal modal logic, assuming just the K-axiom, as well as those of its axiomatic extensions obtained by adding any of the D, T, B, S4 or S5 axioms. Adding distribution can be easily accommodated and, as a side result, we also obtain a new semantic treatment of intuitionistic modal logic.

Logics doi: 10.3390/logics3020002

Authors: Nissim Francez

The paper shows the deductive equivalence between bilateral (and multilateral) proof systems based on signed formulas and proof systems based on multiple derivability relations.

Logics doi: 10.3390/logics3010001

Authors: Chrysafis Hartonas

This article contributes to recent results in the model theory of distribution-free logics (which include a Goldblatt-Thomason theorem and a development of their Sahlqvist theory) by lifting van Benthem’s characterization result for modal logic to the more general setting of the logics of normal lattice expansions. Our proof approach makes use of a fully abstract translation of the language of the logics of interest into the language of sorted residuated modal logic, building on an analogous translation of substructural logics recently published by the author. The article is intended as a demonstration and application of a project of reduction of non-distributive logics to (sorted) residuated modal logics. The reduction makes the proof of a van Benthem characterization of non-distributive logics possible, by adapting, reusing and generalizing existing results.

Logics doi: 10.3390/logics2040007

Authors: Alex Citkin

An algebraic proof of the Nishimura theorem on one-generated Heyting algebras is presented.

Logics doi: 10.3390/logics2040006

Authors: Xiling Luo

Homophily, which means similarity breeds association, is one of the most fundamental principles in social organization. However, in some cases, homophily is not significant, because actors’ perceptions of others differ from the real situation. In this article, we use the term “subjective homophily” to describe the homophily where the perceived similarity of objects is considered. In addition, we also consider social influence, which is closely related to homophily and represents the diffusion of some attributes through associations. In short, the dynamic temporal logic LoSHG,MSC we propose in this article is based on computation tree logic (CTL), which is used to describe the evolution of networks by subjective homophily, and dynamic logic (DL), which provides the dynamic update operator for representing active social influence. Furthermore, we prove that the model checking problem and the validity checking problem for LoSHG,MSC are both PSPACE-complete. Finally, we use an example, named false consensus, to illustrate how logic captures the subjective homophily evolution of networks and the impact of active social influence on evolution and structure.

Logics doi: 10.3390/logics2030005

Authors: Sanderson Molick

Non-Tarskian interpretations of many-valued logics have been widely explored in the logic literature. The development of non-tarskian conceptions of logical consequence set the theoretical foundations for rediscovering well-known (Tarskian) many-valued logics. One may find in distinct authors many novel interpretations of many-valued systems. They are produced through a type of procedure which consists in altering the semantic structure of Tarskian many-valued logics in order to output a non-Tarskian interpretation of these logics. Through this type of transformation the paper explores a uniform way of transforming finitely many-valued Tarskian logics into their non-Tarskian interpretation. Some general properties of carrying out this type of procedure are studied, namely the dualities between these logics and the conditions under which negation-explosive and negation-complete Tarskian logics become non-explosive.

Logics doi: 10.3390/logics2020004

Authors: Giacomo Bergami

This paper considers a specification rewriting meachanism for a specific fragment of Linear Temporal Logic for Finite traces, DECLAREd, working through an equational logic and rewriting mechanism under customary practitioner assumptions from the Business Process Management literature. By rewriting the specification into an equivalent formula which might be easier to compute, we aim to streamline current state-of-the-art temporal artificial intelligence algorithms working on temporal logic. As this specification rewriting mechanism is ultimately also able to determine with the provided specification is a tautology (always true formula) or a formula containing a temporal contradiction, by detecting the necessity of a specific activity label to be both present and absent within a log, this implies that the proved mechanism is ultimately a SAT-solver for DECLAREd. We prove for the first time, to the best of our knowledge, that this fragment is a polytime fragment of LTLf, while all the previously-investigated fragments or extensions of such a language were in polyspace. We test these considerations over formal synthesis (Lydia), SAT-Solvers (AALTAF) and formal verification (KnoBAB) algorithms, where formal verification can be also run on top of a relational database and can be therefore expressed in terms of relational query answering. We show that all these benefit from the aforementioned assumptions, as running their tasks over a rewritten equivalent specification will improve their running times, thus motivating the pressing need of this approach for practical temporal artificial intelligence scenarios. We validate such claims by testing such algorithms over a Cybersecurity dataset.

Logics doi: 10.3390/logics2010003

Authors: Christoph Benzmüller David Fuenmayor Bertram Lomfeld

The logico-pluralist LogiKEy knowledge engineering methodology and framework is applied to the modelling of a theory of legal balancing, in which legal knowledge (cases and laws) is encoded by utilising context-dependent value preferences. The theory obtained is then used to formalise, automatically evaluate, and reconstruct illustrative property law cases (involving the appropriation of wild animals) within the Isabelle/HOL proof assistant system, illustrating how LogiKEy can harness interactive and automated theorem-proving technology to provide a testbed for the development and formal verification of legal domain-specific languages and theories. Modelling value-oriented legal reasoning in that framework, we establish novel bridges between the latest research in knowledge representation and reasoning in non-classical logics, automated theorem proving, and applications in legal reasoning.

Logics doi: 10.3390/logics2010002

Authors: Paul Redding

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.

Logics doi: 10.3390/logics2010001

Authors: J.-Martín Castro-Manzano

In this paper, we produce an extension of Englebretsen’s line diagrams in order to represent modal syllogistic, i.e., we add some diagrammatic objects and rules to his system in order to reason about modal syllogistics in a diagrammatic, linear fashion.

Logics doi: 10.3390/logics1040010

Authors: Uwe Wolter Tam T. Truong

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.

Logics doi: 10.3390/logics1040009

Authors: Haotian Tong Dag Westerståhl

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.

Logics doi: 10.3390/logics1030008

Authors: Nissim Francez

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.

Logics doi: 10.3390/logics1030007

Authors: Jean-Yves Beziau

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.

Logics doi: 10.3390/logics1020006

Authors: Alexandru Baltag Lawrence S. Moss Sławomir Solecki

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 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 L(Σ). In L1(Σ), we drop action repetition; in L0(Σ), 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 L0(Σ) and the weak completeness of one for L1(Σ). We show the finite model property and, hence, decidability of L1(Σ). We translate L1(Σ) into PDL, obtaining a second proof of decidability. We prove results on expressive power, comparing L1(Σ) 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.

Logics doi: 10.3390/logics1020005

Authors: Răzvan Diaconescu

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.

Logics doi: 10.3390/logics1010004

Authors: Wesley H. Holliday

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.

Logics doi: 10.3390/logics1010003

Authors: Valentin Goranko

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.

Logics doi: 10.3390/logics1010002

Authors: Valentin Goranko

Reasoning is one of the most important and distinguished human activities [...]

Logics doi: 10.3390/logics1010001

Authors: Constanze Schelhorn

Logic (from ancient Greek “λογικὴ τέχνη (logiké téchnē)”—“thinking art”, “procedure”) is a multidisciplinary field of research studying the formal principles of reasoning [...]