Search Results (8)

In this note, it is shown, following up on a conjecture of Joyce Friedman and David Warren in 1980, and in contrast with what is commonly assumed, that the reduction of applied $\lambda$-terms in Richard Montague’s intensional logic does not need to conflict with other possible reductions. For a confluent reduction, one only needs to access more rules than just the $\lambda$-reduction alone. In addition, we need to draw from principles, such as $\lambda$-Application Distribution and $\lambda$-Argument Exploitation, and suitably adjust the concept of a normal form. Full article

Richard Montague (1930–1971) is known as a founding figure of natural language semantics, i.e., the formal study of the semantics of natural languages by means of tools from mathematical logic. Less well known is that Montague maintained a strongly skeptical view on the possibility of a systematic logico-philosophical analysis of natural language for most of his short life, adhering to the then-common belief that natural languages are fundamentally different from the languages of logic. Completely unknown, until now, has been how Montague underwent a 180-degree turn in the last few years of his life, in the late 1960s, and pioneered a precise formal analysis of the syntax and semantics of fragments of English in three seminal papers that established the research framework, the methodology, and the formal tools for the new field of study. I provide a precise and documented answer to when, where, and how Montague’s intellectual turn occurred and how it relates to Montague’s previous research interests and work. Full article

This paper introduces the Intensional Conceptualization Model for Open Environments (ICMOE), a formal framework designed to enable semantic integration in dynamic and distributed systems. Grounded in intensional logic and formalized via a domain-specific language (ICMOE-L) built on Description Logic (DL), the model distinguishes between intensional and extensional semantics, allowing structured representation and evolution of concepts, relations, and domain rules under the open world assumption. ICMOE supports advanced semantic reasoning through an interpretation function that bridges relational data and ontological structures. A formal complexity analysis shows that reasoning with ICMOE-L has a worst-case complexity of O(n) ), where n is the total number of TBox and ABox axioms. To validate its effectiveness, ICMOE is evaluated using both qualitative and quantitative metrics. The model achieves a Concept Coverage score of 0.94, Semantic Depth of 0.89, Dynamic Adaptability Index of 0.91, Semantic Rule Density of 0.85, and Ontology Alignment Efficiency of 0.88. These results demonstrate ICMOE’s superior scalability, semantic richness, and adaptability when compared to foundational models such as those by Guarino and Bealer—making it a robust solution for open distributed environments. Full article

Artificial Intelligence (AI) systems are increasingly being deployed in decentralized environments where they interact with other AI systems and humans. In these environments, each participant may have different ways of expressing the same semantics, leading to challenges in communication and collaboration. To address these challenges, this paper presents a novel hybrid model for shared conceptualization in decentralized AI systems. This model integrates ontology, epistemology, and epistemic logic, providing a formal framework for representing and reasoning about shared conceptualization. It captures both the intensional and extensional components of the conceptualization structure and incorporates epistemic logic to capture knowledge and belief relationships between agents. The model’s unique contribution lies in its ability to handle different perspectives and beliefs, making it particularly suitable for decentralized environments. To demonstrate the model’s practical application and effectiveness, it is applied to a scenario in the healthcare sector. The results show that the model has the potential to improve AI system performance in a decentralized context by enabling efficient communication and collaboration among agents. This study fills a gap in the literature concerning the representation of shared conceptualization in decentralized environments and provides a foundation for future research in this area. Full article

This contribution is an essay of formal philosophy—and more specifically of formal ontology and formal epistemology—applied, respectively, to the philosophy of nature and to the philosophy of sciences, interpreted the former as the ontology and the latter as the epistemology of the modern mathematical, natural, and artificial sciences, the theoretical computer science included. I present the formal philosophy in the framework of the category theory (CT) as an axiomatic metalanguage—in many senses “wider” than set theory (ST)—of mathematics and logic, both of the “extensional” logics of the pure and applied mathematical sciences (=mathematical logic), and the “intensional” modal logics of the philosophical disciplines (=philosophical logic). It is particularly significant in this categorical framework the possibility of extending the operator algebra formalism from (quantum and classical) physics to logic, via the so-called “Boolean algebras with operators” (BAOs), with this extension being the core of our formal ontology. In this context, I discuss the relevance of the algebraic Hopf coproduct and colimit operations, and then of the category of coalgebras in the computations over lattices of quantum numbers in the quantum field theory (QFT), interpreted as the fundamental physics. This coalgebraic formalism is particularly relevant for modeling the notion of the “quantum vacuum foliation” in QFT of dissipative systems, as a foundation of the notion of “complexity” in physics, and “memory” in biological and neural systems, using the powerful “colimit” operators. Finally, I suggest that in the CT logic, the relational semantics of BAOs, applied to the modal coalgebraic relational logic of the “possible worlds” in Kripke’s model theory, is the proper logic of the formal ontology and epistemology of the natural realism, as a formalized philosophy of nature and sciences. Full article

Communication is one of the most notable processes in a multi-agent system. For this reason, considerable attention is paid to it—from abstract levels presenting theoretical models describing the basic principles to an implementation level with many details. However, most of these models build on a kind of unchanging basis, which characterizes communication between agents of a multi-agent system as a mutual exchange of messages expressed in a specific language or formal logic system, most often first-order predicate logic. Since most logical systems specialize in a particular area of natural language, the choice of a logical system reduces the communication potential of a multi-agent system. Therefore, we decide to choose the transparent intensional logic, a highly expressive methodology of logical analysis of natural language based on the symmetry between the syntax of expressions and their semantics, which minimizes these limitations and brings a new perspective on the issue of formalization of communication in multi-agent systems. By choosing transparent intensional logic as the central logical apparatus of our solution and postulating the general criterion for the synthesis of the concept of a message, the framework idea of our solution, which is based on hypotheses formulated in the analysis of Singh’s formal theory of communication, we have reached the synthesis of the so-called TIL-Message Formalization System. This system, unlike others based on traditionally used formalisms, simplifies the communication process itself by reducing the level of semantic interpretation of messages formalized by it, and in addition to formalism itself, this system also provides an abstract description of the background of the course of communication, proved by its application on specific examples, by standing out from the order of other formalisms providing only a kind of syntactic standard. Full article

This article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols, internal classes, outer terms and external Boolean-valued classes. Next, we enrich the collection of Boolean-valued research tools with the technique of partial elements and the corresponding joins, mixings and ascents. Passing on to the set-theoretic signature, we prove that bounded formulas are absolute for transitive Boolean-valued subsystems. We also introduce and study intensional, predicative, cyclic and regular Boolean-valued systems, examine the maximum principle, and analyze its relationship with the ascent and mixing principles. The main applications relate to the universe over an arbitrary extensional Boolean-valued system. A close interrelation is established between such a universe and the intensional hierarchy. We prove the existence and uniqueness of the Boolean-valued universe up to a unique isomorphism and show that the conditions in the corresponding axiomatic characterization are logically independent. We also describe the structure of the universe by means of several cumulative hierarchies. Another application, based on the quantifier hierarchy of formulas, improves the transfer principle for the canonical embedding in the Boolean-valued universe. Full article

Although current languages used in ontology-based data access (OBDA) systems allow for mapping source data to instances of concepts and relations in the ontology, several application domains need more flexible tools for inferring knowledge from data, which are able to dynamically acquire axioms about new concepts and relations directly from the data. In this paper we introduce the notion of mapping-based knowledge base (MKB) to formalize the situation where both the extensional and the intensional level of the ontology are determined by suitable mappings to a set of data sources. This allows for making the intensional level of the ontology as dynamic as the extensional level traditionally is. To do so, we resort to the meta-modeling capabilities of higher-order description logics, in particular the description logic $\mathit{Hi}({\mathit{DL-Lite}}_{\mathcal{R}})$ , which allows seeing concepts and relations as individuals, and vice versa. The challenge in this setting is to design efficient algorithms for answering queries posed to MKBs. Besides the definition of MKBs, our main contribution is to prove that answering instance queries posed to MKBs expressed in $\mathit{Hi}({\mathit{DL-Lite}}_{\mathcal{R}})$ can be done efficiently. Full article