Predicate Logic Research Articles

Concept logic trees: enabling user interaction for transparent image classification and human-in-the-loop learning

Interpretable deep learning models are increasingly important in domains where transparent decision-making is required. In this field, the interaction of the user with the model can contribute to the interpretability of the model. In this research work, we present an innovative approach that combines soft decision trees, neural symbolic learning, and concept learning to create an image classification model that enhances interpretability and user interaction, control, and intervention. The key novelty of our method relies on the fusion of an interpretable architecture with neural symbolic learning, allowing the incorporation of expert knowledge and user interaction. Furthermore, our solution facilitates the inspection of the model through queries in the form of first-order logic predicates. Our main contribution is a human-in-the-loop model as a result of the fusion of neural symbolic learning and an interpretable architecture. We validate the effectiveness of our approach through comprehensive experimental results, demonstrating competitive performance on challenging datasets when compared to state-of-the-art solutions.

Paraconsistent logics as a Way of Expressing Objective Contradictions in Science

The article focuses on the problems associated with the need to express in the language of science in a consistent way the movement and various kinds of changes taking place in the objective world by developing logics that would be tolerant of logical contradictions. The paper presents a brief history of the creation of paraconsistent (paraconsistent) logics, in which logical contradictions turned out to be quite permissible. It is noted that the priority in their development belongs to Russian scientists N. A. Vasiliev and I. E. Orlov, as well as Polish philosophers and logicians Y. Lukasevich and S. Yaskovsky; that since the seventies of the last century, the development of paraconsistent logics has assumed an international character; that interest in these logics is not abating at the present time. The possibilities of using paraconsistent logics presented in the works of foreign and domestic philosophers and logicians for the formalization of dialectics are discussed. The positive role of these logics is shown in solving many paradoxes in the foundations of classical propositional calculation and predicate logic, as well as in order to limit undesirable consequences when using IT technologies related to the processing of inconsistent or contradictory information. It is concluded that the use of paranoherent logics in the construction of models of individual fragments of dialectics contributes to its development as logic and as ontology, and indicates that the apparatus of non-classical logics as a whole is a very powerful means of studying and explaining many problems of theoretical cognition.

Semantics of Belief Change Operators for Intelligent Agents

This paper summarises several contributions to the theory of belief change by the authors’ dissertation thesis. First, a relational characterization of belief revision for Tarskian logics is considered, encompassing first-order predicate logic, description logic, modal logics and many monotonic logics with model-theoretic semantics. Those logics where total preorders are the standard semantics for revision are characterized. The second contribution considered is a theory of belief revision that builds upon the idea that agents are limited in what the outcome of a revision is. Furthermore, advancements in principles for iterated belief contraction given in the thesis are outlined.

Agile Logical Semantics for Natural Languages

This paper presents an agile method of logical semantics based on high-order Predicate Logic. An operator of predicate abstraction is introduced that provides a simple mechanism for logical aggregation of predicates and for logical typing. Monadic high-order logic is the natural environment in which predicate abstraction expresses the semantics of typical linguistic structures. Many examples of logical representations of natural language sentences are provided. Future extensions and possible applications in the interaction with chatbots are briefly discussed as well.

Pace Carabelli and Dow, There is No Common Discourse Language Logic in Keynes’s A Treatise on Probability

For 45 years, both A. Carabelli and S. Dow have been arguing that Keynes’s non demonstrative logic in the A Treatise on Probability is a common discourse logic (rhetoric). They provide no textual evidence anywhere in Keynes’s A Treatiseon Probability to support this claim. They have never supported, through the citation of specific pages or paragraphs in Keynes’s book, their claims that Keynes’s logic is NOT a formal logic. What is in the A Treatise on Probability is a version of Boole’s relational, propositional logic that Keynes combined with a first order (predicate) logic. These are mathematical, formal, symbolic logics. They have nothing to do with common discourse logics using the English language. A simultaneous reading of chapters I and II of Keynes’s book and chapters I, XI, and XII of Boole’s The Laws of Thought lead to one, and only one, conclusion- Keynes’s logic is a formal logic derived from G. Boole. The only conclusion that follows from Keynes’s application of Boolean logics is that Carabelli and Dow have been severely confused for 45 years in what a formal logic is and what a common discourse logic is.

Algorithmic properties of modal and superintuitionistic logics of monadic predicates over finite Kripke frames

Abstract We show that the monadic fragment of the modal predicate logic of a single Kripke frame with finitely many possible worlds, but possibly infinite domains, is decidable. This holds true even for multimodal logics with equality, regardless of whether equality is interpreted as identity or as congruence. By the Gödel–Tarski translation, similar results follow for superintuitionistic predicate logics, with or without equality. Using these observations, we establish upper algorithmic bounds, which match the known lower bounds, for monadic fragments of some modal predicate logics. In particular, we prove that, if $L$ is a propositional modal logic contained in $\textbf{S5}$, $\textbf{GL.3}$ or $\textbf{Grz.3}$ and the class of finite Kripke frames validating $L$ is recursively enumerable, then the monadic fragment with equality of the predicate logic of finite Kripke frames validating $L$ is $\varPi ^{0}_{1}$-complete; this, in particular, holds if $L$ is one of the following propositional logics: $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{KTB}$, $\textbf{K4}$, $\textbf{K4.3}$, $\textbf{S4}$, $\textbf{S4.3}$, $\textbf{GL}$, $\textbf{Grz}$, $\textbf{K5}$, $\textbf{K45}$ and $\textbf{S5}$. We also prove that monadic fragments with equality of logics $\textbf{QAlt}^=_{n}$ and $\textbf{QTAlt}^=_{n}$ are decidable. The obtained results are easily extendable to the multimodal versions of the predicate logics we consider and to logics with the Barcan formula.

DimBridge: Interactive Explanation of Visual Patterns in Dimensionality Reductions with Predicate Logic.

Dimensionality reduction techniques are widely used for visualizing high-dimensional data. However, support for interpreting patterns of dimension reduction results in the context of the original data space is often insufficient. Consequently, users may struggle to extract insights from the projections. In this paper, we introduce DimBridge, a visual analytics tool that allows users to interact with visual patterns in a projection and retrieve corresponding data patterns. DimBridge supports several interactions, allowing users to perform various analyses, from contrasting multiple clusters to explaining complex latent structures. Leveraging first-order predicate logic, DimBridge identifies subspaces in the original dimensions relevant to a queried pattern and provides an interface for users to visualize and interact with them. We demonstrate how DimBridge can help users overcome the challenges associated with interpreting visual patterns in projections.

Basics of Second-Order Predicate Logic

Basics of Second-Order Predicate Logic

Gender Semantics of Nominations <i>родитель – родительница – родители</i> in Modern Media Discourse

Purpose. The purpose of the article is to identify manifestation patterns of gender semantics of Russian nouns родитель – родительница – родители (pluralia tantum) in different referent and context conditions. The research is based on newspaper subcorpora of Russian National Corpus. Results. The vocabulary values are marked by gender semes: родитель (father), родительница (mother), родители (father and mother – the separate lexeme). The first two nouns are characterized by a reduced stylistic meaning. They are also evaluated as outdated, but it does not confirm in modern Russian speech. The feminitive родительница retains a gender meaning in all types of usage, it is often used as a neutral synonym for мать (mother). It shows a clear tendency towards stylistic neutrality. At the same time it has a negative connotative background and can be used for reducing the social and personal status of the referent. The word родитель is a male nomination if denotes a particular person. For a generalized referent or if as a logical predicate, the gender seme is not preserved. Colloquial coloration is preserved only when the word referents a particular person. The meaning of the noun родители with semes of the bigender couple is fully realized only if it denotes a particular family or nominates a social role attributed to a particular family. In other cases it has non-gendered semantics, and becomes the plural form of родитель as ‘person with offspring’. Conclusion. We suppose changes in the semantic structure of the noun родитель can affect the lexical and morphological status of the word родители.

Formal model of multi-agent architecture of a software system based on knowledge interpretation

The use of agents across diverse domains within computer science and artificial intelligence is experiencing a notable surge in response to the imperatives of adaptability, efficiency, and scalability. The subject of this study is the application of formal methods to furnish a framework for knowledge interpretation with a specific focus on the agent-based paradigm in software engineering. This study aims to advance a formal approach to knowledge interpretation by leveraging the agent-based paradigm. The objectives are as follows: 1) to examine the current state of the agent-based paradigm in software engineering; 2) to describe the basic concepts of the knowledge interpretation approach; 3) to study the general structure of the rule extraction task; 4) to develop the reference structure of knowledge interpretation; 5) to develop a multi-agent system architecture; 6) and to discuss the research results. This study employs formal methods, including the use of closed path rules and predicate logic. Specifically, the integration of closed path rules contributes to the extraction and explication of facts from extensive knowledge bases. The obtained results encompass the following: 1) a rule mining approach grounded in closed path rules and tailored for processing extensive datasets; 2) a formalization of relevance that facilitates the scrutiny and automated exclusion of irrelevant fragments from the explanatory framework; and 3) the realization of a multi-agent system predicated on the synergy among five distinct types of agents, dedicated to rule extraction and the interpretation of acquired knowledge. This paper provides an example of the application of the proposed formal tenets, demonstrating their practical context. The conclusion underscores that the agent-based paradigm, with its emphasis on decentralized and autonomous entities, presents an innovative framework for handling the intricacies of knowledge processing. It extends to the retrieval of facts and rules. By distributing functions across multiple agents, the framework offers a dynamic and scalable solution to effectively interpret vast knowledge repositories. This approach is particularly valuable in scenarios where traditional methods may struggle to cope with the volume and complexity of information.

THE COMPLEX PROBLEM OF EXPANDING THE CAPABILITIES OF CADASTRAL SYSTEMS AND WAYS TO SOLVE IT

The development of cadastral systems in various countries, including Ukraine, has gone through creating information systems that serve users (stakeholders) in one of the subject areas, such as land administration, real estate construction, water resources, and others. Modern cadastral systems have no capacity to solve complex problems, and, at the same time, their structure does not contain specialised mathematical support in the form of models capable of simulating complex processes. Intelligent cadastres will be able to solve complex problems. Along with the existing types of cadastre support, intelligent cadastres should have well-structured mathematical and heuristic support. It should allow for solving private computing and service tasks, as well as loosely structured ones with a high degree of uncertainty, and provide good visualisation and detailing of objects, processes, and phenomena based on cloud and other modern information technologies. We propose introducing an intelligent cadastral system, presented as a geoinformation system with components of both a decision support system and a system for manipulating thematic layers. The decision support system should contain typical models used during the formalisation of one or another subject area or as a variant of the formal model of the subject areas’ description. The thematic layers manipulation system should allow the integration of attributive data belonging to different subject areas. In this case, ensuring prompt and fruitful work of cognitologists with experts is essential. The shortage of time and the variety of problems to be solved require the development of technology for formalising the knowledge of experts or specific subject areas. Implementing this technology and ensuring its implementation will require the creation of banks of typical knowledge models, which should store standardised procedures for formal representations. The bank of formal representations of heuristic knowledge includes procedures for forming production rules, building semantic and other networks, building frame representations and genetic algorithms, and so on. The bank of formal logical procedures contains procedures for creating logical rules based on the logic of statements, rules of logical inference based on the predicate logic, procedures of other modal logics, in particular, the presence logic, and procedures for building logical constructions based on the theory of categories and functors. Keywords: cadastral system, geoinformation system, decision support system, thematic layer manipulation system, formalisation.

The connection of mathematics and logic in the structure of axiomatized and formalized theories

The aim of the research is to study the relationship between logic and mathematics in the structure of axiomatized and formalized scientific theories. The object of the study is the explication of this connection and its explanation. The subject of the study is syntactic and semantic views on the structure of scientific theories, the relationship between logic and mathematics has not been studied in detail in them. In the syntactic view, the structure of the theory is understood as a linguistic construct build from various logical propositions of the theoretical level, correspondence propositions and observation propositions. The structure of the theory does not take into account the variety of model representations of the theory that generate a variety of language constructs. The semantic view overcomes this disadvantage, and in it the structure of the theory is presented as a hierarchy of models: from axioms to theoretical-level models, experimental models and data models. The structure of the theory, the connection of logic and mathematics were studied using comparative analysis, methods of interpretive analysis and reconstruction of scientific theories. The methods made it possible to explicate mathematical concepts in the structure of the theory and correlate them with logic and natural language. Comparative analysis has shown that in the syntactic view, the connection between logic and mathematics lies in the fact that mathematical concepts of physics are interpreted in the language of logic of first-order predicates with equality. The connection between mathematical concepts is provided by the axiomatic method, which serves as a means of formalizing concepts. Mathematics comes down to logic. In the semantic approach, in order to identify the connection between mathematics and logic, it was necessary to reconstruct the structure of non-relativistic quantum mechanics. With the help of the set-theoretic predicate of Suppes, its axioms were determined, the connection between mathematical structures, postulates of the theory, axioms, and observable quantities was established. Logic and mathematics are related to each other in such a way that metamathematics or linguistics is a part of mathematics. Mathematics includes set theory and model theory, i.e. mathematical logic. The connection of mathematical formalisms with phenomena and with natural language remains problematic, and there is this drawback in the syntactic approach. The novelty lies in the fact that the research contributes to the methodology and logic of science, to the explanation of the connection between logic and mathematics in scientific theory, which was illustrated by various examples from various fields of physics.

Logical–Mathematical Foundations of a Graph Query Framework for Relational Learning

Relational learning has attracted much attention from the machine learning community in recent years, and many real-world applications have been successfully formulated as relational learning problems. In recent years, several relational learning algorithms have been introduced that follow a pattern-based approach. However, this type of learning model suffers from two fundamental problems: the computational complexity arising from relational queries and the lack of a robust and general framework to serve as the basis for relational learning methods. In this paper, we propose an efficient graph query framework that allows for cyclic queries in polynomial time and is ready to be used in pattern-based learning methods. This solution uses logical predicates instead of graph isomorphisms for query evaluation, reducing complexity and allowing for query refinement through atomic operations. The main differences between our method and other previous pattern-based graph query approaches are the ability to evaluate arbitrary subgraphs instead of nodes or complete graphs, the fact that it is based on mathematical formalization that allows the study of refinements and their complementarity, and the ability to detect cyclic patterns in polynomial time. Application examples show that the proposed framework allows learning relational classifiers to be efficient in generating data with high expressiveness capacities. Specifically, relational decision trees are learned from sets of tagged subnetworks that provide both classifiers and characteristic patterns for the identified classes.

Reasoning about logical systems in the Coq proof assistant

The theory of institutions provides an abstract mathematical framework for specifying logical systems and their semantic relationships. Institutions are based on category theory and have deep roots in a well-developed branch of algebraic specification. However, there are no machine-assisted proofs of correctness for institution-theoretic constructions—chiefly satisfaction conditions for institutions and their (co)morphisms—making them difficult to incorporate into mainstream formal methods. This paper therefore provides the details of our approach to formalizing a fragment of the theory of institutions in the Coq proof assistant. We instantiate this framework with the institutions FOPEQ for first-order predicate logic and EVT for the Event-B specification language, and define some institution-independent constructions, all of which serve as an illustration and evaluation of the overall approach.

On duality and model theory for polyadic spaces

This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness properties, and allows for a modular, syntax-free treatment of these model-theoretic results. As further applications of the same method, we prove completeness theorems for constant domain and Gödel-Dummett intuitionistic predicate logics.

On Harrop disjunction property in intermediate predicate logics

On Harrop disjunction property in intermediate predicate logics

AN ALGEBRAIC PROOF OF COMPLETENESS FOR MONADIC FUZZY PREDICATE LOGIC

Abstract Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$ , which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$ . The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$ , we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$ , giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$ , and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

Discrete quantum structures I: Quantum predicate logic

A discrete quantum structure is a discrete quantum space that is equipped with relations and functions of various finite arities. Discrete quantum spaces are identified with hereditarily atomic von Neumann algebras, their relations with projection operators, and their functions with unital normal * -homomorphisms. The propositional quantum logic of Birkhoff and von Neumann has been extended to a predicate quantum logic by Weaver; we investigate this predicate quantum logic as the internal logic of discrete quantum structures. We extend this predicate quantum logic to include function symbols and an equality symbol. Overall, we recover the basic structures of discrete quantum mathematics from physical first principles. More complicated structures will be recovered similarly in part II of this paper.

The suppression task and first‐order predicate calculus

AbstractThe suppression task challenges classical logic. Classical logic is monotonic. However, in the suppression task, an inference with the form of modus ponendo ponens is inhibited by adding a new premise. Several explanations have been given to account for this fact. The present paper indicates three of them as examples: that of the theory of mental models, that based on logic programming and closed world assumption, and that referring to Carnap's concept of state‐descriptions. Besides, the paper offers one more explanation linked to Carnap's idea of reduction. It proposes sentences akin to a reduction sentence, a pair reduction, and a bilateral reduction sentence. Those sentences are related to the initial conditional in the suppression task. This proposal tries to stay within first‐order predicate logic.

Semiproducts, Products, and Modal Predicate Logics: Some Examples

Semiproducts, Products, and Modal Predicate Logics: Some Examples