Language Of First-order Logic Research Articles

Hamilton’s cumular conception of quantifying particles: an exercise in third-order logic

Abstract Sir William Hamilton is remembered for his proposal to extend the four traditional categoricals to eight by quantifying predicate as well as subject terms. He intended the quantifying particles to be understood in a ‘collective’ or ‘cumular’ manner rather than in a ‘distributive’ or ‘exemplar’ one, but commentators from De Morgan onwards have worked primarily from the latter perspective, comforted in the 20th century by the fact that it translates readily into the language of first-order logic with identity. Formal representation of the cumular approach needs more sophisticated resources, and the paper shows how it may be carried out using selection functions in the language of third-order logic. It also reviews a number of variants, some equivalent and others not so, as well as their reductions to second-order logic, and situates historical sources, both before and after Hamilton, with respect to the web of formal constructions.

The Modal Components of Judgements in a Quantum Model of Psychoanalytic Theory.

In the present paper, we develop a theory of thinking based on an attempt to formalize the construction of mental representations as described in psychoanalytic theory. In previous work, we described Freud's and Matte Blanco's structural Unconscious in a formal model in which the properties of unconscious representations are captured by particular sets-infinite singletons-that can be derived in first-order logic language. Here, we afford the issue of the finitization of unconscious representations by assuming that the mind can form an all-purpose modality, originating from abstraction from infinite singletons; in this way, a symmetric prelogical setting for mental representations is formally created, and this is interpreted in a quantum spin model by a modal (necessity) projector. Then, by introducing time, one can describe the links that mental representations can establish with reality, and hence finitize the representations. The modality is so split into finite components, here termed positive, negative and irreal; the splitting of the modality is traced back to the decomposition of the spin observables by means of the Pauli matrices, which can offer a quantum semantics to the method applied. Here, we suggest that the development of the modal approach and its quantum logic implementation can be considered as a proper formalization of some aspect of the psychoanalytic theory of thinking proposed by Bion; namely, we will show that the process of abstraction leading from raw data to preconceptions, and therefore to the definition of the content-container relationship, is adequately captured by our model, and further correspondences can be detected with Bion's theory about links and transformations, implying different ways in which the mind can get in touch with both internal and external reality.

Физическая культура в E-learning

Forming the need for a healthy lifestyle, the transfer of knowledge, skills and abilities in the field of physical culture and sports are an urgent pedagogical problem in the transition from the “classical” forms of organization of the educational process to E-learning. The solution of this problem by means of well-known programming languages is impossible due to the insufficient descriptive potential of the latter, in particular, the lack of the possibility of representing human knowledge, probable or possible future events, actions, results. The purpose of our research is to develop theoretical provisions and to verify in practice the possibilities of formalizing the presentation of the individual educational goal of the student and the conscious need for a healthy lifestyle as its basic component in declarative languages, as well as its implementation by means of the educational platform of the university. The researcher considers the possibility of describing the individual educational goal of a student, including a conscious need for a healthy lifestyle as its basic component, gaining knowledge, forming and developing skills in the field of physical culture and sports, as well as the possibility of building an individual educational trajectory of its achievements in the organization of E-learning at the university in the language of first-order predicate logic. A number of features of the predicate logic language that are essential for the problem under consideration are singled out. A pedagogical experiment was conducted on the basis of the educational platform of Voronezh State Pedagogical University. The study involved 158 students and undergraduates. The descriptions of individual educational goals of the student, formalized by the language of logic of predicates, suitable for implementation by means of computer technology are obtained, on the basis of which the optimal trajectory for achieving this goal is calculated for each participant. The practical implementation of the educational process was carried out, data on a conscious need for a healthy lifestyle, gaining knowledge, forming and developing skills in the field of physical culture and sports were obtained. The obtained results were compared with similar studies and a conclusion was made about the ways to improve the suggested approach to the formation of a conscious need for students to lead a healthy lifestyle, gain knowledge, form and develop their skills in the field of physical culture and sports in the context of the transition from “classical” forms of organization of the educational process to E-learning.

ON DATABASE QUERIES OVER A WEAKLY O-MINIMAL DOMAIN WITH A SMALL COUNTABLE SPECTRUM

In the relational database model introduced by E.F. Codd, the database state is understood as a finite set of relationships between elements. The names of the relationships and their arnts (locations) are fixed and called the database schema. The individual information stored in the relationships of this schema is called the state of the database. Although relational databases were devised for finite sets of data, it is often convenient to assume that there is an infinite domain. We consider relational databases organized over an ordered domain with some additional relations – a typical example is the set of rational numbers together with the relation of linear order and binary operation of addition. If the first-order predicate logic language is used as the query language, then queries can use both database relationships and domain relationships, with variables changing throughout the domain. In the focus of our study are the first-order (FO) queries that are invariant under order-preserving permutations – such queries are called order-generic. It was discovered that for some domains order-generic FO queries fail to express more than pure order queries. Here we prove the collapse result theorem over a weakly o-minimal domain having convexity rank 1 and a small countable spectrum.

Rigorous Design of Fault Tolerance and Recovery Algorithm for Disaster Management and Relief Distribution System using Event-B

India is vulnerable to disasters such as earthquakes, floods, tsunamis, landslides forest fires and cyclones due to its unique socio-economic and geo-climatic conditions. Twenty seven out of thirty-six states and union territories are prone to different types of disasters which cause loss of life, disruption of livelihoods, damage to infrastructure and property which in turn becomes a heavy burden on the national economy. Effective management of relief work is a key step towards normalizing human life post disaster. In this paper, we have presented the formal development and verification of a fault tolerance and recovery algorithm for district level disaster control centers in India which are connected to each other via a communication network. Formal methods help in the verification of critical properties of complex systems by developing mathematical models so that design errors can be detected and removed during the early stages of software development. Event-B, which is a formal method and Rodin platform is used for this work. Event-B is a mathematical language of first-order logic to provide a solution to the complex algorithms formally. In this algorithm a Disaster Control Centre is chosen as the coordinator based on its unique vote value. This vote value is allotted and modified dynamically based on the extent of damage in the area where the center is located. The center having the highest vote value among the currently active centers is elected as the coordinator. The correctness of the algorithm is verified through discharge of proof obligations generated by the Event-B model.

Declarative Programming with Intensional Sets in Java Using JSetL

Abstract Intensional sets are sets given by a property rather than by enumerating their elements. In a previous work, we have proposed a decision procedure for a first-order logic language which provides restricted intensional sets (RISs), i.e. a sub-class of intensional sets that are guaranteed to denote finite—though unbounded—sets. In this paper, we show how RIS can be exploited as a convenient programming tool also in a conventional setting, namely the imperative O-O language Java. We do this by considering a Java library, called JSetL, that integrates the notions of logical variable, (set) unification and constraints that are typical of constraint logic programming languages into the Java language. We show how JSetL is naturally extended to accommodate for RIS and RIS constraints and how this extension can be exploited; on the one hand, to support a more declarative style of programming and, on the other hand, to effectively enhance the expressive power of the constraint language provided by the library.

On the Convexity of a Fragment of Pure Set Theory with Applications within a Nelson-Oppen Framework

The Satisfiability Modulo Theories (SMT) issue concerns the satisfiability of formulae from multiple background theories, usually expressed in the language of first-order predicate logic with equality. SMT solvers are often based on variants of the Nelson-Oppen combination method, a solver for the quantifier-free fragment of the combination of theories with disjoint signatures, via cooperation among their decision procedures. When each of the theories to be combined by the Nelson-Oppen method is convex (that is, any conjunction of its literals can imply a disjunction of equalities only when it implies at least one of the equalities) and decidable in polynomial time, the running time of the combination procedure is guaranteed to be polynomial in the size of the input formula. In this paper, we prove the convexity of a fragment of Zermelo-Fraenkel set theory, called Multi-Level Syllogistic, most of whose polynomially decidable fragments we have recently characterized.

Exorcising the phantom zone

In this paper we introduce a language of first-order hybrid logic in which function symbols are interpreted by partial functions and prove a number of completeness results. Syntactically, our language builds on the basic propositional hybrid language, has a primitive unary predicate symbol DEN which tests whether a term denotes or not, and permits satisfaction operators to rigidify predicate and function symbols. Semantically, our system is actualist, allows terms to be undefined, and has no truth-value gaps. But should we follow Fitting and Mendelsohn and rule out domain elements not belonging to any world, or should we tolerate them? Here we explore both options. As we shall see, while the choice makes no difference when it comes to validity, it has consequences for richer logics.

Automated Reasoning with Restricted Intensional Sets

Intensional sets, i.e., sets given by a property rather than by enumerating elements, are widely recognized as a key feature to describe complex problems (see, e.g., specification languages such as B and Z). Notwithstanding, very few tools exist supporting high-level automated reasoning on general formulas involving intensional sets. In this paper we present a decision procedure for a first-order logic language offering both extensional and (a restricted form of) intensional sets (RIS). RIS are introduced as first-class citizens of the language, and set-theoretical operators on RIS are dealt with as constraints. Syntactic restrictions on RIS guarantee that the denoted sets are finite. The language of RIS, called $${\mathcal {L}}_\mathcal {RIS}$$ , is parametric with respect to any first-order theory $${\mathcal {X}}$$ providing at least equality and a decision procedure for $${\mathcal {X}}$$ -formulas. In particular, we consider the instance of $${\mathcal {L}}_\mathcal {RIS}$$ when $${\mathcal {X}}$$ is the theory of hereditarily finite sets and binary relations. We also present a working implementation of this instance as part of the $$\{log\}$$ tool, and we show through a number of examples and two case studies that, although RIS are a subclass of general intensional sets, they are still sufficiently expressive as to encode and solve many interesting problems. Finally, an extensive empirical evaluation provides evidence that the tool can be used in practice.

Geo-Ontology and Geographical Information System Extended by a first-order logic language: Application to malaria control

Geo-Ontology and Geographical Information System Extended by a first-order logic language: Application to malaria control

Sobre Eliminar a Identidade da lógica

Neste artigo argumentarei que não se pode afirmar sem qualificação que a identidade pode ser removida da lógica como proposto por Wehmeier (2004). Isto é feito ao apontar o fato básico de que a identidade não é definível na linguagem da lógica de primeira ordem. Como sugerido por Béziau (2013), em lógica de primeira ordem temos no máximo a Identidade de Leibniz—uma relação de congruência e não a identidade numérica. Portanto, qualquer tentativa de dispensar a identidade em termos de equivalência lógica entre linguagens de primeira ordem não pode ser bem sucedida. Além disso, apresentamos uma desvantagem metafísica ao mudar para a lógica defendida por Wehmeier (2004).

Geo-Ontology and Geographical Information System Extended by a first-order logic language: Application to malaria control

Geo-Ontology and Geographical Information System Extended by a first-order logic language: Application to malaria control

FINITE DOMAIN STRUCTURES IN THE FRAMEWORK OF THE CONCEPT OF A MODEL-THEORETIC PROPERTY

Abstract. In this work, we follow the algebraic approach using definability by formulas presentable in both existential and universal forms. The class of algebraic Cartesian interpretations of theories is studied presenting a foundation of the finitary first-order combinatorics. Common properties of first-order definability in finite models are studied. Some relations are obtained between automorphism groups of finite models and isomorphisms of Cartesian extensions of their theories. A formal definition of the notion of a model-theoretic property is analyzed based on a separate consideration of cases of theories with finite and infinite models. A description of model-theoretic properties defined via finite domains is found. It is established that the class of all finite models with first-order definable elements as well as the corresponding class of theories of such models forms the only model-theoretic property and, therefore, is of little interest as a database with an interface based on the first-order logic language. Key words: first-order logic, Cartesian extension of a theory, Tarski-Lindenbaum algebra, model-theoreticproperty, computable isomorphism.

Some remarks about nonstandard methods in analysis. I

This and forthcoming articles discuss two of the most known nonstandard methods of analysis---the Robinsons infinitesimal analysis and the Boolean valued analysis, the history of their origination, common features, differences, applications and prospects. This article contains a review of infinitesimal analysis and the original method of forcing. The presentation is intended for a reader who is familiar only with the most basic concepts of mathematical logic---the language of first-order predicate logic and its interpretations. It is also desirable to have some idea of the formal proofs and the Zermelo--Fraenkel axiomatics of the set theory. In presenting the infinitesimal analysis, special attention is paid to formalizing the sentences of ordinary mathematics in a first-order language for a superstructure. The presentation of the forcing method is preceded by a brief review of C.Godels result on the compatibility of the Axiom of Choice and the Continuum Hypothesis with Zermelo--Fraenkels axiomatics. The forthcoming article is devoted to Boolean valued models and to the Boolean valued analysis, with particular attention to the history of its origination.

Characterizing Strongly First Order Dependencies: The Non-Jumping Relativizable Case

Team Semantics generalizes Tarski's Semantics for First Order Logic by allowing formulas to be satisfied or not satisfied by sets of assignments rather than by single assignments. Because of this, in Team Semantics it is possible to extend the language of First Order Logic via new types of atomic formulas that express dependencies between different assignments. Some of these extensions are much more expressive than First Order Logic proper; but the problem of which atoms can instead be added to First Order Logic without increasing its expressive power is still unsolved. In this work, I provide an answer to this question under the additional assumptions (true of most atoms studied so far) that the dependency atoms are relativizable and non-jumping. Furthermore, I show that the global (or Boolean) disjunction connective can be added to any strongly first order family of dependencies without increasing the expressive power, but that the same is not true in general for non strongly first order dependencies.

Solving Quantifier-Free First-Order Constraints Over Finite Sets and Binary Relations

In this paper we present a solver for a first-order logic language where sets and binary relations can be freely and naturally combined. The language can express, at least, any full set relation algebra on finite sets. It provides untyped, hereditarily finite sets, whose elements can be variables, and basically all the classic set and relational operators used in formal languages such as B and Z. Sets are first-class entities in the language, thus they are not encoded in lower level theories. Relations are just sets of ordered pairs. The solver exploits set unification and set constraint solving as primitive features. The solver is proved to be a sound semi-decision procedure for the accepted language. A Prolog implementation is presented and an extensive empirical evaluation provides evidence of its usefulness.

First-order logic and aperiodic languages

A fundamental result about formal languages states: Theorem 1 A regular language is first-order definable if and only if its syntactic monoid contains no nontrivial groups. Rest assured, we will explain in the next section exactly what the various terms in the statement mean!

On the expressive power of first-order modal logic with two-dimensional operators

Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these extensions are often justified only by example rather than by rigorous proof. In this paper, we provide proofs of many of these claims and present a more complete picture of the expressive landscape for such languages.

Improving process conformance with Industry Foundation Processes (IFP)

Conformance with standard corporate and institutional processes and industry best practices are sought because of regulatory requirements and evidence that best practices lead to improved project performance. Automated workflow engine enabled Industry Foundation Processes (IFP) are introduced in this paper that facilitate process conformance through structural transparency, foundation process inheritance, and automated conformance checking. While IFP processes can be customized to suit particular project or corporate conditions, they need to conform to a standard core structure and thus behavior. This has been achieved through defining specific workflow inheritance rules and developing an automated structural process conformance checking algorithm. The algorithm has been developed based on graph theory fundamentals using a first-order logic language, which compares two workflows and detects the conformance of a customized one with its associated IFP. The developed algorithm has been functionally tested and validated with different structural settings of workflows with a number of critical cases. Its functionality has been demonstrated in this paper with an example of the commonly used process of RFI (Request for Information). A new construct is thus contributed that can help improve process conformance to industry best practices particularly in the architectural, engineering, and construction industry, leading to improved project conformance.

A Logic for Trial and Error Classifiers

Trial and error classifiers, corresponding to concepts which change their extensions over time, are introduced and briefly philosophically motivated. A fragment of the language of classical first-order logic is given a new semantics, using $$\omega $$?-sequences of classical models, in order to interpret the basic predicates as classifiers of this kind. It turns out that we can use a natural deduction proof system which differs from classical logic only in the conditions for application of existential elimination. Soundness and completeness theorems are proved for this system.