Generalized Realizability and Basic Logic

Let V be a set of number-theoretical functions. We define a notion of absolute V -realizability for predicate formulas and sequents in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Basic Predicate Calculus is sound with respect to the semantics of absolute V -realizability if V satisfies some natural conditions.

Comparison of Different Knowledge Representations for Complex Structured Objects in Solving AI Problems

The problem of knowledge representation for a complex structured object is one of the actual problems of AI. This is due to the fact that many of the objects under study are not a single indivisible object characterized by its properties, but complex structures whose elements have some known properties and are in some, often multiplace, relations with each other. An approach to the representation of such knowledge based on first-order logic (predicate calculus formulas) is compared in this paper with two currently widespread approaches based on the representation of data information with the use of finite-valued strings or graphs. It is shown that the use of predicate calculus formulas for description of a complex structured object, despite the NP-difficulty of the solved problems arising after formalization, actually have no greater computational complexity than the other two approaches, what is usually not mentioned by their supporters. An algorithm for constructing an ontology is proposed that does not depend on the methodof desc ribing an object, and is based on the selection of the maximum common property of objects from a given set.

Comprehensibility of an ontological model as a characteristic of its quality

There is a tendency to combine professional knowledge within the framework of the creation of cyber-physical sys-tems. This encourages making them available to a wide range of stakeholders. Various models of knowledge represen-tation are considered from the standpoint of historicism and levels of community. It is proposed to add an interdiscipli-nary (general scientific) level to the three levels of knowledge representation according to the degree of its generality – philosophical, narrowly disciplinary (professional) and subject level (knowledge base). It is argued that any model of knowledge representation is based on the model of a formal system and its child model, the predicate calculus. A la-beled graph is a visual model for representing knowledge. A generalized model that reflects the relationship between knowledge and cognition is proposed. Understanding the model of knowledge is interpreted as a means of cognition. The term of comprehensibility of the ontological model is introduced as a property that most characterizes its quality. The comprehensibility of the ontological model is divided into verbal and systemic ones. The factors influencing these components of model comprehensibility are discussed. Indicators for measureing the clarity of the ontological model are proposed.

LOGIC FOR THE THEORY OF CONCEPTS

In this paper, we follow Gödel’s remarks on an envisioned theory of concepts to determine which properties should a logical basis of such a theory have. The discussion is organized around the question of suitability of the classical predicate calculus for this role. Some reasons to think that classical logic is not an appropriate basis for the theory of concepts, will be presented. We consider, based on these reasons, which alternative logical system could fare better as a logical foundation of, in Gödel’s opinion, the most important theory in logic yet to be developed. This paper should, in particular, motivate the study of partial predicates in a certain system of three-valued logic, as a promising starting point for the foundation of the theory of concepts.

Elementary-base cirquent calculus II: Choice quantifiers

Cirquent calculus is a novel proof theory permitting component-sharing between logical expressions. Using it, the predecessor article ‘Elementary-base cirquent calculus I: Parallel and choice connectives’ built the sound and complete axiomatization $\textbf{CL16}$ of a propositional fragment of computability logic. The atoms of the language of $\textbf{CL16}$ represent elementary, i.e. moveless, games and the logical vocabulary consists of negation, parallel connectives and choice connectives. The present paper constructs the first-order version $\textbf{CL17}$ of $\textbf{CL16}$, also enjoying soundness and completeness. The language of $\textbf{CL17}$ augments that of $\textbf{CL16}$ by including choice quantifiers. Unlike classical predicate calculus, $\textbf{CL17}$ turns out to be decidable.