Predicate Logic

An introduction to the formal approach that underlies university mathematics. Topics covered are: Sets and Logic, Relations and Functions, Number Systems, Infinity. The book has arisen out of a first year course taught to Mathematics majors at Shiv Nadar University, with the aim of helping students bridge the different requirements of rigour at school and university.

Abstract. In this paper we introduce a notion of context for Groenendijk & Stokhof's Dynamic Predicate Logic DPL. We use these contexts to give a characterization of the relations on assignments that can be generated by composition from tests and random resettings in the case that ...

In this text we will briefly discuss some examples of the influence of the chosen philosophy and epistemology leading in a natural way to a change in research paradigm and consequently to a change in methods of linguistic description. We will present two types of situations: •  The situation where one arrives at the same types, basically, of descriptive methods in the same discipline starting from different philosophical and epistemological foundations, generally without being aware of it or without having specified them (see, e.g., Predicates-Arguments Structures vs Object-Oriented Approach vs Classes of Objects vs. SketchEngine). •  The situation where one arrives at the same types, basically, of descriptive methods in different disciplines starting from different or similar philosophical and epistemological foundations (see e.g., Object-Oriented Approach, Classes of Objects vs Ecological psychology / Incorporated Radical Cognition). In a “good method”, descriptive methods derive from the philosophical and epistemological foundations adopted consciously or unconsciously. Sometimes, if the choices are more operational and more practical applications oriented, in “normal science”, in T. Kuhn’s terminology, one must go backwards to discover these foundations from the chosen descriptive principles. At the same time, different philosophical foundations can lead to the same types, at least roughly, of descriptive methods of reality. The descriptions made in the framework of different paradigms, like these paradigms, are complementary and, in the last instance, “cumulative”, thus giving a more complete description of the reality, which, while submitted to different visions, does not, naturally, change.

Historical and philosophical background providing justification for an alternative understanding of nothingness that can serve as the basis for an alternative to the empty set and the number 0. Nonexistence arises from using the existential quantifier. A different quantifier could lead to a different notion of nothing such as absence - "Here is [some numbers]" and its negation "here is not [some numbers]". Zero defined as the absence of the Natural numbers in the Peano Axioms for example.

Course Objectives: This course is an intermediate-level study of symbolic logic in which students will learn the methods of direct proof, conditional proof, and indirect proof as they are used in proposition logic and predicate logic. The course will begin with a quick review of the symbolization techniques of propositional logic and of truth-functional operators. This will be followed by a discussion of the methods of direct proof, conditional proof, and indirect proof in propositional logic. This will be followed by discussion of the symbolization techniques of predicate (first-order) logic, which combines the formal content of syllogistic logic and with the content of truth-functional propositional logic. Lastly, Students will learn to apply the methods of direct proof, conditional proof, and indirect proof to arguments in predicate logic. In this context, we will also be covering the quantifier negation rule, the counterexample and the finite universe methods for proving invalidity, relational predicates and overlapping quantifiers, and identity statements, superlatives, numerical statements, and definite descriptions. The course, however, will not cover second-order logic or mathematical logic (set theory, model theory, proof theory, or recursion theory).

Semantic tableau is a proof system used to prove the validity of a formula using Prolog, it can also be used to prove if a formula is a logic consequence of a set of formulas. Semantic tableau is used in both propositional and predicate logic. Tableau is a disjunction normal form. VAMPIRE is a high-performance theorem prover for first-order logic with or without equality. In this paper, a modified version of semantic tableau is presented, experimental results proved the validity of modified tableau. Also, it is shown how Tableau proof system and VAMPIRE can be used in predicate logic.

The paper presents predicates describing abstractions between concepts/classes and between instances/objects. The abstractions considered are:
- Generalization (is-a)
- Aggregation (is-part-of)
- Grouping (is-element-of)
- Instantiation (is-element-of)

Robert Brandom holds that what we mean is best understood in terms of what inferences we are prepared to defend, and that such a defence is best understood in terms of rule-governed social interactions. This manages to explain quite a lot. However, for those who think that there is more to making correct/incorrect inferences than obeying/breaking accepted rules, Brandom’s account fails to adequately capture what it means to reason properly. Thus, in an effort to sketch an alternative that does not rely primarily on peer pressure, I draw on the work of C. S. Peirce. Peirce argued that, when we reason, we manipulate abstract diagrams in order to observe what results. Since some manipulations are barred by the self-same nature of the diagrams, I try to show that this qualitative incompatibility, which I dub “contrapiction,” is a good (non-social) reason to regard some reasoning as bad.

In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles.

In the world of designing network protocols, verification is a crucial step to eliminate weaknesses and inaccuracies of effective network protocols. There are many models and tools to verify network protocols, including, Finite State Machines (FSM), Colored Petri Nets (CP-Nets), Temporal Logic, Predicate Logic, Estelle Specification, Path based Approach etc. This paper presents a survey of various techniques for verifying correctness properties of communications protocol

In Artificial Intelligence, knowledge representation is a combination of data structures and interpretive procedures that leads to knowledgeable behavior. Therefore, it is required to investigate such knowledge representation technique in which knowledge can be easily and efficiently represented in computer. This research paper compares various knowledge representation techniques and proves that integrated approach is a more efficient and more accurate knowledge representation scheme.

Web services-- Web-accessible programs and devices – are a key application area for the Semantic Web. With the proliferation of Web services and the evolution towards the Semantic Web comes the opportunity to automate various Web services tasks. Our objective is to enable markup and automated reasoning technology to describe, simulate, compose, test, and verify compositions of Web services. We take as our starting point the DAML-S DAML+OIL ontology for describing the capabilities of Web services. We define the semantics for a relevant subset of DAML-S in terms of a first-order logical language. With the semantics in hand, we encode our service descriptions in a Petri Net formalism and provide decision procedures for Web service simulation, verification and composition. We also provide an analysis of the complexity of these tasks under different restrictions to the DAML-S composite services we can describe. Finally, we present an implementation of our analysis techniques. This implem...

Temporal specifications are often used when phenomena are modelled where dynamics play a main role. If simulation is one of the aims of modelling, usually a restricted, executable modelling language format is used, based on some form of past to future implications. In this paper a detailed transformation procedure is described that takes any temporal predicate logic specification and generates

Второе издание статьи, опубликовано: Шиян Т.А. О становлении грамматических категорий языка логики предикатов // Философия. Язык. Культура. Вып. 3 / Отв. ред. В. В. Горбатов. СПб.: Алетейя, 2012. С. 78–89. (0,5 а.л.). Первая публикация: Шиян Т.А. О становлении грамматических категорий языка логики предикатов // Логика, язык и формальные модели. Сборник статей и тезисов участников Открытого Российско-Финского коллоквиума по логике – ORFIC-2012 / Под ред. Е.Н. Лисанюк, И.Б. Микиртумова, Ю.Ю. Черноскутова. СПб.: Изд-во С.-Петерб. ун-та, 2012. С. 180–188.
  =====
This paper represents the approach for the history of mathematical logic
which contrasts with the logic mathematization attempts history and
treats it as a history of the making and of the developing inside mathematics such concepts and notations which had been used for constructing the predicate logic. The paper substantiates the necessity of the distinction between variables and parameters and studies the history of the making of the distinction as well as the mathematical single symbols appearance. It is considered in the paper the making of symbolic notations for mathematical operations, the forming of function concept and of meanings for generalized marking for functions. It is briefly considered the making of symbolic notations for mathematical relations and marked some ways of the predicate concept forming. The paper pays any attention to the making of the first symbolic notations for logical connectives and the forming modern conception for quantification.