Negation and Implication in Quasi-Nelson Logic

Quasi-Nelson logic is a recently-introduced generalization of Nelson’s constructive logic with strong negation to a non-involutive setting. In the present paper we axiomatize the negation-implication fragment of quasi-Nelson logic (QNI-logic), which constitutes in a sense the algebraizable core of quasi-Nelson logic. We introduce a finite Hilbert-style calculus for QNI-logic, showing completeness and algebraizability with respect to the variety of QNI-algebras. Members of the latter class, also introduced and investigated in a recent paper, are precisely the negation-implication subreducts of quasi-Nelson algebras. Relying on our completeness result, we also show how the negation-implication fragments of intuitionistic logic and Nelson’s constructive logic may both be obtained as schematic extensions of QNI-logic.

ASSESSMENT OF THE DEPENDENCE OF THE AVERAGE LIFE EXPECTANCY OF THE POPULATION RESIDING IN THE TULA REGION IN THE AREAS WITH A HIGH CONTENT OF HEA VYMETALS IN THE SOIL

The article presents the relevance of the problem, defines the research purpose: to compare the average life expectancy of the population in the areas of the Tula region with different contents of heavy metals in the class of causes of death “Respiratory diseases ”. The authors used the data of the regional mortality register, the results of analyzes of the content of heavy metals (copper, lead, zinc, nickel) in the soil by atomic absorption spectroscopy, and the calculation of the average life expectancy by the algebraic model of constructive logic. The results indicate a decrease in average life expectancy due to the presence of heavy metals in the soil, but the average life expectancy in both contaminated and non-contaminated areas is gradually increasing.

Constructivism in mathematics

Constructivism is not a matter of principles: there are no specifically constructive mathematical axioms which all constructivists accept. Even so, it is traditional to view constructivists as insisting, in one way or another, that proofs of crucial existential theorems in mathematics respect constructive existence: that a crucial existential claim which is constructively admissible must afford means for constructing an instance of it which is also admissible. Allegiance to this idea often demands changes in conventional views about mathematical objects, operations and logic, and, hence, demands reworkings of ordinary mathematics along nonclassical lines. Constructive existence may be so interpreted as to require the abrogation of the law of the excluded middle and the adoption of nonstandard laws of constructive logic and mathematics in its place. There has been great variation in the forms of constructivism, each form distinguished in its interpretation of constructive existence, in its approaches to mathematical ontology and constructive logic, and in the methods chosen to prove theorems, particularly theorems of real analysis. In the twentieth century, Russian constructivism, new constructivism, Brouwerian intuitionism, finitism and predicativism have been the most influential forms of constructivism.

Formalization as the Immanent Part of Logical Solving

The work is devoted to the logical analysis of the problem solving by logical means. It starts from general characteristic of the applied logic as a tool: 1. to bound logic with its applications in theory and practice; 2. to import methods and methodologies from other domains into logic; 3. to export methods and methodologies from logic into other domains. The precise solving of a precisely stated logical problem occupies only one third of the whole process of solving real problems by logical means. The formalizing precedes it and the deformalizing follows it. The main topic when considering formalization is a choice of a logic. The classical logic is usually the best one for a draft formalization. The given problem and peculiarities of the draft formalization could sometimes advise us to use some other logic. If axioms of the classical formalization have some restricted form this is often the advice to use temporal, modal or multi-valued logic. More precisely, if all binary predicates occur only in premises of implications then it is possible sometimes to replace a predicate classical formalization by a propositional modal or temporal in the appropriate logic. If all predicates are unary and some of them occur only in premises then the classical logic maybe can replaced by a more adequate multi-valued. This idea is inspired by using Rosser–Turkette operator $J_i$in the book [22]. If we are interested not in a bare proof but in construction it gives us it is often to transfer to an appropriate constructive logic. Its choice is directed by our main resource (time, real values, money or any other imaginable resource) and by other restrictions.Logics of different by their nature resources are mutually inconsistent (e.g. nilpotent logics of time and linear logics of money). Also it is shown by example how Arnold’s principle works in logic: too “precise” formalization often becomes less adequate than more “rough”. DOI: 10.21146/2074-1472-2018-24-1-129-145

A CONSTRUCTIVE EXAMINATION OF A RUSSELL-STYLE RAMIFIED TYPE THEORY

In this article we examine the natural interpretation of a ramified type hierarchy into Martin-Löf type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of Russell’s reducibility axiom are valid, namely functional reducibility. This is sufficient to make the type hierarchy usable for development of constructive mathematical analysis in the style of Bishop. We present a ramified type theory suitable for this purpose. One may regard the results of this article as an alternative solution to the problem of the proliferation of levels of real numbers in Russell’s theory, which avoids impredicativity, but instead imposes constructive logic. The intuitionistic ramified type theory introduced here also suggests that there is a natural associated notion of predicative elementary topos.

Well-founded Functions and Extreme Predicates in Dafny: A Tutorial

A recursive function is well defined if its every recursive callcorresponds a decrease in some well-founded order. Such a function issaid to be _terminating_ and is in many applications the standard wayto define a function. A boolean function can also be defined asan extreme solution to a recurrence relation, that is, as a least orgreatest fixpoint of some functor. Such _extreme predicates_ areuseful to encode a set of inductive or coinductive inference rulesand are at the core of many a constructive logic. Theverification-aware programming language Dafny supports bothterminating functions and extreme predicates. This tutorialdescribes the difference in general terms, and then describes novelsyntactic support in Dafny for defining and proving lemmas withextreme predicates. Various examples and considerations are given.Although Dafny's verifier has at its core a first-order SMT solver,Dafny's logical encoding makes it possible to reason about fixpointsin an automated way.