On Soft Quantum B-Algebras and Fuzzy Soft Quantum B-Algebras

This paper aims to make a combination between the quantum B-algebras (briefly, X - A s) and two interesting theories (e.g., soft set theory and fuzzy soft set theory). Firstly, we propose the novel notions of soft quantum B-algebras (briefly, S ℚ B - A s), a soft deductive system of ℚ B - A s, and deducible soft quantum B-algebras (briefly, DS ℚ B - A s). Then, we discuss the relationship between S ℚ B - A s and DS ℚ B - A s. Furthermore, we investigate the union and intersection operations of DS ℚ B - A s. Secondly, we introduce the notions of a fuzzy soft quantum B-algebras (briefly, FS ℚ B - A s), a fuzzy soft deductive system of ℚ B - A s, and present some characterizations of FS ℚ B - A s, along with several examples. Finally, we explain the basic properties of homomorphism image of FS ℚ B - A s.

Factor generalized BE-semigroups through deductive systems

In this paper, we investigate some properties of homomorphisms and deductive systems of generalized BE-semigroups. In particular, we show that through every deductive system, we may get factor generalized BE-semigroup.

The Development Trend of the Manufacturing Industry and the "One Belt And One Road" Model Research from the Historical Perspective

As the United States subprime mortgage crisis triggered by the global economic crisis, in the face of the plight of world economic recovery is weak, in order to seek better way out of this dilemma, this article emphatically from the world market on the history of the development of vertical comb, to conduct a comprehensive in-depth theory to the essence of the manufacturing industry movement rule and its development trend. Through the integrated use of literature research, exploratory study, information research methods and scientific thinking method, inductive deductive system science method, and analogical reasoning, abstract generalization analysis of a series of existing theory of manufacturing industry substance and on the basis of comprehensive study to clarify the essence of the manufacturing industry movement rule and its development trend. Therefore, against the background of sluggish world economic recovery, we must start a new engine of world economic development under the original engine of world economic development. "One Belt And One Road" reflects and complies with the historical trend of economic globalization.

Analyticity and modulation

Acceptable analyticities, i.e. contradictions or tautologies, constitute problematic evidence for the idea that language includes a deductive system. In recent discussion, two accounts have been presented in the literature to explain the available evidence. According to one of the accounts, grammatical analyticities are accessible to the system but a pragmatic strengthening repair mechanism can apply and prevent the structures from being actually interpreted as contradictions or tautologies. The proposed data, however, leaves it open whether other versions of the meaning modulation operation are required. Novel evidence we present argues that a loosening version of the repair mechanism must be available. Our observation concerns acceptable lexical contradictions that cannot be rescued if only a strengthening version of the pragmatic strategy is available.

Fuzzy deductive systems of RM algebras

The theory of fuzzy deductive systems in RM algebras is developed. Various characterizations of fuzzy deductive systems are given. It is proved that the set of all fuzzy deductive systems of a RM algebra 𝒜 is a complete lattice (it is distributive if 𝒜 is a pre-BBBCC algebra). Some characterizations of Noetherian RM algebras by fuzzy deductive systems are obtained. In pre-BBBZ algebras, the fuzzy deductive system generated by a fuzzy set is constructed. Finally, closed fuzzy deductive systems are defined and studied. It is showed that in finite CI and pre-BBBZ algebras, every fuzzy deductive system is closed. Moreover, the homomorphic properties of (closed) fuzzy deductive systems are provided.

Interactive proof-search for equational reasoning

Equational reasoning arises in many areas of mathematics and computer science. It is a cornerstone of algebraic reasoning and results essential in tasks of specification and verification in functional programming, where a program is mainly a set of equations. The usual manipulation of identities while conducting informal proofs obviates many intermediate steps that are neccesary while developing them using a formal system, such as the equationally complete Birkhoff calculus ${\mathcal{B}}$. This deductive system does not fit in the common manner of doing mathematical proofs, and it is not compatible with the mechanisms of proof assistants. The aim of this work is to provide a deductive system ${\mathcal{B}}^{\textrm{GOAL}}$ for equality, equivalent to ${\mathcal{B}}$ but suitable for constructing equational proofs in a backward fashion. This feature makes it adequate for interactive proof-search in the approach of proof assistants. This will be achieved by turning ${\mathcal{B}}^{\textrm{GOAL}}$ into a transition system of formal tactics in the style of Edinburgh LCF, such transformation allows us to give a direct formal definition of backward proof in equational logic.

Aristotle’s Syllogistic as a Deductive System

Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the Aristotelian system of syllogistic and Łukasiewicz’s reconstruction of it based on the tools of modern formal logic. We pay special attention to the notion of completeness of a deductive system as discussed by both authors. We describe in detail how completeness can be defined and proved with the use of an axiomatic refutation system. Finally, we apply this methodology to different axiomatizations of syllogistic presented by Łukasiewicz, Lemmon and Shepherdson.