The paper continues a study of extensional formulas and their representation by normal forms. The notion of extensional formula is extended to the case when extensionality is considered with respect to a binary reflexive relation. It has been proved that extensional formulas can be represented by normal forms of infinite type and “approximated” by normal forms of finite type. All proofs are giv...

Probability judgments about logical propositions have raised substantial doubts about human rationality. Here we explore the idea that people’s probability judgments often may not refer to the relative frequency of a set, but instead to the probability of an explanatory logical pattern given the data. This idea has been formalized by Bayesian logic (BL), predicting a system of frequency-based l...

Because of the reason that the individuals’ thoughts have the complicated and uncertain characteristics in the field of social science, this study makes the most of the fuzzy sets, which solve uncertain and imprecise conditions, to transform the opinions belonging to human beings into the quantitative numbers. Advertising involvement is regarded as the object for this empirical study which comp...

This paper is a study about the relationships among topologies and intuitionistic fuzzy topology induced, respectively, by approximation operators and an intuitionistic fuzzy approximation operator associated with an approximation space (X,R), when the relation R on X is precisely reflexive and transitive. In particular, we consider an intuitionistic fuzzy approximation operator on an approxima...

An often raised objection to fuzzy logic can be articulated as follows. Fuzzy sets are intended to represent vague collections of objects. On the one hand, boundaries of such collections are inherently imprecise. On the other hand, fuzzy sets are defined by membership degrees which are precise. There is, therefore, an obvious discrepancy between the imprecision of a collection and the precision...

Fuzzy logic and fuzzy systems have recently been receiving a lot of attention, both from the media and scientific community, yet the basic techniques were originally developed in the mid-sixties. Fuzzy logic provides a formalism for implementing expert or heuristic rules on computers, and while this is the main goal in the field of expert or knowledge-based systems, fuzzy systems have had consi...

The immune system has unique defense mechanisms such as innate, humoral and cellular immunity. These mechanisms are closely related to prevent pathogens from spreading in the host and to clear them effectively. To get a comprehensive understanding of the immune system, it is necessary to integrate the knowledge through modeling. Many immune models have been developed based on differential equat...

This paper studies the class of automaton semigroups from two perspectives: closure under constructions, and examples of semigroups that are not automaton semigroups. We prove that (semigroup) free products of finite semigroups always arise as automaton semigroups, and that the class of automaton monoids is closed under forming wreath products with finite monoids. We also consider closure under...