In this paper, we shall firstly illustrate why we should introduce set-valued stochastic integrals, and then we shall discuss some properties of set-valued stochastic processes and the relation between a set-valued stochastic process and its selection set. After recalling the Aumann type definition of stochastic integral, we shall introduce a new definition of Lebesgue integral of a set-valued ...

Kleene’s strong three-valued logic extends naturally to a four-valued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it four-valued analogs of Kleene’s weak three-valued logic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bi...

This paper studies Lebesgue integral of a fuzzy closed set-valued stochastic process with respect to the time t. Firstly, a progressively measurable fuzzy closed set-valued stochastic process is discussed and an almost everywhere problem in the former Aumann type Lebesgue integral of the level-set process is pointed out. Secondly, a new definition of the Lebesgue integral by decomposable closur...

In his well-known paper \How computer should think" ((Be77b]) Belnap argues that four valued semantics is a very suitable setting for computerized reasoning. In this paper we vindicate this thesis by showing that the logical role that the four-valued structure has among Ginsberg's well-known bilattices is similar to the role that the two-valued algebra has among Boolean algebras. Speciically, w...

Axiomatic bases of admissible rules are obtained for fragments of the substructural logic R-mingle. In particular, it is shown that a “modus-ponens-like” rule introduced by Arnon Avron forms a basis for the admissible rules of its implication and implication-fusion fragments, while a basis for the admissible rules of the full multiplicative fragment requires an additional countably infinite set...

Information technology (IT) outsourcing has been recognized as a new methodology in many organizations. Yet making an appropriate decision with regard to selection and use of these methodologies may impose uncertainties and risks. Estimating the occurrence probability of risks and their impacts organizations goals may reduce their threats. In this study, an extended analytical hierarchical proc...

In this paper, under some appropriate conditions, we prove some $Delta$ and strong convergence theorems of endpoints for multi-valued nonexpansive mappings using modified Agarwal-O'Regan-Sahu iterative process in the general setting of 2-uniformly convex hyperbolic spaces. Our results extend and unify some recent results of the current literature.

Logical systems with classical negation and means for sentential or propositional self-reference involve, in some way, paradoxical statements such as the liar. However, the paradox disappears if one replaces classical by an appropriate non-classical negation such as a paraconsistent one (no paradox arises if the liar is both true and false). We consider a non-Fregean logic which is a revised an...

ACP is combined with Belnap’s four-valued logic via conditional composition (if–then–else). We show that the operators of ACP can be seen as instances of more general, conditional operators. For example, both the choice operator + and δ (deadlock) can be seen as instances of conditional composition, and the axiom x + δ = x follows from this perspective. Parallel composition is generalized to th...