Abstract Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest generic approaches to behavioural distances. In particular, coalgebraic methods capture variations system type (nondeterministic, probabilistic, game-based etc.), and notion of quantale abstracts over actual values take, thus covering,...

Three-valued accounts of conditionals frequently promise (a) to conform to the probabilistic view that conditionals are evaluated by conditional probabilities, and (b) to yield a plausible account of compounds of conditionals. However, McGee (1981) shows that probabilistic validity, the conception of validity most naturally associated with the probabilistic view, cannot be characterized by a fi...

Probabilistic logic and credibilistic logic are two branches of multi-valued logic for dealing with random knowledge and fuzzy knowledge, respectively. In this paper, a hybrid logic is introduced for dealing with random knowledge and fuzzy knowledge simultaneously. First, a hybrid formula is introduced on the basis of random proposition and fuzzy proposition. Furthermore, a hybrid truth value i...

In 1989, Kent and Richardson [Ordered probabilistic metric spaces, J. Austral. Math. Soc. Ser. A 46(1) (1989), 88-99, MR0966286 (90b:54022)] introduced the class of probabilistic quasi-metric spaces which offers a wider framework than that of metric spaces. The aim of this paper is to prove common fixed point theorems for single-valued and set-valued weakly compatible mappings in Menger probabi...

We present n-valued rst-order logics with a purely proba-bilistic semantics. We then introduce a new probabilistic semantics of n-valued rst-order logics that lies between the purely probabilistic semantics and the truth-functional semantics of the n-valued Lukasiewicz logics Ln. Within this semantics, closed formulas of classical rst-order logics that are logically equivalent in the classical ...

where s is a “type parameter”, usually s = −i or s = 1, s = −i standing for “oscillatory”, s = 1 for “probabilistic type”. Φ is a real-valued function, in the probabilistic case assumed to be lower bounded; it is called “phase function”; f is a complex-valued function, called “amplitude function”; γ is thought to be a member of some space Γ “on which the integration extends”; dγ is a heuristic ...

Given a measurable mapping f from a nonatomic Loeb probability space (T; T ; P ) to the space of Borel probability measures on a compact metric space A, we show the existence of a measurable mapping g from (T; T ; P ) to A itself such that f and g yield the same values for the integrals associated with a countable class of functions on T A. A corollary generalizes the classical result of Dvoret...

Probabilistic programming and statistical computing are vibrant areas in the development of Julia language, but underlying infrastructure dramatically predates recent developments. The goal MeasureTheory.jl is to provide with right vocabulary tools for these tasks. In package we introduce a well-chosen set notions from foundations probability together powerful combinators transforms, giving gen...