Abstract
Let S be a set of states of a physical system and p(s) the probability of an occurrence of an event when the system is in state s∈S. The function p from S to [0,1] is called a numerical event, multidimensional probability or, more precisely, S-probability. If a set of numerical events is ordered by the order of real functions one obtains a partial ordered set P in which the sum and difference of S-probabilities are related to their order within P. According to the structure that arises, this further opens up the opportunity to decide whether one deals with a quantum mechanical situation or a classical one. In this paper we focus on the situation that P is generated by a given set of measurements, i.e. S-probabilities, without assuming that these S-probabilities can be complemented by further measurements or are embeddable into Boolean algebras, assumptions that were made in most of the preceding papers. In particular, we study the generation by S-probabilities that can only assume the values 0 and 1, thus dealing with so called concrete logics. We characterize these logics under several suppositions that might occur with measurements and generalize our findings to arbitrary S-probabilities, this way providing a possibility to distinguish between potential classical and quantum situations and the fact that an obtained structure might not be sufficient for an appropriate decision. Moreover, we provide some explanatory examples from physics.
Highlights
Let S be a set of states of a physical system and p ( s) the probability of an occurrence of an event when the system is in state s ∈ S
Because for all j ∈ N, G consists of all unions of some of the pairwise disjoint sets BI, I ∈ 2N and is a Boolean algebra having at most 22n elements
Maczyński introduced the notion of a numerical event, i.e. S-probability
Summary
Let S be a set of states of a physical system and p ( s) the probability of the oc-. D. (For example, think of the probability that the numerical value of an observable is inside a given set of values.) consider p ( s) for all s ∈ S This way one obtains a function p from S to [0,1]. In many cases the set S will be finite so that one obtains an n-tuple of probabilities which can be thought of as an event arising when measuring an observable In their papers of 1991 and 1993 ([1] and [2]) E. We will assume that there will only be two outcomes to measurements, namely that an S-probability might be either 0 or 1, and we will generalize some of the obtained results to arbitrary numerical events. To explain our results we provide some examples of physical experiments
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