Logic, states, and quantum probabilities

Gunn Quznetsov

This is the logical foundation for for Relativity Theory, Probability Theory, and for Quantum Theory. Contents is the following: 1 Introduction. 2 Classical logic. 3 Time and space. 3.1 Recorders. 3.2 Time. 3.3 Space. 3.4 Relativity. 4. Probability. 4.1 B-functions. 4.2 Independent tests. 4.3 Function of probability. 4.4 Conditional probability. 4.5 Classical probability 4.6 B-functions and classical propositional logic. 4.7 Consistency of the probability function. 4.7.1 Nonstandard numbers. 4.7.2 Model. 5. Quantum theory. 5.1 Events and the moving equations. 5.2 Rotations of the x5Ox4 system and B-bosonn. 5.3 Masses. 5.4 The one-mass states, particles and antiparticles. 5.5 The two-masses states. 5.5.1 Neutrinno. 5.5.2 Electroweak transformations. 5.6 Rotations of the Cartesian coordinates system and quarrks. 5.7 Gustatory pentads. 5.8 Two events. 5.9 The dimension of physical space. 5.10 The events-probability interpretation of Quantum Theory. 6. Conclusion.

Meir Hemmo, Orly Shenker

Dietmar Dorninger, Helmut Länger

Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s in S. Such a function p from S to [0,1] is known as a numerical event or more accurately an S-probability. A set P of numerical events including the constant functions 0 and 1 and 1-p with every p in P becomes a poset when ordered by the order of real functions and can serve as a general setting for quantum logics. We call such a poset P a general set of events (GSE). The thoroughly investigated algebras of S-probabilities (including Hilbert logics), concrete logics and Boolean algebras can all be represented within this setting. In this paper we study various classes of GSEs, in particular those that are orthoposets and their interrelations and connections to known logics. Moreover, we characterize GSEs as posets by means of states and discuss the situation for GSEs to be lattices.

Patrick Suppes

The aim of this paper is to state the single most powerful argument for use of a non-classical logic in quantum mechanics. In outline the argument is the following. The working logic of a science is the logic of the events and propositions to which probabilities are assigned. A probability should be assigned to every element of the algebra of events. In the case of quantum mechanics probabilities may be assigned to events but not, without restriction, to the conjunction of two events. The conclusion is that the working logic of quantum mechanics is not classical. The nature of the logic that is appropriate for quantum mechanics is examined.

Gregg S. Jaeger

Roberto Leporini