Propensities are presented as a generalization of classical determinism. They describe a physical reality intermediary between Laplacian determinism and pure randomness, such as in quantum mechanics. They are characterized by the fact that their values are determined by the collection of all actual properties. It is argued that they do not satisfy Kolmogorov axioms; other axioms are proposed.

In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int (...) which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus. (shrink)

In the present paper we discuss: a) how Hilbert’s unification program failed completely, and b) we outline a new electron model based on Helmholtz’s electron vortex and Kolmogorov theory of turbulence. Novelty aspect: we discuss among other things, electron capture event, and von Karman vortex street. We also discuss a new model of origination of charge and matter. This paper is a sequel to a preceding paper on similar theme.

Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov’s probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice–Bob EPR paradigm, non-contextuality means that the identity of Alice’s spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis $\alpha _{i}$ chosen by Alice, irrespective of Bob’s axis $\beta _{j}$ (and vice versa). Here, we study contextual KPT models, with two properties: (1) (...) Alice’s and Bob’s spins are identified as $A_{ij}$ and $B_{ij}$ , even though their distributions are determined by, respectively, $\alpha _{i}$ alone and $\beta _{j}$ alone, in accordance with the no-signaling requirement; and (2) the joint distributions of the spins $A_{ij},B_{ij}$ across all values of $\alpha _{i},\beta _{j}$ are constrained by fixing distributions of some subsets thereof. Of special interest among these subsets is the set of probabilistic connections, defined as the pairs $\left( A_{ij},A_{ij'}\right) $ and $\left( B_{ij},B_{i'j}\right) $ with $\alpha _{i}\not =\alpha _{i'}$ and $\beta _{j}\not =\beta _{j'}$ (the non-contextuality assumption is obtained as a special case of connections, with zero probabilities of $A_{ij}\not =A_{ij'}$ and $B_{ij}\not =B_{i'j}$ ). Thus, one can achieve a complete KPT characterization of the Bell-type inequalities, or Tsirelson’s inequalities, by specifying the distributions of probabilistic connections compatible with those and only those spin pairs $\left( A_{ij},B_{ij}\right) $ that are subject to these inequalities. We show, however, that quantum-mechanical (QM) constraints are special. No-forcing theorem says that if a set of probabilistic connections is not compatible with correlations violating QM, then it is compatible only with the classical–mechanical correlations. No-matching theorem says that there are no subsets of the spin variables $A_{ij},B_{ij}$ whose distributions can be fixed to be compatible with and only with QM-compliant correlations. (shrink)

ABSTRACT In 1933 Gödel introduced an axiomatic system, currently known as S4, for a logic of an absolute provability, i.e. not depending on the formalism chosen ([God 33]). The problem of finding a fair provability model for S4 was left open. The famous formal provability predicate which first appeared in the Gödel Incompleteness Theorem does not do this job: the logic of formal provability is not compatible with S4. As was discovered in [Art 95], this defect of the formal provability (...) predicate can be bypassed by replacing hidden quantifiers over proofs by proof polynomials in a certain finite basis. The resulting Logic of Proofs enjoys a natural arithmetical semantics and provides an intended provability model for S4, thus answering a question left open by Gödel in 1933. Proof polynomials give an intended semantics for some other constructions based on the concept of provability, including intuitionistic logic with its Brouwer- Heyting- Kolmogorov interpretation, ?-calculus and modal ?-calculus. In the current paper we demonstrate how the intuitionistic propositional logic Int can be directly realized by proof polynomials. It is shown, that Int is complete with respect to this proof realizability. (shrink)

t. 1. Filosofii︠a︡ ; Izbrannye predislovii︠a︡ ; I︠A︡zykoznanie -- t. 2. Filologii︠a︡ ; Vospominanii︠a︡ i nabli︠u︡denii︠a︡ ; Pami︠a︡ti uchiteleĭ i kolleg ; Prilozhenie : A.N. Kolmogorov, semioticheskie poslanii︠a︡.

In a pair of very important papers, namely “Space, Time and Individuals” in the Journal of Philosophy for October 1955 and “The Indestructibility and Immutability of Substances” in Philosophical Studies for April 1956, Professor N. L. Wilson began something which badly needed beginning, namely the construction of a logically rigorous “substance-language” in which we talk about enduring and changing individuals as we do in common speech, as opposed to the “space-time” language favoured by very many mathematical logicians, perhaps most notably (...) by Quine. This enterprise of Wilson's is one with which I could hardly sympathize more heartily than I do; and one wishes for this logically rigorous “substance-language” not only when one is reading Quine but also when one is reading many other people. How fantastic it is, for instance, that Kotarbinski1 should call his metaphysics “Reism” when the very last kind of entity it has room for is things —instead of them it just has the world-lines or life-histories of things; “fourdimensional worms”, as Wilson says. Wilson, moreover, has at least one point of superiority to another rebel against space-time talk, P. F. Strawson; namely he does seriously attempt to meet formalism with formalism—to show that logical rigour is not a monopoly of the other side. At another point, however, Strawson seems to me to see further than Wilson; he is aware that substance-talk cannot be carried on without tenses, whereas Wilson tries to do without them. Wilson, in short, has indeed brought us out of Egypt; but as yet has us still wandering about the Sinai Peninsula; the Promised Land is a little further on than he has taken us. (shrink)

Proceedings of the Steklov Institute of Mathematics is a cover-to-cover translation of the Trudy Matematicheskogo Instituta imeni V.A. Steklova of the Russian Academy of Sciences. Each issue ordinarily contains either one book-length article or a collection of articles pertaining to the same topic.