An Explication of the Concent “Analogy”

Mathematical Structures in Computer Science, 2014

In this paper, we discuss the crucial but little-known fact that, as Kolmogorov himself claimed, the mathematical theory of probabilities cannot be applied to factual probabilistic situations. This is because it is nowhere specified how, for any given particular random phenomenon, we should construct, effectively and without circularity, the specific and stable distribution law that gives the individual numerical probabilities for the set of possible outcomes. Furthermore, we do not even know what significance we should attach to the simple assertion that such a distribution law “exists”. We call this problem Kolmogorov's aporia†.We provide a solution to this aporia in this paper. To do this, we first propose a general interpretation of the concept of probability on the basis of an example, and then develop it into a non-circular and effective general algorithm of semantic integration for the factual probability law involved in a specific factual probabilistic situation. The dev...

Studia Logica, 1975

The main aim of this paper is to study the logic of a binary sentential operator 'z=', with the intended meaning 'is at least as probable as'. The object language will be simple; to an ordinary language for truth-functional connectives we add '&' as the only intensional operator. Our choice of axioms is heavily dependent of a theorem due to Kraft et al. [8], which states necessary and sufficient conditions for an ordering of the elements of a finite subset algebra to be compatible with some probability measure. Following a construction due to Segerberg [ 121, we show that these conditions can be translated into our language.

Advances in Pure Mathematics, 2012

Classical statistics and Bayesian statistics refer to the frequentist and subjective theories of probability respectively. Von Mises and De Finetti, who authored those conceptualizations, provide interpretations of the probability that appear incompatible. This discrepancy raises ample debates and the foundations of the probability calculus emerge as a tricky, open issue so far. Instead of developing philosophical discussion, this research resorts to analytical and mathematical methods. We present two theorems that sustain the validity of both the frequentist and the subjective views on the probability. Secondly we show how the double facets of the probability turn out to be consistent within the present logical frame.

Arxiv preprint quant-ph/0205165, 2002

arXiv: Quantum Physics, 2009

The crucial but very confidential fact is brought into evidence that, as Kolmogorov himself repeatedly claimed, the mathematical theory of probabilities cannot be applied to physical, factual probabilistic situations because the factual concept of probability is not defined : it is nowhere specified how to construct, for a given physical random phenomenon, the specific numerical distribution of relative frequencies of outcomes from the universe of elementary events produced by that phenomenon, that constitutes the factual probability law to be asserted on this universe; nor is it known what significance to associate to the assertion of mere 'existence' of such a factual probability law. An algorithm of semantic integration of the factual probability law to be asserted in any given factual probabilistic situation, is then constructed. This algorithm, developed inside a general method of relativized conceptualization, involves a quantification of the factual concept of probabi...

Erkenntnis, 1981

The first part of this note contains some critical remarks about the fore going paper ofNiiniluoto (1981) (to which I shall always refer unless specified otherwise). These remarks arrive at the conclusion that the systems proposed by Niiniluoto, failing to satisfy symmetry, positive instantial relevance, and the Reichenbach axiom, are more defective than seems acceptable. At least the latter two properties may be regained for analogical inference, as I try to show in the second part. The note closes with some general and skeptical remarks to the effect that by trying to accommodate analogy considerations, inductive logic regresses to the subjectivistic position. I. CRITICISM Apart from general doubts about inductive logic Carnap's A-continuum, though working weH for enumerative induction, faced mainly two criticisms: that it deals with generalizations in an inadequate way, i.e. that it is unable to account for eliminative induction, and that it leaves no room for considerations from analogy. The first difliculty is shown to be solvable in aseries of papers by Hintikka; and the K-dimensional system proposed by Hintikka and Niiniluoto (1976) may be said to be the perfect generalization of the A-continuum with respect to eliminative induction. The second difficulty is less attended to, at least on a technicallevel, and in fact unsolved. The aim of the foregoing paper of Niiniluoto is to overcome this problem or at least to pave the way towards overcoming it; and his general strategy to this effect is to modify the K-dimensional system so as to leave room for analogy considerations without losing any or too many of the basic properties of the K-dimensional system. This results in two (families of) probability measures Ps and Pes presented in sections 5 and 6 of his paper. I want to argue in the sequel that, after all, these modifications encroach on the K-dimensional system too heavily to be acceptable: (a) About the system p.: As Niiniluoto himself states on p. 14, the probability measure Ps is identical with the K-dimensional system for exempli

Canadian mathematical bulletin, 1967

Kolmogorov’s Heritage in Mathematics

2016

Résumé: La théorie de la décision contemporaine accorde une importance considérable à une famille de résultats mathématiques qu’on appelle les théorèmes de représentations. Ces théorèmes relient des critères pour évaluer les options qui s’offrent au décideur (comme le critère de l’espérance d’utilité) à des axiomes qui portent sur ses préférences (comme l’axiome de transitivité). Plusieurs raisons ont été avancées pour expliquer ou défendre l’importance de ces résultats. L’objectif de cet article est d’évaluer leur rôle sémantique: dans cette perspective, les théorèmes de représentation ont pour fonction de fournir des définitions des concepts décisionnels mobilisés dans les critères d’évaluation (comme ceux d’utilité ou de probabilité subjective, qui sont mobilisés par le critère de l’espérance subjective d’utilité). Nous examinerons cette fonction en comparant les théorèmes de représentation aux théories philosophiques de la signification des termes dits théoriques. Mots-Clés: thé...

Studies in Fuzziness and Soft Computing, 2013

We have discussed the potential and possibility spaces and their relationships to the knowledge-production process as viewed from the theory of the knowledge square. These conceptual relationships were connected to the possible-world space, the explanatory science, prescriptive science and the defective knowledge structure. On the basis of the defective knowledge structure, the roles of explanatory theory and the corresponding explanatory rationality were projected. Similarly, the prescriptive theory and the corresponding prescriptive rationality were discussed .We shall now turn our attention to the probability space and show how it is connected to the possibility space and then to the universal object set in a backward regressive process of the knowledge-production process. The possibility space and its construct on the basis of the fuzzy paradigm to deal with the problem of possibilistic uncertainty provide us with the analytical structure of possibilistic reasoning under conditions of defective information structure that is constrained by quality and quantity deficiencies about epistemic elements with neutrality of time.