In this paper, we present valuation semantics for the Propositional Intuitionistic Calculus (also called Heyting Calculus) and three important subcalculi: the Implicative, the Positive and the Minimal Calculus (also known as Kolmogoroff or Johansson Calculus). Algorithms based in our definitions yields decision methods for these calculi. DOI:10.5007/1808-1711.2010v14n1p125.

This is a series of lectures on formal decision theory held at the University of Bayreuth during the summer terms 2008 and 2009. It largely follows the book from Michael D. Resnik: Choices. An Introduction to Decision Theory, 5th ed. Minneapolis London 2000 and covers the topics: -/- Decisions under ignorance and risk Probability calculus (Kolmogoroff Axioms, Bayes' Theorem) Philosophical interpretations of probability (R. v. Mises, Ramsey-De Finetti) Neuman-Morgenstern Utility Theory Introductory Game Theory Social Choice Theory (Sen's Paradox of (...) Liberalism, Arrow's Theorem) . (shrink)

ZusammenfassungDa die zuletzt von Shimony, Lehman und Kemeny ausgebaute Begründung der Wahrscheinlichkeitstheorie durch Zurückführung auf den Begriff der Wette im wesentlichen nur die Axiome von Kolmogoroff liefert, welche für die Anwendungen nicht ausreichen, muss man nach einem neuen Prinzip suchen, um weitere Axiome zu gewinnen. Als solches bietet sich an ein Einfachheitsprinzip, welches besagt, dass eine einfachere Hypothese die wahrscheinlichere ist. Es wird kritisch berichtet über verschiedene Versuche, die in Münster insbesondere von Kiesow und W. Oberschelp unternommen worden sind, (...) um dieses Einfachheitsprinzip zu präzisieren durch eine exakte Definition der Einfachheit, welche im wesentlichen als sprachliche Einfachheit aufgefasst wird.RésuméRécemment Shimony, Lehman et Kemeny ont développé une fondation de la théorié de la probabilite au moyen d'une réduction au concept de la gageure, qui permet de comprendre essentiellement les axiomes donnés par Kolmogoroff, mais pas d'autres axiomes dont on se sert souvent pour légitimer les applications de la théorie de la probabilité. Il est done nécessaire de chercher un nouveau principe pour motiver ces autres axiomes. On peut réussir en utilisant un principe de simplicité disant que la probabilité d'une hypothése diminue avec sa complication. Un récit et une critique sont donnés concernant plusieurs recherches faites à MuUnster, spécialement par M. Kiesow et M. W. Oberschelp, pour préciser la notion de la simplicité en la réduisant à certaines propriétés syntactiques des propositions.Shimony, Lehman and Kemeny recently developed a foundation of the theory of confirmation by reduction to the concept of rational betting. This procedure yields essentially only the axioms first stated by Kolmogoroff. It is well known that these are not sufficient for application. Thus it is necessary to search for a new principle if one wants to motivate new axioms. This can be done by a principle of simplicity, which expresses that the probability of a hypothesis increases with its degree of simplicity. A critical survey is given about several attempts which have been tried in Münster especially by Kiesow and W. Oberschelp with the aim to make the notion of simplicity precise. The simplicity is reduced to syntactical properties of propositions. (shrink)

We introduce the concept of partial event as a pair of disjoint sets, respectively the favorable and the unfavorable cases. Partial events can be seen as a De Morgan algebra with a single fixed point for the complement. We introduce the concept of a measure of partial probability, based on a set of axioms resembling Kolmogoroff’s. Finally we define a concept of conditional probability for partial events and apply this concept to the analysis of the two-slit experiment in quantum (...) mechanics. (shrink)

Nach W. Stegmüller besteht die "wahrscheinlichkeitstheoretische Antinomie der Quantenmechanik" darin, daß die Quantenmechanik einerseits den Wahrscheinlichkeitskalkül nach Kolmogoroff verwendet, andererseits aber zu Ergebnissen zu gelangen scheint, die dem Wahrscheinlichkeitskalkül widersprechen: so werden Orts- und Impulsobservable als Zufallsvariable interpretiert, aber es existiert nach der Quantenmechanik keine gemeinsame Verteilungsfunktion dieser Zufallsvariablen. Mit einer wahrscheinlichkeitstheoretischen Umformulierung einiger Ausführungen von G. Ludwig in [2] gelingt der Nachweis, daß Orts- und Impulsobservable Zufallsvariable auf verschiedenen Wahrscheinlichkeitsräumen sind. Daher steht die Aussage der Quantenmechanik, daß keine (...) gemeinsame Verteilungsfunktion für Orts- und Impulsobservable existiert, nicht im Widerspruch zum Wahrscheinlichkeitskalkül nach Kolmogoroff. (shrink)

We propose an alternative resolution of Simpson's paradox in multiple classification experiments, using a different maximum likelihood estimator. In the center of our analysis is a formal representation of free choice and randomization that is based on the notion of incompatible measurements.We first introduce a representation of incompatible measurements as a collection of sets of outcomes. This leads to a natural generalization of Kolmogoroff's axioms of probability. We then discuss the existence and uniqueness of the maximum likelihood estimator for (...) a probability weight on such a generalized sample space, given absolute frequency data.As a first example, we discuss an estimation problem with censured data that classically admits only biased ad hoc estimators.Next, we derive an explicit solution of the maximum likelihood estimation problem for a large class of experiments that arise from various kids of compositions of sample spaces. We identify the (categorical) direct sum of sample spaces as a representation of “free choice,” and the (categorical) direct product as a representation of “randomization.”Finally, we apply the foregoing discussion to the case of multiple classification experiments in order to show that there is no Simpson's paradox if the difference between free choice and randomization is recognized in the structure of the experiment.A comparison between our new estimator and the “usual” calculation can be summarized as follows: Pooling the data over one classification factor in the “usual” way in fact destroys or ignores the information contained in it, whereas our proposed maximum likelihood estimator is a proper marginal over this factor that “averages out” the information contained in it. The estimators agree with each other in the case of proportional sample sizes. (shrink)

As regards cognitive capacities, the point of view of classical Artificial Intelligence has been much challenged by the so-called emergentist point of view. This paper attempts to outline,on the basis of logical considerations dealing with practical feasibility, a general theory of incompressible unfolding that is consonant with an old Leibnizian stance rather with the contemporary theory of complexity. I defend a variant of emergentism according to which any process that leads to endow a system with cognitive capacities is such an (...) incompressible unfolding. (shrink)