Foundations of a new system of probability theory

The aim of my book is to explain the content of the different notions of probability.Based on a concept of logical probability, which is modified as compared with Carnap, we succeed by means of the mathematical results of de Finetti in defining the concept of statistical probability.The starting point is the fundamental concept that certain phenomena are of the same kind, that certain occurrences can be repeated, that certain experiments are identical. We introduce for this idea the notion: concept K of similarity. From concept K of similarity we derive logically some probability-theoretic conclusions:If the events E() are similar —of the same kind - on the basis of such a concept K, it holds good that intersections of n of these events are equiprobable on the basis of K; in formulae: E(₁)...E( _{n K} E('₁)...E(' _n , _{i j} ,' _j ' _j for ij On the basis of some further axioms a partial comparative probability structure results from K, which forms the starting point of our further investigations and which we call logical probability on the basis of K.We investigate a metrisation of this partial comparative structure, i.e. normed -additive functions m _K, which are compatible with this structure; we call these functions m _K measure-functions in relation to K.The measure-functions may be interpreted as subjective probabilities of individuals, who accept the concept K.Now it holds good: For each measure-function there exists with measure one the limit of relative frequencies in a sequence of the E().In such an event, where all measure-functions coincide, we speak of a quantitative logical probability, which is the common measure of this event. In formulae we have: l _K (h _n lim h _n )=1 in words: There is the quantitative logical probability one that the limit of the relative frequencies exists. Another way of saying this is that the event ^* (h_n lim h _n) is a maximal element in the comparative structure resulting from K.Therefore we are entitled to introduce this limit and call it statistical probability P.With the aid of the measure-functions it is possible to calculate the velocity of this convergence. The analog of the Bernoulli inequation holds true: m _K (¦h _n –P¦)1–1/4n².It is further possible in the work to obtain relationships for the concept of statistical independence which are expressed in terms of the comparative probability.The theory has a special significance for quantum mechanics: The similarity of the phenomena in the domain of quantum mechanics explains the statistical behaviour of the phenomena.The usual mathematical statistics are explained in my book. But it seems more expedient on the basis of this new theory to use besides the notion of statistical probability also the notion of logical probability; the notion of subjective probability has only a heuristic function in my system.The following dualism is to be noted: The statistical behaviour of similar phenomena may be described on the one hand according to the model of the classical probability theory by means of a figure called statistical probability, on the other hand we may express all formulae by means of a function, called statistical probability function. This function is defined as the limit of the relative frequencies depending on the respective state of the universe. The statistical probability function is the primary notion, the notion of statistical probability is derived from it; it is defined as the value of the statistical probability function for the true unknown state of the universe.As far as the Hume problem, the problem of inductive inference, is concerned, the book seems to give an example of how to solve it.The developed notions such as concept, measure-function, logical probability, etc. seem to be important beyond the concept of similarity.The present work represents a summary of my book Grundzüge zu einem neuen Aufbau der Wahrscheinlich-keitstheorie [5], For this reason, I have frequently dispensed with providing proof and in this connection refer the interested reader to my book.     

Providing Foundations for Coherentism

We prove that four theses commonly associated with coherentism are incompatible with the representation of a belief state as a logically closed set of sentences. The result is applied to the conventional coherence interpretation of the AGM theory of belief revision, which appears not to be tenable. Our argument also counts against the coherentistic acceptability of a certain form of propositional holism. We argue that the problems arise as an effect of ignoring the distinction between derived and non-derived beliefs, and we suggest that the kind of coherence relevant to epistemic justification is the coherence of non-derived beliefs. This revised version was published online in July 2006 with corrections to the Cover Date.     

Logical Foundations for Belief Representation

This essay presents a philosophical and computational theory of the representation of de re, de dicto, nested, and quasi-indexical belief reports expressed in natural language. The propositional Semantic Network Processing System (SNePS) is used for representing and reasoning about these reports. In particular, quasi-indicators (indexical expressions occurring in intentional contexts and representing uses of indicators by another speaker) pose problems far natural-language representation and reasoning systems, because—unlike pure indicators—they cannot be replaced by coreferential NPs without changing the meaning of the embedding sentence. Therefore, the referent of the quasi-indicator must be represented in such a way that no invalid coreferential claims are entailed. The importance of quasi-indicators is discussed, and it is shown that all four of the above categories of belief reports can be handled by a single representational technique using belief spaces containing intensional entities. Inference rules and belief-revision techniques for the system ore also examined.     

The Indispensability Argument and Multiple Foundations for Mathematics

One recent trend in the philosophy of mathematics has been to approach the central epistemological and metaphysical issues concerning mathematics from the perspective of the applications of mathematics to describing the world, especially within the context of empirical science. A second area of activity is where philosophy of mathematics intersects with foundational issues in mathematics, including debates over the choice of set–theoretic axioms, and over whether category theory, for example, may provide an alternative foundation for mathematics. My central claim is that these latter issues are of direct relevance to philosophical arguments connected to the applicability of mathematics. In particular, the possibility of there being distinct alternative foundations for mathematics blocks the standard argument from the indispensable role of mathematics in science to the existence of specific mathematical objects.     

Moral Foundations Theory and Its Implications for Counseling

Moral foundations theory has recently emerged from the field of social psychology as a means for understanding cultural, political, and religious differences between individuals and groups. The authors review the moral foundations theory literature to help counselors in understanding the roots of their clients' and their own moral belief systems and how these systems contribute to decision-making pertaining to morally laden topics. Lastly, the authors present a model for self-awareness around moral issues along with a fictional case study intended to illustrate the use of the model.     

Modal Provability Foundations for Argumentation Networks

Given an argumentation network we associate with it a modal formula representing the ‘logical content’ of the network. We show a one-to-one correspondence between all possible complete Caminada labellings of the network and all possible models of the formula.     

Factor analysis and personality theory

This article deals with the factor analytic approach to personality. More specifically, it deals with problems in factor analyses of personality questionnaires which contribute to factorial confusion. It is stated that in fact the factorial results make better sense than is usually admitted. The apparent disparity of results can be accounted for by technical defects in the chosen factor analytic method. Furthermore, it is shown that another source of disagreement lies in the interpretation of what are essentially the same factors.     

Between proof and truth

We consider two versions of truth as grounded in verification procedures: Dummett??s notion of proof as an effective way to establish the truth of a statement and Hintikka??s GTS notion of truth as given by the existence of a winning strategy for the game associated with a statement. Hintikka has argued that the two notions should be effective and that one should thus restrict one??s attention to recursive winning strategies. In the context of arithmetic, we show that the two notions do not coincide: on the one hand, proofs in PA do not yield recursive winning strategies for the associated game; on the other hand, there is no sound and effective proof procedure that captures recursive GTS truths. We then consider a generalized version of Game Theoretical Semantics by introducing games with backward moves. In this setting, a connection is made between proofs and recursive winning strategies. We then apply this distinction between two kinds of verificationist procedures to a recent debate about how we recognize the truth of G?delian sentences.     

Old Foundations for the Logic of Agency and Action

The paper presents an infinite hierarchy of sound and complete axiomatic systems for Two-Dimensional Modal Tense Logic with Historical Necessity, Agents and Acts. A main novelty of these logics is their capacity to represent formally (i) basic action-sentences asserting that such and such an act is performed/omitted by an agent, as well as (ii) causative action-sentences asserting that by performing/omitting a certain act, an agent causes that such and such a state-of-affairs is realized (e.g. comes about/ceases/remains/remains absent). We illustrate how the formal machinery of our systems can be used to reconstruct a number of interesting ideas in the Logic of Agency and Action that have been proposed by authors like von Wright, von Kutschera, Belnap and Segerberg.