Keynes’s Theory, Based on an Imprecise, Interval Valued Approach to Probability, Rejected Ramsey’s Emphasis on the Importance of the Use of Mathematical Expectations in Decision Theory

Keynes spent chapter 12 of the General Theory emphasizing two major points that were extremely important in long run decision making, confidence and expectations. Keynes saw that the technical analysis of the role of confidence in decision making had been overlooked in economics. Keynes corrected this lacuna in the General Theory. Confidence was defined as a function of Keynes’s evidential weight of the argument, V, where V=V (a/h) =w,0?w?1,just as the term “very uncertain”, used three times on p.148 of the General Theory, was defined as a function of a slight amount of information, a definition that is identical to Keynes’s definition of “very uncertain” on p.310 of his A Treatise on Probability. w equaled the degree of the completeness of the relevant information upon which the probabilities were based. Keynes’s definition of V can be found on p.315 of his A Treatise on Probability in chapter 26, titled “The application of probability to conduct”. His discussion of the completeness of the evidence can be found on pp.313-315.The second major point was Keynes’s completely overlooked discussion of the reasonable calculation of probabilities, based on approximate and inexact measures like interval valued probability and his decision weight, c, which he called a conventional coefficient of weight and risk, c, versus the unreasonable calculation of probabilities based on strict or exact mathematical expectations calculations as advocated by Frank Ramsey. The heart of Ramsey’s theory is a reliance on betting quotients and mathematical expectations based on precise probability

G\"odel's Notre Dame Course

This is a companion to a paper by the authors entitled "G\"odel's natural deduction", which presented and made comments about the natural deduction system in G\"odel's unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to G\"odel and general proof theory, one can find a brief summary of G\"odel's notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.Comment: 18 pages. minor additions, arXiv admin note: text overlap with arXiv:1604.0307

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Convergence and quantale-enriched categories

Generalising Nachbin's theory of "topology and order", in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $\mathcal{V}$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category

Logics of Imprecise Comparative Probability

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures

Achievable hierarchies in voting games with abstention

It is well known that he influence relation orders the voters the same way as the classical Banzhaf and Shapley-Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the influence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of influence relation. (C) 2013 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author’s final draft