Abstract
In this chapter we will see that coherent probability assessments on (not necessarily yes–no) events, such as those given by the measurement of physical observables, are convex combinations of valuations in Łukasiewicz propositional logic Ł\(_\infty.\)Besides familiarity with [1], the only prerequisite for this chapter is some acquaintance with the very basic properties of convex sets in euclidean space.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cignoli, R. L. O., D’Ottaviano, I. M. L., Mundici, D. (2000). Algebraic foundations of many-valued reasoning. Volume 7 of Trends in Logic. Dordrecht: Kluwer
de Finetti, B. (1993). Sul significato soggettivo della probabilitá. (in Italian), Fundamenta Mathematicae, 17, 298–329, 1931. Translated into English as "On the Subjective Meaning of Probability (pp. 291–321)". In P. Monari & D. Cocchi (Eds.), Probabilitá e Induzione. Bologna: Clueb.
de Finetti, B. (1980). La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut H. Poincaré, 7 (pp. 1–68), 1937. Translated into English by Henry E. Kyburg Jr., as “Foresight: Its Logical Laws, its Subjective Sources”. In H. E. Kyburg Jr. & H. E. Smokler (Eds.), Studies in subjective probability. Wiley: New York, 1964. Second edition published by Krieger, New York, pp. 53–118.
de Finetti, B. (1974). Theory of Probability, Vol. 1. Chichester: Wiley.
Mundici, D. (1986). Interpretation of AF \(C^{\ast}\)-algebras in Łukasiewicz sentential calculus. Journal of Functional Analysis, 65, 15–63.
Mundici, D. (2009). Interpretation of De Finetti coherence criterion in Łukasiewicz logic. Annals of Pure and Applied Logic, 161, 235–245.
Hay, L.S. (1963). Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic, 28, 77–86.
Wójcicki, R. (1973). On matrix representations of consequence operations of Łukasiewicz sentential calculi, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 19, 239–247, Reprinted in: [255], p. 101–111.
Pogorzelski, W. A. (1964). A survey of deduction theorems for sentential calculi, (Polish) Studia Logica, 15, 163–178.
Wójcicki, R. (1988). Theory of Logical Calculi: Basic Theory of Consequence Operations, Synthese Library, Vol. 199. Dordrecht: Kluwer.
Mundici, D. (1988). The derivative of truth in Łukasiewicz sentential calculus, Contemporary Mathematics, American Mathematical Society, 69, 209–227.
Emch, G. G. (1984). Mathematical and conceptual foundations of 20th century physics, Notas de Matemática, Vol. 100, Amsterdam: North-Holland.
Conway, J. B. (1985). A course in functional analysis, Graduate Texts in Mathematics, Vol. 96, New York: Springer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Mundici, D. (2011). Prologue: de Finetti Coherence Criterion and Łukasiewicz Logic. In: Advanced Łukasiewicz calculus and MV-algebras. Trends in Logic, vol 35. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0840-2_1
Download citation
DOI: https://doi.org/10.1007/978-94-007-0840-2_1
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0839-6
Online ISBN: 978-94-007-0840-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)