In [Chapter XVI, 4, p. 246] of his “Investigation of the laws of thought,” (Walton and Maberley, London, 1854, reprinted Dover, NY, 1958), Boole writes:
the object of the theory of probabilities might be thus defined. Given the probabilities of any events, of whatever kind, to find the probability of some other event connected with them.
The first question concerns the consistency of these probability assignments. We will deal with this logical problem, both for yes-no boolean events and for continuous events in Łukasiewicz infinite-valued logic.
We will assume no prerequisites in probability theory and in many-valued logic.
In standard textbooks on probability, the additivity of probability for disjunctions of incompatible events is an axiom, while the multiplicativity of probability for conjunctions of independent events is a definition. Interestingly, on page 168 of his 1905 lecture notes (cited below), one finds the following remark by Hilbert:
in the present state of development, especially the terms axiom and definition are still a bit confused.
Recalling the well known adage “old theorems never die: they turn into definitions”, we will show that both the additivity axiom and the definition of independence by the product law are in fact corollaries of a deeper notion of consistency, going back to de Finetti. If time allows, working in the context of Łukasiewicz infinite-valued logic we will cast light to the vexata quaestio of countable vs. finite additivity.
D. Hilbert, Logische Prinzipien des mathematischen Denkens. Sommersemester 1905. (Logical principles of mathematical thinking.) Lecture notes taken by M. Born. Niedersächsische Staats-und Universitätsbibliothek Göttingen, Handschriftenabteilung, Cod. Ms. D. Hilbert 558; Lecture notes taken by E. Hellinger. University of Göttingen, Library of the Mathematical Institut.
B. de Finetti, Sul significato soggettivo della probabilità. Fundamenta Mathematicae, 17 (1931) 298-329. Translated into English as “On the Subjective Meaning of Probability”. In: P. Monari et al. (Eds.), Probabilità e Induzione, CLUEB, Bologna, pp. 291-321, 1993.
__________ La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut H. Poincaré, 7 (1937) 1-68. English translation by Henry E. Kyburg Jr., as “Foresight: Its Logical Laws, its Subjective Sources.” In: H. E. Kyburg Jr. et al. (Eds.), “Studies in Subjective Probability”, J. Wiley, New York, pp. 93-158, 1964. Second edition published by Krieger, New York, pp. 53-118, 1980.
R. Cignoli, I.M.L. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer, 2000. Reprinted by Springer Science & Business Media, 2013.
D. Mundici, Bookmaking over infinite-valued events, International J. of Approximate Reasoning, 43 (2006) 223-240.
_________ Interpretation of de Finetti coherence criterion in Łukasiewicz logic, Ann. Pure Applied Logic, 161 (2009) 235-245.
_________ Coherence of the product law for independent continuous events. Chapter 10, pp 207-212, in: Contradictions, from Consistency to Inconsistency, W. Carnielli and J. Malinowski (Eds.), Trends in Logic, Vol. 47, Springer International Publishing, Springer Nature Switzerland AG, 2018.
__________ Coherence of de Finetti coherence, Synthese, 194 (2017) 4055-4063. Ibid., 196 (2019) 265-271.