Abstract
Leibniz seems to have been the first to suggest a logical interpretation of probability, but there have always seemed formidable mathematical and interpretational barriers to implementing the idea. De Finetti revived it only, it seemed, to reject it in favour of a purely decision-theoretic approach. In this paper I argue that not only is it possible to view (Bayesian) probability as a continuum-valued logic, but that it has a very close formal kinship with classical propositional logic.
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Notes
The penalty is a quadratic function of the difference, added over the (finitely many) quantities estimated.
In the same book de Finetti proved that coherence is equivalent to invulnerability to certain loss or gain in finitely many simultaneous bets.
‘… il faudrait une nouvelle espèce de logique, qui traiterait des degrés de probabilité.’ (Nouveaux Essais, I, IV c. 16).
Even natural language arguably demands a second-order view: if Jack is tall and Jill is tall then there is a property they share; while branching quantifiers, which are claimed to formalise some natural-language sentences, seem to be second order (their Skolemization is second order).
Wenmackers and Horsten (2013): they call the condition SUM.
The point is made in Milne (1990).
There is a more extended defence of the case—essentially de Finetti’s—against countable additivity in Howson (2014).
For example Maher in his own discussion of the example: ‘de Finetti cannot consistently reject countable additivity’ (1993, p. 2000).
Finetti (1972, p. 91).
Finetti (1972, p. 77).
Finetti (1972, p. 91).
As is explained clearly in Weintraub (2001).
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Acknowledgments
I am grateful to two anonymous referees for their very helpful comments.
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Howson, C. A Continuum-Valued Logic of Degrees of Probability. Erkenn 79, 1001–1013 (2014). https://doi.org/10.1007/s10670-013-9586-5
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DOI: https://doi.org/10.1007/s10670-013-9586-5