Triviality Results for Probabilistic Modals

Abstract

In recent years, a number of theorists have claimed that beliefs about probability are transparent. To believe probably p is simply to have a high credence that p. In this paper, I prove a variety of triviality results for theses like the above. I show that such claims are inconsistent with the thesis that probabilistic modal sentences have propositions or sets of worlds as their meaning. Then I consider the extent to which a dynamic semantics for probabilistic modals can capture theses connecting belief, certainty, credence, and probability. I show that although a dynamic semantics for probabilistic modals does allow one to validate such theses, it can only do so at a cost. I prove that such theses can only be valid if probabilistic modals do not satisfy the axioms of the probability calculus.

Probabilities of conditionals: Updating Adams

“…This prediction seems to be borne out. In the recent literature, it has been argued, independently of triviality results, that sentences involving epistemic modals like might and probably are non‐factual (see, a.o., Yalcin, 2011; Swanson, 2011; Willer, 2013): and indeed, as expected, in the literature we find triviality results about the probabilities of such sentences (Russell & Hawthorne, 2016; Goldstein, 2019b). Similarly, though more controversially, Edgington (2008) argued that counterfactual conditionals are also non‐factual, and indeed we find corresponding triviality results for counterfactuals (Williams, 2012; Leitgeb, 2012; Santorio, 2022a; Schultheis, 2022).…”

“…This enterprise involves not only assigning suitable semantic values to such sentences, but also formulating more general accounts of logical notions like consequence, speech acts like assertion, attitudes like acceptance, credence, and desire, and mental acts like supposition. For some recent contributions see, among others: (Gillies, 2004;Yalcin, 2007Yalcin, , 2015Swanson, 2011Swanson, , 2016Bledin, 2014Bledin, , 2020Starr, 2014;Moss, 2015;Goldstein, 2019aGoldstein, , 2019bCiardelli, 2020;Punčochář & Gauker, 2020;Hawke & Steinert-Threlkeld, 2021;Cantwell, 2021;Santorio, 2022b;Goldstein & Santorio, 2021).…”

“…One can replicate Lewis's proof using much weaker assumptions that concern only certainty about conditionals (Bradley, 2000), and which seem overwhelmingly plausible. One can also obtain triviality results from plausible assumptions which do not concern conditionals at all, but which constrain the probabilities of sentences involving, for instance, the epistemic modals might and probably (Russell & Hawthorne, 2016; Goldstein, 2019b). These findings suggest that the problem discovered by Lewis is a deep and general one.…”

Probabilistic Antecedents and Conditional Attitudes

“…Finally, there are formal triviality results that can be interpreted to suggest that Fertz's thought is not a belief with a probabilistic content (Russell and Hawthorne 2016;Schroeder 2018;Goldstein 2019). While the results merely show that several plausible principles are incompatible, one conclusion we might draw goes as follows: take for granted a notion of being 60 percent confident in a proposition, p. Then there is no other proposition, q, that a person must be completely or very confident in just in case one is 60 percent confident in p. This might be interpreted to show that, for instance, there is no proposition that it is likely that it will be sunny tomorrow in Phoenix, which can be the object of Fertz's belief.…”

Trivializing Informational Consequence

“…A very incomplete list of helpful references on triviality may includeLewis 1976, Hájek and Hall 1994, Bradley 2000, Bradley 2007, Russell and Hawthorne 2016, Goldstein 2019; seeKhoo and Santorio 2018 for an introductory overview.19 This notion of triviality is obviously different from other notions of triviality appearing in the literature, but this is irrelevant. What matters is that we get collapse of one notion of consequence onto another.…”