Abstract
In addition to beliefs, people have attitudes of confidence called credences. Combinations of credences, like combinations of beliefs, can be inconsistent. It is common to use tools from probability theory to understand the normative relationships between a person’s credences. More precisely, it is common to think that something is a consistency norm on a person’s credal state if and only if it is a simple transformation of a truth of probability (a transformation that merely changes the statement from one about probability to one about credences). Though it is common to challenge the right-to-left direction of this biconditional, I argue in this paper that the left-to-right direction is false for standard versions of probability theory. That is, I make the case that there are consistency constraints on credal states that are not simple transformations of truths of standard versions of probability theory. I do so by drawing on a newly discovered type of credal attitude, a quantificational credence, and by showing how the consistency norms on this attitude can’t be represented as simple transformations of truths of standard versions of probability theory. I conclude by showing that a probability theory that could avoid the result would have to be strikingly different from the standard versions—so different that I suspect many would hesitate to call it a theory of probability at all.
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Notes
Frank Ramsey expresses this sentiment in his seminal “Truth and Probability”: “...a precise account of the nature of partial belief reveals that the laws of probability are laws of consistency, an extension to partial beliefs of formal logic, the logic of consistency” (1926, p. 182).
Until that section, I will use “probability theory” to refer to the version outlined above.
Hájek (2003) provides a number reasons to think that (Cond) does not hold in general and that we should take it merely as a constraint, rather than a definition of, analysis of, or characteristic fact about conditional probability (see Easwaran (forthcoming) for a reply to many of these reasons). Hájek, therefore, suggests an axiomatization of probability theory different from the one I will mention in footnote 6 and further discuss in §7. Nonetheless, the result I prove holds for extant alternative axiomatizations, like those of Popper (1955, 1959) and Rényi (1955). See note 21 for more on this.
It is important to note that the CfP transformation of (Cond) is a norm on synchronic consistency. It says what our credences and conditional credences must be like at a particular time in order to be consistent. It is common to think that conditional credences are also important in determining how a credal state should evolve across time (through a diachronic norm called conditionalization). For example, if I have a conditional credence of strength .5 in p given q and learn q, then I should come to have a credence of strength .5 in p. While such a norm is plausible, I won’t be concerned with diachronic consistency in this paper.
One way to give some structure to the picture suggested by (CNTP) involves offering a few fundamental consistency norms and deriving others from them. This is in the same way that all of the truths of probability theory follow from a few axioms. More precisely, we can axiomatize probability theory (including conditional probability) with (Add), (Cond), and the following two axioms:
- (Pos):
-
For any proposition, the probability of that proposition is non-negative.
- (Taut):
-
For any tautology, the probability of that tautology is one.
From these four axioms, all of the truths of probability follow (for example, (Neg) follows from (Taut) and (Add)). In a similar way, we could take the four basic consistency norms on credal states to be CfP transformations of these axioms and could derive all the other norms on credal states from them.
Here is a quick sketch of two ways that (SCNTP) is often challenged. First, as mentioned in the previous footnote, the following is a truth of probability:
- (Taut):
-
For any tautology, the probability of that tautology is one.
Given (SCNTP), this implies that an agent must, in order to be rational, have a credence of one in all tautologies. But we might think that it is perfectly consistent to fail to have full credence in complicated tautologies. Another common worry notes that probabilities are measured in real numbers. Given (SCNTP), this implies that in order to be rational, all of one’s credences must have real number strengths. But many think that agents can be uncertain in a less precise way. For instance, I may be somewhat confident that Oklahoma wins the 2017 college football playoff without having exactly 25 % confidence or 26.4926 % confidence or any other exact degree of confidence (see Levi 1980; Jeffrey 1983; Sturgeon 2008; Moss 2015 for defenses of the view that it can be rational to have imprecise credences and White 2010; Elga 2010 for arguments that it is never rational to have imprecise credences). Though I sympathize with these challenges to (SCNTP), I will not pursue them here. (Furthermore, those who argue against (SCNTP) have the burden of showing what is wrong with the common arguments in favor of it—Representation Theorems like those of Ramsey (1926) and Jeffrey (1990), Dutch Books like in de Finetti (1937), and Joyce’s (1998) accuracy dominance argument).
This is only a special case of the principle. The general version does not require the agent to be certain about the proposition about objective chance. It says that the following is a necessary condition on the rationality of a credal state:
If X is the proposition that the objective chance that p is true is x, then your conditional credence in p given X is x.
This statement is from Lewis (1981) (though I am ignoring his explicit mention of times and already assuming that your conditional credence is relative to your current evidence).
Strictly speaking, she must also have credences in the Boolean combinations of these propositions, but we can assume that these also obey all relevant \(\hbox {CfP}_{\mathrm{Leah}}\) transformations of the truths of probability.
de Finetti (1937) expresses this position very explicitly.
There can be cases where we have ordinary credences and generalize to quantificational credences. Perhaps, Holly has a credence of strength .5 that the coin lands heads and a credence of strength .5 that the coin lands tails. She might, on this basis, come to think that each side of the coin is 50 % likely to be flipped (a quantificational credence). But such cases don’t, I think, shed much light onto why quantificational credences are useful (just as focusing on cases where we infer an existential proposition from a de re one don’t shed much light onto why having beliefs with existential contents are useful).
Lennertz (2015) attempted to generalize this sort of norm to the case where one doesn’t learn that x is A but merely has some credence that it is. Because this generalization is complicated, controversial, and not essential for my sketch of quantificational credences, I won’t discuss or evaluate it here. The reader should also note that even if this is a norm, it is a diachronic norm rather than a synchronic one (which are our concern here). This part of the role of quantificational credences should be familiar from the conditional credence case. See note 5.
It’s worth reminding the reader here that this result holds for a standard version of probability theory. In Sect. 7, I show just how non-standard probability theory would have to be in order to block the result.
An additional reason to doubt (NoQC2) is that it doesn’t respect one of our important conclusions from the discussion in §4—that, typically, it is nonspecific evidence that leads an agent to think that everyone who is a top-four fundraiser is somewhat likely to win the election. It is mysterious how such nonspecific evidence could, in a case where the agent is ignorant of the candidates and their fundraising, lead her to have four ordinary credences, each about one of the top four fundraisers (which is what is required, according to (NoQC2), in order to think that everyone who is a top-four fundraiser is somewhat likely to win the election).
Thanks to an anonymous reviewer for suggesting that I confront \((\hbox {NoQC}_{3})\). Lennertz (2015) compares the problem raised in the text to the impossibility of giving a linguistic analysis of quantifiers like ‘half’ without using restricted quantification.
I’m ignoring worries here about cases where P \(({\hbox {p}}_{2}) = 0\). Note 4 contained a brief discussion of this issue.
I’m treating ‘\(\vee \)’ as a connective that can also operate on properties to create disjunctive properties, in the way that the English word ‘or’ can (strictly speaking it operates on predicates which pick out properties, but this use-mention issue doesn’t create any problems here or in the text).
That is, perhaps in order to be consistent, for any F and G, I ought to think that q of the Gs are at least 0 % likely to be F. Furthermore, perhaps I ought to think that, q of the Gs are 100 % likely to be T’.
Indeed, it would have to be quite different from other popular axiomatizations of probability theory as well—e.g. those of Popper (1938, 1955, 1959) and Rényi (1955). Because of the non-uniqueness of purported quantificational probabilities, all of these systems would strictly speaking not allow for quantificational probabilities. Furthermore, these systems have axioms quantificational versions of which would fail for the same sort of reason that (QAdd) fails. If we look at the axiomatization from Popper (1938, p. 276), a quantificational version of Axiom B2 (Addition and Lower Limit) would fail. A quantificational version of the Complement axiom would fail for both his formal system of absolute probabilities (1955, p. 53) and his formal system of relative probabilities (1955, p. 56; 1959, p. 332). Finally, a quantificational version of Axiom 2 from Rényi’s (1955, p. 289) system would fail. Thanks to an anonymous reviewer for the suggestion to consult these views. Easwaran (forthcoming) offers a helpful discussion of the motivations for and consequences of Popper’s and Rényi’s systems (among others).
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Acknowledgments
Thanks to Mark Schroeder, David Gray, Reid Blackman, Scott Soames, Shyam Nair, and two anonymous reviewers for comments on and/or discussion of earlier drafts of this paper.
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Lennertz, B. Probabilistic consistency norms and quantificational credences. Synthese 194, 2101–2119 (2017). https://doi.org/10.1007/s11229-016-1039-7
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DOI: https://doi.org/10.1007/s11229-016-1039-7