Knowledge from Probability

Salow

2023

Philosophical Review

We offer a general framework for theorizing about the structure of knowledge and belief in terms of the comparative normality of situations compatible with one’s evidence. The guiding idea is that, if a possibility is sufficiently less normal than one’s actual situation, then one can know that that possibility does not obtain. This explains how people can have inductive knowledge that goes beyond what is strictly entailed by their evidence. We motivate the framework by showing how it illuminates knowledge about the future, knowledge of lawful regularities, knowledge about parameters measured using imperfect instruments, the connection between knowledge, belief, and probability, and the dynamics of knowledge and belief in response to new evidence.

Section: Dynamicsmentioning

confidence: 90%

“…In Goodman and Salow (2021) we show how this basic thought can be fleshed out into an attractive probabilist theory of comparative normality. For present purposes we can focus on the following three features of the theory.…”

Section: Comparabilitymentioning

confidence: 99%

Epistemology Normalized

Salow

2023

Philosophical Review

“…We can think of the members of the question Q as its answers; we write [s] Q for the member of Q containing s. The proposal in [5] then boils down to claiming that s ∈ B(E) if and only if s ∈ E and the answers to Q that are more probable than [s] Q have total probability less than the threshold t. Writing Pr E for Pr(•|E), this can be formalized as follows:…”

Section: Probability Structuresmentioning

confidence: 99%

“…Pr (the prior) is a probability distribution over S, and 5. t ∈ [0, 1] (the threshold)Propositions are modeled as subsets of S, where p is true in s if and only if s ∈ p. We say that E ′ ∈ E is the result of discovering p in E ∈ E just in case E ′ = E ∩ p; this will allow us to talk about how beliefs evolve in response to changes in one's evidence.Which propositions an agent believes is a function of their evidence and is also given by a set of states, so that an agent with evidence E believes p if and only if B(E) ⊆ p. This ensures that their beliefs are closed under entailment, and thus already marks a departure from popular 'Lockean' accounts of belief [4], according to which one believes a proposition if and only if its probability exceeds a particular threshold. But it is compatible with the more plausible direction of Lockeanism, namely: THRESHOLD: You believe p only if p is sufficiently probable given your evidence.If B(E) ⊆ p, then Pr(p|E) ≥ t.We can think of the members of the question Q as its answers; we write [s] Q for the member of Q containing s. The proposal in [5] then boils down to claiming that s ∈ B(E) if and only if s ∈ E and the answers to Q that are more probable than [s] Q have total probability less than the threshold t. Writing Pr E for Pr(•|E), this can be formalized as follows:…”

mentioning

confidence: 99%

Belief Revision from Probability

Electron. Proc. Theor. Comput. Sci.

Salow

2023

Self Cite

In previous work ([5, 6]), we develop a question-relative, probabilistic account of belief. On this account, what someone believes relative to a given question is (i) closed under entailment, (ii) sufficiently probable given their evidence, and (iii) sensitive to the relative probabilities of the answers to the question. Here we explore the implications of this account for the dynamics of belief. We show that the principles it validates are much weaker than those of orthodox theories of belief revision like AGM [1], but still stronger than those valid according to the popular Lockean theory of belief [4], which equates belief with high subjective probability. We then consider a restricted class of models, suitable for many but not all applications, and identify some further natural principles valid on this class. We conclude by arguing that the present framework compares favorably to the rival probabilistic accounts of belief developed by Leitgeb [13,14] and Lin and Kelly [17]. Probability StructuresWe will work with the following simplification of the models in [5]: Definition 1.1. A probability structure is a tuple S, E , Q, Pr,t such that:1. S is a non-empty set (of states), 2. E ⊆ P(S)\{ / 0} (the possible bodies of evidence), 3. Q (the question) is a partition of S, 4. Pr (the prior) is a probability distribution over S, and 5. t ∈ [0, 1] (the threshold)Propositions are modeled as subsets of S, where p is true in s if and only if s ∈ p. We say that E ′ ∈ E is the result of discovering p in E ∈ E just in case E ′ = E ∩ p; this will allow us to talk about how beliefs evolve in response to changes in one's evidence.Which propositions an agent believes is a function of their evidence and is also given by a set of states, so that an agent with evidence E believes p if and only if B(E) ⊆ p. This ensures that their beliefs are closed under entailment, and thus already marks a departure from popular 'Lockean' accounts of belief [4], according to which one believes a proposition if and only if its probability exceeds a particular threshold. But it is compatible with the more plausible direction of Lockeanism, namely: THRESHOLD: You believe p only if p is sufficiently probable given your evidence.If B(E) ⊆ p, then Pr(p|E) ≥ t.We can think of the members of the question Q as its answers; we write [s] Q for the member of Q containing s. The proposal in [5] then boils down to claiming that s ∈ B(E) if and only if s ∈ E and the answers to Q that are more probable than [s] Q have total probability less than the threshold t. Writing Pr E for Pr(•|E), this can be formalized as follows:

“…Note also that the question sensitivity of thinking would then induce a corresponding question-sensitivity in knowledge. Goodman and Salow (2021) develop a framework for thinking about question-sensitivity of knowledge and belief marching in step; while we think their notion of 'belief' is best understood as being sure, their framework deploys probabilistic orderings of questions' answers in a way that would harmonize easily with the theory of thinking in Holguín (forthcoming). See Goodman (in preparation) for details.…”

mentioning

confidence: 99%

Thinking and being sure^*

Philos Phenomenol Research

Holguín

2022

Self Cite

How is what we believe related to how we act? That depends on what we mean by ‘believe’. On the one hand, there is what we're sure of: what our names are, where we were born, whether we are sitting in front of a screen. Surety, in this sense, is not uncommon — it does not imply Cartesian absolute certainty, from which no possible course of experience could dislodge us. But there are many things that we think that we are not sure of. For example, you might think that it will rain sometime this month, but not be sure that it will. Both what we're sure of and what we think have important normative connections to action. But the connections are quite different. This paper explores these issues with respect to assertion, inquiry, and decision making. We conclude by arguing that there is no theoretically significant notion of ‘full belief’ intermediate in strength between thinking and being sure.