From probabilities to categorical beliefs: Going beyond toy models (original) (raw)
How to Believe Long Conjunctions of Beliefs: Probability, Quasi-Dogmatism and Contextualism
Erkenntnis
According to the so-called Lockean thesis, a rational agent believes a proposition just in case its probability is sufficiently high, i.e., greater than some suitably fixed threshold. The Preface paradox is usually taken to show that the Lockean thesis is untenable, if one also assumes that rational agents should believe the conjunction of their own beliefs: high probability and rational belief are in a sense incompatible. In this paper, we show that this is not the case in general. More precisely, we consider two methods of computing how probable must each of a series of propositions be in order to rationally believe their conjunction under the Lockean thesis. The price one has to pay for the proposed solutions to the paradox is what we call “quasi-dogmatism”: the view that a rational agent should believe only those propositions which are “nearly certain” in a suitably defined sense.
Belief as Qualitative Probability
2014
In this paper we argue that all-or-nothing belief may be understood as a kind of qualitative (subjective) probability. We present the basics of a joint theory of belief and degrees of belief to that effect, and we compare it with classic work on qualitative probability (by de Finetti, Suppes, and others).
Two Views of Belief: Belief as Generalized Probability and Belief as Evidence
Artificial Intelligence, 1992
Belief functions are mathematical objects defined to satisfy three axioms that look somewhat similar to the Kolmogorov axioms defining probability functions. We argue that there are (at least) two useful and quite different ways of understanding belief functions. The first is as a generalized probability function (which technically corresponds to the inner measure induced by a probability function). The second is as a way of representing evidence. Evidence, in turn, can be understood as a mapping from probability functions to probability functions. It makes sense to think of updating a belief if we think of it as a generalized probability. On the other hand, it makes sense to combine two beliefs (using, say, Dempster's rule of combination) only if we think of the belief functions as representing evidence. Many previous papers have pointed out problems with the belief function approach; the claim of this paper is that these problems can be explained as a consequence of confounding these two views of belief functions.
Journal of Mathematical Psychology, 2003
A theory of belief is presented in which uncertainty has two dimensions. The two dimensions have a variety of interpretations. The article focusses on two of these interpretations. The first is that one dimension corresponds to probability and the other to ''definiteness,'' which itself has a variety of interpretations. One interpretation of definiteness is as the ordinal inverse of an aspect of uncertainty called ''ambiguity'' that is often considered important in the decision theory literature. (Greater ambiguity produces less definiteness and vice versa.) Another interpretation of definiteness is as a factor that measures the distortion of an individual's probability judgments that is due to specific factors involved in the cognitive processing leading to judgments. This interpretation is used to provide a new foundation for support theories of probability judgments and a new formulation of the ''Unpacking Principle'' of Tversky and Koehler. The second interpretation of the two dimensions of uncertainty is that one dimension of an event A corresponds to a function that measures the probabilistic strength of A as the focal event in conditional events of the form AjB; and the other dimension corresponds to a function that measures the probabilistic strength of A as the context or conditioning event in conditional events of the form CjA: The second interpretation is used to provide an account of experimental results in which for disjoint events A and B; the judge probabilities of AjðA,BÞ and BjðA,BÞ do not sum to 1. The theory of belief is axiomatized qualitatively in terms of a primitive binary relation h on conditional events. (AjBhCjD is interpreted as ''the degree of belief of AjB is greater than the degree of belief of CjD:'') It is shown that the axiomatization is a generalization of conditional probability in which a principle of conditional probability that has been repeatedly criticized on normative grounds may fail. Representation and uniqueness theorems for the axiomatization demonstrate that the resulting generalization is comparable in mathematical richness to finitely additive probability theory.
Probability, logic and the cognitive foundations of rational belief
Journal of Applied Logic, 2003
Since Pascal introduced the idea of mathematical probability in the 17th century discussions of uncertainty and "rational" belief have been dogged by philosophical and technical disputes. Furthermore, the last quarter century has seen an explosion of new questions and ideas, stimulated by developments in the computer and cognitive sciences. Competing ideas about probability are often driven by different intuitions about the nature of belief that arise from the needs of different domains (e.g., economics, management theory, engineering, medicine, the life sciences etc). Taking medicine as our focus we develop three lines of argument (historical, practical and cognitive) that suggest that traditional views of probability cannot accommodate all the competing demands and diverse constraints that arise in complex real-world domains. A model of uncertain reasoning based on a form of logical argumentation appears to unify many diverse ideas. The model has precursors in informal discussions of argumentation due to Toulmin, and the notion of logical probability advocated by Keynes, but recent developments in artificial intelligence and cognitive science suggest ways of resolving epistemological and technical issues that they could not address. Crown
Varieties of Belief and Probability
2015
For reasoning about uncertain situations, we have probability theory, and we have logics of knowledge and belief. How does elementary probability theory relate to epistemic logic and the logic of belief? The paper focuses on the notion of betting belief, and interprets a language for knowledge and belief in two kinds of models: epistemic neighbourhood models and epistemic probability models. It is shown that the first class of models is more general in the sense that every probability model gives rise to a neighbourhood model, but not vice versa. The basic calculus of knowledge and betting belief is incomplete for probability models. These formal results were obtained in Van Eijck and Renne [9].
Springer eBooks, 2003
The aims of this paper are (i) to summarize the semantics of (the propositional part of) a unified epistemic/doxastic logic as it has been developed at greater length in Lenzen [1980] and (ii) to use some of these principles for the development of a semi-formal pragmatics of epistemic sentences. While a semantic investigation of epistemic attitudes has to elaborate the truth-conditions for, and the analytically true relations between, the fundamental notions of belief, knowledge, and conviction, a pragmatic investigation instead has to analyse the specific conditions of rational utterance or utterability of epistemic sentences. Some people might think that both tasks coincide. According to Wittgenstein, e.g., the meaning of a word or a phrase is nothing else but its use (say, within a certain community of speakers). Therefore the pragmatic conditions of utterance of words or sentences are assumed to determine the meaning of the corresponding expressions. One point I wish to make here, however, is that one may elaborate the meaning of epistemic expressions in a way that is largely independent of-and, indeed, even partly incompatible with-the pragmatic conditions of utterability. Furthermore, the crucial differences between the pragmatics and the semantics of epistemic expressions can satisfactorily be explained by means of some general principles of communication. In the first three sections of this paper the logic (or semantics) of the epistemic attitudes belief, knowledge, and conviction will be sketched. In the fourth section the basic idea of a general pragmatics will be developed which can then be applied to epistemic utterances in particular. 1 The Logic of Conviction Let 'C(a,p)' abbreviate the fact that person a is firmly convinced that p, i.e. that a considers the proposition p (or, equivalently, the state of affairs expressed by that proposition) as absolutely certain; in other words, p has maximal likelihood or probability for a. Using 'Prob' as a symbol for subjective probability functions, this idea can be formalized by the requirement: (PROB-C) C(a,p) ↔ Prob(a,p)=1. Within the framework of standard possible-worlds semantics <I,R,V>, C(a,p) would have to be interpreted by the following condition: (POSS-C) V(i,C(a,p))=t ↔ ∀j(iRj → V(j,p)=t). Here I is a non-empty set of (indices of) possible worlds; R is a binary relation on I such that iRj holds iff, in world i, a considers world j as possible; and V is a valuation-function assigning to each proposition p relative to each world i a truth-value V(i,p)∈{t,f}. Thus C(a,p) is true (in world i∈I) iff p itself is true in every possible world j which is considered by a as possible (relative to i). The probabilistic "definition" POSS-C together with some elementary theorems of the theory of subjective probability immediately entails the validity of the subsequent laws of conjunction and non-contradiction. If a is convinced both of p and of q, then a must also be convinced that p and q: (C1) C(a,p) ∧ C(a,q) → C(a,p∧q). For if both Prob(a,p) and Prob(a,q) are equal to 1, then it follows that Prob(a,p∧q)=1, too. Furthermore, if a is convinced that p (is true), a cannot be convinced that ¬p, i.e. that p is false: (C2) C(a,p) → ¬C(a,¬p). For if Prob(a,p)=1, then Prob(a,¬p)=0, and hence Prob(a,¬p)≠1. Just like the alethic modal operators of possibility, ◊, and necessity, , are linked by the relation ◊p ↔ ¬ ¬p, so also the doxastic modalities of thinking p to be possible-formally: P(a,p)-and of being convinced that p, C(a,p), satisfy the relation (Def. P) P(a,p) ↔ ¬C(a,¬p). Thus, from the probabilistic point of view, P(a,p) holds iff a assigns to the proposition p (or to the event expressed by that proposition) a likelihood greater than 0: (PROB-P) V(P(a,p))=t ↔ Prob(a,p)>0. Within the framework of possible-worlds semantics, one obtains the following condition: (POSS-P) V(i,P(a,p))=t ↔ ∃j(iRj ∧ V(j,p)=t), according to which P(a,p) is true in world i iff there is at least one possible world j-i.e. a world j accessible from i-in which p is true. 1 Cf., e.g., Hintikka [1970]. 2 Clearly, since C(a,p) ∨ ¬C(a,p) holds tautologically, C10 and C11 entail that C(a,C(a,p)) ∨ C(a,¬C(a,p)) is epistemic-logically true. So either way there exists a q such that C(a,q).
I give original arguments for the thesis that one believes a proposition only if one's credence in it is 1, as well as respond to several objections to that thesis. Belief implies having a credence of 1 because of the connections between assertoric representation, possibility, and probability.
Journal of Mathematical Psychology 47 (2003) 1–31 A theory of belief
1999
A theory of belief is presented in which uncertainty has two dimensions. The two dimensions have a variety of interpretations. The article focusses on two of these interpretations. The first is that one dimension corresponds to probability and the other to ‘‘definiteness,’ ’ which itself has a variety of interpretations. One interpretation of definiteness is as the ordinal inverse of an aspect of uncertainty called ‘‘ambiguity’ ’ that is often considered important in the decision theory literature. (Greater ambiguity produces less definiteness and vice versa.) Another interpretation of definiteness is as a factor that measures the distortion of an individual’s probability judgments that is due to specific factors involved in the cognitive processing leading to judgments. This interpretation is used to provide a new foundation for support theories of probability judgments and a new formulation of the ‘‘Unpacking Principle’ ’ of Tversky and Koehler. The second interpretation of the tw...