Online research in philosophy

Explains how to use a trivalent semantics to explain what is often called Adam’s Thesis, the thesis that the probability of a conditional is the conditional probability of the consequent given the antecedent.

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

There is no set Δ of probability axioms that meets the following three desiderata: (1) Δ is vindicated by a Dutch book theorem; (2) Δ does not imply regularity (and thus allows, among other things, updating by conditionalization); (3) Δ constrains the conditional probability q(·,z) even when the unconditional probability p(z) (=q(z,T)) equals 0. This has significant consequences for Bayesian epistemology, some of which are discussed.

The Swamping Problem is one of the standard objections to reliabilism. If one assumes, as reliabilism does, that truth is the only non instrumental epistemic value, then the worry is that the additional value of knowledge over true belief cannot be adequately explained, for reliability only has instrumental value relative to the non instrumental value of truth. Goldman and Olsson reply to this objection that reliabilist knowledge raises the objective probability of future true beliefs and is thus more valuable (...) than mere true belief. I argue against their proposed solution to the Swamping Problem that the conditional probability of future true beliefs given knowledge is not clearly higher than given mere true belief. (shrink)

In this paper we examine the thesis that the probability of the conditional is the conditional probability. Previous work by a number of authors has shown that in standard numerical probability theories, the addition of the thesis leads to triviality. We introduce very weak, comparative conditional probability structures and discuss some extremely simple constraints. We show that even in such a minimal context, if one adds the thesis that the probability of a (...) conditional is the conditional probability, then one trivializes the theory. Another way of stating the result is that the conditional of conditional probability cannot be represented in the object language on pain of trivializing the theory. (shrink)

The Swamping Problem is one of the standard objections to reliabilism. If one assumes, as reliabilism does, that truth is the only non-instrumental epistemic value, then the worry is that the additional value of knowledge over true belief cannot be adequately explained, for reliability only has instrumental value relative to the non-instrumental value of truth. Goldman and Olsson reply to this objection that reliabilist knowledge raises the objective probability of future true beliefs and is thus more valuable than mere (...) true belief. I argue against their proposed solution to the Swamping Problem that the conditional probability of future true beliefs given knowledge is not clearly higher than given mere true belief. (shrink)

This paper (based on joint work with M.J.Schervish and J.B.Kadane) discusses some differences between the received theory of regular conditional distributions, which is the countably additive theory of conditional probability, and a rival theory of conditional probability using the theory of finitely additive probability. The focus of the paper is maximally "improper" conditional probability distributions, where the received theory requires, in effect, that P{a: P(a|a) = 0} = 1. This work builds upon (...) the results of Blackwell and Dubins (1975). (shrink)

Recent research (e.g., Evans & Over, 2004) has provided support for the hypothesis that people evaluate the probability of conditional statements of the form if p then q as the conditional probability of q given p , P( q / p ). The present paper extends this approach to pragmatic conditionals in the form of inducements (i.e., promises and threats) and advice (i.e., tips and warnings). In so doing, we demonstrate a distinction between the truth status (...) of these conditionals and their effectiveness as speech acts. Specifically, while probability judgements of the truth of conditional inducements and advice are highly correlated with estimates of P( q / p ), their perceived effectiveness in changing behaviour instead varies as a function of the conditional probability of q given not-p , P( q / ∼p ). Finally, we show that the conditional probability approach can be extended to predicting inference rates on a conditional reasoning task. (shrink)

Knowledge is more valuable than mere true belief. Many authors contend, however, that reliabilism is incompatible with this item of common sense. If a belief is true, adding that it was reliably produced doesn't seem to make it more valuable. The value of reliability is swamped by the value of truth. In Goldman and Olsson (2009), two independent solutions to the problem were suggested. According to the conditional probability solution, reliabilist knowledge is more valuable in virtue of being (...) a stronger indicator than mere true belief of future true belief. This article defends this solution against some objections. (shrink)

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey–Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of non-monotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).Expectation is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form (...) of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. (shrink)

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached (...) with certainty. This goal is achieved by taking Adams" (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams" logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of ¬ C given B are both close to one but the bound on the former"s deviation from 1 is much smaller than the latter"s, then it may be concluded that in all likelihood the conditional probability of C given A B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor"s rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl"s System-Z +. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their random-structures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL. (shrink)

An important field of probability logic is the investigation of inference rules that propagate point probabilities or, more generally, interval probabilities from premises to conclusions. Conditional probability logic (CPL) interprets the common sense expressions of the form “if . . . , then . . . ” by conditional probabilities and not by the probability of the material implication. An inference rule is probabilistically informative if the coherent probability interval of its conclusion is not (...) necessarily equal to the unit interval [0, 1]. Not all logically valid inference rules are probabilistically informative and vice versa. The relationship between logically valid and probabilistically informative inference rules is discussed and illustrated by examples such as the modus ponens or the affirming the consequent. We propose a method to evaluate the strength of CPL inference. (shrink)

In standard treatments of probability, Pr (A|B) is defined as the ratio of Pr (A∩B) to Pr (B), provided that Pr (B) > 0. This account of conditional probability suggests a psychological question, namely, whether estimates of Pr (A|B) arise in the mind via implicit calculation of Pr (A ∩ B)/Pr (B). We tested this hypothesis (Experiment 1) by presenting brief visual scenes composed of forms, and collecting estimates of relevant probabilities. Direct estimates of conditional (...) class='Hi'>probability were not well predicted by Pr (A ∩ B)/Pr (B). Direct estimates were also closer to the objective probabilities defined by the stimuli, compared to estimates computed from the foregoing ratio. The hypothesis that Pr (A|B) arises from the ratio Pr (A ∩ B)/[Pr (A ∩ B) + Pr (A ∩ B)] fared better (Experiment 2). In a third experiment, the same hypotheses were evaluated in the context of subjective estimates of the chance of future events. (shrink)

A logic for classical conditional events was investigated by Dubois and Prade. In their approach, the truth value of a conditional event may be undetermined. In this paper we extend the treatment to many-valued events. Then we support the thesis that probability over partially undetermined events is a conditional probability, and we interpret it in terms of bets in the style of de Finetti. Finally, we show that the whole investigation can be carried out in (...) a logical and algebraic setting, and we find a logical characterization of coherence for assessments of partially undetermined events. (shrink)

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey-Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of nonmonotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).'Expectation' is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form (...) of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. (shrink)

I consider the notions of logical probability, degree of belief, and objective chance, and argue that a different formalism for conditional probability is appropriate for each.

in Probability is the Very Guide of Life: The Philosophical Uses of Chance, eds. Henry Kyburg, Jr. and Mariam Thalos, Open Court. Abridged version in Proceedings of the International Society for Bayesian Analysis 2002.

Draft of a paper for the Sinn und Bedeutung 14 conference. Explains how to capture the link between conditionals the probability of indicative conditionals and conditional probability using a classical semantics for conditionals. (Note: some introductory material is shared with a twin paper, "Capturing the Relationship Between Conditionals and Conditional Probability with a Trivalent Semantics".).

Dynamic update of information states is a new paradigm in logicalsemantics. But such updates are also a traditional hallmark ofprobabilistic reasoning. This note brings the two perspectives togetherin an update mechanism for probabilities which modifies state spaces.

The present chapter describes a probabilistic framework of human reasoning. It is based on probability logic. While there are several approaches to probability logic, we adopt the coherence based approach.

The so-called ‘Adams’ Thesis’ is often understood as the claim that the assertibility of an indicative conditional equals the corresponding conditional probability—schematically: $${({\rm AT})}\qquad\qquad\quad As(A\rightarrow B)=P({B|A}),{\rm provided}\quad P(A)\neq 0.$$ The Thesis is taken by many to be a touchstone of any theorizing about indicative conditionals. Yet it is unclear exactly what the Thesis is . I suggest some precise statements of it. I then rebut a number of arguments that have been given in its favor. Finally, I (...) offer a new argument against it. I appeal to an old triviality result of mine against ‘Stalnaker’s Thesis’ that the probability of a conditional equals the corresponding conditional probability. I showed that for all finite-ranged probability functions, there are strictly more distinct values of conditional probabilities than there are distinct values of probabilities of conditionals, so they cannot all be paired up as Stalnaker’s Thesis promises. Conditional probabilities are too fine-grained to coincide with probabilities of conditionals across the board. If the assertibilities of conditionals are to coincide with conditional probabilities across the board, then assertibilities must be finer-grained than probabilities. I contend that this is implausible—it is surely the other way round. I generalize this argument to other interpretations of ‘ As ’, including ‘acceptability’ and ‘assentability’. I find it hard to see how any such figure of merit for conditionals can systematically align with the corresponding conditional probabilities. (shrink)

Journal of Philosophical Logic 34, 97-119, 2005.

Friedman and Putnam have argued (Friedman and Putnam 1978) that the quantum logical interpretation of quantum mechanics gives us an explanation of interference that the Copenhagen interpretation cannot supply without invoking an additional ad hoc principle, the projection postulate. I show that it is possible to define a notion of equivalence of experimental arrangements relative to a pure state φ , or (correspondingly) equivalence of Boolean subalgebras in the partial Boolean algebra of projection operators of a system, which plays a (...) role in the Copenhagen explanation of interference analogous to the role played by the material equivalence, given φ , of certain propositions in the Friedman-Putnam quantum logical analysis. I also show that the quantum logical interpretation and the Copenhagen interpretation are equally capable of avoiding the paradoxical conclusion of the Einstein-Podolsky-Rosen argument (Einstein, Podolsky, and Rosen 1935). Thus, neither interference phenomena nor the correlations between separated systems provide a test case for distinguishing between the relative acceptability of the Copenhagen interpretation and the quantum logical interpretation as explanations of quantum effects. (shrink)

An interesting link between two very different physical aspects of quantum mechanics is revealed; these are the absence of third-order interference and Tsirelson’s bound for the nonlocal correlations. Considering multiple-slit experiments—not only the traditional configuration with two slits, but also configurations with three and more slits—Sorkin detected that third-order (and higher-order) interference is not possible in quantum mechanics. The EPR experiments show that quantum mechanics involves nonlocal correlations which are demonstrated in a violation of the Bell or CHSH inequality, but (...) are still limited by a bound discovered by Tsirelson. It now turns out that Tsirelson’s bound holds in a broad class of probabilistic theories provided that they rule out third-order interference. A major characteristic of this class is the existence of a reasonable calculus of conditional probability or, phrased more physically, of a reasonable model for the quantum measurement process. (shrink)

Set-valued choice functions provide a framework that is general enough to encompass a wide variety of theories that are significant to the study of rationality but, at the same time, offer enough structure to articulate consistency conditions that can be used to characterize some of the theories within this encompassed variety. Nonetheless, two-tiered choice functions, such as those advocated by Isaac Levi, are not easily characterized within the framework of set-valued choice functions. The present work proposes conditional choice functions (...) as the proper carriers of synchronic rationality. The resulting framework generalizes the familiar one mentioned above without emptying it and, moreover, provides a natural setting for two-tiered choice rules. (shrink)

Numerous triviality results have been directed at a collection of views that tie the probability of a conditional sentence to the conditional probability of the consequent on its antecedent. -/- In this paper I argue that this identification makes little sense if conditional sentences are context sensitive. The best alternative, I argue, is a version of the thesis which states that if your total evidence is E then the evidential probability of a conditional (...) evaluated in a context where E is salient is the probability of the consequent given the antecedent. The biggest challenge to this thesis comes from the 'static' triviality arguments developed by Stalnaker, Hajek and Hall. It is argued that these arguments rely on invalid principles of conditional logic and that the thesis is consistent with a reasonably strong logic that does not include the principles in question. (shrink)

In this paper we discuss the new Tweety puzzle. The original Tweety puzzle was addressed by approaches in non-monotonic logic, which aim to adequately represent the Tweety case, namely that Tweety is a penguin and, thus, an exceptional bird, which cannot fly, although in general birds can fly. The new Tweety puzzle is intended as a challenge for probabilistic theories of epistemic states. In the first part of the paper we argue against monistic Bayesians, who assume that epistemic states can (...) at any given time be adequately described by a single subjective probability function. We show that monistic Bayesians cannot provide an adequate solution to the new Tweety puzzle, because this requires one to refer to a frequency-based probability function. We conclude that monistic Bayesianism cannot be a fully adequate theory of epistemic states. In the second part we describe an empirical study, which provides support for the thesis that monistic Bayesianism is also inadequate as a descriptive theory of cognitive states. In the final part of the paper we criticize Bayesian approaches in cognitive science, insofar as their monistic tendency cannot adequately address the new Tweety puzzle. We, further, argue against monistic Bayesianism in cognitive science by means of a case study. In this case study we show that Oaksford and Chater’s (2007, 2008) model of conditional inference—contrary to the authors’ theoretical position—has to refer also to a frequency-based probability function. (shrink)

While there is now considerable experimental evidence that, on the one hand, participants assign to the indicative conditional as probability the conditional probability of consequent given antecedent and, on the other, they assign to the indicative conditional the ?defective truth-table? in which a conditional with false antecedent is deemed neither true nor false, these findings do not in themselves establish which multi-premise inferences involving conditionals participants endorse. A natural extension of the truth-table semantics pronounces (...) as valid numerous inference patterns that do seem to be part of ordinary usage. However, coupled with something the probability account gives us?namely that when conditional-free ? entails conditional-free ?, ?if ? then ?? is a trivial, uninformative truth?we have enough logic to derive the paradoxes of material implication. It thus becomes a matter of some urgency to determine which inference patterns involving indicative conditionals participants do endorse. Only thus will we be able to arrive at a realistic, systematic semantics for the indicative conditional. (shrink)

In a previous paper I described a range of nonmonotonic conditionals that behave like conditional probability functions at various levels of probabilistic support. These conditionals were defined as semantic relations on an object language for sentential logic. In this paper I extend the most prominent family of these conditionals to a language for predicate logic. My approach to quantifiers is closely related to Hartry Field''s probabilistic semantics. Along the way I will show how Field''s semantics differs from a (...) substitutional interpretation of quantifiers in crucial ways, and show that Field''s approach is closely related to the usual objectual semantics. One of Field''s quantifier rules, however, must be significantly modified to be adapted to nonmonotonic conditional semantics. And this modification suggests, in turn, an alternative quantifier rule for probabilistic semantics. (shrink)

I will describe the logics of a range of conditionals that behave like conditional probabilities at various levels of probabilistic support. Families of these conditionals will be characterized in terms of the rules that their members obey. I will show that for each conditional, , in a given family, there is a probabilistic support level r and a conditional probability function P such that, for all sentences C and B, C->B holds just in case P[B|C] is (...) greater than or equal to r. Thus, each conditional in a given family behaves like conditional probability above some specific support level. (shrink)

This paper rejects a traditional epistemic interpretation of conditional probability. Suppose some chance process produces outcomes X, Y,..., with probabilities P(X), P(Y),... If later observation reveals that outcome Y has in fact been achieved, then the probability of outcome X cannot normally be revised to P(X|Y) ['P&Y)/P(Y)]. This can only be done in exceptional circumstances - when more than just knowledge of Y-ness has been attained. The primary reason for this is that the weight of a piece (...) of evidence varies with the means by which it is provided, so knowledge of Y-ness does not have uniform impact on the probability of X. A better updating of the probability of X is provided by P(X|Y*), where Y* is not an outcome of the chance process being observed, but the sentence 'the outcome Y has been observed', an 'outcome' of the subsequent observation process. This alternative formula is widely endorsed in practice, but not well recognized in theory, where the oversight has generated some unsatisfactory consequences. (shrink)

The conditional probability of h given e is commonly claimed to be equal to the probability that h would have if e were learned. Here I contend that this general claim about conditional probabilities is false. I present a counter-example that involves probabilities of probabilities, a second that involves probabilities of possible future actions, and a third that involves probabilities of indicative conditionals. In addition, I briefly defend these counter-examples against charges that the probabilities they involve (...) are illegitimate. (shrink)

According to what is now commonly referred to as “the Equation” in the literature on indicative conditionals, the probability of any indicative conditional equals the probability of its consequent of the conditional given the antecedent of the conditional. Philosophers widely agree in their assessment that the triviality arguments of Lewis and others have conclusively shown the Equation to be tenable only at the expense of the view that indicative conditionals express propositions. This study challenges the (...) correctness of that assessment by presenting data that cast doubt on an assumption underlying all triviality arguments. (shrink)

We show that the implicational fragment of intuitionism is the weakest logic with a non-trivial probabilistic semantics which satisfies the thesis that the probabilities of conditionals are conditional probabilities. We also show that several logics between intuitionism and classical logic also admit non-trivial probability functions which satisfy that thesis. On the other hand, we also prove that very weak assumptions concerning negation added to the core probability conditions with the restriction that probabilities of conditionals are conditional (...) probabilities are sufficient to trivialize the semantics. (shrink)

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1,..., N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering (...) asymptotic conditional probabilities. As shown by Liogon'kii [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable. (shrink)

The aim of the paper is to draw a connection between a semantical theory of conditional statements and the theory of conditional probability. First, the probability calculus is interpreted as a semantics for truth functional logic. Absolute probabilities are treated as degrees of rational belief. Conditional probabilities are explicitly defined in terms of absolute probabilities in the familiar way. Second, the probability calculus is extended in order to provide an interpretation for counterfactual probabilities--conditional (...) probabilities where the condition has zero probability. Third, conditional propositions are introduced as propositions whose absolute probability is equal to the conditional probability of the consequent on the antecedent. An axiom system for this conditional connective is recovered from the probabilistic definition. Finally, the primary semantics for this axiom system, presented elsewhere, is related to the probabilistic interpretation. (shrink)

This is a 'state of the art' collection of essays on the relation between probabilities, especially conditional probabilities, and conditionals. It provides new negative results which sharply limit the ways conditionals can be related to conditional probabilities. There are also positive ideas and results which will open up new areas of research. The collection is intended to honour Ernest W. Adams, whose seminal work is largely responsible for creating this area of inquiry. As well as describing, evaluating, and (...) applying Adams' work the contributions extend his ideas in directions he may or may not have anticipated, but that he certainly inspired. In addition to a wide range of philosophers of science, the volume should interest computer scientists and linguists. (shrink)

The two main psychological theories of the ordinary conditional were designed to account for inferences made from assumptions, but few premises in everyday life can be simply assumed true. Useful premises usually have a probability that is less than certainty. But what is the probability of the ordinary conditional and how is it determined? We argue that people use a two stage Ramsey test that we specify to make probability judgements about indicative conditionals in natural (...) language, and we describe experiments that support this conclusion. Our account can explain why most people give the conditional probability as the probability of the conditional, but also why some give the conjunctive probability. We discuss how our psychological work is related to the analysis of ordinary indicative conditionals in philosophical logic. (shrink)

Two major themes in the literature on indicative conditionals are (1) that the content of indicative conditionals typically depends on what is known;1 (2) that conditionals are intimately related to conditional probabilities.2 In possible world semantics for counterfactual conditionals, a standard assumption is that conditionals whose antecedents are metaphysically impossible are vacuously true.3 This aspect has recently been brought to the fore, and defended by Tim Williamson, who uses it in to characterize alethic necessity by exploiting such equivalences as: (...) A⇔¬A A. One might wish to postulate an analogous connection for indicative conditionals, with indicatives whose antecedents are (in some relevant sense) epistemically impossible being vacuously true: and indeed, the modal account of indicative conditionals of Brian Weatherson has exactly this feature.4 This allows one to characterize an epistemic modal by the equivalence A⇔¬A→A. For simplicity, in what follows we write A as KA and think of it as expressing that subject S knows that A.5 The connection to probability has received much attention. Stalnaker (1970) suggested, as a way of articulating the ‘Ramsey Test’, the following very general schema for indicative conditionals relative to some probability function P: P(A→B) = P(B|A) 1For example, Nolan (2003); Weatherson (2001); Gillies (2007). 2For example Stalnaker (1970); McGee (1989); Adams (1975). 3Lewis (1973). See Nolan (1997) for criticism. 4‘epistemically possible’ here means incompatible with what is known (where ‘what is known’ is to be cashed out in some relevant sense). 5This idea was suggested to me in conversation by John Hawthorne. I do not know of it being explored in print. The plausibility of this characterization will depend on the exact sense of ‘epistemically possible’ in play—if it is compatibility with what a single subject knows, then can be read ‘the relevant subject knows that p’. If it is more delicately formulated, we might be able to read as the epistemic modal ‘must’.. (shrink)

Discusses how to capture the link between the probability of indicative conditionals and conditional probability using a classical semantics for conditionals.

According to probabilistic theories of reasoning in psychology, people's degree of belief in an indicative conditional `if A, then B' is given by the conditional probability, P(B|A). The role of language pragmatics is relatively unexplored in the new probabilistic paradigm. We investigated how consequent relevance aects participants' degrees of belief in conditionals about a randomly chosen card. The set of events referred to by the consequent was either a strict superset or a strict subset of the set (...) of events referred to by the antecedent. We manipulated whether the superset was expressed using a disjunction or a hypernym. We also manipulated the source of the dependency, whether in long-term memory or in the stimulus. For subset-consequent conditionals, patterns of responses were mostly conditional probability followed by conjunction. For superset-consequent conditionals, conditional probability responses were most common for hypernym dependencies and least common for disjunction dependencies, which were replaced with responses indicating inferred consequent irrelevance. Conditional probability responses were also more common for knowledge-based than stimulus-based dependencies. We suggest. (shrink)

Some propositions add more information to bodies of propositions than do others. We start with intuitive considerations on qualitative comparisons of information added . Central to these are considerations bearing on conjunctions and on negations. We find that we can discern two distinct, incompatible, notions of information added. From the comparative notions we pass to quantitative measurement of information added. In this we borrow heavily from the literature on quantitative representations of qualitative, comparative conditional probability. We look at (...) two ways to obtain a quantitative conception of information added. One, the most direct, mirrors Bernard Koopman’s construction of conditional probability: by making a strong structural assumption, it leads to a measure that is, transparently, some function of a function P which is, formally, an assignment of conditional probability (in fact, a Popper function). P reverses the information added order and mislocates the natural zero of the scale so some transformation of this scale is needed but the derivation of P falls out so readily that no particular transformation suggests itself. The Cox–Good–Aczél method assumes the existence of a quantitative measure matching the qualitative relation, and builds on the structural constraints to obtain a measure of information that can be rescaled as, formally, an assignment of conditional probability. A classical result of Cantor’s, subsequently strengthened by Debreu, goes some way towards justifying the assumption of the existence of a quantitative scale. What the two approaches give us is a pointer towards a novel interpretation of probability as a rescaling of a measure of information added. (shrink)

The swamping problem is the problem of explaining why reliabilist knowledge (reliable true belief) has greater value than mere true belief. Swamping problem advocates see the lack of a solution to the swamping problem (i.e., the lack of a value-difference between reliabilist knowledge and mere true belief) as grounds for rejecting reliabilism. My aims here are (i) to specify clear requirements for a solution to the swamping problem that are as congenial to reliabilism's critics as possible, (ii) to clear away (...) various existing reliabilist solutions on the basis of these requirements, and (iii) to present a reliabilist solution that succeeds in meeting all of them. To meet all the requirements, my solution develops a more nuanced understanding of the epistemic end than is currently discussed, and with it a novel way of individuating beliefs. I close with a brief discussion of the question whether reliabilism's critics might impose further demands which reliabilism cannot possibly meet. (shrink)

Conditional information can be equally asserted in the forms if p, then q (e.g., ?if I am ill, I will miss work tomorrow?) and q, if p (e.g., ?I will miss work tomorrow, if I am ill?). While this type of clause order manipulation has previously been found to have no influence on the ultimate conclusions participants draw from conditional rules, we used self-paced reading to examine how it affects the real time incremental processing of everyday conditional (...) statements. Experiment 1 revealed that clause order interacts with presuppositional congruency as readers hypothetically represent counterfactual statements. When if p, then q counterfactuals contained a presupposition that was incongruent with prior context, these statements took longer to read than when the presupposition was congruent, but for q, if p conditionals there was no such congruency effect. Experiment 2 revealed that reading times were influenced by the subjective probability of an indicative conditional regardless of clause order, with a penalty observed for low-probability statements relative to high-probability statements in both conditional clause orders. These data reveal a dissociation whereby clause order mediates the effect of suppositional congruency on reading times, but does not mediate the effect of subjective probability. (shrink)

THE INDICATIVE CONDITIONAL. A PROBABILISTIC CRITERION OF SOUNDNESS FOR DEDUCTIVE INFERENCES Our objective in this section is to establish a prima facie case ...

I describe /mindset semantics/, a semantical framework built around a conception of entailment as preservation of /support/ (implicit acceptance undergirded by competence) together with a /classical modal/ semantics for declarative sentences---with the central application of showing how a language could integrate discourse that is expressive with discourse that is informative (namely, of solving the 'Frege-Geach problem'). (The approach owes much to the work of Veltman and Yalcin, and, less proximally, of Stalnaker.) I provide a range of philosophical, technical, and pedagogical (...) arguments for mindset semantics. And I apply mindset semantics to as wide a range of phenomena as I have been able to think of (some more familiar, others less familiar): epistemic modals, avowals of belief, avowals of what it's like, /ascriptions/ of all these, 'looks'-sentences, avowals of presumption, avowals of subjective credence and statements of objective chance, indicative and subjunctive conditionals, conditional probability, deontic modals, questions, avowals of wonderment, metaphysical modals, metaphysical indeterminacy, avowals and ascriptions of knowledge, 'ought'-claims, and avowals of various practical positions (intending, trying, needing). (shrink)

Today philosophical discussion on indicative conditionals is dominated by the so called Lewis Triviality Results, according to which, tehere is no binary connective '-->' (let alone truth-functional) such that the probability of p --> q equals the probability of q conditionally on p, so that P(p --> q)= P(q|p). This tenet, that suggests that conditonals lack truth-values, has been challenged in 1991 by Goodman et al. who show that using a suitable three-valued logic the above equation may be (...) restored. In this paper it is first analysed a long neglected paper by Bruno de Finetti, written in 1935, where the essentials of Goodman's theory was clearly outlined. It is also stressed that de Finetti anticipated Kleene's as well as Bochvar and Blamey ideas. In the second part of the paper it is argued that the de Finetti-Goodman's original theory is defective and leads to absurd results. However, a new semantics, called semantics of hypervaluations, is here defined, that avoids the defects of the original theory. This appears to be a powerful challenge to Lewis Triviality results and to the thesis by which conditionals lack truth-values as well. (shrink)

A simple argument is given that shows that conditional probabilities do not supervene on unconditional probabilities. In particular, one cannot in general define conditional probabilities using the ratio formula P ( U | V ) = P ( U & V )/ P ( U ), or using any more sophisticated method based on unconditional probabilities.

I’ll describe a range of systems for nonmonotonic conditionals that behave like conditional probabilities above a threshold. The rules that govern each system are probabilistically sound in that each rule holds when the conditionals are interpreted as conditional probabilities above a threshold level specific to that system. The well-known preferential and rational consequence relations turn out to be special cases in which the threshold level is 1. I’ll describe systems that employ weaker rules appropriate to thresholds lower than (...) 1, and compare them to these two standard systems. (shrink)

The orthodoxy that conditional probabilities reflect what are for a subject evidential bearings is seconded. This significance suggests that there should be principles equating rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. Several principles, two of which are endorsed, are considered. A book is made against a violator of these, and it is argued that there must be something wrong with a person against whom such books can be made. Appendices comment on Popper-functions, (...) elaborate on bets and odds, and relate dutch books and strategies to conditions of inconsistency (Ramsey's idea) and imperfection. (shrink)

Relations between conditional probabilities, revisions of probabilities in the light of new information, and conditions of ideal rationality are discussed herein. The formal character of conditional probabilities, and their significance for epistemic states of agents is taken up. Then principles are considered that would, under certain conditions, equate rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. And lastly the possibility of kinds of 'books' against known non-conditionalizers is explored, and the question (...) is taken up, What, if anything, would be wrong with a person against whom such a book could be made? (shrink)

This is a study in the meaning of natural language probability operators, sentential operators such as probably and likely. We ask what sort of formal structure is required to model the logic and semantics of these operators. Along the way we investigate their deep connections to indicative conditionals and epistemic modals, probe their scalar structure, observe their sensitivity to contex- tually salient contrasts, and explore some of their scopal idiosyncrasies.

Kaufmann has recently argued that the thesis according to which the probability of an indicative conditional equals the conditional probability of the consequent given the antecedent under certain specifiable circumstances deviates from intuition. He presents a method for calculating the probability of a conditional that does seem to give the intuitively correct result under those circumstances. However, the present paper shows that Kaufmann’s method is inconsistent in that it may lead one to assign different (...) probabilities to a single conditional at the same time. (shrink)

Adams' famous thesis that the probabilities of conditionals are conditional probabilities is incompatible with standard probability theory. Indeed it is incompatible with any system of monotonic conditional probability satisfying the usual multiplication rule for conditional probabilities. This paper explores the possibility of accommodating Adams' thesis in systems of non-monotonic probability of varying strength. It shows that such systems impose many familiar lattice theoretic properties on their models as well as yielding interesting logics of conditionals, (...) but that a standard complementation operation cannot be defined within them, on pain of collapsing probability into bivalence. (shrink)

Timothy Williamson argues against the tactic of criticizing confidence in a theory by identifying a logical consequence of the theory whose probability is not raised by the evidence. He dubs it "the consequence fallacy". In this paper we will show that Williamson's formulation of the tactic in question is ambiguous. On one reading of Williamson's formulation, the tactic is indeed a fallacy, but it is not a commonly used tactic; on another reading, it is a commonly used tactic (or (...) at least more often used than the former tactic), but it is not a fallacy. (shrink)

There are narrowest bounds for P(h) when P(e) = y and P(h/e) = x, which bounds collapse to x as y goes to 1. A theorem for these bounds -- bounds for probable modus ponens -- entails a principle for updating on possibly uncertain evidence subject to these bounds that is a generalization of the principle for updating by conditioning on certain evidence. This way of updating on possibly uncertain evidence is appropriate when updating by ’probability kinematics’ or ’Jeffrey-conditioning’ (...) is, and apparently in countless other cases as well. A more complicated theorem due to Karl Wagner -- bounds for probable modus tollens -- registers narrowest bounds for P(not h) when P(not e) = y and P(e/h) = x. This theorem serves another principle for updating on possibly uncertain evidence that might be termed ’contraditioning’, though it is for a way of updating that seems in practice to be frequently not appropriate. It is definitely not a way of putting down a theory -- for example, a random-chanc. (shrink)

This paper explores the interaction of well-motivated (if controversial) principles governing the probability conditionals, with accounts of what it is for a sentence to be indefinite. The conclusion can be played in a variety of ways. It could be regarded as a new reason to be suspicious of the intuitive data about the probability of conditionals; or, holding fixed the data, it could be used to give traction on the philosophical analysis of a contentious notion—indefiniteness. The paper outlines (...) the various options, and shows that ‘rejectionist’ theories of indefiniteness are incompatible with the results. Rejectionist theories include popular accounts such as supervaluationism, non-classical truth-value gap theories, and accounts of indeterminacy that centre on rejecting the law of excluded middle. An appendix compares the results obtained here with the ‘impossibility’ results descending from Lewis ( 1976 ). (shrink)

The aim of this paper is to distinguish between, and examine, three issues surrounding Humphreys's paradox and interpretation of conditional propensities. The first issue involves the controversy over the interpretation of inverse conditional propensities — conditional propensities in which the conditioned event occurs before the conditioning event. The second issue is the consistency of the dispositional nature of the propensity interpretation and the inversion theorems of the probability calculus, where an inversion theorem is any theorem of (...) probability that makes explicit (or implicit) appeal to a conditional probability and its corresponding inverse conditional probability. The third issue concerns the relationship between the notion of stochastic independence which is supported by the propensity interpretation, and various notions of causal independence. In examining each of these issues, it is argued that the dispositional character of the propensity interpretation provides a consistent and useful interpretation of the probability calculus. (shrink)

The philosophy of probability has been alive and well for several decades in Australia and New Zealand. Some distinctive lines of thought have emerged, resonating with broader themes that have come to be associated with Australasian philosophers: realist/objectivist accounts of various theoretical entities; an ongoing concern with logic, including the development of non­classical logics; and enthusiasm for conceptual analysis, rooted in commonsense but informed by science. In this article I concentrate on work by philosophers on the interpretation of (...) class='Hi'>probability, its logical foundations, and its philosophical applications.1 My nomination for the earliest major Australasian philosopher of probability may surprise some readers: Karl Popper. He counts as Australasian by dint of his employment at the University of Canterbury from 1937 until the end of World War II; he counts as a major philosopher of probability by any estimation. Two of his contributions have initiated research programs in the foundations of probability that are still thriving: his (1959a) axiomatization of primitive conditional probability functions (so­called ‘Popper functions’), and his ‘propensity’ interpretation of probability (1959b), intended to illuminate single­case attributions of objective probabilities, as are putatively found in quantum mechanics. (shrink)

Jenann’s central metaphysical thesis is that there is an objective conditional probability function PrG(A/B), the domain of which includes a great many, perhaps all, pairs of contingent propositions. This pair can be synchronic or diachronic: both can concern how things are at the same time, or not. Jenann’s central epistemological thesis is antiskepticism about PrG, in the following sense: prima facie, the subjective credence functions of epistemically reasonable agents converge on PrG: roughly, if you’ve done a lot of (...) science, for all A, B, your C(A/B) is similar to PrG(A/B). (Compare antiskepticism about perceptual knowledge: prima facie, if circumstances are good and one’s visual experience represents that p, p.) These theses have two cool consequences: first, the possibility of a novel approach to objective Bayesianism; second, a way of doing away with dynamical laws. (shrink)

We advance a theory of inductive inference designed to predict the conditional probability that certain natural categories satisfy a given predicate given that others do (or do not). A key component of the theory is the similarity of the categories to one another. We measure such similarities in terms of the overlap of metabolic activity in voxels of various posterior regions of the brain in response to viewing instances of the category. The theory and similarity measure are tested (...) against averaged probability judgments elicited from a separate group of subjects. Fruit serve as categories in the present experiment; results are compared to earlier work with mammals. (shrink)

We introduce the concept of partial event as a pair of disjoint sets, respectively the favorable and the unfavorable cases. Partial events can be seen as a De Morgan algebra with a single fixed point for the complement. We introduce the concept of a measure of partial probability, based on a set of axioms resembling Kolmogoroff’s. Finally we define a concept of conditional probability for partial events and apply this concept to the analysis of the two-slit experiment (...) in quantum mechanics. (shrink)

Two experiments were conducted to investigate the roles of covariation and of causality in people's readiness to believe a conditional. The experiments used a probabilistic truth-table task (Oberauer & Wilhelm, 2003) in which people estimated the probability of a conditional given information about the frequency distribution of truth-table cases. For one group of people, belief in the conditional was determined by the conditional probability of the consequent, given the antecedent, whereas for another group it (...) depended on the probability of the conjunction of antecedent and consequent. There was little evidence that covariation, expressed as the probabilistic contrast or as the pCI rule (White, 2003), influences belief in the conditional. The explicit presence of a causal link between antecedent and consequent in a context story had a weak positive effect on belief in a conditional when the frequency distribution of relevant cases was held constant. (shrink)

As the paper explains, it is crucial to epistemology in general and to the theory of causation in particular to investigate the properties of conditional independence as completely as possible. The paper summarizes the most important results concerning conditional independence with respect to two important representations of epistemic states, namely (strictly positive) probability measures and natural conditional (or disbelief or ranking) functions. It finally adds some new observations.

This paper examines definitions of independence for events and variables in the context of full conditional measures; that is, when conditional probability is a primitive notion and conditioning is allowed on null events. Several independence concepts are evaluated with respect to graphoid properties; we show that properties of weak union, contraction and intersection may fail when null events are present. We propose a concept of “full” independence, characterize the form of a full conditional measure under full (...) independence, and suggest how to build a theory of Bayesian networks that accommodates null events. (shrink)

We study a probabilistic logic based on the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence). We examine probabilistic conditional knowledge bases associated with imprecise probability assessments defined on arbitrary families of conditional events. We introduce a notion of conditional interpretation defined directly in terms of precise probability assessments. We also examine a property of strong satisfiability which is related to the notion of toleration well known in default reasoning. In (...) our framework we give more general definitions of the notions of probabilistic consistency and probabilistic entailment of Adams. We also recall a notion of strict p-consistency and some related results. Moreover, we give new proofs of some results obtained in probabilistic default reasoning. Finally, we examine the relationships between conditional probability rankings and the notions of g-coherence and g-coherent entailment. (shrink)

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable (...) as Popper functions, and that every Popper function is representable in terms of the standard real values of some infinitesimal measure.Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-question-begging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30]. (shrink)

Following the pioneer work of Bruno De Finetti, conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's.

The nature of quantum mechanical probability has often seemed mysterious. To shed some light on this topic, the present paper analyzes the logical form of probability assignment in quantum mechanics. To begin the paper, I set out and criticize several attempts to analyze the form. I go on to propose a new form which utilizes a novel, probabilistic conditional and argue that this proposal is, overall, the best rendering of the quantum mechanical probability assignments. Finally, quantum (...) mechanics aside, the discussion here has consequences for counterfactual logic, conditional probability, and epistemic probability. (shrink)

Exhumation and study of the 1945 paradox of confirmation brings out the defect of its formulation. In the context of quantifier conditional-probability logic it is shown that a repair can be accomplished if the truth-functional conditional used in the statement of the paradox is replaced with a connective that is appropriate to the probabilistic context. Description of the quantifier probability logic involved in the resolution of the paradox is presented in stages. Careful distinction is maintained between (...) a formal logic language and its semantics, as the same language may be outfitted with different semantics. An acquaintance with sections 1? 5 of Hailperin (2006) covering the sentential aspects of probability logic is assumed as background information for quantifier probability logic. (shrink)

Logical argument forms are investigated by second order probability density functions. When the premises are expressed by beta distributions, the conclusions usually are mixtures of beta distributions. If the shape parameters of the distributions are assumed to be additive (natural sampling), then the lower and upper bounds of the mixing distributions (P´olya-Eggenberger distributions) are parallel to the corresponding lower and upper probabilities in conditional probability logic.

We advance a theory of inductive inference designed to predict the conditional probability that certain natural categories satisfy a given predicate given that others do (or do not). A key component of the theory is the similarity of the categories to one another. We measure such similarities in terms of the overlap of metabolic activity in voxels of various posterior regions of the brain in response to viewing instances of the category. The theory and similarity measure are tested (...) against averaged probability judgments elicited from a separate group of subjects. Fruit serve as categories in the present experiment; results are compared to earlier work with mammals. (shrink)

This book shows how these developments have led researchers to view people's conditional reasoning behaviour more as succesful probabilistic reasoning rather ...

Conditional attitudes are not the attitudes an agent is disposed to acquire in event of learning that a condition holds. Rather they are the components of agent's current attitudes that derive from the consideration they give to the possibility that the condition is true. Jeffrey's decision theory can be extended to include quantitative representation of the strength of these components. A conditional desirability measure for degrees of conditional desire is proposed and shown to imply that an agent's (...) degrees of conditional belief are conditional probabilities. Rational conditional preference is axiomatised and by application of Bolker's representation theorem for rational preferences it is shown that conditional preference rankings determine the existence of probability and desirability measures that agree with them. It is then proven that every conditional desirability function agrees with an agent's conditional preferences and, under certain assumptions, every desirability function agreeing with an agent's conditional preferences is a conditional desirability function agreeing with her unconditional preferences. (shrink)

In hisStudy of War, Q. Wright considered a model for the probability of warP during a period ofn crises, and proposed the equationP=1–(1–p) n , wherep is the probability of war escalating at each individual crisis. This probability measure was formally derived recently by Cioffi-Revilla (1987), using the general theory of political reliability and an interpretation of the n-crises problem as a branching process. Two new, alternate solutions are presented here, one using D. Bernoulli''s St. Petersburg (...) Paradox as an analogue, the other based on the logic of conditional probabilities. Analysis shows that, while Wright''s solution is robust with regard to the general overall behavior ofp andn, some significant qualitative and quantitative differences emerge from the alternative solutions. In particular,P converges to 1 only in a special case (Wright''s) and not generally. (shrink)

Mental probability logic is a psychological competence theory about how humans interpret and reason about common-sense conditionals. Probability logic is proposed as an appropriate standard of reference for evaluating the rationality of human inferences. Common-sense conditionals are interpreted as “high” conditional probabilities, P(B|A) > .5. Probability logical accounts of nonmonotonic reasoning and inference rules like the modus ponens are explored. Categorical syllogisms with comparative and quantitative quantifiers are investigated. A series of eight experiments on human probabilistic (...) reasoning in the framework of the basic nonmonotonic system p corroborate the psychological plausibility of the proposed approach. (shrink)

In this paper we investigate Accardi's claim that the "quantum paradoxes" have their roots in probability theory and that, in particular, they can be evaded by giving up Bayes' rule, concerning the relation between composite and conditional probabilities. We reach the conclusion that, although it may be possible to give up Bayes' rule and define conditional probabilities differently, this contributes nothing to solving the philosophical problems which surround quantum mechanics.

On the basis of impossibility results on probability, belief revision, and conditionals, it is argued that conditional beliefs differ from beliefs in conditionals qua mental states. Once this is established, it will be pointed out in what sense conditional beliefs are still conditional, even though they may lack conditional contents, and why it is permissible to still regard them as beliefs, although they are not beliefs in conditionals. Along the way, the main logical, dispositional, representational, (...) and normative properties of conditional beliefs are studied, and it is explained how the failure of not distinguishing conditional beliefs from beliefs in conditionals can lead philosophical and empirical theories astray. (shrink)