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)

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)

The value problem for knowledge is the problem of explaining why knowledge is more valuable than mere true belief. The problem arises for reliabilism in particular, i.e., the externalist view that knowledge amounts to reliably acquired true belief. Goldman and Olsson argue that knowledge, in this sense, is more valuable than mere true belief due to the higher likelihood of future true beliefs (produced by the same reliable process) in the case of knowledge. They maintain that their solution works (...) given four empirical assumptions, which they claim hold “normally.” However, they do not show that their assumptions are externalistically acceptable; nor do they provide detailed evidence for their normality claim. We address these remaining gaps in Goldman and Olsson’s solution in a constructive spirit. In particular, we suggest an externalist interpretation of the assumptions such that they essentially spell out what it means for a broad range of organisms capable of belief-like representations to be epistemically adapted to their environment. Our investigation also sheds light on the circumstances in which the assumptions fail to hold and knowledge therefore fails to have extra value in Goldman and Olsson’s sense. The upshot is a deeper understanding of their solution as a contribution to naturalized epistemology and a strengthened case for its empirical validity. (shrink)

The problem of probabilities of conditionals is one of the long-standing puzzles in philosophy of language. We defend and update Adams' solution to the puzzle: the probability of an epistemic conditional is not the probability of a proposition, but a probability under a supposition. -/- Close inspection of how a triviality result unfolds in a concrete scenario does not provide counterexamples to the view that probabilities of conditionals are conditional probabilities: instead, it supports the (...) conclusion that probabilities of conditionals violate standard probability theory. -/- This does not call into question probability theory per se; rather, it calls for a more careful understanding of its role: probability theory is a theory of probabilities of propositions; but as conditionals do not express propositions, their probabilities are not subject to the standard laws. -/- We argue that both conditional probabilities and probabilities of conditionals are best understood in terms of the dynamics of supposing, modeled as a restriction operation on a probability space. This version of the suppositionalist view allows us to connect Adams' Thesis to the widely held restrictor view of the semantics of conditionals. -/- We address two common objections to Adams' view: that the relevant probabilities are ‘probabilities only in name’, and that giving up conditional propositions puts us at a disadvantage when it comes to interpreting compounds. -/- Finally, we argue that some putative counterexamples to Adams' Thesis can be diagnosed as fallacies of probabilistic reasoning: they arise from applying to conditionals laws of standard probability theory which are invalid for them. (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– n, wherep is the probability of war escalating at each individual crisis. This probability measure was formally derived recently by Cioffi -Revilla, 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 and not generally. (shrink)

Dynamic approaches to understanding probability in the non-fundamental sciences turn on certain properties of physical processes that are apt to produce “probabilistically patterned” outcomes. The dynamic properties on their own, however, seem not quite sufficient to explain the patterns; in addition, some sort of assumption about initial conditions must be made, an assumption that itself typically takes a probabilistic form. How should such a posit be understood? That is the problem of initial conditions. Reichenbach, in his doctoral dissertation, floated (...) a Kantian solution to the problem. In this paper I provide a Reichenbachian alternative. (shrink)

Two open questions of inductive reasoning are solved: (1) does the principle of maximum entropy (pme) give a solution to the obverse Majerník problem; and (2) is Wagner correct when he claims that Jeffrey’s updating principle (jup) contradicts pme? Majerník shows that pme provides unique and plausible marginal probabilities, given conditional probabilities. The obverse problem posed here is whether pme also provides such conditional probabilities, given certain marginal probabilities. The theorem developed to solve the obverse Majerník problem (...) demonstrates that in the special case introduced by Wagner pme does not contradict jup, but elegantly generalizes it and offers a more integrated approach to probability updating. (shrink)

According to Goldman and Olsson's so‐called conditional probability solution to the extra‐value‐of‐knowledge problem, knowledge is more valuable than mere true belief because having the former makes the acquisition of further similar true beliefs in the future more likely than having the latter does. Unfortunately, however, several philosophers have rejected the comparative probability claim Goldman and Olsson's solution is based on. In this paper, I present a new argument in defence of this claim. More precisely, I (...) point out a highly plausible necessary and sufficient condition for the latter and argue for its truth by bringing together confirmation‐theoretic and reliabilist considerations. I also explain where a number of objections against Goldman and Olsson's probabilistic claim seem to go wrong. (shrink)

One of central problems in the theory of conditionals is the construction of a probability space, where conditionals can be interpreted as events and assigned probabilities. The problem has been given a technical formulation by van Fraassen (23), who also discussed in great detail the solution in the form of Stalnaker Bernoulli spaces. These spaces are very complex – they have the cardinality of the continuum, even if the language is finite. A natural question is, therefore, whether a (...) technically simpler (in particular finite) partial construction can be given. In the paper we provide a new solution to the problem. We show how to construct a finite probability space $$\mathrm {S}^\#=\left(\mathrm\Omega^\#,\mathrm\Sigma^\#,\mathrm P^\#\right)$$ S # = Ω #, Σ #, P # in which simple conditionals and their Boolean combinations can be interpreted. The structure is minimal in terms of cardinality within a certain, naturally defined class of models – an interesting side-effect is an estimate of the number of non-equivalent propositions in the conditional language. We demand that the structure satisfy certain natural assumptions concerning the logic and semantics of conditionals and also that it satisfy PCCP. The construction can be easily iterated, producing interpretations for conditionals of arbitrary complexity. (shrink)

This paper discusses and relates two puzzles for indicative conditionals: a puzzle about indeterminacy and a puzzle about triviality. Both puzzles arise because of Ramsey's Observation, which states that the probability of a conditional is equal to the conditional probability of its consequent given its antecedent. The puzzle of indeterminacy is the problem of reconciling this fact about conditionals with the fact that they seem to lack truth values at worlds where their antecedents are false. The (...) puzzle of triviality is the problem of reconciling Ramsey's Observation with various triviality proofs which establish that Ramsey's Observation cannot hold in full generality. In the paper, I argue for a solution to the indeterminacy puzzle and then apply the resulting theory to the triviality puzzle. On the theory I defend, the truth conditions of indicative conditionals are highly context dependent and such that an indicative conditional may be indeterminate in truth value at each possible world throughout some region of logical space and yet still have a nonzero probability throughout that region. (shrink)

This article proposes a new theory of rational decision, distinct from both causal decision theory (CDT) and evidential decision theory (EDT). First, some intuitive counterexamples to CDT and EDT are presented. Then the motivation for the new theory is given: the correct theory of rational decision will resemble CDT in that it will not be sensitive to any comparisons of absolute levels of value across different states of nature, but only to comparisons of the differences in value between the available (...) options within states of nature; however, the correct theory will also resemble EDT in that it will rely on conditional probabilities (not unconditional probabilities). The new theory gives a prominent role to the notion of a “benchmark” for each state of nature, by comparison with which the value of the available options in that state of nature are measured, and so it has been called the Benchmark Theory (BT). It is argued that BT gives the right verdict on the cases that seem to be counterexamples to CDT and EDT. Finally, some objections to BT are considered and answered. (shrink)

The λ-continuum of inductive methods was derived from an assumption, called λ-condition, which says that the probability of finding an individual having property $x_{j}$ depends only on the number of observed individuals having property $x_{j}$ and on the total number of observed individuals. So, according to that assumption, all individuals with properties which are different from $x_{j}$ have equal weight with respect to that probability and, in particular, it does not matter whether any individual was observed having some (...) property similar to $x_{j}$ (the most complete proof of this result is presented in Carnap, 1980). The problem thus remained open to find some general condition, weaker than the λ-condition, which would allow for the derivation of probability functions which might be sensitive to similarity. Carnap himself suggested a weakening of the λ-condition which might allow for similarity sensitive probability functions (Carnap, 1980, p. 45) but he did not find the family of probability functions derivable from that principle. The aim of this paper is to present the family of probability functions derivable from Carnap's suggestion and to show how it is derived. In Section 1 the general problem of analogy by similarity in inductive logic is presented, Section 2 outlines the notation and the conceptual background involved in the proof, Section 3 gives the proof, Section 4 discusses Carnap's principle and the result, Section 5 is a brief review of the solutions which have previously been proposed. (shrink)

It is natural and important to have a formal representation of plain belief, according to which propositions are held true, or held false, or neither. (In the paper this is called a deterministic representation of epistemic states). And it is of great philosophical importance to have a dynamic account of plain belief. AGM belief revision theory seems to provide such an account, but it founders at the problem of iterated belief revision, since it can generally account only for one step (...) of revision. The paper discusses and rejects two solutions within the confines of AGM theory. It then introduces ranking functions (as I prefer to call them now; in the paper they are still called ordinal conditional functions) as the proper (and, I find, still the best) solution of the problem, proves that conditional independence w.r.t. ranking functions satisfies the so-called graphoid axioms, and proposes general rules of belief change (in close analogy to Jeffrey's generalized probabilistic conditionalization) that encompass revision and contraction as conceived in AGM theory. Indeed, the parallel to probability theory is amazing. Probability theory can profit from ranking theory as well since it is also plagued by the problem of iterated belief revision even if probability measures are conceived as Popper measures (see No. 11). Finally, the theory is compared with predecessors which are numerous and impressive, but somehow failed to explain the all-important conditional ranks in the appropriate way. (shrink)

Uncertainty and vagueness/imprecision are not the same: one can be certain about events described using vague predicates and about imprecisely specified events, just as one can be uncertain about precisely specified events. Exactly because of this, a question arises about how one ought to assign probabilities to imprecisely specified events in the case when no possible available evidence will eradicate the imprecision (because, say, of the limits of accuracy of a measuring device). Modelling imprecision by rough sets over an approximation (...) space presents an especially tractable case to help get one’s bearings. Two solutions present themselves: the first takes as upper and lower probabilities of the event X the (exact) probabilities assigned X ’s upper and lower rough-set approximations; the second, motivated both by formal considerations and by a simple betting argument, is to treat X ’s rough-set approximation as a conditional event and assign to it a point-valued (conditional) probability. (shrink)

This doctoral dissertation investigates what influence indicative conditionals have on belief updating and how learning from conditionals may be modelled in a probabilistic framework. Because the problem is related to the interpretation of conditionals, we first assess different semantics of indicative conditionals. We propose that conditionals should be taken as primary concepts. This allows us to defend a claim that learning a conditional is equivalent to learning that the relevant conditional probability is 1. This implies that learning (...) a conditional can be modelled as learning a material conditional. We then turn to a few competing approaches to the problem of learning from a conditional -- an approach that models these situations with regards to changes in the explanatory status of the antecedent, as minimising probability distances between the prior and posterior probability distributions, and, particularly, an approach based on epistemic entrenchment. It is argued that although all approaches are important for their pioneering contributions, none of them ultimately provides a fully satisfactory solution. We then formulate a proposal according to which learning a conditional may be modelled as learning a material conditional in conjunction with other contextually inferred or observed information. We show that the view has an intuitive appeal on the grounds of pragmatics and empirical research in psychology of reasoning. We also show that the proposed approach successfully resolves all standard cases from the literature. The proposal is then extended to left-nested conditionals. To do so, we first inspect some preliminary empirical data about actual reasoning with nested conditionals and then show that the proposed approach also successfully models these cases. We then investigate how the approach may be extended to the so-called imperative conditionals. We introduce a tripartite typology of imperative conditionals and argue that although imperatives cannot be probabilistically modelled in any straight-forward way, at least some general insights of the proposed method help us understand the influence that the three different types of imperative conditionals have on listeners’ beliefs. We conclude with a summary and some open questions that should be addressed in future work (e.g. the influence of counterfactuals on belief updating). (shrink)

I describe in this paper an ontological solution to the Sleeping Beauty problem. I begin with describing the Entanglement urn experiment. I restate first the Sleeping Beauty problem from a wider perspective than the usual opposition between halfers and thirders. I also argue that the Sleeping Beauty experiment is best modelled with the Entanglement urn. I draw then the consequences of considering that some balls in the Entanglement urn have ontologically different properties form normal ones. The upshot is that (...) I endorse the halfer conclusion on the probability of Heads once beauty is awaken and the thirder conclusion on conditional probabilities, and that original conclusions ensue on the probability of waking on Monday. (shrink)

In “Process Reliabilism and the Value Problem” I argue that Erik Olsson and Alvin Goldman's conditional probability solution to the value problem in epistemology is unsuccessful and that it makes significant internalist concessions. In “Kinds of Learning and the Likelihood of Future True Beliefs” Olsson and Martin Jönsson try to show that my argument does “not in the end reduce the plausibility” of Olsson and Goldman's account. Here I argue that, while Olsson and Jönsson clarify and amend (...) the conditional probability approach in a number of helpful ways, my case against it remains intact. I conclude with a constructive proposal as to how their account may be steered in a more promising direction. (shrink)

Alvin Goldman and Erik Olsson have recently proposed a novel solution to the value problem in epistemology, i.e., to the question of how to account for the apparent surplus value of knowledge over mere true belief. Their “conditional probability solution” maintains that even simple process reliabilism can account for the added value of knowledge, since forming true beliefs in a reliable way raises the objective probability that the subject will have more true belief of a (...) similar kind in the future. I argue that this proposal confronts significant internal problems and implicitly invokes higher-level epistemic conditions that run against the spirit of externalism. (shrink)

According to the permissibility solution to the lottery paradox, the paradox can be solved if we conceive of epistemic justification as a species of permissibility. Clayton Littlejohn has objected that the permissibility solution draws on a sufficient condition for permissible belief that has implausible consequences and that the solution conflicts with our lack of knowledge that a given lottery ticket will lose. The paper defends the permissibility solution against Littlejohn's objections.

The probability ‘measure’ for measurements at two consecutive mo- ments of time is non-additive. These probabilities, on the other hand, may be determined by the limit of relative frequency of measured events, which are by nature additive. We demonstrate that there are only two ways to resolve this problem. The first solution places emphasis on the precise use of the concept of conditional probability for successive mea- surements. The physically correct conditional probabilities define additive probabilities (...) for two-time measurements. These probabilities depend ex- plicitly on the resolution of the physical device and do not, therefore, correspond to a function of the associated projection operators. It follows that quantum theory distinguishes between physical events and proposi- tions about events, the latter are not represented by projection operators and that the outcomes of two-time experiments cannot be described by quantum logic. The alternative explanation is rather radical: it is conceivable that the relative frequencies for two-time measurements do not converge, unless a particular consistency condition is satisfied. If this is true, a strong revision of the quantum mechanical formalism may prove necessary. We stress that it is possible to perform experiments that will distinguish the two alternatives. (shrink)

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) (...) is high, provided that (ii) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

The well known Monty Hall-problem has a clear solution if one deals with a long enough series of individual games. However, the situation is different if one switches to probabilities in a single case. This paper presents an argument for Monty Hall situations with two players (not just one, as is usual). It leads to a quite general conclusion: One cannot apply probabilistic considerations (for or against any of the strategies) to isolated single cases. If one does that, one (...) cannot but violate a very plausible non-arbitrariness condition and is led into a Moore-paradoxical incoherence. Even though arguments for switching are correct as applied to series of games, they don’t say anything useful about what rationality demands in a single case. (shrink)

We explore ways in which purely qualitative belief change in the AGM tradition throws light on options in the treatment of conditional probability. First, by helping see why it can be useful to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying what is at stake in different ways of doing that. Third, by suggesting novel forms of conditional probability corresponding to familiar variants of qualitative belief change, and conversely. Likewise, (...) we explain how recent work on the qualitative part of probabilistic inference leads to a very broad class of 'proto-probability' functions. (shrink)

Sometimes we receive evidence in a form that standard conditioning (or Jeffrey conditioning) cannot accommodate. The principle of maximum entropy (MAXENT) provides a unique solution for the posterior probability distribution based on the intuition that the information gain consistent with assumptions and evidence should be minimal. Opponents of objective methods to determine these probabilities prominently cite van Fraassen’s Judy Benjamin case to undermine the generality of maxent. This article shows that an intuitive approach to Judy Benjamin’s case supports (...) maxent. This is surprising because based on independence assumptions the anticipated result is that it would support the opponents. It also demonstrates that opponents improperly apply independence assumptions to the problem. (shrink)

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

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)

Philosophers investigating the interpretation and use of conditional sentences have long been intrigued by the intuitive correspondence between the probability of a conditional `if A, then C' and the conditional probability of C, given A. Attempts to account for this intuition within a general probabilistic theory of belief, meaning and use have been plagued by a danger of trivialization, which has proven to be remarkably recalcitrant and absorbed much of the creative effort in the area. (...) But there is a strategy for avoiding triviality that has been known for almost as long as the triviality results themselves. What is lacking is a straightforward integration of this approach in a larger framework of belief representation and dynamics. This paper discusses some of the issues involved and proposes an account of belief update by conditionalization. (shrink)

In this paper I explore a version of standard (expected utility) decision theory in which the probability parameter is interpreted as an objective chance believed by agents to obtain and values of this parameter are fixed by indicative conditionals linking possible actions with possible outcomes. After reviewing some recent developments centering on the common-cause counterexamples to the standard approach, I introduce and briefly discuss the key notions in my own approach. (This approach has essentially the same results as the (...) causal approach in common-cause cases.) I then discuss the Rule of Dominance and find, in the context of the present proposal, that it cannot serve as an independent source of action justification. Turning next to Newcomb''s Problem, I argue that the much discussed issue of back-tracking counterfactuals is something of a red herring for decision theory. Once the twin distractions of back-tracking counterfactuals and Dominance Reasoning are set aside the 1-box solution emerges as a natural consequence of the present proposal. It is of interest that this proposal agrees with the causal approach in the standard common-cause examples and the expected-utility approach in the Newcomb case: one can be smart and rich and keep on smoking. (shrink)

Clark (2018) worries that predictive processing accounts of perception introduce a puzzling disconnect between the content of personal-level perceptual states and their underlying subpersonal representations. According to PP, in perception, the brain encodes information about the environment in conditional probability density distributions over causes of sensory input. But it seems perceptual experience only presents us with one way the world is at a time. If perception is at bottom probabilistic, shouldn’t this aspect of subpersonally represented content show up (...) in consciousness? To address this worry, Clark argues that representations underlying personal-level content are constrained by the need to provide a single action-guiding take on the environment. However, this proposal rests a conception of the nature of agency, famously articulated by Davidson (1980a,b), that is inconsistent with a view of the mind as embodied-extended. Since Clark and other enactivist PP theorists present the extended mind as an important consequence of the predictive framework, the proposal is in tension with his complete view. I claim that this inconsistency could be resolved either by retaining the Davidsonian view of action and abandoning the extended-embodied approach, or by adopting a more processual, world-involving account of agency and perceptual experience than Clark currently endorses. To solve the puzzle he raises, Clark must become a radical enactivist or a consistent internalist. (shrink)

Rational agents seem more confident in any possible event than in an impossible event. But if rational credences are real-valued, then there are some possible events that are assigned 0 credence nonetheless. How do we differentiate these events from impossible events then when we order events? de Finetti (1975), Hájek (2012) and Easwaran (2014) suggest that when ordering events, conditional credences and subset relations are as relevant as unconditional credences. I present a counterexample to all their proposals in this (...) paper. While their proposals order possible and impossible events correctly, they deliver the wrong verdict for disjoint possible events assigned equal positive credence. (shrink)

This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicative conditionals has been subject to severe criticism. The majority view is now apparently in favour of a subjective conditional probability interpretation. (...) A comparative model fitting exercise is presented that shows that a conditional probability model can explain as much of the data on abstract indicative conditional reasoning tasks as psychological theories that supplement material implication with various rationally unjustified processing assumptions. Consequently, when people are asked to solve laboratory reasoning tasks, they can be seen as simply generalising their everyday probabilistic reasoning strategies to this novel context. (shrink)

There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P be the probability of B on the subjunctive supposition that A. Is P equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do (...) not succeed. On the other hand, I argue that even if we can equate P with P, we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped. (shrink)

This paper argues that the technical notion of conditional probability, as given by the ratio analysis, is unsuitable for dealing with our pretheoretical and intuitive understanding of both conditionality and probability. This is an ontological account of conditionals that include an irreducible dispositional connection between the antecedent and consequent conditions and where the conditional has to be treated as an indivisible whole rather than compositional. The relevant type of conditionality is found in some well-defined group of (...) conditional statements. As an alternative, therefore, we briefly offer grounds for what we would call an ontological reading: for both conditionality and conditional probability in general. It is not offered as a fully developed theory of conditionality but can be used, we claim, to explain why calculations according to the RATIO scheme does not coincide with our intuitive notion of conditional probability. What it shows us is that for an understanding of the whole range of conditionals we will need what John Heil (2003), in response to Quine (1953), calls an ontological point of view. (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)

We reply to Christoph Jäger's criticism of the conditional probability solution (CPS) to the value problem for reliabilism due to Goldman and Olsson (2009). We argue that while Jäger raises some legitimate concerns about the compatibility of CPS with externalist epistemology, his objections do not in the end reduce the plausibility of that solution.

This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicative conditionals has been subject to severe criticism. The majority view is now apparently in favour of a subjective conditional probability interpretation. (...) A comparative model fitting exercise is presented that shows that a conditional probability model can explain as much of the data on abstract indicative conditional reasoning tasks as psychological theories that supplement material implication with various rationally unjustified processing assumptions. Consequently, when people are asked to solve laboratory reasoning tasks, they can be seen as simply generalising their everyday probabilistic reasoning strategies to this novel context. (shrink)

The correspondence principle states that the quantum system will approach the classical system in high quantum numbers. Indeed, the average of the quantum probability density distribution reflects a classical-like distribution. However, the probability of finding a particle at the node of the wave function is zero. This condition is recognized as the nodal issue. In this paper, we propose a solution for this issue by means of complex quantum random trajectories, which are obtained by solving the stochastic (...) differential equation derived from the optimal guidance law. It turns out that point set A, which is formed by the intersections of complex random trajectories with the real axis, can represent the quantum mechanical compatible distribution of the quantum harmonic oscillator system. Meanwhile, the projections of complex quantum random trajectories on the real axis form point set B that gives a spatial distribution without the appearance of nodes, and approaches the classical compatible distribution in high quantum numbers. Furthermore, the statistical distribution of point set B is verified by the solution of the Fokker–Planck equation. (shrink)

Motivated by a recent proof of free choices in linking equations to the experiments they describe, I clarify some relations among purely mathematical entities featured in quantum mechanics (probabilities, density operators, partial traces, and operator-valued measures), thereby allowing applications of these entities to the modeling of a wider variety of physical situations. I relate conditional probabilities associated with projection-valued measures to conditional density operators identical, in some cases but not in others, to the usual reduced density operators. While (...) a fatal obstacle precludes associating conditional density operators with general non-projective measures, tensor products of general positive operator-valued measures (POVMs) are associated with conditional density operators. This association together with the free choice of probe particles allows a postulate of state reductions to be replaced by a theorem. An application shows an equivalence between one form of quantum key distribution and another with respect to certain eavesdropping attacks. (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)

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.

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.

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)

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)

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.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Eternal Life and Human Happiness in Heaven: Philosophical Problems, Thomistic Solutions by Christopher M. BrownElizabeth C. Shaw and Staff*BROWN, Christopher M. Eternal Life and Human Happiness in Heaven: Philosophical Problems, Thomistic Solutions. Washington, D.C.: The Catholic University of America Press, 2021. xiii + 487 pp. Cloth, $75.00The contents of the book are straightforwardly announced by the title. Christopher Brown entertains four apparent problems about eternal life in heaven (...) considered by contemporary philosophers and theologians and offers solutions to them from the thought of St. Thomas Aquinas. The book consists of seventeen chapters together with an introduction and conclusion and is divided into four parts. In the first part Brown lays out the problems and the solutions to them proposed by contemporary philosophers and theologians. In the second part he explains St. Thomas's distinction between essential and accidental reward in heaven and goes on to treat of the former—which consists in the beatific vision, that is, our beholding of God—and its proper accidents—which are delight, joy, and charity. In the third part he then deals with the accidental reward, which has to do with the perfected condition of our glorified bodies, society with other persons, and so on. Finally, in the fourth part Brown, drawing on his expositions of St. Thomas's doctrine in the two previous parts, details Thomistic solutions to the four problems discussed in the first part and argues that these solutions are superior to those proposed by the contemporary philosophers and theologians he has engaged.So, what are the four problems that Brown addresses? Let's present them as questions. (1) Is heaven just a private communion between a human person and God, or is it a perfect community centered on God that also involves angels, other human persons, and other creatures? (2) Is heaven just a spiritual affair, or are our resurrected bodies a part of it? (3) Is heaven a static or dynamic reality? (4) Is heaven boring?The answers are as follows. Ad primum: Heaven is a perfect community centered on God that also involves angels, other human persons, and other creatures. Ad secundum: Heaven is a spiritual and physical affair that includes our souls and our resurrected bodies. Ad tertium: Heaven is a dynamic reality. Ad quartum: Heaven is not boring!I cannot cover here all of the specifics of the way Brown shows how St. Thomas arrives at these answers. People who know St. Thomas well, even if they have not reflected much on these problems before, probably will [End Page 135] not have great difficulty thinking through how he might get to the first and second answers, at least in general terms. But the third and fourth, I think, call for further comment. My comments will necessarily be brief.If I understand Brown correctly, he wants to say (and contends that St. Thomas holds) that heaven is dynamic insofar as in heaven we are in act. But we can be in act in two senses: We can be in act in an unchanging way or in a changing way. In heaven my capacities to know God and to love him are both fully perfected, that is, they are fully actualized and undergo no change. But in heaven we will enjoy our resurrected bodies as well, and our bodies will also be active and interacting with other bodies. This activity will entail change. So, either way we will be active in heaven and, therefore, heaven will be something dynamic.What, then, about the problem of boredom? Because we will be perfectly happy in heaven, argues Brown (interpreting St. Thomas), heaven won't be boring. But the matter doesn't end there. Brown points out that Brian Ribeiro, developing an idea of Bernard Williams, objects that a person who is perfectly happy in the way that we are supposed to be in heaven, cannot be personally identical with any person in this life because the conditions of the beatified person and the nonbeatified person are just too radically different for there to be continuity between the two. Brown replies that St. Thomas convincingly shows that God can, by grace, prepare persons in this... (shrink)

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

Conditional probability is often used to represent the probability of the conditional. However, triviality results suggest that the thesis that the probability of the conditional always equals conditional probability leads to untenable conclusions. In this paper, I offer an interpretation of this thesis in a possible worlds framework, arguing that the triviality results make assumptions at odds with the use of conditional probability. I argue that these assumptions come from a (...) theory called the operator theory and that the rival restrictor theory can avoid these problematic assumptions. In doing so, I argue that recent extensions of the triviality arguments to restrictor conditionals fail, making assumptions which are only justified on the operator theory. (shrink)