In a recent pair of publications, Richard Bradley has offered two novel no-go theorems involving the principle of Preservation for conditionals, which guarantees that one’s prior conditional beliefs will exhibit a certain degree of inertia in the face of a change in one’s non-conditional beliefs. We first note that Bradley’s original discussions of these results—in which he finds motivation for rejecting Preservation, first in a principle of Commutativity, then in a doxastic analogue of the rule of modus ponens —are problematic (...) in a significant number of respects. We then turn to a recent U-turn on his part, in which he winds up rescinding his commitment to modus ponens, on the grounds of a tension with the rule of Import-Export for conditionals. Here we offer an important positive contribution to the literature, settling the following crucial question that Bradley leaves unanswered: assuming that one gives up on full-blown modus ponens on the grounds of its incompatibility with Import-Export, what weakened version of the principle should one be settling for instead? Our discussion of the issue turns out to unearth an interesting connection between epistemic undermining and the apparent failures of modus ponens in McGee’s famous counterexamples. (shrink)

In this paper I present a precise version of Stalnaker's thesis and show that it is both consistent and predicts our intuitive judgments about the probabilities of conditionals. The thesis states that someone whose total evidence is E should have the same credence in the proposition expressed by 'if A then B' in a context where E is salient as they have conditional credence in the proposition B expresses given the proposition A expresses in that context. The thesis is formalised (...) rigorously and two models are provided that demonstrate that the new thesis is indeed tenable within a standard possible world semantics based on selection functions. Unlike the Stalnaker-Lewis semantics the selection functions cannot be understood in terms of similarity. A probabilistic account of selection is defended in its place. -/- I end the paper by suggesting that this approach overcomes some of the objections often leveled at accounts of indicatives based on the notion of similarity. (shrink)

What distinguishes indicative conditionals from subjunctive conditionals, according to one popular view, is that the so-called Adams’ thesis holds for the former kind of conditionals but the so-called Skyrms’ thesis for the latter. According to a plausible metaphysical view, both conditionals and chances supervene on non-modal facts. But since chances do not supervene on facts about particular events but facts about event-types, the past as well as the future is chancy. Some philosophers have worried that this metaphysical view is incompatible (...) with the aforementioned view on the probability of conditionals. This paper however shows that there is no need to worry, as these views can all be simultaneously satisfied within the so-called Multidimensional Possible World Semantics for conditionals. (shrink)

This is part A of a paper in which we defend a semantics for counterfactuals which is probabilistic in the sense that the truth condition for counterfactuals refers to a probability measure. Because of its probabilistic nature, it allows a counterfactual ‘ifAthenB’ to be true even in the presence of relevant ‘Aand notB’-worlds, as long such exceptions are not too widely spread. The semantics is made precise and studied in different versions which are related to each other by representation theorems. (...) Despite its probabilistic nature, we show that the semantics and the resulting system of logic may be regarded as a naturalistically vindicated variant of David Lewis’ truth-conditional semantics and logic of counterfactuals. At the same time, the semantics overlaps in various ways with the non-truth-conditional suppositional theory for conditionals that derives from Ernest Adams’ work. We argue that counterfactuals have two kinds of pragmatic meanings and come attached with two types of degrees of acceptability or belief, one being suppositional, the other one being truth based as determined by our probabilistic semantics; these degrees could not always coincide due to a new triviality result for counterfactuals, and they should not be identified in the light of their different interpretation and pragmatic purpose. However, for plain assertability the difference between them does not matter. Hence, if the suppositional theory of counterfactuals is formulated with sufficient care, our truth-conditional theory of counterfactuals is consistent with it. The results of our investigation are used to assess a claim considered by Hawthorne and Hájek, that is, the thesis that most ordinary counterfactuals are false. (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 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)

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)

The connection between the probabilities of conditionals and the corresponding conditional probabilities has long been explored in the philosophical literature, but its implementation faces both technical obstacles and objections on empirical grounds. In this paper I ?rst outline the motivation for the probabilistic turn and Lewis’ triviality results, which stand in the way of what would seem to be its most straightforward implementation. I then focus on Richard Jeffrey’s ’random-variable’ approach, which circumvents these problems by giving up the notion that (...) conditionals denote propositions in the usual sense. Even so, however, the random-variable approach makes counterintuitive predictions in simple cases of embedded conditionals. I propose to address this problem by enriching the model with an explicit representation of causal dependencies. The addition of such causal information not only remedies the shortcomings of Jeffrey’s conditional, but also opens up the possibility of a uni?ed probabilistic account of indicative and counterfactual conditionals. (shrink)

The Newcomb problem is analysed here as a type of common cause problem. In relation to such problems, if you take the dominated option your expected outcome will be good and if you take the dominant option your expected outcome will be not so good. As is explained, however, these arenot conventional conditional expected outcomes but `conditional evidence expected outcomes' and while in the deliberation process, the evidence on which they are based is only hypothetical evidence.Conventional conditional expected outcomes are (...) more sensitive to your current epistemic state in that they are based purely on actual evidence which is available to you during the deliberation process. So although they are conditional on a certain act being performed, they are not based on evidence that you would have only if that act is performed. Moreover, for any given epistemic state during the deliberation process, your conventional conditional expected outcome for the dominant option will be better than that for the dominated option. The principle of dominance is thus in perfect harmony with the conventional conditional expected outcomes. In relation to the Newcomb problem then, the evidence unequivocally supports two-boxing as the rational option. Yet what is advanced here is not simply a two-boxing strategy. To see why, two stages to the problem need to be recognised. The first stage is that which occurs before the information used by the predictor in making his predictions has been gained. The second stage is after this point. Provided that you are still in the first stage, you have an opportunity to influence whether or not the predictor places the $1m in the opaque box. To maximise the probability that it is, you need to commit yourself to one-boxing. (shrink)

It is often claimed that credences are not reducible to ordinary beliefs about probabilities. Such a reduction appears to be decisively ruled out by certain sorts of triviality results–analogous to those often discussed in the literature on conditionals. I show why these results do not, in fact, rule out the view. They merely give us a constraint on what such a reduction could look like. In particular they show that there is no single proposition belief in which suffices for having (...) a particular credence, regardless of one’s evidence. But if we allow such propositions to vary with evidence–as we should–then the results do not rule out a reduction. So, at least on this count, credences might very well just be beliefs about probabilities. (shrink)

Conditionals with conditional constituents pose challenges for _the Thesis_, the idea that the probability of a conditional is the corresponding conditional probability. This note is concerned with two proposals for overcoming those challenges, both inspired by early work of van Fraassen: the _Bernoulli Semantics_ associated with Stalnaker and Jeffrey, and augmented with a mechanism for obtaining “local probabilities” by Kaufmann; and a proposal by Bacon which I dub _Ordinal Semantics_. Despite differences in mathematical details and emphasis of presentation, both proposals (...) lend themselves for use as a basis for a modal-theoretic interpretation of embedded conditionals. The goal of this note is to compare the two frameworks by implementing a model for the interpretation of conditionals in each, based on the same underlying probability model for non-conditional sentences. I show that in the Ordinal model, certain sentences are assigned probabilities that do not accord with intuitions. This problem is familiar from the literature on Bernoulli models and can be addressed by introducing Kaufmann-style local probabilities into Ordinal models. I then show that Bernoulli Semantics has other limitations, in that it assigns probabilities in violation of the Thesis to certain very complex formulas. The upshot is that a fusion of the theories may be our best shot at getting the predictions right. (shrink)

The chapter is devoted to the probability and acceptability of indicative conditionals. Focusing on three influential theses, the Equation, Adams’ thesis, and the qualitative version of Adams’ thesis, Sikorski argues that none of them is well supported by the available empirical evidence. In the most controversial case of the Equation, the results of many studies which support it are, at least to some degree, undermined by some recent experimental findings. Sikorski discusses the Ramsey Test, and Lewis’s triviality proof, with special (...) attention dedicated to the popular ways of blocking it. Sikorski concludes that the role of the three theses in future studies of conditionals should be re-thought, and he presents alternative proposals. (shrink)

This paper calls for a re-appraisal of McGee's analysis of the semantics, logic and probabilities of indicative conditionals presented in his 1989 paper Conditional probabilities and compounds of conditionals. The probabilistic measures introduced by McGee are given a new axiomatisation built on the principle that the antecedent of a conditional is probabilistically independent of the conditional and a more transparent method of constructing such measures is provided. McGee's Dutch book argument is restructured to more clearly reveal that it introduces a (...) novel contribution to the epistemology of semantic indeterminacy, and shows that its more controversial implications are unavoidable if we want to maintain the Ramsey Test along with the standard laws of probability. Importantly, it is shown that the counterexamples that have been levelled at McGee's analysis|generating a rather wide consensus that it yields `unintuitive' or `wrong' probabilities for compounds |fail to strike at their intended target; for to honour the intuitions of the counterexamples one must either give up the Ramsey Test or the standard laws of probability. It will be argued that we need to give up neither if we take the counterexamples as further evidence that the indicative conditional sometimes allows for a non-epistemic `causal' interpretation alongside its usual epistemic interpretation. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

On an influential line of thinking tracing back to Ramsey, conditionals are closely linked to the attitude of supposition. When applied to counterfactuals, this view suggests a subjunctive version of the so-called Ramsey test: the probability of a counterfactual If A, would B ought to be equivalent to the probability of B, under the subjunctive supposition that A. I present a collapse result for any view that endorses the subjunctive version of the Ramsey test. Starting from plausible assumptions, the result (...) shows that one’s rational credence in a would-counterfactual and in the corresponding might-counterfactual have to be identical. (shrink)

‘If I were to toss a coin 1000 times, then it would land heads exactly n times’. Is there a specific value of n that renders this counterfactual true? According to an increasingly influential view, there is. A precursor of the view goes back to the Molinists; more recently it has been inspired by Stalnaker, and versions of it have been advocated by Hawthorne, Bradley, Moss, Schulz, and Stefánsson. More generally, I attribute to these authors what I call Counterfactual Plenitude:For (...) any antecedent A, there is a world wisuch that A ☐→ wiis true.Moreover, some of these authors are also committed to Primitive Counterfacts Realism:There exist primitive modal facts that serve as truth-makers for counterfactual claims.Call the conjunction of these italicized theses counterfactism. I clarify it and suggest some of its virtues, while ultimately rejecting it.Stefánsson’s counterfactism is motivated by and targeted at my “counterfactual skepticism”—I argue that most counterfactuals are false—and counterfactism has various other sources of support. I briefly defend that skepticism, and I seek to undercut those sources of support. I then argue more directly against counterfactism, especially on grounds of its ontological profligacy, and its leading to another kind of skepticism about counterfactuals that I believe is more problematic than my kind. In the process, I discuss how Bradley’s multidimensional semantics bears on counterfactism; I offer some new considerations against some central theses regarding conditionals ; and I reflect more generally on the epistemology of modality and the choice of primitives in our theorizing. (shrink)

This paper argues that two widely accepted principles about the indicative conditional jointly presuppose the falsity of one of the most prominent arguments against epistemological iteration principles. The first principle about the indicative conditional, which has close ties both to the Ramsey test and the “or‐to‐if” inference, says that knowing a material conditional suffices for knowing the corresponding indicative. The second principle says that conditional contradictions cannot be true when their antecedents are epistemically possible. Taken together, these principles entail that (...) it is impossible to be in a certain kind of epistemic state: namely, a state of ignorance about which of two partially overlapping bodies of knowledge corresponds to one's actual one. However, some of the more popular “margin for error” style arguments against epistemological iteration principles suggest that such states are not only possible, but commonplace. I argue that the tension between these views runs deep, arising just as much for non‐factive attitudes like belief, presupposition, and certainty. I also argue that this is worse news for those who accept the principles about the indicative conditional than it is for those who reject epistemological iteration principles. (shrink)

There is strong disagreement about whether indicative conditionals have truth values. In this paper, I present a new argument for the conclusion that indicative conditionals have truth values based on the claim that some true statements entail indicative conditionals. I then address four arguments that conclude that indicative conditionals lack truth values, showing them to be inadequate. Finally, I present further benefits to having a worldly view of conditionals, which supports the assignment of truth values to indicative conditionals. I conclude (...) that certain types of account of indicative conditionals, which have been ignored in the literature partly on the basis of assigning truth values to indicative conditionals, deserve consideration. (shrink)

Many recent theories of epistemic discourse exploit an informational notion of consequence, i.e. a notion that defines entailment as preservation of support by an information state. This paper investigates how informational consequence fits with probabilistic reasoning. I raise two problems. First, all informational inferences that are not also classical inferences are, intuitively, probabilistically invalid. Second, all these inferences can be exploited, in a systematic way, to generate triviality results. The informational theorist is left with two options, both of them radical: (...) they can either deny that epistemic modal claims have probability at all, or they can move to a nonstandard probability theory. (shrink)

This Open Access book addresses the age-old problem of infinite regresses in epistemology. How can we ever come to know something if knowing requires having good reasons, and reasons can only be good if they are backed by good reasons in turn? The problem has puzzled philosophers ever since antiquity, giving rise to what is often called Agrippa's Trilemma. The current volume approaches the old problem in a provocative and thoroughly contemporary way. Taking seriously the idea that good reasons are (...) typically probabilistic in character, it develops and defends a new solution that challenges venerable philosophical intuitions and explains why they were mistakenly held. Key to the new solution is the phenomenon of fading foundations, according to which distant reasons are less important than those that are nearby. The phenomenon takes the sting out of Agrippa's Trilemma; moreover, since the theory that describes it is general and abstract, it is readily applicable outside epistemology, notably to debates on infinite regresses in metaphysics. (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)

The Ramseyan thesis that the probability of an indicative conditional is equal to the corresponding conditional probability of its consequent given its antecedent is both widely confirmed and subject to attested counterexamples (e.g., McGee 2000, Kaufmann 2004). This raises several puzzling questions. For instance, why are there interpretations of conditionals that violate this Ramseyan thesis in certain contexts, and why are they otherwise very rare? In this paper, I raise some challenges to Stefan Kaufmann's account of why the Ramseyan thesis (...) sometimes fails, and motivate my own theory. On my theory, the proposition expressed by an indicative conditional is partially determined by a background partition, and hence its probability depends on the choice of such a partition. I hold that this background partition is contextually determined, and in certain conditions is set by a salient question under discussion in the context. I show how the resulting theory offers compelling answers to the puzzling questions raised by failures of the Ramseyan thesis. (shrink)

Adams’ thesis is generally agreed to be linguistically compelling for simple conditionals with factual antecedent and consequent. We propose a derivation of Adams’ thesis from the Lewis- Kratzer analysis of if-clauses as domain restrictors, applied to probability operators. We argue that Lewis’s triviality result may be seen as a result of inexpressibility of the kind familiar in generalized quantifier theory. Some implications of the Lewis- Kratzer analysis are presented concerning the assignment of probabilities to compounds of conditionals.

This paper argues that just as full beliefs can constitute knowledge, so can properties of your credence distribution. The resulting notion of probabilistic knowledge helps us give a natural account of knowledge ascriptions embedding language of subjective uncertainty, and a simple diagnosis of probabilistic analogs of Gettier cases. Just like propositional knowledge, probabilistic knowledge is factive, safe, and sensitive. And it helps us build knowledge-based norms of action without accepting implausible semantic assumptions or endorsing the claim that knowledge is interest-relative.

This note relates the Lewis/Kratzer view of conditionals as restrictors to the philosophical debate over the meaning of conditionals.

One very popular kind of semantics for subjunctive conditionals is aclosest-worlds account along the lines of theories given by David Lewisand Robert Stalnaker. If we could give the same sort of semantics forindicative conditionals, we would have a more unified account of themeaning of ``if ... then ...'' statements, one with manyadvantages for explaining the behaviour of conditional sentences. Such atreatment of indicative conditionals, however, has faced a battery ofobjections. This paper outlines a closest-worlds account of indicativeconditionals that does better (...) than some of its cousins in explaining thebehaviour of such conditionals. The paper then discusses objectionsoffered by Dorothy Edgington and Frank Jackson to closest-worldsaccounts of indicative conditionals, and shows that these objections canbe met by the account outlined. (shrink)

Less‐than‐certain conditional judgments pose notorious problems for strict analyses of conditionals: across their various incarnations, these analyses have trouble making sense of how conditionals could have non‐trivial probabilities in the first place; minimal constraints on how such probabilities are to be assigned, moreover, lead to results that seem at odds with a strict outlook on the semantics of conditionals, most notably the validity of Conditional Excluded Middle. I demonstrate that a strict analysis can overcome the trouble if couched in a (...) bilateral dynamic setting that properly extends the familiar Ramsey test for accepting conditionals to other iffy attitudes, most importantly the one of rejecting a conditional. The resulting framework accommodates the appeal of Stalnaker's thesis as well as of Conditional Excluded Middle in a strict setting. A discussion of how to handle the probability of epistemically modalized conditionals and of compounds of conditionals is provided. (shrink)

In this paper it is shown that Lewis' MWD (might/would duality) and imaging principles lead to wildly implausible probability assignments for would counterfactuals.

The Desire-as-Belief thesis (DAB) states that any rational person desires a proposition exactly to the degree that she believes or expects the proposition to be good. Many people take David Lewis to have shown the thesis to be inconsistent with Bayesian decision theory. However, as we show, Lewis's argument was based on an Invariance condition that itself is inconsistent with the (standard formulation of the) version of Bayesian decision theory that he assumed in his arguments against DAB. The aim of (...) this paper is to explore what impact the rejection of Invariance has on the DAB thesis. Without assuming Invariance, we first refute all versions of DAB that entail that there are only two levels of goodness. We next consider two theses according to which rational desires are intimately connected to expectations of (multi-levelled) goodness, and show that these are consistent with Bayesian decision theory as long as we assume that the contents of 'value propositions' are not fixed. We explain why this conclusion is independently plausible, and show how to construct such propositions. (shrink)

The literature on conditionals is rife with alternate formulations of the abstract semantics of conditional logic. Each formulation has its own advantages in terms of applications and generalizations; nevertheless, they are for the most part equivalent, in the sense that they underwrite the same range of logical systems. The purpose of the present note is to bring under this umbrella the partition semantics introduced by Brian Skyrms in (Skyrms, 1984).

This paper investigates the link between informational consequence and credence. I first suggest a natural constraint, namely that informational consequence should preserve certainty: on any rational credence distribution, when the premises of an informational inferences have credence 1, the conclusion also has credence 1. Then I show that the certainty‐preserving constraint leads to triviality. In particular, the following three claims are incompatible: (i) informational consequence is extensionally distinct from classical consequence; (ii) informational inferences preserve certainty; (iii) credences obey (a subset (...) of) classical Bayesian constraints. The proof is straightforward, but the theoretical implications are substantial. Informational theorists need to either give up the idea that credence applies to epistemic discourse, or develop a nonclassical theory of credence and credal update. Moreover, the result also suggests that there is a connection between informational consequence and triviality results, including classical triviality results like Lewis's. (shrink)

According to Stalnaker’s Thesis, the probability of a conditional is the conditional probability. Under some mild conditions, the thesis trivialises probabilities and conditionals, as initially shown by David Lewis. This article asks the following question: does still lead to triviality, if the probability function in is replaced by a probability-like function? The article considers plausibility functions, in the sense of Friedman and Halpern, which additionally mimic probabilistic additivity and conditionalisation. These quasi probabilities comprise Friedman–Halpern’s conditional plausibility spaces, as well as (...) other known representations of conditional doxastic states. The paper proves Lewis’ triviality for quasi probabilities and discusses how this has implications for three other prominent strategies to avoid Lewis’ triviality: Adams’ thesis, where the probability function on the left in is replaced by a probability-like function, abandoning conditionalisation, where probability conditionalisation on the right in is replaced by another propositional update procedure and the approximation thesis, where equality in is replaced by approximation. The paper also shows that Lewis’ triviality result is really about ‘additiveness’ and ‘conditionality’. (shrink)

There is a renewed debate about modus ponens. Strikingly, the recent counterexamples in Cantwell, Dreier and MacFarlane and Kolodny are generated by restricted readings of the ‘if’-clause. Moreover, it can be argued on general grounds that the restrictor view of conditionals developed in Kratzer and Lewis leads to counterexamples to modus ponens. This paper provides a careful analysis of modus ponens within the framework of the restrictor view. Despite appearances to the contrary, there is a robust sense in which modus (...) ponens is valid, owing to the fact that conditionals do not only allow for restricted readings but have bare interpretations, too. (shrink)

This paper argues that several leading theories of subjunctive conditionals are incompatible with ordinary intuitions about what credences we ought to have in subjunctive conditionals. In short, our theory of subjunctives should intuitively display semantic humility, i.e. our semantic theory should deliver the truth conditions of sentences without pronouncing on whether those conditions actually obtain. In addition to describing intuitions about subjunctive conditionals, I argue that we can derive these ordinary intuitions from justified premises, and I answer a possible worry (...) for my derivation by refuting a subjunctive triviality result modeled on (Lewis 1976). I conclude that the debate over the correct theory of subjunctive conditionals requires settling meta-philosophical questions about the relative value of various virtues of first-order theories of subjunctive conditionals. (shrink)

Various semantics for studying the square of opposition have been proposed recently. So far, only [14] studied a probabilistic version of the square where the sentences were interpreted by (negated) defaults. We extend this work by interpreting sentences by imprecise (set-valued) probability assessments on a sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square in terms of acceptability and show how to construct probabilistic versions of the square (...) of opposition by forming suitable tripartitions. Finally, as an application, we present a new square involving generalized quantifiers. (shrink)

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)

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

Suppose you’re certain that a claim—say, ‘Frida is tall’—does not have a determinate truth value. What attitude should you take towards it? This is the question of the cognitive role of indeterminacy. This paper presents a puzzle for theories of cognitive role. Many of these theories vindicate a seemingly plausible principle: if you are fully certain that A, you are rationally required to be fully certain that A is determinate. Call this principle ‘Certainty’. We show that Certainty, in combination with (...) some minimal side premises, entails a very implausible claim: whenever you’re certain that it’s indeterminate whether A, it is rationally required that you reject A. This is a surprising result, which requires abandoning at least some intuitive views about indeterminacy and cognitive role. (shrink)

The strongest possible Lewisian triviality result for the indicative conditional is proven.

This work explores the hypothesis that natural language is a tool for changing a language user's state of mind and, more specifically, the hypothesis that a sentence's meaning is constituted by its characteristic role in fulfilling this purpose. This view contrasts with the dominant approach to semantics due to Frege, Tarski and others' work on artificial languages: language is first and foremost a tool for representing the world. Adapted to natural language by Davidson, Lewis, Montague, et. al. this dominant approach (...) has crystalized as truth-conditional semantics: to know the meaning of a sentence is to know the conditions under which that sentence is true. Chapter 1 details the animating ideas of my alternative approach and shows that the representational function of language can be understood in terms of the more general function of changing representational mental states. Chapters 2-4 argue that the additional resources of this more general conception of meaning allow us to explain certain phenomena involving conditionals and grammatical mood that truth-conditional semantics does not. In the analysis of these specific phenomena and the articulation of the general approach on offer, it emerges that this approach combines insights and benefits from both use-theoretic and truth-theoretic work on meaning. (shrink)

The thesis that probabilities of conditionals are conditional probabilities has putatively been refuted many times by so-called ‘triviality results’, although it has also enjoyed a number of resurrections. In this paper I assault it yet again with a new such result. I begin by motivating the thesis and discussing some of the philosophical ramifications of its fluctuating fortunes. I will canvas various reasons, old and new, why the thesis seems plausible, and why we should care about its fate. I will (...) look at some objections to Lewis’s famous triviality results, and thus some reasons for the pursuit of further triviality results. I will generalize Lewis’s results in ways that meet the objections. I will conclude with some reflections on the demise of the thesis—or otherwise. (shrink)

We show how to extend any finite probability space into another finite one which satisfies the conditional construal of conditional probability for the original propositions, given some maximal allowed degree of nesting of the conditional. This mitigates the force of the well-known triviality results.

We explore which types of probabilistic updating commute with convex IP pooling. Positive results are stated for Bayesian conditionalization, imaging, and a certain parameterization of Jeffrey conditioning. This last observation is obtained with the help of a slight generalization of a characterization of externally Bayesian pooling operators due to Wagner :336–345, 2009). These results strengthen the case that pooling should go by imprecise probabilities since no precise pooling method is as versatile.

According to the PCCP thesis, the probability of a conditional A → C is the conditional probability P. This claim is undermined by Lewis’ triviality results, which purport to show that apart from trivial cases, PCCP is not true. In the present article we show that the only rational, “Dutch Book-resistant” extension of the agent’s beliefs concerning non-conditional sentences A and C to the conditional A → C is by assuming that P = P. In other cases a diachronic Dutch (...) Book against the agent can be constructed. There is a tension between our findings and Lewis’ results, which needs to be explained. Therefore, we present a probability space which corresponds in a natural way to the diachronic Dutch Book—and which allows the conditional A → C to be interpreted as an event in a mathematically sound way. It also allows to formalize the notion of conditionalizing A → C on ¬C which plays a crucial role in Lewis’ proof. Our conclusion is that Lewis’ proof is circular, so it cannot be considered to be a sound argument against PCCP. (shrink)