This paper develops a trivalent semantics for the truth conditions and the probability of the natural language indicative conditional. Our framework rests on trivalent truth conditions first proposed by Cooper (1968) and Belnap (1973) and it yields two logics of conditional reasoning: (i) a logic C of certainty-preserving inference; and (ii) a logic U for uncertain reasoning that preserves the probability of the premises. We show systematic correspondences between trivalent and probabilistic representations of inferences in either framework, and we use (...) the distinction between the two systems to cast light on the validity of inferences such as Modus Ponens, Or-To-If, and Conditional Excluded Middle. Specifically, the conditional behaves monotonically in C, but non-monotonically in U; Modus Ponens is valid in C, but valid in U only for non-nested conditionals. The result is a unified account of the semantics and epistemology of indicative conditionals that can be fruitfully applied to analyzing the validity of conditional inferences. (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)

The classic formulation of causal decision theory (CDT) appeals to counterfactuals. It says that you should aim to choose an option that would have a good outcome, were you to choose it. However, this version of CDT faces trouble if the laws of nature are deterministic. After all, the standard theory of counterfactuals says that, if the laws are deterministic, then if anything—including the choice you make—were different in the present, either the laws would be violated or the distant past (...) would be changed. And as several authors have shown, it's easy to transform this upshot of the standard theory of counterfactuals into full-blown counterexamples to CDT. In response to these counterexamples, I argue here that the problem lies, not so much with CDT's guiding idea—that it's the expected causal consequences of your actions that matter for rational decision-making—but with the fact that the classic formulation of CDT doesn't pay sufficient attention to the context-sensitivity of counterfactuals. I develop a contextualist version of CDT which better accounts for this context-sensitivity. And I show that my theory avoids the problems faced by the classic formulation of CDT in determinstic worlds. (shrink)

This paper presents a unified theory of the truth conditions and probability of indicative conditionals and their compounds in a trivalent framework. The semantics validates a Reduction Theorem: any compound of conditionals is semantically equivalent to a simple conditional. This allows us to validate Stalnaker's Thesis in full generality and to use Adams's notion of $p$-validity as a criterion for valid inference. Finally, this gives us an elegant account of Bayesian update with indicative conditionals, establishing that despite differences in meaning, (...) it is tantamount to learning a material conditional. (shrink)

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)

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)

This paper extends a trivalent semantics for indicative conditionals to a language including the modal operators "might" and "must". Specifically, we combine Cooper's (1968) truth-functional, trivalent analysis of the conditional connective with Kratzer's (1986, 2012) idea that if-clauses restrict modal operators. By hard-wiring both trivalence and the restriction operation into the truth conditions of conditional-modal expressions, we obtain an attractive theory that yields plausible predictions for the interaction of conditionals and modals, explains the intuitive appeal of the Restrictor View and (...) avoids the typical problems of Kratzer-style accounts, especially regarding the probability of conditional-modal expressions. (shrink)

The material conditional and the suppositional analysis of the indicative conditional are based on different philosophical foundations and they leave important successes of their competitor unexplained. This paper unifies both accounts within a truth-functional, trivalent model of the suppositional analysis. In this model, we observe that the material and the suppositional conditional exhibit the same logical behavior while they have different truth conditions and different probabilities. The result is a unified semantic analysis that closes an important gap in the suppositional (...) story and explains the persistent appeal of the material conditional analysis for philosophers and psychologists of reasoning. (shrink)

Counterfactual skepticism says that most ordinary counterfactuals are false. While few endorse counterfactual skepticism, the precise costs of the view are disputed and not generally well-understood. I have two aims in this paper. My first and primary aim is to establish, on grounds acceptable to all parties, that counterfactual skepticism is not benign. I argue it leads to significant skepticism about the future: if counterfactual skepticism is true, then we can have only very limited knowledge about the future. I give (...) three arguments to this conclusion, taking as premises principles connecting knowledge of indicative conditionals to knowledge of disjunctions; and principles linking knowledge of indicative conditionals to knowledge of counterfactual conditionals.My second aim is to examine the consequences for non-skeptical theories. My arguments indicate that not only are many ordinary counterfactuals true, but that we also know them. How can this be reconciled with the very real tension between counterfactuals and chance? I consider two solutions, one purely epistemic and one contextualist. Together they suggest something like contextualism about knowledge is required to fully resolve the skeptical threat. (shrink)

We introduce an expected-value theory of linguistic modality that makes reference to expected utility and a likelihood-based confirmation measure for deontics and epistemics, respectively. The account is a probabilistic semantics for deontics and epistemics, yet it proposes that deontics and epistemics share a common core modal semantics, as in traditional possible-worlds analysis of modality. We argue that this account is not only theoretically advantageous, but also has far-reaching empirical consequences. In particular, we predict modal versions of reasoning fallacies from the (...) heuristics and biases literature. Additionally, we derive the modal semantics in an entirely transparent manner, as it is based on the compositional semantics of Korean modal expressions that are morphosyntactically decomposed into a conditional and an evaluative predicate. (shrink)

// tl;dr A Proposition is a Way of Thinking // -/- This chapter is about type-theoretic approaches to propositional content. Type-theoretic approaches to propositional content originate with Hintikka, Stalnaker, and Lewis, and involve treating attitude environments (e.g. "Nate thinks") as universal quantifiers over domains of "doxastic possibilities" -- ways things could be, given what the subject thinks. -/- This chapter introduces and motivates a line of a type-theoretic theorizing about content that is an outgrowth of the recent literature on epistemic (...) modality, according to which contentful thought is broadly "informational" in its nature and import. The general idea here is that an object of thought is not a way *the world* could be, but rather a way *one's perspective* could be (with respect to a relevant representational question). I will spend the middle part of this chapter motivating and developing a version of this strategy that is, I’ll argue, well-suited to explaining clear phenomena concerning the attribution of perspectival attitudes -- in particular, attitudes towards loosely information-sensitive propositions -- with which extant approaches struggle. My overarching goal here will be to motivate a distinctive version of the "informational" approach -- the "Flexible Types" approach, which is based on the theory proposed in Charlow (2020). According to the Flexible Types approach, propositional attitude verbs are quantifiers over sets of possibilities, but a possibility is a type-flexible notion -- sometimes a possible world, sometimes a perspective, sometimes a set of possible worlds, sometimes a set of perspectives. -/- After introducing the Flexible Types approach, this chapter circles back to more traditional concerns for the analysis of propositions as types of possibilities -- Frege's Puzzle and the problem of Logical Omniscience. Here too the Flexible Types approach bears fruit. Although there are certainly significant differences -- I note some in the concluding section -- the gist of this theory is Hinitkkan or Lewisian in spirit (if not quite in letter). We can make progress on addressing the challenges for the analysis of propositional content in terms of types of possibilities, through empirically driven refinement of our notion of what kind of thing a "doxastic possibility" is. (shrink)

This paper concerns a controversy between two compelling and popular claims in the theory of ability. One is the claim that ability requires control. The other is the claim that success entails ability, that is, that φ-ing entails that you are able to φ. Since actually φ-ing obviously does not entail that φ is in your control, these two claims cannot both be true. I introduce a new form of evidence to help adjudicate this controversy: judgments about the possibility and (...) probability of ability ascriptions. I argue that these judgments provide evidence in favor of the thesis that success entails ability, and against the thesis that ability requires control. Moreover, I argue that these judgments support an analysis of ability in terms of conditionals. (shrink)

In ‘The logic of partial supposition’ (Analysis vol. 81), Benjamin Eva and Stephan Hartmann investigate partial imaging, a credence-revision method which combines the partiality familiar from Jeffrey Conditioning(The Logic of Decision, 1983) with the formal notion of imaging familiar from Lewis’s ‘Causal decision theory’ (1981). They argue that because partial imaging is non-monotonic, it ‘fail[s] to provide a plausible account of the norms of partial subjunctive suppositions’.In this note, I present a notion of partial imaging that does satisfy monotonicity, and (...) discuss some of the applications and ramifications. The account frames conditioning as a form of imaging, and rejects Gärdenfors’s principle of linearity in ‘Imaging and conditionalization’ (1982). (shrink)

Recent literature on Stalnaker's Thesis, which seeks to vindicate it from Lewis (1976)'s triviality results, has featured linguistic data that is prima facie incompatible with Conditionalization in iterated cases (McGee 1989, 2000; Kaufmann 2015; Khoo & Santorio, 2018). In a recent paper (2021), Goldstein & Santorio make a bold claim: they hold that these departures light the way to a new, non‐conditionalizing theory of rational update.Here, I consider whether this new form of update is subject to a Dutch book. On (...) the official, invariantist version of the theory, I show that the answer is “yes”. On a competing, contextualist theory of indicative conditionals (Bacon, 2015), the answer is “no”, for reasons that have familiar connections to the limits of textbook Bayesianism. After presenting a concrete case, I explore the dialectical ramifications. The upshot is some hard choices for theories that seek to save the linguistic phenomena. (shrink)

I give a semantics for a logic with two pairs of doxastic modals and an indicative conditional connective that all nest without restriction. Sentences are evaluated as accepted, rejected, or neither. Certainty is the necessity-like modality of acceptance. Inferences may proceed from premises that are certain, or merely accepted, or a mix of both. This semantic setup yields some striking results. Notably, the existence of inferences that preserve certainty but not acceptance very directly implies both failure of modus ponens for (...) the indicative conditional in the logic of acceptance and failure of the deduction theorem for the material conditional in the logic of certainty. The latter failure dissolves, in the logic of certainty, the much - discussed tension between modus ponens and the law of import-export. (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)

Indicative and subjunctive conditionals are in non-complimentary distribution: there are conversational contexts at which both are licensed (Stalnaker 1975; Karttunen and Peters 1979; von Fintel 1998). This means we can ask an important, but under-explored, question: in contexts which license both, what relations hold between the two? In this paper, I’ll argue for an initially surprising conclusion: when attention is restricted to the relevant contexts, indicatives and subjunctives are co-entailing. §1 introduces the indicative/subjunctive distinction, along with a discussion of the (...) relevant notion of entailment; §2 presents the main argument of the paper, and §3 considers some of the philosophical implications of the argument in §2. Finally, §4 argues that we can reconcile the equivalence of indicatives and subjunctives with apparently conflicting judgements. (shrink)

This paper clarifies the relationship between the Triviality Results for the conditional and the Restrictor Theory of the conditional. On the understanding of Triviality proposed here, it is implausible—pace many proponents of the Restrictor Theory—that Triviality rests on a syntactic error. As argued here, Triviality arises from simply mistaking the feature a claim has when that claim is logically unacceptable for the feature a claim has when that claim is unsatisfiable. Triviality rests on a semantic confusion—one which some semantic theories, (...) but not others, are prone to making. On the interpretation proposed here, Triviality Results thus play a theoretically constructive role in the project of natural language semantics. (shrink)

Recent work in epistemology defends the unorthodox theses that belief is (1) an evidentially weak, and (2) question-sensitive attitude, and (3) that forming beliefs is sometimes a matter of guessing. What motivates these theses are examples of rationally permissible belief-ascriptions that exhibit these traits. The main aim of this paper is to outline a semantic account of categorical and conditional belief-ascriptions that captures the motivating data. We then survey some consequences of the proposed semantics, particularly with respect to the question (...) of whether closure under rules of inference is rationally required for weak, question-sensitive belief. (shrink)