The standard use of the propositional calculus ('P.C.?) in analyzing the validity of inferences involving conditionals leads to fallacies, and the problem is to determine where P.C. may be ?safely? used. An alternative analysis of criteria of reasonableness of inferences in terms of conditions of justification rather than truth of statements is proposed. It is argued, under certain restrictions, that P. C. may be safely used, except in inferences whose conclusions are conditionals whose antecedents are incompatible with the premises in (...) the sense that if the antecedent became known, some of the previously asserted premises would have to be withdrawn. (shrink)

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

It is now well known that, on pain of triviality, the probability of a conditional cannot be identified with the corresponding conditional probability [27]. This surprising impossibility result has a qualitative counterpart. In fact, Peter Gardenfors showed in [13] that believing 'If A then B' cannot be equated with the act of believing B on the supposition that A.

Recent work has shown that in spite of these negative results, the question 'how to accept a conditional?' has a clear answer. Even if conditionals are not truth-carriers, they do have precise acceptability conditions. Nevertheless most epistemic models of conditionals do not provide acceptance conditions for iterated conditionals. One of the main goals of this essay is to provide a comprehensive account of the notion of epistemic conditionality covering all forms of iteration.

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The notion of probability occupies a central role in contemporary epistemology and cognitive science. Nevertheless, the classical notion of probability is hard to reconcile with the central notions postulated by the epistemological tradition.

It is now well known that, on pain of triviality, the probability of a conditional cannot be identified with the corresponding conditional probability [25]. This surprising impossibility result has a qualitative counterpart. In fact, Peter Gärdenfors showed in [13] that believing ‘If A then B’ cannot be equated with the act of believing B on the supposition that A — as long as supposing obeys minimal Bayesian constraints.Recent work has shown that in spite of these negative results, the question ‘how (...) to accept a conditional?’ has a clear answer. Even if conditionals are not truth-carriers, they do have precise acceptability conditions. Nevertheless most epistemic models of conditionals do not provide acceptance conditions for iterated conditionals. One of the main goals of this essay is to provide a comprehensive account of the notion of epistemic conditionality covering all forms of iteration.First we propose an account of the basic idea of epistemic conditionality, by studying the conditionals validated by epistemic models where iteration is permitted but not constrained by special axioms. Our modeling does not presuppose that epistemic states should be represented by belief sets (we only assume that to each epistemic state corresponds an associated belief state). A full encoding of the basic epistemic conditionals (encompassing all forms of iteration) is presented and a representation result is proved.In the second part of the essay we argue that the notion of change involved in the evaluation of conditionals is suppositional, and that such notion should be distinguished from the notion of updating (modelled by AGM and other methods). We conclude by considering how some of the recent modellings of iterated change fare as methods for iterated supposing. (shrink)

Horacio Arlo-Costa. Belief Revision Conditionals: Models of Suppositional Reasoning.

Consider a rational agent X at certain point of time t. X's epistemic state can be represented in different ways.

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

We present a semantic analysis of the Ramsey test, pointing out its deep underlying flaw: the tension between the “static” nature of AGM revision (which was originally tailored for revision of only purely ontic beliefs, and can be applied to higher-order beliefs only if given a “backwards-looking” interpretation) and the fact that, semantically speaking, any Ramsey conditional must be a modal operator (more precisely, a dynamic-epistemic one). Thus, a belief about a Ramsey conditional is in fact a higher-order belief, hence (...) the AGM revision postulates are not applicable to it, except in their “backwards-looking” interpretation. But that interpretation is consistent only with a restricted (weak) version of Ramsey’s test (in-applicable to already revised theories). The solution out of the conundrum is twofold: either accept only the weak Ramsey test; or replace the AGM revision operator ∗ by a truly “dynamic” revision operator ⊗, which will not satisfy the AGM axioms, but will do something better: it will “keep up with reality”, correctly describing revision with higher-order beliefs. (shrink)

A conditional takes the form ‘If A, then C’. On the truth-conditional view of conditionals, conditional statements state things with truth-conditions. On the suppositional view, conditional statements involve the expression of a supposition. I develop and defend a view on which conditional statements both state things with truth-conditions and express suppositions. On this view, something is fundamentally right about standard truth-conditional and standard suppositional views. Considerations in favor of conditional contents lead us to attribute truth-conditional contents to conditional statements; considerations (...) in favor of the suppositional view then lead us to an unexpected account of these contents. The resulting view has a number of benefits, including a unified treatment of conditional speech acts, a plausible account of our practice of ascribing truth-values to conditional statements, a simple explanation of the apparent equivalence between probabilities of conditionals and conditional probabilities, an intuitive treatment of ‘Gibbardian stand-offs’, a plausible logic of conditionals, and an explanation of why theorizing about conditionals has proved so difficult. (shrink)

According to the Ramsey Test hypothesis the conditional claim that if A then B is credible just in case it is credible that B, on the supposition that A. If true the hypothesis helps explain the way in which we evaluate and use ordinary language conditionals. But impossibility results for the Ramsey Test hypothesis in its various forms suggest that it is untenable. In this paper, I argue that these results do not in fact have this implication, on (...) the grounds that similar results can be proved without recourse to the Ramsey test hypothesis. Instead they show that a number of well entrenched principles of rational belief and belief revision do not apply to conditionals. (shrink)

The paper presents a non-monotonic inference relation on a language containing a conditional that satisfies the Ramsey Test. The logic is a weakening of classical logic and preserves many of the ‘paradoxes of implication’ associated with the material implication. It is argued, however, that once one makes the proper distinction between supposing that something is the case and accepting that it is the case, these ‘paradoxes’ cease to be counterintuitive. A representation theorem is provided where conditionals are given a non-bivalent (...) semantics and epistemic states are represented via preferential models. (shrink)

It is argued that indicative conditionals are best viewed as having truth conditions (and so they are in part factual) but that these truth conditions are ‘gappy’ which leaves an explanatory gap that can only be filled by epistemic considerations (and so indicative conditionals are in part epistemic). This dual nature of indicative conditionals gives reason to rethink the relationship between logic viewed as a descriptive discipline (focusing on semantics) and logic viewed as a discipline with a normative import (focusing (...) on epistemic notions such as ‘reasoning’, ‘beliefs’ and ‘assumptions’). In particular, it is argued that the development of formal models for epistemic states can serve as a starting point for exploring logic when viewed as a normative discipline. (shrink)

Frank Ramsey (1931) wrote: If two people are arguing 'if p will q?' and both are in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q. We can say that they are fixing their degrees of belief in q given p. Let us take the first sentence the way it is often taken, as proposing the following test for the acceptability of an indicative conditional: ‘If p then q’ (...) is acceptable to a subject S iff, were S to accept p and consider q, S would accept q. Now consider an indicative conditional of the form (1) If p, then I believe p. Suppose that you accept p and consider ‘I believe p’. To accept p while rejecting ‘I believe p’ is tantamount to accepting the Moore-paradoxical sentence ‘p and I do not believe p’, and so is irrational. To accept p while suspending judgment about ‘I believe p’ is irrational for similar reasons. So rationality requires that if you accept p and consider ‘I believe p’, you accept ‘I believe p’. (shrink)

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

One of the main applications of the logic of theory change is to the epistemic analysis of conditionals via the so-called Ramsey test. In the first part of the present note this test is studied in the limiting case where the theory being revised is inconsistent, and it is shown that this case manifests an intrinsic incompatibility between the Ramsey test and the AGM postulate of success. The paper then analyses the use of the postulate of success, and a weakening (...) of it, generating axioms of conditional logic via the test, and it is shown that for certain purposes both success and weak success are quite superfluous. This suggests the proposal of abandoning both success and weak success entirely, thus permitting retention of the postulate of preservation discarded by Gärdenfors. (shrink)

In "Against the Indicative," AUSTRALASIAN JOURNAL OF PHILOSOPHY 72 (1994): 17-26, and more recently in "Classifying `Conditionals': the Traditional Way is Wrong", ANALYSIS 60 (2000): 147, V.H. Dudman argues that (a) `If Oswald didn't shoot Kennedy then someone else did' and (b) `If Oswald doesn't shoot Kennedy then someone else will' should not be classified together as "indicative conditionals." Dudman relies on the assumption that (a) is entailed by (c) `Someone shot Kennedy', whereas (b) is not entailed by (d) `Someone (...) will shoot Kennedy'. I argue that the same reasoning which shows that (d) does not entail (b) also shows that (c) does not entail (a). One upshot is that Dudman's and Mellor's respective interpretations of so-called past indicative conditionals cannot be correct. (shrink)

Peter Gärdenfors has proved (Philosophical Review, 1986) that the Ramsey rule and the methodologically conservative Preservation principle are incompatible given innocuous-looking background assumptions about belief revision. Gärdenfors gives up the Ramsey rule; I argue for preserving the Ramsey rule and interpret Gärdenfors's theorem as showing that no rational belief-reviser can avoid reasoning nonmonotonically. I argue against the Preservation principle and show that counterexamples to it always involve nonmonotonic reasoning. I then construct a new formal model of belief revision that does (...) accommodate nonmonotonic reasoning. (shrink)

According to so-called epistemic theories of conditionals, the assertability/acceptability/acceptance of a conditional requires the existence of an epistemically significant relation between the conditional’s antecedent and its consequent. This paper points to some linguistic data that our current best theories of the foregoing type appear unable to explain. Further, it presents a new theory of the same type that does not have that shortcoming. The theory is then defended against some seemingly obvious objections.

Test for the rational acceptance of conditionals and it still incites much of the interest in conditional reasoning. For instance, the test has been considered as a good starting point for several formal semantics for conditionals. Furthermore, its ramifications have important implications for several disciplines, from logic and artificial intelligence to decision theory and psychology. This volume presents a small but fine sample of the state of the art of such multifarious area of research.

Abstract I analyse the relationship between the Ramsey Test (RT) for the acceptance of indicative conditionals and the so-called problem of decision-instability. In particular, I argue that the situations which allegedly bring about this problem are troublesome just in case the relevant conditionals are evaluated by non-suppositional versions, e.g. causal/evidential, of the test. In contrast, a suppositional RT, by highlighting the metacognitive nature of the evaluation of indicative conditionals, allows an agent to run a simulation of such evaluation, without yet (...) committing her to the acceptance of such conditionals. I conclude that a suppositional interpretation of RT is superior to its nonsuppositional counterparts and by briefly showing that a suppositional RT is compatible with a deliberational decision theory. (shrink)

Section 1 briefly examines three theories of indicative conditionals. The Suppositional Theory is defended, and shown to be incompatible with understanding conditionals in terms of truth conditions. Section 2 discusses the psychological evidence about conditionals reported by Over and Evans (this volume). Section 3 discusses the syntactic grounds offered by Haegeman (this volume) for distinguishing two sorts of conditional.

In this paper, I discuss conditionals as illocutionary speech acts whose interpretation depends upon the whole of the social context in which they are uttered and whose purpose is to affect the opinions and actions of others. I argue for a suppositional approach to conditional statements based in what philosophers call the Ramsey test and developing the psychological theory that conditionals elicit a process of hypothetical thinking in their listeners. By reference to the experimental psychological literature on conditionals, I show (...) that in general conditionals, even ones that are basic or abstract in nature, are not treated as truth-functional or material by ordinary people. Drawing upon the suppositional nature of conditionals and the influence of pragmatic implicature, I discuss uses of conditionals as advice, inducement, persuasions and dissuasion, arguing that speakers use conditionals to try to influence the beliefs and actions of their listeners by shaping their hypothetical thought about possibilities. (shrink)

There is an important class of conditionals whose assertibility conditions are not given by the Ramsey test but by an inductive extension of that test. Such inductive Ramsey conditionals fail to satisfy some of the core properties of plain conditionals. Associated principles of nonmonotonic inference should not be assumed to hold generally if interpretations in terms of induction or appeals to total evidence are not to be ruled out.

The book presents the results of the joint annual conference of the four Operations Research Societies DGOR, GM\OR, \GOR and SVOR, held in Vienna in 1990.

HANDBOOK OF DEFEASIBLE REASONING AND UNCERTAINTY MANAGEMENT SYSTEMS EDITORS: DOV M. ... and A. Hunter Volume 3: Belief Change Edited by D. Dubois and H. Prade HANDBOOK OF DEFEASIBLE REASONING AND ...

The purpose of this note is to formulate some weaker versions of the so called Ramsey test that do not entail the following unacceptable consequenceIf A and C are already accepted in K, then if A, then C is also accepted in K. and to show that these versions still lead to the same triviality result when combined with a preservation criterion.

The so called Ramsey test is a semantic recipe for determining whether a conditional proposition is acceptable in a given state of belief. Informally, it can be formulated as follows: (RT) Accept a proposition of the form "if A, then C" in a state of belief K, if and only if the minimal change of K needed to accept A also requires accepting C. In Gärdenfors (1986) it was shown that the Ramsey test is, in the context of some other (...) weak conditions, on pain of triviality incompatible with the following principle, which was there called the preservation criterion: (P) If a proposition B is accepted in a given state of belief K and the proposition A is consistent with the beliefs in K, then B is still accepted in the minimal change of K needed to accept A. (RT) provides a necessary and sufficient criterion for when a 'positive' conditional should be included in a belief state, but it does not say anything about when the negation of a conditional sentence should be accepted. A very natural candidate for this purpose is the following negative Ramsey test: (NRT) Accept the negation of a proposition of the form "if A, then C" in a consistent state of belief K, if and only if the minimal change of K needed to accept A does not require accepting C. This note shows that (NRT) leads to triviality results even in the absence of additional conditions like (P). (shrink)

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

According to operator theories, "if" denotes a two-place operator. According to restrictor theories, "if" doesn't contribute an operator of its own but instead merely restricts the domain of some co-occurring quantifier. The standard arguments (Lewis 1975, Kratzer 1986) for restrictor theories have it that operator theories (but not restrictor theories) struggle to predict the truth conditions of quantified conditionals like -/- (1) a. If John didn't work at home, he usually worked in his office. b. If John didn't work at (...) home, he must have worked in his office. -/- Gillies (2010) offers a context-shifty conditional operator theory that predicts the right truth conditions for epistemically modalized conditionals like (1b), thus undercutting one standard argument for restrictor theories. I explore how we might generalize Gillies' theory to adverbially quantified conditionals like (1a) and deontic conditionals, and argue that a natural generalization of Gillies' theory -- following his strategy for handling epistemically modalized conditionals -- won't work for these other conditionals because a crucial assumption that epistemic modal bases are closed (used to neutralize the epistemic quantification contributed by "if") doesn't have plausible analogs in these other domains. (shrink)

This book by one of the world's foremost philosophers in the fields of epistemology and logic offers an account of suppositional reasoning relevant to practical deliberation, explanation, prediction and hypothesis testing. Suppositions made 'for the sake of argument' sometimes conflict with our beliefs, and when they do, some beliefs are rejected and others retained. Thanks to such belief contravention, adding content to a supposition can undermine conclusions reached without it. Subversion can also arise because suppositional reasoning is ampliative. These two (...) types of nonmonotonic logic are the focus of this book. A detailed comparison of nonmonotonicity appropriate to both belief contravening and ampliative suppositional reasoning reveals important differences that have been overlooked. (shrink)

Epistemic conditionals have often been thought to satisfy the Ramsey test (RT): If A, then B is acceptable in a belief state G if and only if B should be accepted upon revising G with A. But as Peter Gärdenfors has shown, RT conflicts with the intuitively plausible condition of Preservation on belief revision. We investigate what happens if (a) RT is retained while Preservation is weakened, or (b) vice versa. We also generalize Gärdenfors' approach by treating belief revision (...) as a relation rather than as a function.In our semantic approach, the same relation is used to model belief revision and to give truth-conditions for conditionals. The approach validates a weak version of the Ramsey Test (WRR) — essentially, a restriction of RT to maximally consistent belief states. (shrink)

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

We introduce two new belief revision axioms: partial monotonicity and consequence correctness. We show that partial monotonicity is consistent with but independent of the full set of axioms for a Gärdenfors belief revision sytem. In contrast to the Gärdenfors inconsistency results for certain monotonicity principles, we use partial monotonicity to inform a consistent formalization of the Ramsey test within a belief revision system extended by a conditional operator. We take this to be a technical dissolution of the well-known Gärdenfors (...) dilemma.In addition, we present the consequential correctness axiom as a new measure of minimal revision in terms of the deductive core of a proposition whose support we wish to excise. We survey several syntactic and semantic belief revision systems and evaluate them according to both the Gärdenfors axioms and our new axioms. Furthermore, our algebraic characterization of semantic revision systems provides a useful technical device for analysis and comparison, which we illustrate with several new proofs. (shrink)

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

The semantic theory of expressivism has been applied within metaethics to evaluative words like ‘good’ and ‘wrong’, within epistemology to words like ‘knows’, and within the philosophy of language, to words like ‘true’, to epistemic modals like ‘might’, ‘must’, and ‘probably’, and to indicative conditionals. For each topic, expressivism promises the advantage of giving us the resources to say what sentences involving these words mean by telling us what it is to believe these things, rather than by telling us what (...) it would be for them to be true. This, in turn, absolves these theories of the burden of holding that there is any general answer to what it is for these sentences to be true. However, expressivism is famously subject to a deep and general problem about how to account for the meanings of complex sentences – a problem variously known as the ‘Frege-Geach’ or ‘embedding’ problem. In this paper I will be interested in whether there are reasons to think that the embedding problem looks less difficult for some of these applications for expressivism, than for others. In particular, in this paper I will be interested in the prospects for expressivism about what I will call epistemics – a class which I take to include epistemic modals like ‘might’ and ‘must’, sentential adverbs like ‘probably’, adjectives like ‘likely’ and ‘improbable’, and so-called ‘open’ indicative conditionals like ‘if the Fed doesn’t intervene, then the economy will enter a deflationary spiral’. There are several reasons to be particularly interested in expressivism about epistemics, relating both to the philosophical payoffs of such a view, and relating to the technical prospects for making it work. In other work I’ve touched on the especially interesting philosophical payoffs which make expressivism about epistemics interesting; in this paper I will be interested primarily in evaluating the possibility that there are better prospects for making expressivism about epistemics work than there are for making expressivism work about other topics. There are two main reasons why one might suspect that expressivism about epistemics will have better prospects than expressivism about many other topics, including in metaethics.. (shrink)

A paper by Schulz (Philos Stud 149:367–386, 2010) describes how the suppositional view of indicative conditionals can be supplemented with a derived view of epistemic modals. In a recent criticism of this paper, Willer (Philos Stud 153:365–375, 2011) argues that the resulting account of conditionals and epistemic modals cannot do justice to the validity of certain inference patterns involving modalised conditionals. In the present response, I analyse Willer’s argument, identify an implicit presupposition which can plausibly be denied and show that (...) accepting it would blur the difference between plain assumptions and their epistemic necessitations. (shrink)

Peter G¨ ardenfors proved a theorem purporting to show that it is impossible to adjoin to the AGM -postulates for belief-revision a principle of monotonicity for revisions. The principle of monotonicity in question is implied by the Ramsey test for conditionals. So G¨.

I set out and defend a view on indicative conditionals that I call “indexical relativism”. The core of the view is that which proposition is (semantically) expressed by an utterance of a conditional is a function of (among other things) the speaker’s context and the assessor’s context. This implies a kind of relativism, namely that a single utterance may be correctly assessed as true by one assessor and false by another.

This paper presents a new theory of the truth conditions for indicative conditionals. The theory allows us to give a fairly unified account of the semantics for indicative and subjunctive conditionals, though there remains a distinction between the two classes. Put simply, the idea behind the theory is that the distinction between the indicative and the subjunctive parallels the distinction between the necessary and the a priori. Since that distinction is best understood formally using the resources of two-dimensional modal logic, (...) those resources will be brought to bear on the logic of conditionals. (shrink)

In contemporary discussions of the Ramsey Test for conditionals, it is commonly held that (i) supposing the antecedent of a conditional is adopting a potential state of full belief, and (ii) Modus Ponens is a valid rule of inference. I argue on the basis of Thomason Conditionals (such as ‘If Sally is deceiving, I do not believe it’) and Moore’s Paradox that both claims are wrong. I then develop a double-indexed Update Semantics for conditionals which takes these two results (...) into account while doing justice to the key intuitions underlying the Ramsey Test. The semantics is extended to cover some further phenomena, including the recent observation that epistemic modal operators give rise to something very like, but also very unlike, Moore’s Paradox. (shrink)

When should we believe a indicative conditional, and how much confidence in it should we have? Here’s one proposal: one supposes actual the antecedent; and sees under that supposition what credence attaches to the consequent. Thus we suppose that Oswald did not shot Kennedy; and note that under this assumption, Kennedy was assassinated by someone other than Oswald. Thus we are highly confident in the indicative: if Oswald did not kill Kennedy, someone else did.

Chalmers and Hájek argue that on an epistemic reading of Ramsey’s test for the rational acceptability of conditionals, it is faulty. They claim that applying the test to each of a certain pair of conditionals requires one to think that one is omniscient or infallible, unless one forms irrational Moore-paradoxical beliefs. I show that this claim is false. The epistemic Ramsey test is indeed faulty. Applying it requires that one think of anyone as all-believing and if one is rational, to (...) think of anyone as infallible-if-rational. But this is not because of Moore-paradoxical beliefs. Rather it is because applying the test requires a certain supposition about conscious belief. It is important to understand the nature of this supposition. (shrink)

This paper defends a counterexample to Modus Tollens, and uses it to draw some conclusions about the logic and semantics of indicative conditionals and probability operators in natural language. Along the way we investigate some of the interactions of these expressions with 'knows', and we call into question the thesis that all knowledge ascriptions have truth-conditions.

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