Conditionals

I have read enough of your paper uploaded today, Ciardelli and Ommundsen! I don't accept it. Consider the proposition "If Beryl is in Paris, she often visits the Louvre." They say that it is ambiguous and the following is a legitimate interpretation of it: on many of the occasions on which Beryl is in Paris, she visits the Louvre. I don't agree. This is not a legitimate interpretation. It is called "the O reading," and I can see why: because you (...) are going to get beaten severely if you rely on this interpretation. The legitimate interpretation is: from the supposition that she is in Paris, we can conclude that she visits the Louvre often. (shrink)

This paper is concerned with disjunctions of universal modals, such as ‘You have to clean your room or you have to walk the dog’, and disjunctions of conditionals, such as ‘If Alice dances, Charlie will dance, or if Bob dances, Charlie will dance’. We aim to provide a uniform account of their surprising behaviour. Our proposal combines three independently attested components. Firstly, disjunction's dynamic effect, familiar from presupposition projection: when we evaluate a disjunction ‘A or B’, we typically interpret B (...) assuming the negation of A, and optionally, also interpret A assuming the negation of B. Secondly, the fact that disjunctions often receive a conjunctive interpretation, familiar from free choice phenomena. Thirdly, that modal bases are restricted to local contexts, what Mandelkern has called ‘bounded modality’. (shrink)

The propositional connectives – and, or, not, if–then, etc. – are fundamental building blocks in formal as well as natural languages. In the Western tradition, they were first studied as such by the Stoics, and Propositional Logic is the fundament of practically all current systems of logic; every beginning logic course starts with it. Still, the proof theory and semantics of systems of propositional logic are far from trivial, and have been studied intensely by logicians in the last one and (...) a half centuries, not least in recent decades. It is actually a vast area of research, as witnessed by Lloyd Humberstone’s 1500 page tome The Connectives (2011), which overviews much of that research. Perhaps the most familiar recent work in this area concerns conditionals in formal and natural languages. The 4th Tsinghua Interdisciplinary Workshop on Logic, Language, and Meaning (TLLM 2024) also focused on the apparently simpler connectives expressing (various versions of) conjunction, disjunction, and negation. -/- Researchers working from a cross-linguistic perspective also study how the connectives are encoded in different languages, and ask whether classical logic is capable of capturing the variations and universals exhibited. Even in well-studied languages such as English, there are intricate phenomena that remain challenging for classical logic, including free choice disjunction, non-Boolean conjunction, and metalinguistic negation, to name just a few. There is also growing interest in the acquisition and processing of natural language connectives. In the context of the hotly discussed Large Language Models (LLMs), understanding connectives presents novel challenges that deserve in-depth exploration. (shrink)

The aim of the paper is to discuss the problem of the assignment of the degree of belief to conditional sentences by a rational agent. After presenting the general methodological framework, we analyze two possible approaches. In both cases we assume that we have a non-conditional probabilistic system of beliefs expressed in a language L0, and modeled in some initial probability space S, which allows us to assign probabilities to sentences from L0. Our aim is to extend this system of (...) beliefs to a given class of sentences Φ containing conditionals. The first approach is what we call the “credence-like” approach: for a given class Φ, we define credence as a function defined directly on linguistic objects. The second approach consists in assuming the existence of a standard probability space, in which the sentences from the set Φ are interpreted as events. In this case, the degree of belief of α is defined as the probability of the corresponding event in the probability space SΦ. We present both of these approaches, indicating what their advantages and disadvantages are. The thesis of the article is that the probabilistic approach to the analysis of degrees of belief of sentences containing the conditional connective is by far the more universal one and preferable method, particularly in the context of more complex conditionals. (shrink)

This paper raises some questions about the formalization of sentences containing ‘if’ or similar expressions. In particular, it focuses on three kinds of sentences that resemble conditionals in some respects but exhibit distinctive logical features that deserve separate consideration: whether-or-not sentences, biscuit conditionals, and concessive conditionals. As will be suggested, the examples discussed show in different ways that an adequate formalization of a sentence must take into account the content expressed by the sentence. This upshot is arguably what one should (...) expect on the view that logical form is determined by truth conditions. (shrink)

The dominant analyses of the logical form of natural-language conditionals take them to be “suppositional conditionals”. The latter are true or accepted if the consequent is true/accepted on the supposition of the antecedent. But this can happen although the antecedent is completely irrelevant (or even somewhat adverse) to the consequent. In natural-language conditionals, however, the antecedent is typically meant to support or be evidence for the consequent. The logical form of conditionals will thus be more complex than the suppositional theory (...) would have it. Recently some suggestions as to what this logical form might look like have been made. In this paper, I critically discuss Vincenzo Crupi and Andrea Iacona’s account of “evidential conditionals”, including its recent amendments. (shrink)

Franz Huber’s book addresses beliefs and counterfactuals and their relationship. Using a normative approach, the author explores how agents should believe and change their beliefs, given they have...

The epistemic thesis is the thesis that a 'might' counterfactual like 'If Matt had gone to the parade, David might have gone to the parade' has the same meaning as 'Maybe, if Matt had gone to the parade, David would have gone to the parade.' I offer a new theory of the counterfactual interpretation of the modal 'might' on which 'might' has the same meaning as 'maybe would'. And I show that, when coupled with a plausible semantics for 'if' clauses, (...) my theory validates the epistemic thesis. (shrink)

Volume 46, Issue 1, February 2025, Page 198-200.

Causal pre-emption scenarios are problematic for the counterfactual framework of causation (CFC) because people judge an action to be the actual cause of an outcome although the outcome would have occurred anyway due to the action of a pre-empted alternative cause. We propose that commonsense causal questions typically probe specific events that actually happened as and how they did, and show that counterfactuals that probe specific events match selections of actual cause, and dissociations only occur with non-specific counterfactuals. In addition, (...) the pre-empted action is often selected as the or an actual cause when it causes the pre-empting action (auto-pre-emption). Judgements of an action’s responsibility for the outcome track judgements of actual cause following the legal sine qua non principle. Agent reproach is also influenced by the agent's intention. The effects of causal dependency structure and counterfactual question type are robust across the intentionality of the pre-empting action and scenario content. (shrink)

Lee Archie argued that if any truth-values are consistently assigned to a natural language conditional, where modus ponens and modus tollens are valid argument forms, and affirming the consequent is invalid, this conditional will have the same truth-conditions as a material implication. This argument is simple and requires few and relatively uncontroversial assumptions. We show that it is possible to extend Archie’s argument to three- and five-valued logics and vindicate a slightly weaker conclusion, but one that is still important: Even (...) if you do not believe in bivalence and the classical negation operator, you would still have good reasons to accept that natural language conditionals and the material implication share truth-conditions. (shrink)

This paper is about progressive aspect. We defend a new principle that we call ‘Progressive Specificity’: if you are Ving and to V is to X or to Y, then you are Xing or you are Ying. We offer six arguments for Progressive Specificity. We then suggest that those six arguments extend to the futurative progressive. We conclude by comparing Progressive Specificity to Conditional Excluded Middle. -/- .

Causal Modeling Semantics (CMS, e.g., Galles and Pearl 1998; Pearl 2000; Halpern 2000) is a powerful framework for evaluating counterfactuals whose antecedent is a conjunction of atomic formulas. We extend CMS to an evaluation of the probability of counterfactuals with disjunctive antecedents, and more generally, to counterfactuals whose antecedent is an arbitrary Boolean combination of atomic formulas. Our main idea is to assign a probability to a counterfactual (A ∨ B) € C at a causal model M as a weighted (...) average of the probability of C in those submodels that truthmake A ∨ B (Briggs 2012; Fine 2016, 2017). The weights of the submodels are given by the inverse distance to the original model M, based on a distance metric proposed by Eva, Stern, and Hartmann (2019). Apart from solving a major problem in the epistemology of counterfactuals, our paper shows how work in semantics, causal inference and formal epistemology can be fruitfully combined. (shrink)

In this article, I analyze historical counterfactuals. Historical counterfactuals are conditional statements in which the antecedent refers to some change in the past. We ask what would have happened, had that change occurred. I discuss the nature of such counterfactuals. I then identify important functions that historical counterfactuals have. I point out that they are at the heart of explanations and, therefore, reveal issues related to contingency and actual history. I then discuss counterfactual reasoning in historiography. I argue that the (...) problem of suitable antecedent conditions has been exaggerated, and more serious issues concern the tracking of counterfactual scenarios. Throughout the paper, I argue that the interventionist way of thinking about historical counterfactuals clarifies both historical explanations and the nature of historical counterfactuals and should be adopted as the standard. I conclude by noting that historical counterfactuals may not fundamentally differ from more familiar forms of historiography. (shrink)

According to Molinism, logically prior to his creative decree, God knows via middle knowledge the truth value of the counterfactuals or conditionals of creaturely freedom (CFs) and thus what any possible person would do in any given circumstance. Critics of Molinism have pointed out that the Molinist God gets lucky that the CFs allow him to actualize either a world of his liking or even a good-enough world at all. In this paper, I advance and strengthen the popular critique in (...) two ways. First, I specify the kind of luck to which God is subject, which is circumstantial moral luck. Second, and more importantly, I argue that exposure to this luck is problematic because something external to God and beyond his control determines the degree of praiseworthiness he deserves in relevant possible worlds. My main contention is that divine subjection to circumstantial moral luck raises the theoretical costs of Molinism. (shrink)

This paper is about four inferences patterns governing conditionals: Transitivity, Simplification, Contraposition, and Antecedent Strengthening. Transitivity, Simplification, and Contraposition are intuitively compelling. Although Antecedent Strengthening may seem less attractive at first, close attention to the full range of data reveals that it too has considerable appeal. An adequate theory of conditionals should account for these facts. The strict theory does so by validating them. But the variably strict theory invalidates them. So the variably strict theorist faces a question: why do (...) we find these inferences so compelling? In this paper, I suggest an answer on her behalf: that they are reasonable inferences in the sense introduced by Stalnaker (1975). -/- . (shrink)

Abstract: I argue that practically all vague language is context-sensitive in a covert and unfamiliar way. I first outline a novel puzzle concerning the interaction of conditionals and vagueness. I then argue that the best way of resolving the puzzle is through positing context-sensitive penumbral connections between sundry parts of language. I argue that these penumbral connections shift through a distinct form of Lewisian accommodation. The upshot is that meaning is a far shiftier thing than has typically been thought.

This book provides a fully developed account of Frank Ramsey's theory of conditionals. No proper study of Ramsey's view of conditionals has been carried out so far. This work aims at filling this gap. The book is divided in two parts and in four chapters. The first part of the book is historical, investigating Ramsey’s texts and recovering his view on conditionals. The second part is systematic, updating and developing a unified account of conditionals following Ramsey’s ideas.

Today there is a wealth of fascinating studies of connexive logical systems. But sometimes it looks as if connexive logic is still in search of a convincing interpretation that explains in intuitive terms _why_ the connexive principles should be valid. In this paper I argue that difference-making conditionals as presented in Rott (_Review of Symbolic Logic_ 15, 2022) offer one principled way of interpreting connexive principles. From a philosophical point of view, the idea of difference-making demands full, unrestricted connexivity, because (...) neither logical truths nor contradictions or other absurdities can ever ‘make a difference’ (i.e., be relevantly connected) to anything. However, difference-making conditionals have so far been only partially connexive. I show how the existing analysis of difference-making conditionals can be reshaped to obtain full connexivity. The classical AGM belief revision model is replaced by a conceivability-limited revision model that serves as the semantic base for the analysis. The key point of the latter is that the agent should never accept any absurdities. (shrink)

We present two approaches to investigate the validity of connexive principles and related formulas and properties within coherence-based probability logic. Connexive logic emerged from the intuition that conditionals of the formif not-A,thenA, should not hold, since the conditional’s antecedentnot-Acontradicts its consequentA. Our approaches cover this intuition by observing that the only coherent probability assessment on the conditional event$${A| \overline{A}}$$A|A¯is$${p(A| \overline{A})=0}$$p(A|A¯)=0. In the first approach we investigate connexive principles within coherence-based probabilistic default reasoning, by interpreting defaults and negated defaults in terms (...) of suitable probabilistic constraints on conditional events. In the second approach we study connexivity within the coherence framework of compound conditionals, by interpreting connexive principles in terms of suitable conditional random quantities. After developing notions of validity in each approach, we analyze the following connexive principles:Aristotle’s theses,Aristotle’s Second Thesis,Abelard’s First Principle, andBoethius’ theses. We also deepen and generalize some principles and investigate further properties related to connexive logic (likenon-symmetry). Both approaches satisfyminimalrequirements for a connexive logic. Finally, we compare both approaches conceptually. (shrink)

Indicative conditionals - e.g. ‘If I was born in Glasgow, then I was born in Scotland’ - seem to express propositions. In other words, they seem to express thoughts that can be shared and evaluated as either true or false. For the past hundred years or so, analytic philosophers have commonly interpreted them as equivalent to disjunctions, e.g. ‘Either I was not born in Glasgow, or I was born in Scotland’. Those who dig a little deeper tend to agree that (...) this is not quite right, but they often come to the conclusion that indicative conditionals do not express propositions at all. I argue for an alternative theory according to which indicative conditionals express a type of strict implication. What this means is that they are modal statements - e.g. ‘Necessarily, either I was not born in Glasgow, or I was born in Scotland’ - an idea that used to be quite popular despite striking many contemporary philosophers as radical. More specifically, I defend the idea that indicative conditionals express metaphysical necessity. I argue that this view respects our intuitions in an important way, and I show that it can be incorporated into a promising view of natural language conditionals more generally. Lastly, I argue that it also has plausible consequences for the related concepts of conditional probability, causal explanation, and risk. (shrink)

The article exposes (in brief) the theories of the counterfactuals in analytical philosophy (metalinguistic, ontological and suppositionist) and discusses the conditions of assertability of counterfactuals in historiography. The conviction of some historians that the counterfactuals are essentially forbidden in the scientific historiography is critised on the ground that counterfactuals are implied by the causal statements and statements of the relative causal importance. The admission is made that the traditional (historist) historiography lacks adequate conceptual infrastructure for the detailed analysis of the (...) objectively possible historical alternatives because its only method for validation of such analysis is inference by analogy. The procedure for validation of counterfactuals in economtric history is discussed citing as an exemple the work of Robert Fogel about the impact of the railroads on the economic growth in USA. The irrelevance of Jon Elster's "basic paradox of counterfactuals" for the working historian is asserted on the ground that as a matter of fact there is no empirical social theory capable to endogenize all initial conditions in explanatory arguments where it is instrumental for deduction of explanandum. (shrink)

This paper develops the idea that valid arguments are equivalent to true conditionals by combining Kripke’s theory of truth with the evidential account of conditionals offered by Crupi and Iacona. As will be shown, in a first-order language that contains a naïve truth predicate and a suitable conditional, one can define a validity predicate in accordance with the thesis that the inference from a conjunction of premises to a conclusion is valid when the corresponding conditional is true. The validity predicate (...) so defined significantly increases our expressive resources and provides a coherent formal treatment of paradoxical arguments. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:A Báñezian Grounding for Counterfactuals of Creaturely Freedom:A Response to James Dominic Rooney, O.P.Taylor Patrick O'NeillIntroductionIn a recently published article, James Rooney, O.P., critiques a fundamental aspect of Réginald Garrigou-Lagrange's articulation of the relation between divine causality and creaturely freedom, which I also defended in my recent book.1 Specifically, Rooney argues that at least some of what Garrigou-Lagrange holds is rooted in a Molinist rather than Báñezian understanding of (...) human freedom and divine causality. The aim of this paper is to articulate the basic objections raised by Rooney and to show why the objections fail. We shall not provide an exegetical or textual analysis of Báñez himself, but rather will establish a speculative grounding and agreement between basic principles of Báñezianism and Garrigou-Lagrange's articulation, ultimately affirming the necessity of positing creaturely counterfactuals for Báñezianism.Rooney makes two distinct (but related) objections to Garrigou-Lagrange's treatment. First, he claims that Báñezianism as a system rejects the notion of "alternative possibilities" when considering the creaturely will and its relation to divine causality, especially in regard to physical premotion. I shall argue that this is incorrect, and indeed misses one of the constitutive elements of Báñezianism—in fact the very grounding for its claim that infallibly moved actions remain free. Second, Rooney claims that Garrigou-Lagrange posits [End Page 651] a circular argument regarding the reality of sin. Garrigou-Lagrange's argument would be circular if it corresponded to a single order of causality or if counterfactuals of creaturely freedom were incompatible with Báñezianism. But we shall argue that they are not. As such, the argument is not only not circular, but rather elucidates a constitutive element of the Báñezian system, thereby refuting the objection. In short, Garrigou-Lagrange's articulation of Báñezian premotion admits of counterfactuals without relying upon divine passivity, and it remains internally coherent.Rooney's Critique of the So-called "Garrigou Manoeuvre"Rooney begins the essay by outlining a few of the major differences between a Molinist and Báñezian view of human freedom. Rooney's argument is that there is a potential "grounding problem" for Báñezianism, or at least for Báñezianism as articulated by Garrigou-Lagrange. As Rooney notes, Báñezianism posits that God's efficacious grace is ontologically antecedent to our good actions (and his permission of sin is ontologically antecedent to our sin). But how does this position retain free choice, if divine causation or permission is infallibly determinative and antecedent to creaturely choice? Rooney says:The Banezians need to account for how it is possible that a creature can do otherwise in the special case when God moves the creature to act through grace. Specifically, what grounds the truth of the claim that, even when God determines human free choices from eternity such that a human being has only one course of action open to them, those human beings are morally responsible for their actions? If the Banezian view entails that human beings, acting under grace, strictly lack any ability to do otherwise, then the view is incoherent.2As such, we can see that "the ability to do otherwise" is an essential, grounding element for defending Báñezianism against radically compatibilist or double-predestinarian views of human freedom, which have been condemned by the Church at Trent and in subsequent condemnations of Jansenism.3 Of course, the phrase "ability to do otherwise" is metaphysically [End Page 652] ambiguous, which Rooney admits. Various articulations of a more metaphysically precise grounding for the preservation of free choice under the influence of efficacious grace (or while efficacious grace is withheld) have been put forth, and Rooney focuses on a particular "strategy" which he associates with Garrigou-Lagrange and myself.4 He terms this strategy the "Garrigou Manoeuvre."Rooney notes that "Garrigou-Lagrange claims that there is something in virtue of which God denies someone efficacious grace, and that this is in the control of the sinner."5 He quotes Garrigou-Lagrange as saying that a person "is deprived of efficacious grace because by sinning he resists sufficient grace.... [Therefore,] God... (shrink)

Timothy Williamson contends that our primary cognitive heuristic for prospectively assessing conditionals, i.e., the suppositional procedure, is provably inconsistent. Our diagnosis is that stipulations about the nature of suppositional rejection are the likely source of these results. We show that on at least one alternative, and quite natural, understanding of the suppositional attitudes, the inconsistency results are blocked. The upshot is an increase in the reliability of our suppositional heuristics across a wider range of contexts. One interesting consequence of the (...) increased reliability is a proportional decrease in the plausibility of an error-theory that Williamson employs against widespread intuitions about the truth values of counterpossible conditionals. (shrink)

This paper concerns the semantics of coordinated if-clauses, as in (1)-(2). It is argued that the meanings of such sentences are explained straightforwardly on theories of conditionals that tie their non- monotonic behaviour to the if-clause itself (e.g. Schlenker 2004, but not theories that tie it to a (covert) modal operator (e.g. Kratzer 1981; 1991). Coordinated if-clauses are revealing of the fine-grained compositional semantics of conditionals.

It has been more than sixty years since R. D. Collison Black published his Economic Thought and the Irish Question, a book which ever since has been widely regarded as a classic in the history of e...

Kevin Dorst (2019) uses the ‘manifest unassertability’ of conditionals of the form ‘If I don’t know p, then p’ as a new motivation for the KK thesis. In this paper we show that his argumentation is misguided. Plausible heuristics offer a compelling and nuanced explanation of the relevant infelicity data. Meanwhile, Dorst relies on tools that, quite independently of KK, turn out to be rather poor predictors of the infelicity of indicative conditionals.

Impossible antecedents entered the scene of medieval logic around the 1120s and soon started to dominate this scene, becoming one of the most debated issues from the second half of the twelfth century onwards. This article focuses on theories of counterpossibles from this period and aims to offer an overview of the different responses offered by twelfth-century logicians on whether everything, something, or nothing follows from an impossible statement. Rather than trying to historically reconstruct the positions of the different authors (...) – an operation that may be premature given the insufficient notions we still have of the authorship and dating of sources – I aim to provide a provisional map of the main arguments that were advanced in favour of or against the ex impossibili quodlibet principle. The article also aims to show that the strategies developed by twelfth-century logical schools in their approach to counterpossibles survive in logical discussions of the same topic during the thirteenth and the early fourteenth century, especially in the syncategoremata and sophistaria traditions. (shrink)

I present an intractable puzzle for the currently popular view that belief is weak—the view that expressions like ‘S believes p’ ascribe to S a doxastic attitude towards p that is rationally compatible with low credence that p. The puzzle concerns issues that arise on considering beliefs in conditionals. I show that proponents of weak belief either cannot consistently apply their preferred methodology when accommodating beliefs in conditionals, or they must deny that beliefs in conditionals can be used in reasoning.

We bring together two ideas that are widely accepted in the literature: (i) conditionals with modal consequents are systematically ambiguous between a reading on which the if-clause restricts the modal, and a reading on which the if-clause introduces an epistemic supposition and the modal is unrestricted; (ii) ‘will’ is a modal. Together, (i) and (ii) imply that ‘will’-conditionals are systematically ambiguous between a reading as restricted historical claims, and one as unrestricted historical claims made under a supposition. We argue that (...) this prediction is correct and that the existence of the relevant ambiguity has far-reaching repercussions for several important debates concerning conditionals. (shrink)

“If the tower is any taller than 320 ms, it may collapse,” Eiffel thinks out loud. Although understanding this counterfactual poses no trouble, the most successful interventionist semantics struggle to model it because the antecedent can come about in infinitely many ways. My aim is to provide a semantics that will make modeling such counterfactuals easy for philosophers, computer scientists, and cognitive scientists who work on causation and causal reasoning. I first propose three desiderata that will guide my theory: it (...) should be general, yet conservative, yet useful. Next, I develop a formalization of events in the form of an algebra. I identify an event with all the ways in which it can be brought about and provide rules for determining the referent of an arbitrary event description. I apply this algebra to counterfactuals expressed using underdeterministic causal models-models that encode non-probabilistic causal indeterminacies. Specifically, I develop semaphore interventions, which represent how the target system may be modified from without in a coordinated fashion. This, in turn, allows me to bring about any event within a single model. Finally, I explain the advantages of this semantics over other interventionist competitors. (shrink)

Conditional statements are ubiquitous, from promises and threats to reasoning and decision making. By now, logicians have studied them from many different angles, both semantic and proof-theoretic. This paper suggests two more perspectives on the meaning of conditionals, one dynamic and one geometric, that may throw yet more light on a familiar and yet in some ways surprisingly elusive and many-faceted notion.1.

In this paper two paradoxes of infinity are considered through the lens of counterfactual logic, drawing heavily on a result of Fine (2012b). I will argue that a satisfactory resolution of these paradoxes will have wide ranging implications for the logic of counterfactuals. I then situate these puzzles in the context of the wider role of counterfactuals, connecting them to indicative conditionals, probabilities, rationality and the direction of causation, and compare my own resolution of the paradoxes to alternatives inspired by (...) the theories of Lewis and Fine. Here is a quick overview of the paper. Sections 17.1 and 17.2 introduce two paradoxes of infinity that rest on certain principals concerning the logic of counterfactuals. Section 17.3 considers three possible ways to weaken the counterfactual logic that would resolve these paradoxes, and examines three concrete theories of conditionals which make those weakenings — Lewis (1973), Fine (2012a) and Bacon (2015). With the first two theories found wanting, Sect. 17.4 develops my preferred solution in (Bacon, 2015) a little further, and provides some simple models to show that the paradoxes can indeed be resolved in that framework. (shrink)

I defend the truthmaker approach to counterfactuals against certain objections that Andrew Bacon has made in his paper.

Stalnaker's Thesis about indicative conditionals is, roughly, that the probability one ought to assign to an indicative conditional equals the probability that one ought to assign to its consequent conditional on its antecedent. The thesis seems right. If you draw a card from a standard 52-card deck, how confident are you that the card is a diamond if it's a red card? To answer this, you calculate the proportion of red cards that are diamonds -- that is, you calculate the (...) probability of drawing a diamond conditional on drawing a red card. Skyrms' Thesis about counterfactual conditionals is, roughly, that the probability that one ought to assign to a counterfactual equals one's rational expectation of the chance, at a relevant past time, of its consequent conditional on its antecedent. This thesis also seems right. If you decide not to enter a 100-ticket lottery, how confident are you that you would have won had you bought a ticket? To answer this, you calculate the prior chance--that is, the chance just before your decision not to buy a ticket---of winning conditional on entering the lottery. The central project of this article is to develop a new uniform theory of conditionals that allows us to derive a version of Skyrms' Thesis from a version of Stalnaker's Thesis, together with a chance-deference norm relating rational credence to beliefs about objective chance. (shrink)

We propose a novel approach to capturing the acceptability conditions of indicative conditionals using awareness logic. Specifically, we posit that the relevance between the antecedent and consequent of a conditional is determined by the concomitant occurrence of their awareness. We provide a sound and complete axiomatization of the logic. We compare the properties of the indicative conditionals in our framework with other existing theories.

Optical media were instrumental in transforming the conception of facts, objectivity, and the »real.« This paper considers their role in structuring understandings of counterfactuals and states that could not be real. By returning to Ernst Mach’s photographic ballistics experiments, writing on thought experiments (a term he coined), and his dispute with Max Planck about the nature of the Weltbild, the article shows that, despite his legacy as a positivist, Mach’s epistemology of mechanical images opened a legitimate space of indeterminacy, contingency, (...) and counterfactuality. (shrink)

Conditionals frequently convey that the antecedent is relevant to the consequent. Recently many authors have argued that this relevance is part of the conventional meaning of conditionals, but this approach fails to account for many examples where a conditional is used to conveyirrelevance of antecedent to consequent. Both types of conditionals are best explained by a conventional meaning with no relevance requirement, and a separate process of coherence establishment among successive clauses in discourse. This account is supported by the distribution (...) of discourse particles and is able to account for experimental studies used to support the conventionalist position. (shrink)

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

This paper presents a weak Kleene approach to conditionals that preserves some salient formal features of conditionals, particularly their interdefinability with Boolean logical connectives. I argue that such an approach fares better than other proposed weak logics of conditionals in this regard. In particular, it fares better than the logics proposed by Cooper, Cantwell, Farrell, De Finetti, Égré, Rossi, and Sprenger.

The basis of the paper is a logic of analytical consequential implication, CI.0, which is known to be equivalent to the well-known modal system KT thanks to the definition A → B = df A ⥽ B ∧ Ξ (Α, Β), Ξ (Α, Β) being a symbol for what is called here Equimodality Property: (□A ≡ □B) ∧ (◊A ≡ ◊B). Extending CI.0 (=KT) with axioms and rules for the so-called circumstantial operator symbolized by *, one obtains a system CI.0*Eq (...) in whose language one can define an operator ↠ suitable to formalize context-dependent conditionals (so to counterfactual conditionals) via the definition (Def ↠) A ↠ B = df *A ⥽ B ∧ Ξ (Α, Β).. The central problem of the paper is to identify inside CI.0*Eq + Def ↠ a set of axioms yielding the fragment consisting of all and only theorems in which only truth-functional operators and the two operators → and ↠ occur. This system, here called CI.0, is introduced in §3. In view of the intended purpose, it is introduced a complete and tableau-decidable system KTw, which is an extension of KT with axioms for the so-called quasi-variables w(A), w(B)… Three translation functions among the three languages used in the paper are then introduced. The first, Tr, maps every wff of form *A into wffs of form w(A) ∧ A and translates theorems of CI.0*Eq into theorems of KTw. The second, t°, translates theorems of CI.0 into theorems of CI.0*Eq by applying Def↠. A third one, t, translates theorems of CI.0 into theorems of KTw. As a consequence, it follows that t°A = TrtA. In §4 it is proved that CI.0 is complete w.r.t. the class of CI.0-models of form where f is a selection function and R an access relation. It is then proved (i) that CI.0 models may be converted into KTw-models; (ii) that the truth-value of a proposition in a world of a CI.0-model is preserved in the same world of the derived KTw-model; (iii) that A is a CI.0-thesis iff its translation tA is KTw-thesis. It follows that, if A is not a CI.0-thesis, tA is not a KTw-thesis, so also that TrtA is not a CI.0*Eq-thesis. Since t°A = TrtA and t°A is a function built on Def↠, this proves that every t°-translation of a CI.0-wff that is a CI.0*Eq-theorem is a CI.0-theorem. Since the converse proposition is proved in §2, their conjunction establishes the desired result. In the final section, it is proved that CI.0 is tableau-decidable, that ↠ is not a trivial operator and that ↠ enjoys the positive and negative properties required in §1 for context-dependent conditionals. Some final remarks suggest that studying the relations between CI.0 and systems of classical conditional logic may be a promising line of investigation. (shrink)

Although it receives less attention, (Lewis in Noûs 13:455–476, 1979. https://doi.org/10.2307/2215339) admitted that the branching-time(-like) model fits a wide range of counterfactuals, including (Nix) ‘If Nixon had pressed the button, there would have been a nuclear war’, which was raised by (Fine in Mind 84:451–458, 1975). However, Lewis then claimed that similarity analysis is more general than temporality analysis. In this paper, we do not scrutinise his claim. Instead, we re-analyse (Nix) not only model-theoretically but also proof-theoretically from the ‘meaning-as-use’ (...) and ‘inferentialist’ points of view. Then, we re-formalise (Nix) in a natural extension of hybrid tense logic, which we refer to as hybrid tense logic for temporal conditionals (HTL$$_{TC}$$ TC ). Consequently, we find that not only among counterfactuals, but also among indicatives, there is a wide range of conditionals whose formalisation in HTL$$_{TC}$$ TC is appropriate. We refer to these conditionals as temporal conditionals. This suggests a new logical generality that temporality analysis has but similarity analysis does not, from which emerges a new logical perspective on conditionals in general: temporal ones and others. (shrink)