This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious program by Yli-Vakkuri and Hawthorne, who seek to establish that mathematics is committed to its own necessity. I demonstrate that their assumptions collapse the counterfactual conditional into the material conditional. This collapse entails the success of counterfactual strengthening, which is controversial within counterfactual logic, and which has counterexamples within (...) pure and applied mathematics. I close by discussing the dispensability of counterfactual conditionals within the language of mathematics. (shrink)

The purpose of this paper is to argue that the hybrid formalism fits naturally in the context of David Lewis’s counterfactual logic and that its introduction into this framework is desirable. This hybridization enables us to regard the inference “The pig is Mary; Mary is pregnant; therefore the pig is pregnant” as a process of updating local information (which depends on the given situation) by using global information (independent of the situation). Our hybridization also has the following technical (...) advantages: (i) it preserves the completeness and decidability of Lewis’s logic; (ii) it allows us to characterize the Limit Assumption as a proof-rule with some side-conditions; and (iii) it enables us to establish a general Kripke completeness result by using the proof-rule corresponding to the Limit Assumption. Keywords Counterfactual logic - David Lewis - Contextually definite description - Hybrid logic - Arthur Prior - The limit assumption - Strong completeness - Decidability - Bisimulation - Pure completeness. (shrink)

This is an extended critique of comments made by Abner Shimony and Howard Stein on Henry Stapp’s proof of the non-locality of quantum mechanics. Although I claim that ultimately Stapp’s proof does not establish its purported conclusion, yet Shimony and Stein’s criticism contains a number of weak points, which need to be clarified.

Lewis (Noûs 13:455–476, 1979) claimed that branching-time(-like) models can be derived from his sphere models. However, he did not present any specific construction of branching-time(-like) models from his sphere models formally. Meanwhile, Hosokawa (in: Modern logic of modality and its philosophical range: counterfactuals, Gettier problem, and information flow, Tokyo Metropolitan University, 2018) presented a logico-mathematically strict manner in which sphere models can be reconstructed from branching-time models. Subsequently, Hosokawa (J Logic Lang Inf 32:677–706, 2023) presented a proof-theoretically refined (...) version of hybrid tense logic for a certain type of conditionals, which is referred to as hybrid tense logic for temporal conditionals ( $$\textbf{HTL}_{TC}$$ ). In this paper, we interpret a hybrid version $${\textbf{V}}_{\mathcal {HC(@, \downarrow )}}$$ of Lewis’s counterfactual logic $${\textbf{V}}$$ into $$\textbf{HTL}_{TC}$$, and then prove that $$\textbf{HTL}_{TC}$$ is more expressive than $${\textbf{V}}_{\mathcal {HC(@, \downarrow )}}$$ on the class of temporal sphere models, i.e., sphere models derived from a type of branching-time models. (shrink)

In this paper we present labelled sequent calculi and labelled natural deduction calculi for the counterfactual logics CK + {ID, MP}. As for the sequent calculi we prove, in a semantic manner, that the cut-rule is admissible. As for the natural deduction calculi we prove, in a purely syntactic way, the normalization theorem. Finally, we demonstrate that both calculi are sound and complete with respect to Nute semantics [12] and that the natural deduction calculi can be effectively transformed into (...) the sequent calculi. (shrink)

This paper investigates the metaphysics in higher-order counterfactual logic. I establish the necessity of identity and distinctness and show that the logic is committed to vacuism, which entails that all counteridenticals are true. I prove the Barcan, Converse Barcan, Being Constraint and Necessitism. I then show how to derive the Identity of Indiscernibles in counterfactual logic. I study a form of maximalist ontology which has been claimed to be so expansive as to be inconsistent. I (...) show that it is equivalent to the collapse of the counterfactual into the material conditional---which is itself equivalent to the modal logic TRIV. TRIV is consistent, from which it follows that maximalism is, surprisingly, consistent. I close by arguing that stating the limit assumption requires a higher-order logic. (shrink)

We present a generalization of Segerberg's onion semantics for belief revision, in which the linearity of the spheres need not occur. The resulting logic is called broccoli logic. We provide a minimal relational logic, with a bi-modal neighborhood semantics. We then show that broccoli logic is a well-known conditional logic, the Burgess-Veltman minimal conditional logic.

In this work, we investigate the representation of counterfactual conditionals using the vector logic, a matrix-vector formalism for logical functions and truth values. Inside this formalism, the counterfactuals can be transformed in complex matrices preprocessing an implication matrix with one of the square roots of NOT, a complex matrix. This mathematical approach puts in evidence the virtual character of the counterfactuals. This happens because this representation produces a valuation of a counterfactual that is the superposition of the (...) two opposite truth values weighted, respectively, by two complex conjugated coefficients. This result shows that this procedure gives an uncertain evaluation projected on the complex domain. After this basic representation, the judgement of the plausibility of a given counterfactual allows us to shift the decision towards an acceptance or a refusal. This shift is the result of applying for a second time one of the two square roots of NOT. (shrink)

This paper analyzes the logical truths as (very roughly) those truths that would still have been true under a certain range of counterfactual perturbations.What’s nice is that the relevant range is characterized without relying (overtly, at least) upon the notion of logical truth. This approach suggests a conception of necessity that explains what the different varieties of necessity (logical, physical, etc.) have in common, in virtue of which they are all varieties of necessity. However, this approach places the (...) class='Hi'>counterfactual conditionals in an unfamiliar foundational role. (shrink)

This paper will introduce and explore a set of quantified counterfactual temporal alethic-deontic systems, that is, systems that combine counterfactual temporal alethic-deontic logic with predicate logic. I will consider three types of systems: constant, variable and constant and variable domain systems. Every system can be combined with either necessary or contingent identity. All logics are described both semantically and proof theoretically. I use a kind of possible world semantics, inspired by the so-called T x W semantics, (...) to characterise them semantically and semantic tableaux to characterise them proof theoretically. Our models contain several different accessibility relations and a similarity relation between possible worlds, which are used in the definitions of the truth conditions for the various operators. Soundness results are obtained for every tableau system and completeness results for a subclass of these. (shrink)

Causal models show promise as a foundation for the semantics of counterfactual sentences. However, current approaches face limitations compared to the alternative similarity theory: they only apply to a limited subset of counterfactuals and the connection to counterfactual logic is not straightforward. This paper addresses these difficulties using exogenous interventions, where causal interventions change the values of exogenous variables rather than structural equations. This model accommodates judgments about backtracking counterfactuals, extends to logically complex counterfactuals, and validates familiar (...) principles of counterfactual logic. This combines the interventionist intuitions of the causal approach with the logical advantages of the similarity approach. (shrink)

We examine a formal semantics for counterfactual conditionals due to Judea Pearl, which formalizes the interventionist interpretation of counterfactuals central to the interventionist accounts of causation and explanation. We show that a characteristic principle validated by Pearl’s semantics, known as the principle of reversibility, states a kind of irreversibility: counterfactual dependence (in David Lewis’s sense) between two distinct events is irreversible. Moreover, we show that Pearl’s semantics rules out only mutual counterfactual dependence, not cyclic dependence in general. (...) This, we argue, suggests that Pearl’s logic is either too weak or too strong. (shrink)

The 2021 Nobel Prize in Economics recognizes a type of causal model known as the Rubin causal model, or potential outcome framework, which deserves far more attention from philosophers than it currently receives. To spark philosophers' interest, I develop a dialectic connecting the Rubin causal model to the Lewis-Stalnaker debate on a logical principle of counterfactuals: Conditional Excluded Middle (CEM). I begin by playing good cop for CEM, developing a new argument in its favor---a Quine-Putnam-style indispensability argument. This argument is (...) based on the observation that CEM seems to be indispensable to the Rubin causal model, which underpins our best scientific theory of causal inference in health and social sciences---a Nobel Prize-winning theory. Indeed, CEM has long remained a core assumption of the Rubin causal model, despite challenges from within the statistics and economics communities over twenty years ago. I then switch sides to play bad cop for CEM, undermining the indispensability argument by developing a new theory of causal inference that dispenses with CEM while preserving the successes of the original theory (thanks to a new theorem proved here). The key, somewhat surprisingly, is to integrate two approaches to causal modeling: the Rubin causal model, more familiar in health and social sciences, and the causal Bayes net, more familiar in philosophy. The good cop/bad cop dialectic is concluded with a connection to broader philosophical issues, including intertheory relations, the revisability of logic, and the role of background assumptions in justifying scientific inference. (shrink)

In the artificial intelligence literature a promising approach to counterfactual reasoning is to interpret counterfactual conditionals based on causal models. Different logics of such causal counterfactuals have been developed with respect to different classes of causal models. In this paper I characterize the class of causal models that are Lewisian in the sense that they validate the principles in Lewis’s well-known logic of counterfactuals. I then develop a system sound and complete with respect to this class. The (...) resulting logic is the weakest logic of causal counterfactuals that respects Lewis’s principles, sits in between the logic developed by Galles and Pearl and the logic developed by Halpern, and stands to Galles and Pearl’s logic in the same fashion as Lewis’s stands to Stalnaker’s. (shrink)

Leibniz denied, famously, that any possible individual exists in more than one possible world, so that a man who in fact never marries could not have married and still been himself. He claimed that this follows from his thesis that the predicate of every true affirmative proposition is contained in some way in the concept of its subject and his associated thesis that the definitive concept of each individual substance is complete. This chapter argues that the purely formal aims of (...) logical theory involved in the predicate containment and complete concept theses do not in fact require the denial of counterfactual or transworld identity. (shrink)

Moritz Schulz explores counterfactual thought and language: what would have happened if things had gone a different way. Counterfactual questions may concern large scale derivations or small scale evaluations of minor derivations. A common impression, which receives a thorough defence in the book, is that oftentimes we find it impossible to know what would have happened. However, this does not mean that we are completely at a loss: we are typically capable of evaluating counterfactual questions probabilistically: we (...) can say what would have been likely or unlikely to happen. Schulz describes these probabilistic ways of evaluating counterfactual questions and turns the data into a novel account of the workings of counterfactual thought. (shrink)

The paper gives a physicist's view on the framework of branching space-time, 385--434). Branching models are constructed from physical state assignments. The models are then employed to give a formal semantics for the modal operators ``possibly'' and ``necessarily'' and for the counterfactual conditional. The resulting formal language can be used to analyze quantum correlation experiments. As an application sketch, Stapp's premises LOC1 and LOC2 from his purported proof of non-locality, 300--304) are analyzed.

What are counterfactuals and what is their point? In many cases, none at all. It may be true that if kangaroos didn't have tails, they would fall over, but they do have tails and if they didn't they wouldn't be kangaroos (or would they?). This is the sort of thing that can give counterfactuals a bad name, as inhabitants of a La La Land of the mind. On the other hand, counterfactuals do useful service across a broad range of disciplines (...) in both the sciences and the humanities, including philosophy, history, cosmology, biology, cognitive psychology, jurisprudence, economics, art history, literary theory. They are also richly, albeit sometimes treacherously, present in the everyday human realm of how our lives are both imagined and lived: in the 'crossroads' scenario of decision-making, the place of regret in retrospective assessments of paths taken and not taken, and, at the outer limit, as the wish not to have been born. Christopher Prendergast take us on a dizzying exploratory journey through some of these intellectual and human landscapes, mobilizing a wide range of reference from antiquity to the present, and sustained by the belief that, whether as help or hindrance, and with many variations across cultures, counterfactual thinking and imagining are fundamental to what it is to be human. (shrink)

How are causal judgements such as 'The ice on the road caused the traffic accident' connected with counterfactual judgements such as 'If there had not been any ice on the road, the traffic accident would not have happened'? This volume throws new light on this question by uniting, for the first time, psychological and philosophical approaches to causation and counterfactuals. Traditionally, philosophers have primarily been interested in connections between causal and counterfactual claims on the level of meaning or (...) truth-conditions. More recently, however, they have also increasingly turned their attention to psychological connections between causal and counterfactual understanding or reasoning. At the same time, there has been a surge in interest in empirical work on causal and counterfactual cognition amongst developmental, cognitive, and social psychologists--much of it inspired by work in philosophy. In this volume, twelve original contributions from leading philosophers and psychologists explore in detail what bearing empirical findings might have on philosophical concerns about counterfactuals and causation, and how, in turn, work in philosophy might help clarify the issues at stake in empirical work on the cognitive underpinnings of, and relationships between, causal and counterfactual thought. (shrink)

We point out a simple but hitherto ignored link between the theoryof updates, the theory of counterfactuals, and classical modal logic: update is a classicalexistential modality, counterfactual is a classical universalmodality, and the accessibility relations corresponding to these modalities are inverses. The Ramsey Rule (often thought esoteric) is simply an axiomatisation of this inverse relationship. We use this fact to translate between rules for updates andrules for counterfactuals. Thus, Katsuno and Mendelzons postulatesU1--U8 are translated into counterfactual rules (...) C1--C8(Table VII), and many of the familiar counterfactual rulesare translated into rules for updates (Table VIII). Ourconclusions are summarised in Table V. (shrink)

The present thesis focuses on counterfactuals. Specifically, we will address new questions and open problems that arise for the standard semantic accounts of counterfactual conditionals. The first four chapters deal with the Lewisian semantic account of counterfactuals. On a technical level, we contribute by providing an equivalent algebraic semantics for Lewis' variably strict conditional logics, which is notably absent in the literature. We introduce a new kind of algebra and differentiate between local and global versions of each of Lewis' (...) variably strict conditional logics. We study the algebraic properties of Lewis' logics and the structure theory of our newly introduced algebras. Additionally, we employ a new algebraic construction, based on the framework of Boolean algebras of conditionals, to provide an alternative semantics for Lewisian counterfactual conditionals. This semantic account allows us to establish new truth conditions for Lewisian counterfactuals, implying that Lewisian counterfactuals are definable conditionals, and each counterfactual can be characterized as a modality of a corresponding probabilistic conditional. We further extend these results by demonstrating that each Lewisian counterfactual can also be characterized as a modality of the corresponding Stalnaker conditional. The resulting formal semantic framework is much more expressive than the standard one and, in addition to providing new truth conditions for counterfactuals, it also allows us to define a new class of conditional logics falling into the broader framework of weak logics. On the philosophical side, we argue that our results shed new light on the understanding of Lewisian counterfactuals and prompt a conceptual shift in this field: Lewisian counterfactual dependence can be understood as a modality of probabilistic conditional dependence or Stalnakerian conditional dependence. In other words, whether a counterfactual connection occurs between A and B depends on whether it is "necessary" for a Stalnakerian/probabilistic dependence to occur between A and B. We also propose some ways to interpret the kind of necessity involved in this interpretation. The remaining two chapters deal with the probability of counterfactuals. We provide an answer to the question of how we can characterize the probability that a Lewisian counterfactual is true, which is an open problem in the literature. We show that the probability of a Lewisian counterfactual can be characterized in terms of belief functions from Dempster-Shafer theory of evidence, which are a super-additive generalization of standard probability. We define an updating procedure for belief functions based on the imaging procedure and show that the probability of a counterfactual A > B amounts to the belief function of B imaged on A. This characterization strongly relies on the logical results we proved in the previous chapters. Moreover, we also solve an open problem concerning the procedure to assign a probability to complex counterfactuals in the framework of causal modelling semantics. A limitation of causal modelling semantics is that it cannot account for the probability of counterfactuals with disjunctive antecedents. Drawing on the same previous works, we define a new procedure to assign a probability to counterfactuals with disjunctive antecedents in the framework of causal modelling semantics. We also argue that our procedure is satisfactory in that it yields meaningful results and adheres to some conceptually intuitive constraints one may want to impose when computing the probability of counterfactuals. (shrink)

We analyze G.M. Hardegree's interpretation of the Sasaki hook as a Stalnaker conditional and explain how he makes use of the basic conceptual machinery of OQL, i.e. the operational quantum logic which originated with the Geneva Approach to the foundations of physics. In particular we focus on measurements which are ideal and of the first kind, since these encode the content of the so-called Sasaki projections within the Geneva Approach. The Sasaki projections play a fundamental role when analyzing the (...) condition under which the properties expressed by Sasaki hooks can be considered as actual. We finish with a note on how the Sasaki hook can be conceived as ``assigning causes for properties to be actual", which links the interpretation of G.M. Hardegree to what has been called ``dynamic OQL". (shrink)

The purpose of this paper is to develop a class of semantic tableau systems for some counterfactual logics. All in all I will discuss 1024 systems. Possible world semantics is used to interpret our formal languages. Soundness results are obtained for every tableau system and completeness results for a large subclass of these.

Counterfactual thinking has become an established method to evaluate decisions in a range of disciplines, including history, psychology and literature. Elke Reinhuber argues it also has valuable applications in the Fine Arts and popular media. A fascination with the path not taken is a logical consequence of a world saturated with choices. Art which provokes and explores these tendencies can help to recognise and contextualise the impulse to avoid or endlessly revisit individual or collective decisions. Reinhuber describes the term (...) in broad strokes through the disciplines, to show how counterfactualism finds shape in contemporary art forms, especially in photography, film, immersive and interactive media art. (such as 360° content, virtual reality and augmented reality).She analyses the different stages of counterfactuals with examples where artists experience counterfactual thoughts in the process of art production, explore these thoughts in their artwork, or where the artwork itself evokes counterfactual thoughts in the audience. A fascinating exploration for scholars and students of Art, Media and the Humanities, and anybody else with an interest in choices, the art of decision making and - counterfactualism. (shrink)

The semantics for counterfactuals due to David Lewis has been challenged by appealing to miracles. Miracles may skew a given similarity order in favour of those possible worlds which exhibit them. Lewis responded with a system of priorities that mitigates the significance of miracles when constructing similarity relations. We propose a prioritised ceteris paribus analysis of counterfactuals inspired by Lewis’ system of priorities. By analysing the couterfactuals with a ceteris paribus clause one forces out, in a natural manner, those possible (...) worlds which do not satisfy the requirements of the clause, thus excluding miracles. If no world can satisfy the ceteris paribus clause in its entirety, then prioritisation is triggered to select worlds that maximise agreement on those things which are favoured most. (shrink)

It is human nature to wonder how things might have turned out differently--either for the better or for the worse. For the past two decades psychologists have been intrigued by this phenomenon, which they call counterfactual thinking. Specifically, researchers have sought to answer the "big" questions: Why do people have such a strong propensity to generate counterfactuals, and what functions does counterfactual thinking serve? What are the determinants of counterfactual thinking, and what are its adaptive and psychological (...) consequences? This important work brings together a collection of thought-provoking papers by social and cognitive psychologists who have made important theoretical and empirical contributions to our understanding of this topic. The essays in this volume contain novel theoretical insights, and, in many cases descriptions of previously unpublished empirical studies. The Psychology of Counterfactual Thinking provides an excellent overview of this fascinating topic for researchers, as well as advanced undergraduates and graduates in psychology--particularly those with an interest in social cognition, social judgment, decision judgment, decision making, thinking and reasoning. (shrink)

Many have accepted that ordinary counterfactuals and might counterfactuals are duals. In this paper, I show that this thesis leads to paradoxical results when combined with a few different unorthodox yet increasingly popular theses, including the thesis that counterfactuals are strict conditionals. Given Duality and several other theses, we can quickly infer the validity of another paradoxical principle, ‘The Counterfactual Direct Argument’, which says that ‘A> ’ entails ‘A> ’. First, I provide a collapse theorem for the ‘counterfactual (...) direct argument’. The counterfactual direct argument entails the logical equivalence of the subjunctive and material conditional, given a variety of assumptions. Second, I provide a semantics that validates the counterfactual direct argument without collapse. This theory further develops extant dynamic accounts of conditionals. I give a new semantics for disjunction, on which A or B is only true in a context when A and B are both unsettled. The resulting framework validates CDA while invalidating other commonly accepted principles concerning the conditional and disjunction. (shrink)

The accessibility relation between possible worlds can be defined in the metalanguage of counterfactual semantics. As a result, counterfactuals can ground the whole of standard modal logic.

This article aims to propagate Logic Programming as a formal tool to deal with non-monotonic reasoning. In philosophy and linguistics non-monotonic reasoning is modelled using Minimal Models as standard, i.e., by imposing an order (or selection function) on the class of all models and then by defining entailment as only caring about the minimal models of the premises with respect to the order. In this article we investigate the question whether instead of minimal models we should use logic (...) programming to model non-monotonic reasoning. Logic programming is an attractive alternative to a minimal models approach in that it makes concrete predictions in an efficient and transparent way. We study this question by focusing on one particular phenomenon that gives rise to non-monotonic inferences: conditional sentences. (shrink)

Counterfactuals is David Lewis's forceful presentation of and sustained argument for a particular view about propositions which express contrary-to-fact conditionals, including his famous defense of realism about possible worlds. Since its original publication in 1973, it has become a classic of contemporary philosophy, and is essential reading for anyone interested in the logic and metaphysics of counterfactuals. The book also includes an appendix of related writings by Lewis.

In this paper, I will discuss some examples of the so-called contrary-to-duty paradox, a well-known puzzle in deontic logic. A contrary-to-duty obligation is an obligation telling us what ought to be the case if something forbidden is true, for example: ‘If she is guilty, she should confess’. Contrary-to-duty obligations are important in our moral and legal thinking. Therefore, we want to be able to find an adequate symbolisation of such obligations in some logical system, a task that has turned (...) out to be difficult. This is shown by the so-called contrary-to-duty paradox. I will investigate and evaluate one kind of solution to this problem that has been suggested in the literature, which I will call the ‘counterfactual solution’. I will use some recent systems that combine not only counterfactual logic and deontic logic, but also temporal logic, in my analysis of the paradox. I will argue that the counterfactual solution has many attractive features and that it can give a fairly satisfactory answer to some examples of the contrary-to-duty paradox, but that it nevertheless has some serious problems. The conclusion is that, notwithstanding the many attractive features of the solution, there seem to be other approaches to the paradox that are more promising. (shrink)

The natural interpretation of counterfactuals with disjunctive antecedents involves selecting from each of the disjuncts the worlds that come closest to the world of evaluation. It has been long noticed that capturing this interpretation poses a problem for a minimal change semantics for counterfactuals, because selecting the closest worlds from each disjunct requires accessing the denotation of the disjuncts from the denotation of the disjunctive antecedent, which the standard boolean analysis of or does not allow (Creary and Hill, Philosophy of (...) Science 43:341–344, 1975; Nute, Journal of Philosophy 72:773–778, 1975; Fine, Mind 84(335):451–458, 1975; Ellis et al. Journal of Philosophical Logic 6:335–357, 1977). This paper argues that the failure to capture the natural interpretation of disjunctive counterfactuals provides no reason to abandon a minimal change semantics. It shows that the natural interpretation of disjunctive counterfactuals is expected once we refine our assumptions about the semantics of or and the logical form of conditionals, and (i) we assume that disjunctions introduce propositional alternatives in the semantic derivation, in line with independently motivated proposals about the semantics of or (Aloni, 2003a; Simons, Natural Language Semantics 13:271–316, 2005; Alonso-Ovalle, Disjunction in Alternative Semantics. PhD thesis, 2006); and (ii) we treat conditionals as correlative constructions, as advocated in von Fintel (1994), Izvorski (Proceedings of NELS 26, 1996), Bhatt and Pancheva (2006), and Schlenker (2004). (shrink)

We present an analysis of counterfactuals in terms of stories and combine it with an account similar to Walton’s account of truth in fiction to yield truth conditions for counterfactuals. We discuss unusual features of this account, and compare it to other main approaches. In particular, we argue that our analysis succeeds in accounting for counterpossibles and counterfactuals with true antecedents while the other two main approaches fail, and we give reasons for thinking that it is important to have an (...) adequate account of these two areas. (shrink)

The paper provides an explanation of our knowledge of metaphysical modality, or modal knowledge, from our ability to evaluate counterfactual conditionals. The latter ability lends itself to an evolutionary explanation since it enables us to learn from mistakes. Different logical principles linking counterfactuals to metaphysical modality can be employed to extend this explanation to the epistemology of modality. While the epistemological use of some of these principles is either philosophically implausible or empirically inadequate, the equivalence of ‘Necessarily p’ with (...) ‘For all q, if q were the case, p would be the case’ is a suitable starting-point for an explanation of modal knowledge. (shrink)

We analyze the causal-observational languages that were introduced in Barbero and Sandu (2018), which allow discussing interventionist counterfactuals and functional dependencies in a unified framework. In particular, we systematically investigate the expressive power of these languages in causal team semantics, and we provide complete natural deduction calculi for each language. Furthermore, we introduce a generalized semantics which allows representing uncertainty about the causal laws, and we analyze the expressive power and proof theory of the causal-observational languages over this enriched semantics.