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.

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)

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 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 counterfactual conditionals (...) in an unfamiliar foundational role. (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)

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)

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)

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)

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)

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)

The 2021 Nobel Prize in Economics recognized an epistemology of causal inference based on the Rubin causal model (Rubin 1974), which merits broader attention in philosophy. This model, in fact, presupposes a logical principle of counterfactuals, Conditional Excluded Middle (CEM), the locus of a pivotal debate between Stalnaker (1968) and Lewis (1973) on the semantics of counterfactuals. Proponents of CEM should recognize that this connection points to a new argument for CEM---a Quine-Putnam indispensability argument grounded in the Nobel-winning (...) applications of the Rubin model in health and social sciences. To advance the dialectic, I challenge this argument with an updated Rubin causal model that retains its successes while dispensing with CEM. This novel approach combines the strengths of the Rubin causal model and a causal model familiar in philosophy, the causal Bayes net. The takeaway: deductive logic and inductive inference, often studied in isolation, are deeply interconnected. (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)

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)

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)

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)

A number of recent authors (Galles and Pearl, Found Sci 3 (1):151–182, 1998; Hiddleston, Noûs 39 (4):232–257, 2005; Halpern, J Artif Intell Res 12:317–337, 2000) advocate a causal modeling semantics for counterfactuals. But the precise logical significance of the causal modeling semantics remains murky. Particularly important, yet particularly under-explored, is its relationship to the similarity-based semantics for counterfactuals developed by Lewis (Counterfactuals. Harvard University Press, 1973b). The causal modeling semantics is both an account of the truth conditions (...) of counterfactuals, and an account of which inferences involving counterfactuals are valid. As an account of truth conditions, it is incomplete. While Lewis's similarity semantics lets us evaluate counterfactuals with arbitrarily complex antecedents and consequents, the causal modeling semantics makes it hard to ascertain the truth conditions of all but a highly restricted class of counterfactuals. I explain how to extend the causal modeling language to encompass a wider range of sentences, and provide a sound and complete axiomatization for the extended language. Extending the truth conditions for counterfactuals has serious consequences concerning valid inference. The extended language is unlike any logic of Lewis's: modus ponens is invalid, and classical logical equivalents cannot be freely substituted in the antecedents of conditionals. (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)

Stephen Barker argues that a possible worlds semantics for the counterfactual conditional of the sort proposed by Stalnaker and Lewis cannot accommodate certain examples in which determinism is true and a counterfactual Q > R is false, but where, for some P, the compound counterfactual P > (Q > R) is true. I argue that the completeness theorem for Lewis’s system VC of counterfactual logic shows that Stalnaker–Lewis semantics does accommodate Barker’s example, and I argue that its doing so (...) should be understood as showing that the example is an exception to Lewis’s Time’s Arrow requirements. (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)

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)

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.

This article is a selective review of David Lewis's Counterfactuals, a challenging, provocative, absorbingly interesting attempt to analyze statements of the form “If it were the case that P, then it would be the case that Q.” I shall follow Lewis in calling these “counterfactuals,” and shall nearly follow him in abbreviating them to the form P→Q.Chapter 1, which is nearly a third of the whole, gives the analysis and proves that it endows counterfactuals with some properties (...) which they evidently do have. Chapter 2 presents some “alternative formulations” of the analysis—a logical jeu d'esprit which I shall not discuss except for the section about “cotenability.”. (shrink)

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)

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.

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

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)

Priest has provided a simple tableau calculus for Chellas's conditional logic Ck. We provide rules which, when added to Priest's system, result in tableau calculi for Chellas's CK and Lewis's VC. Completeness of these tableaux, however, relies on the cut rule.