Conditional probability is often used to represent the probability of the conditional. However, triviality results suggest that the thesis that the probability of the conditional always equals conditional probability leads to untenable conclusions. In this paper, I offer an interpretation of this thesis in a possible worlds framework, arguing that the triviality results make assumptions at odds with the use of conditional probability. I argue that these assumptions come from a theory called the operator theory and that the rival restrictor (...) theory can avoid these problematic assumptions. In doing so, I argue that recent extensions of the triviality arguments to restrictor conditionals fail, making assumptions which are only justified on the operator theory. (shrink)

Based on a crowdsourced truth value judgment experiment, we provide empirical evidence challenging two classical views in semantics, and we develop a novel account of counterfactuals that combines ideas from inquisitive semantics and causal reasoning. First, we show that two truth-conditionally equivalent clauses can make different semantic contributions when embedded in a counterfactual antecedent. Assuming compositionality, this means that the meaning of these clauses is not fully determined by their truth conditions. This finding has a clear explanation in inquisitive semantics: (...) truth-conditionally equivalent clauses may be associated with different propositional alternatives, each of which counts as a separate counterfactual assumption. Second, we show that our results contradict the common idea that the interpretation of a counterfactual involves minimizing change with respect to the actual state of affairs. We propose to replace the idea of minimal change by a distinction between foreground and background for a given counterfactual assumption: the background is held fixed in the counterfactual situation, while the foreground can be varied without any minimality constraint. (shrink)

In this paper I present a limitation to what may be called strictly-interventionistic causal-model semantic theories for subjunctive conditionals. And I offer a line of response to Briggs’ counterexample to Modus Ponens—given within a strictly-interventionistic framework—for the subjunctive conditional. The paper also contains some discussion of backtracking counterfactuals and backtracking interpretations. The limitation inherent to strict interventionism is brought out via a class of counterexamples. A causal-model semantics is strictly interventionistic just in case the procedure it gives for evaluating a (...) subjunctive conditional requires making the values of the variables implicated in the antecedent independent from the values of the parents of these antecedent variables. Most causal-model semantic theories that have gained attention are strictly interventionistic. (shrink)

Written by one of the preeminent researchers in the field, this book provides a comprehensive exposition of modern analysis of causation. It shows how causality has grown from a nebulous concept into a mathematical theory with significant applications in the fields of statistics, artificial intelligence, economics, philosophy, cognitive science, and the health and social sciences. Judea Pearl presents and unifies the probabilistic, manipulative, counterfactual, and structural approaches to causation and devises simple mathematical tools for studying the relationships between causal connections (...) and statistical associations. Cited in more than 2,100 scientific publications, it continues to liberate scientists from the traditional molds of statistical thinking. In this revised edition, Judea Pearl elucidates thorny issues, answers readers' questions, and offers a panoramic view of recent advances in this field of research. Causality will be of interest to students and professionals in a wide variety of fields. Dr Judea Pearl has received the 2011 Rumelhart Prize for his leading research in Artificial Intelligence and systems from The Cognitive Science Society. (shrink)

When people want to identify the causes of an event, assign credit or blame, or learn from their mistakes, they often reflect on how things could have gone differently. In this kind of reasoning, one considers a counterfactual world in which some events are different from their real-world counterparts and considers what else would have changed. Researchers have recently proposed several probabilistic models that aim to capture how people do (or should) reason about counterfactuals. We present a new model and (...) show that it accounts better for human inferences than several alternative models. Our model builds on the work of Pearl (2000), and extends his approach in a way that accommodates backtracking inferences and that acknowledges the difference between counterfactual interventions and counterfactual observations. We present six new experiments and analyze data from four experiments carried out by Rips (2010), and the results suggest that the new model provides an accurate account of both mean human judgments and the judgments of individuals. (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)

I argue that the suppositional view of conditionals, which is quite popular for indicative conditionals, extends also to subjunctive or counterfactual conditionals. According to this view, conditional judgements should not be construed as factual, categorical judgements, but as judgements about the consequent under the supposition of the antecedent. The strongest evidence for the view comes from focusing on the fact that conditional judgements are often uncertain; and conditional uncertainty, which is a well-understood notion, does not function like uncertainty about matters (...) of fact. I argue that the evidence for this view is as strong for subjunctives as it is for indicatives. (shrink)

Counterfactuals are inherently ambiguous in the sense that the same counterfactual may be true under one mode of counterfactualization but false under the other. Many have regarded the ambiguity of counterfactuals as consisting in the distinction between forward-tracking and backtracking counterfactuals. This is incorrect since the ambiguity persists even in cases not involving backtracking counterfactualization. In this paper, I argue that causal modeling semantics has the resources enough for accounting for the ambiguity of counterfactuals. Specifically, we need to distinguish two (...) types of causal manipulation, which I call “intervention” and “extrapolation” respectively. To intervene in a causal model M is to change M’s structural equations in some specific ways, while to extrapolate M is to change the value assignment of M’s variables in some specific ways. I argue that intervention and extrapolation offer a natural explanation for the ambiguity of counterfactuals. (shrink)

I discuss three observations about backtracking counterfactuals not predicted by existing theories, and then motivate a theory of counterfactuals that does predict them. On my theory, counterfactuals quantify over a suitably restricted set of historical possibilities from some contextually relevant past time. I motivate each feature of the theory relevant to predicting our three observations about backtracking counterfactuals. The paper concludes with replies to three potential objections.

This paper explores some issues about the choice of variables for causal representation and explanation. Depending on which variables a researcher employs, many causal inference procedures and many treatments of causation will reach different conclusions about which causal relationships are present in some system of interest. The assumption of this paper is that some choices of variables are superior to other choices for the purpose of causal analysis. A number of possible criteria for variable choice are described and defended within (...) a broadly interventionist approach to causation. (shrink)

Adams’ thesis is generally agreed to be linguistically compelling for simple conditionals with factual antecedent and consequent. We propose a derivation of Adams’ thesis from the Lewis- Kratzer analysis of if-clauses as domain restrictors, applied to probability operators. We argue that Lewis’s triviality result may be seen as a result of inexpressibility of the kind familiar in generalized quantifier theory. Some implications of the Lewis- Kratzer analysis are presented concerning the assignment of probabilities to compounds of conditionals.

This book offers a discussion about how people think, talk, learn, and explain things in causal terms in terms of action and manipulation. Sloman also reviews the role of causality, causal models, and intervention in the basic human cognitive functions: decision making, reasoning, judgement, categorization, inductive inference, language, and learning.

This paper studies the causal interpretation of counterfactual sentences using a modifiable structural equation model. It is shown that two properties of counterfactuals, namely, composition and effectiveness, are sound and complete relative to this interpretation, when recursive (i.e., feedback-less) models are considered. Composition and effectiveness also hold in Lewis's closest-world semantics, which implies that for recursive models the causal interpretation imposes no restrictions beyond those embodied in Lewis's framework. A third property, called reversibility, holds in nonrecursive causal models but not (...) in Lewis's closest-world semantics, which implies that Lewis's axioms do not capture some properties of systems with feedback. Causal inferences based on counterfactual analysis are exemplified and compared to those based on graphical models. (shrink)

Understanding causal structure is a central task of human cognition. Causal learning underpins the development of our concepts and categories, our intuitive theories, and our capacities for planning, imagination and inference. During the last few years, there has been an interdisciplinary revolution in our understanding of learning and reasoning: Researchers in philosophy, psychology, and computation have discovered new mechanisms for learning the causal structure of the world. This new work provides a rigorous, formal basis for theory theories of concepts and (...) cognitive development, and moreover, the causal learning mechanisms it has uncovered go dramatically beyond the traditional mechanisms of both nativist theories, such as modularity theories, and empiricist ones, such as association or connectionism. (shrink)

I develop an account of counterfactual conditionals using “causal models”, and argue that this account is preferable to the currently standard account in terms of “similarity of possible worlds” due to David Lewis and Robert Stalnaker. I diagnose the attraction of counterfactual theories of causation, and argue that it is illusory.

A counterlegal is a counterfactual conditional containing an antecedent that is inconsistent with some set of laws. A backtracker is a counterfactual that tells us how things would be at a time earlier than that of its antecedent, were the antecedent to obtain. Typically, theories that evaluate counterlegals appropriately don’t evaluate backtrackers properly, and vice versa. Two cases in point: Lewis’ ordering semantics handles counterlegals well but not backtrackers. Hiddleston’s :632–657, 2005) causal-model semantics nicely handles backtrackers but not counterlegals. Taking (...) Hiddleston’s account as a starting point, I offer steps toward a theory capable of handling both counterlegals and backtrackers. The core contribution of this paper is a means for evaluating counterlegals relative to minimally-illegal models. (shrink)

Galles & Pearl (l998) claimed that s [possible-worlds] framework.s framework. Recursive models are shown to correspond precisely to a subclass of (possible-world) counterfactual structures. On the other hand, a slight generalization of recursive models, models where all equations have unique solutions, is shown to be incomparable in expressive power to counterfactual structures, despite the fact that the Galles and Pearl arguments should apply to them as well. The problem with the Galles and Pearl argument is identified: an axiom that they (...) viewed as irrelevant, because it involved disjunction (which was not in their language), is not irrelevant at all. (shrink)

Bayes nets are formal representations of causal systems that many psychologists have claimed as plausible mental representations. One purported advantage of Bayes nets is that they may provide a theory of counterfactual conditionals, such as If Calvin had been at the party, Miriam would have left early. This article compares two proposed Bayes net theories as models of people's understanding of counterfactuals. Experiments 1-3 show that neither theory makes correct predictions about backtracking counterfactuals (in which the event of the if-clause (...) occurs after the event of the then-clause), and Experiment 4 shows the same is true of forward counterfactuals. An amended version of one of the approaches, however, can provide a more accurate account of these data. (shrink)

We consider the dispute between causal decision theorists and evidential decision theorists over Newcomb-like problems. We introduce a framework relating causation and directed graphs developed by Spirtes et al. (1993) and evaluate several arguments in this context. We argue that much of the debate between the two camps is misplaced; the disputes turn on the distinction between conditioning on an event E as against conditioning on an event I which is an action to bring about E. We give the essential (...) machinery for calculating the effect of an intervention and consider recent work which extends the basic account given here to the case where causal Knowledge is incomplete. (shrink)

This paper investigates what happens when we merge two different lines of theorizing about counterfactuals. One is the comparative closeness view, which was developed by Stalnaker and Lewis in the framework of possible worlds semantics. The second is the interventionist view, which is part of the causal models framework developed in statistics and computer science. Common lore and existing literature have it that the two views can be easily fit together, aside from a few details. I argue that, on the (...) contrary, transplanting causal-models-inspired ideas in a possible worlds framework yields a new semantics. The difference is grounded in different algorithms for handling inconsistent information, hence it touches on issues that are at the very heart of a semantics for contrary-to-fact conditionals. Roughly, Stalnaker/Lewis semantics requires us to evaluate the consequent of a counterfactual at all closest antecedent-verifying possibilities. Causal-models-based semantics also does this, but in addition uses the information contained in the antecedent, together with background causal information, to shift what worlds count as closest. This makes systematically different predictions and generates a new logic. The upshot is that we have a new semantics to study, and a substantial theoretical choice to make. (shrink)

The rise of causality and the attendant graph-theoretic modeling tools in the study of counterfactual reasoning has had resounding effects in many areas of cognitive science, but it has thus far not permeated the mainstream in linguistic theory to a comparable degree. In this study I show that a version of the predominant framework for the formal semantic analysis of conditionals, Kratzer-style premise semantics, allows for a straightforward implementation of the crucial ideas and insights of Pearl-style causal networks. I spell (...) out the details of such an implementation, focusing especially on the notions of intervention on a network and backtracking interpretations of counterfactuals. (shrink)