Abstract
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’ (Noûs 13:455–476, 1979a) ordering semantics handles counterlegals well but not backtrackers. Hiddleston’s (Noûs 39(4):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.
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Notes
I take it that such sentences often include those that are law-like according to a favored scientific theory. Consequently, I take it that the project at hand bears on the problem of interpreting counterfactuals containing antecedents that are nomically impossible according to a favored scientific theory. I do not, however, have feelings of attachment to any traditional, weighty conception of law-likeness from the philosophy of science of past decades. I am, for example, happy to understand talk of scientificlaw-likeness in terms of invariance across possible manipulations, where “invariance across possible manipulations” is understood along the lines of a Woodward (2003)-style account.
Some of Bennett’s remarks about counterlegals are reproduced at the conclusion of this paper.
Characterizations of what it is to be a backtracker typically suffer from lack of precision in order to avoid falsity. The characterization I give is no exception. The clauses of conditionals do not typically describe events so precisely as to single out a unique time or period of time at which the implicated events must (have) occur(red). So, there is often no such thing as the time of the event described in the antecedent or the consequent. Thanks to Adam Sennet for this point. Nevertheless, there are backtracking counterfactuals and I hope my characterization points well enough to the paradigm cases.
Lewis (1979a) appears to think that the falsity of backtrackers with false consequents is delivered by his similarity metric (reproduced in Sect. 1 below) for evaluating counterfactuals that he offers in the “Times Arrow” paper. And indeed the similarity-metric does typically weight similarity of earlier events much more than later ones. Still, Lewis’ belief (subject to caveats to do with time travel cases and the like) that a backtracker is true on its standard resolution just in case its consequent is true seems to be largely motivated by his belief that past events do not counterfactually depend on future events. It is not too hard to see that holding this view can make it attractive to think that any counterfactual saying that an earlier time would be the same were some later time different will be true, while any counterfactual saying that some earlier time would be different were some later time different will be false.
They are “stripped down” in that they are free of various constraints imposed by Pearl and SGS: The models need not satisfy either the Markov or Faithfulness conditions. And the sentences in E may describe relations that are probabilistic and/or non-functional. Pearl (2000) characterizes the Causal Markov Condition on page 30 (see also page 19) and faithfulness (there called ‘stability’) on page 48.
The restriction to finite graphs is to ensure that, given a model M, the set of \(\varphi \)-minimally altered models (as defined below) relative to M will always be non-empty. Thanks to Alex Kocurek for highlighting that without appropriate constraints on the class of models, some causal models will fail to have a non-empty set of \(\varphi \)-minimally altered models.
Further constraints on membership conditions for the law sets of these causal models are introduced via the Nomicality constraint offered in Sect. 3.
Where I think it will not cause confusion, I let linguistic constructions act as names for themselves.
Hiddleston directs us to read ‘\(\Rightarrow \)’ as a strict conditional. (Cf. Hiddleston 2005 p. 651.) I see no reason not to read it as the material conditional.
The list of definitions is Hiddleston’s, though my formulations vary from his in places.
Here I reproduce Hiddleston’s endnote 9: “Disjunctive antecedents pose problems for all theories of counterfactuals and I do not have anything to add concerning them. See, e.g., Loewer (1976)”.
Hiddleston points this out in his endnote 7 (Hiddleston 2005, p. 655).
I believe it is correct to think that the context, and hence the operative laws, change as the Mr. D’Arcy and Elizabeth story plays out. The points I make are intended to stand relative to the single context we are left with at the close of the story.
Where \(\Rightarrow \) is the material conditional of our metalanguage.
The imperfectness of the fuse as a cause of explosions was important for Hiddleston but is unimportant for my purposes. I have simply chosen to leave this feature of the case unchanged. This is why the probability statement appears in the consequent of (\(\alpha \)); it may be ignored.
This is a right-nested conditional, and Hiddleston (2005) does not consider such conditionals. In considering nested conditionals we may be going beyond the class of conditionals Hiddleston had in mind to treat with his semantics.
Thanks to Adam Edwards and Adam Sennet for making me think more carefully about these points.
This is not to say that we must work with linguistic characterizations of the relations, we could consider changes to the relations directly.
Recall that for a model M we call the pair containing its graph and law set a schema and denote it with an ‘S’.
CTC for counterlegal truth conditions or causal theory of counterlegals, depending on your taste.
For careful characterizations of interventionistic semantic theories, see, e.g., Pearl (2000; esp pp. 33–38, 202–215) and Briggs (2012). See, e.g., Rips (2009) for a lengthier informal characterization of intervention semantic theories, there called “pruning theory”. For a largely critical discussion of interventionist theories see Fisher (2016).
Notice that even if it is correct to say that causal relations determine which counterfactuals are true and which are false, it does not follow that our best (epistemic) methods for discovering such relations will also turn out to be our best (semantic) methods for evaluating counterfactuals.
Note too that a typical Lewis-style possible-worlds semantics will agree with CTC on (1)–(3). This is enough, in the present context I hope, to calm suspicions that (1) is a backtracker or somehow unfairly non-standard as counterfactuals go. For further argumentation that counterfactuals like (1) are not backtrackers and often come out intuitively false, contrary to what interventionist semantic theories would predict, see Fisher (2016).
I take Glymour’s The Mind’s Arrows to be an elaboration and defense of an idea along these lines (Glymour 2001).
References
Bennett, J. (2003). A philosophical guide to conditionals. Oxford: Clarendon Press.
Briggs, R. (2012). Interventionist counterfactuals. Philosophical Studies, 160, 139–166.
Fine, K. (1975). Review of Lewis’ counterfactuals. Mind, 84(335), 451–458.
Fisher, T. (2016). Causal counterfactuals are not interventionist counterfactuals. Synthese. doi:10.1007/s11229-016-1183-0.
Galles, D., & Pearl, J. (1998). An axiomatic characterization of causal counterfactuals. Foundations of Science, 3(1), 151–182.
Glymour, C. (2001). The Mind’s arrows: Bayes nets and graphical causal models in psychology. Cambridge, MA: MIT Press.
Halpern, J. Y. (2000). Axiomatizing causal reasoning. Journal of Artificial Intelligence Research, 12, 317–337.
Hiddleston, E. (2005). A causal theory of counterfactuals. Noûs, 39(4), 632–657.
Lewis, D. (1973). Counterfactuals. Blackwell Publishing. First published in 1973.
Lewis, D. (1979a). Counterfactual dependence and time’s arrow. Noûs, 13, 455–476.
Lewis, D. (1979b). Scorekeeping in a language game. Journal of Philosophical Logic, 8(1), 339–359.
Loewer, B. M. (1976). Counterfactuals with disjunctive antecedents. Journal of Philosophy, 73, 531–537.
Pearl, J. (1999). Probabilities of causation: Three counterfactual interpretations and their identification. Synthese, 121(1–2), 93–149.
Pearl, J. (2000). Causality: Models, reasoning, and inferences (1st ed.). Cambridge: Cambridge University Press.
Pollock, J. (1976). Subjunctive reasoning. Dordrecht: Reidel.
Pollock, J. (1981). A refined theory of counterfactuals. Journal of Philosophical Logic, 10, 239–266.
Rips, L. J. (2009). Two causal theories of counterfactual conditionals. Cognitive Science, 34(2), 175–221.
Spirtes, P., Glymour, C. N., & Scheines, R. (2000). Causation, prediction, and search (2nd ed.). Cambridge, MA: MIT Press.
Tichý, P. (1976). A counterexample to the Stalnaker–Lewis analysis of counterfactuals. Philosophical Studies, 29(4), 271–273.
Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford: Oxford University Press.
Acknowledgments
Thanks to Aldo Antonelli, Adam Edwards, Bas van Fraassen, Bernard Molyneux, and Ted Shear for their comments on earlier versions of this paper. Thanks to Alex Kocurek for some especially detailed and helpful comments on an earlier version. And most of all, thanks to my adviser, Adam Sennet, for all his help.
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Fisher, T. Counterlegal dependence and causation’s arrows: causal models for backtrackers and counterlegals. Synthese 194, 4983–5003 (2017). https://doi.org/10.1007/s11229-016-1189-7
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DOI: https://doi.org/10.1007/s11229-016-1189-7