Synthese (2012) 184:299–317DOI 10.1007/s11229-010-9814-3

What could be caused must actually be caused

Christopher Gregory Weaver

Received: 1 November 2009 / Accepted: 1 September 2010 / Published online: 23 September 2010© Springer Science+Business Media B.V. 2010

Abstract I give two arguments for the claim that all events which occur at the actualworld and are such that they could be caused, are also such that they must actuallybe caused. The first argument is an improvement of a similar argument advanced byAlexander Pruss, which I show to be invalid. It uses Pruss’s Brouwer Analog for coun-terfactual logic, and, as a consequence, implies inconsistency with Lewis’s semanticsfor counterfactuals. While (I suggest) this consequence may not be objectionable, theargument founders on the fact that Pruss’s Brouwer Analog has a clear counterexam-ple. I thus turn to a second, “Lewisian” argument, which requires only an affirmationof one element of Lewis’s analysis of causation and one other, fairly weak possi-bility claim about the nature of wholly contingent events. The final section of thepaper explains how both arguments escape objections from supposed indeterminacyin quantum physics.

Keywords Causation · Counterfactuals · Conditionals · Counterfactual logic ·Natural laws · Quantum physics

1 Introduction

Events which occur and are such that they could be caused, are also events which couldnot have occurred without at least a singular cause. Understanding events this way is abit audacious, but there are two interesting arguments for just such an understanding.

This, my writing sample to PhD programs in Philosophy, I dedicate to the memory of my Father GregoryGlenn Weaver.

C. G. Weaver (B)Rutgers University, New Brunswick, USAe-mail: christophergweaver@gmail.com

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The first argument appropriates an analog of the Brouwer Axiom in modal logic forcounterfactual logic (the Brouwer Analog). Alexander R. Pruss is the theoreticianresponsible for motivating arguments for the Brouwer Analog. His work on the natureof causation and explanation will be foundational for much of what will be the firstargument (hence my labeling it the “Pruss-argument”) for the thesis above. The sec-ond argument does not utilize Pruss’s Brouwer Analog, but instead requires that anecessary condition for causation be counterfactual dependence, and that all whollycontingent events be such that they could be caused.

I begin in Sect. 2 with an explication of the Pruss-argument, subsequently show-ing in Sect. 2.1 how an appropriation of the Brouwer-Analog leads to absurdity onLewis’s semantics for counterfactuals. In Sect. 2.2, I present a counterexample toPruss’s Brouwer Analog, thereby properly motivating the articulation of what I callthe “Lewisian” argument for my main thesis in Sect. 3. I then (in Sect. 3.1) explainhow the Lewisian argument escapes the counter-example of Sect. 2.2. The final task(Sect. 4) of the paper is to engage objections to the conclusions of both argumentsfrom quantum physics.

2 The Pruss-argument

Suppose that, by the phrase Rxy, we mean to pick out the relation x caused or is thecausal explanation of event y.1 The predicate assignment Ox will mean that some eventx occurs or occurred. Assume that, by the expression (p ⇒ q), we mean �(p ⊃ q),an entailment relation. Suppose further that, by (p � → q), we mean to pick out the‘would’-counterfactual, if it were the case that p, then it would be the case that q.Let us also assume that (p ♦ → q) picks out the ‘might’-counterfactual, if it were thecase that p, then it might be the case that q.2 Understand the relationship between twocounterfactual conditionals in the following standard way:

(p � → q) ↔ ∼(p ♦ → ∼q) (1)

(p ♦ → q) ↔ ∼(p � → ∼q) (2)

I will also need to assume: (a) K-semantics for possible worlds and propositionalmodal logic (including first-order alethic modal logic)3, (b) the S5 axiomatiziation forpropositional modal logic (and first-order alethic modal logic) where the accessibilityrelation between worlds is understood as an equivalence relation4, (c) actualism as the

1 I will assume that the types of things which stand in causal relations are events. Furthermore, I will assumethat events are property exemplifications. This view is both explicated and defended in: Kim (1976); reprint(1993).2 This notation is similarly used in Lewis (1973); cf. Harper (1981, pp. 24–32).3 cf. Schurz (1999), Schurz (2002, pp. 443–444)); cf. Plantinga (1982, p. 2).4 (Fitting and Mendelsohn, 1998, p. 71); Humberstone (2005, p. 571. no. 44 cf. pp. 572–573 esp. no. 46).

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applied semantics and/or interpretation of quantified modal logic5, and (d) Lewis’ssemantics for counterfactual logic.6

With the assumptions above, we can explicate a very interesting argument for ageneral causal principle. Assuming that our specified domain or universe of discourseis events, the first premise of the argument states that:

∀x∀y [Rxy ⇒ (∼∃zRzy �→ ∼Oy)] [Premise] (3)

Assuming that we have applied the inference rule universal instantiation to (3), wecan insert the following constants when relevant [x = e, y = e∗]. This premise isintuitively plausible, for it says that e’s causing e∗ entails that if it were not the casethat there is at least one event z such that z caused e∗, then e∗ would not have occurred.Premise (3) is consistent with overdetermination, and so in no way precludes that someother cause besides e stands in the relevant causal relationship with e∗. Now considerthe second premise:

∀y(Oy ⊃ ♦∃xRxy) [Premise]7 (4)

Premise (4) affirms that if e∗ occurs, then it is possibly the case that there is at leastone event x such that x caused e∗. There is no question that there are objections to(4), and I will turn my attention to those objections when I employ a similar premisein the Lewisian version of the argument. I should point out that my thesis is not that

5 Actualism is the thesis that there neither are nor could have been non-existent objects. This is defendedin not a few places: (van Inwagen, 2001, pp. 206–242); Plantinga (2003, pp. 103–121); Stalnaker (2003, pp.25–54) [though he holds on to counter-part theory]; Melia (2008, pp. 143–145); Adams (1974, pp. 211–231);Kripke (1980, pp. 15–20; 43–53).6 Lewis (1973); cf. Sider (2010, pp. 252–256); Williamson (2007, p. 293ff). I will also be assuming thatthere is a fixed domain of possible events, that the very same events are possible in each possible world,and that the aggregation of events is a species of conjunction. cf. Oppy (2006, pp. 125–126).7 Some might reject this premise on grounds that it might not hold with respect to all events which occurand are caused by some other event, if there are alien dispositions. Truth-makers for such statements aboutalien dispositions are alien properties. Alien properties are properties that are not possessed by any actualentities, and are not obtainable by means of a conjunction, interpolation, or extrapolation, of some actualproperties. See Lewis (1986: pp. 159–165); Heller (1998, pp. 298–308); Armstrong (1989a, pp. 57–63);Divers (1999); Divers and Melia (2002). It is thought that such alien properties could serve as truth-makersfor alien disposition statements. But David Lewis reduced disposition statements to counterfactual causalstatements, “then to purely counterfactual statements, and ultimately to statements about events in andsimilarity relations among possible worlds” McKitrick (2009, p. 42). I think Lewis’s view can accountfor conditional causal statements, and it can answer counter-examples from finkish dispositions cf. Lewis(1999) I agree with his reductive strategy. On Lewis’s view then, alien disposition statements are nothingabove and beyond alien counterfactual causal statements. The truth-makers for such statements are associ-ated with events in and similarity relations among possible worlds. If there are alien properties which makealien disposition statements true, and such disposition statements entail that certain events are essentiallysuch that they do not have causes. Fine. My argument here is not concerned with such events, since theycannot stand in the Rxy relation. Furthermore, as per the Lewisian argument below, I’m concerned withwholly contingent events. If there are alien properties and alien disposition statements about those propertiesinstantiated by objects, such a fact will be inconsequential to my argument. Since predicating to events thatthey are wholly contingent entails that such events are in no way alien.

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every event has a cause, but that every event that could be caused has a cause. Premise(4) is therefore not as robust as one might have gathered from an initial glance.

Two axioms of Lewis’s semantics for counterfactual logic will provide the Pruss-argument with premises (5) and (6):

[(p ⇒ q) ⇒ (p � → q)] [Premise] (5)

That is to say, p entails q, entails that if it were the case that p, then it would be the casethat q. Premise (5) suggests that entailment relations are stronger than counterfactualconditionals, assuming that the converse of (5) is false.

[(p � → q)&(p � → ∼q)] ⇒ ∼♦ p [Premise] (6)

Or, if it were the case that p, then it would be the case that q, and if it were the casethat p, then it would be the case that not-q, entails that p is impossible.

Premises (7) and (8) will be the Brouwer Axiom from propositional modal logic,and the Brouwer Analog for counterfactual logic:

(p ⊃ �♦ p) [Brouwer Axiom] (7)

(q & p & ♦∼p) ⊃ [∼p � → (p ♦ → q)] [Brouwer Analog for CFL] (8)

Now pick out any event that occurs. Let q be the true proposition that that event, callit e, occurred. Suppose we let p be the proposition that there is nothing which causese i.e., ∼∃x(Rxe). The rest of the argument can now run as follows:

Assume that p. [Assumption] (9)

From the fact that premise (3) states that every event which occurs is such that it couldhave had a cause, we can infer that p in our deduction is only contingently true. So:

(p & ♦∼p) [from (4) and (9)] (10)

(q & p & ♦∼p) [from our supposition that event e occurred conjoined with (10)](11)

∼p �→(p ♦→q) [MP (8), (11)] (12)

Suppose though that w is any possible world at which not-p is the case (as the ante-cedent of (12) has it), then w is also a world at which the event in question (vi z. e)occurs by virtue of being caused by some event x . This follows because nonexistentevents cannot cause other events to occur, and cannot themselves be caused to occur.Thus, e occurs at w, and so does the cause of e. Given (3), it follows that it will betrue at w that, were no cause of e to have existed, e would not have occurred (i.e., it istrue at w that (p � → ∼q)). Since this is true at every world at which e has a cause,

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(or at every world at which not-p is the case), it follows that:

∼p ⇒ (p � → ∼q) [from (3), (12)] (13)

(p � → ∼q) ↔ ∼(p ♦ → q) [from Lewis’s semantics for CFs] (14)

Recall premise (5) above. That premise states that the proposition p entails q, entailsthat if it were the case that p, then it would be the case that q.8 When we consider(13), with not-p as one proposition, the entailment relation, and then (p � → ∼q) asthe other proposition, it should follow (by entailment) that ∼p � → (p � → ∼q). Butthen, by our replacement rule from Lewis’s semantics, we get ∼p � → ∼(p ♦ → q).So we write:

∼p � → ∼(p ♦ → q) [from (5), (13) and (14)] (15)

Now it seems we have a proof that not-p is impossible, since the second of the two axi-oms we took from Lewis’s semantics for counterfactuals (premise (6) above) impliessuch an impossibility just as soon as we conjoin (12) and (15) into (16):

[∼p � → (p ♦ → q)] & [∼p � → ∼(p ♦ → q)] [Conj. (12), (15)] (16)

And now our conclusion follows in the following way:

∼♦∼p [from (6), (16)] (17)

(♦∼p & ∼♦∼p) [from Simp. and Conj. (10), (17)] (18)

p ⊃ (♦∼p & ∼♦∼ p) [CP from (9–18)] (19)

∴ ∼p [Reductio (19)] (20)

So, given the principles specified above, and the occurrence of any event, the argumenthere can run so as to give one the conclusion, that that event could not have been suchthat it did not have a cause. Of course, this only follows if that event can stand in theRxy relation, i.e., the event one plugs in must have a cause at the actual world only ifit could be caused.

It turns out that Pruss’s original formulation of an argument for (20), similar to theone I have given here, is logically invalid. Pruss appropriated premises (6), (12), and(15) above. The problem is that Pruss, in more than one place, supposed that, from(6), (12), and (15), one could infer9:

∼♦p (21)

But this is confused. Lewis’s axiom states that when the antecedent of two distinct‘would’-counterfactuals gets you (by counterfactual implication) inconsistent propo-sitions, one can infer that that antecedent is not even possibly true. What is important

8 Again, on the assumption that the converse is false.9 You see this from him here: Pruss (2006, p. 243); and Pruss (2009a, p. 67).

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to point out here though, is that the antecedent of (12) and (15) is not-p, not theproposition that p. Thus, what we can infer from the relevant lines is (17), not themistakenly derived (21). So Pruss’s argument is invalid, and the Pruss-formulationshould be preferred.

2.1 The Brouwer-analog and Lewis’s semantics

Pruss has argued for the plausibility of (8).10 For convenience, let us recall that, forpropositional modal logic, the Brouwer Axiom is the following:

(p ⊃ �♦ p) [Brouwer Axiom] (7)

And Pruss’s Brouwer Analog for counterfactual logic is:

(q & p & ♦∼p) ⊃ [∼p � → (p ♦ → q)] [Brouwer Analog for CFL] (8)

Suppose that p were the event of my hitting a baseball with a baseball bat. Supposethat q were the event of the baseball shattering a nearby window. Proposition (8)would then be suggesting that, if I really did hit a baseball, and the window really didshatter, then, in a nearby possible world w at which I did not hit the baseball, whatactually occurs (i.e., me hitting the baseball) is relevant for any truth evaluation of thecounterfactual claim about what occurs (or would occur) at w.

Lewis’s semantics for counterfactuals assumes the following with respect to the‘would’-counterfactual11:

(p � → q) ↔ [�∼p v w(p & q) <@ w1(p & ∼q)]12 (22)

This means that the proposition, if it were the case that p, then it would be the casethat q, holds, just in case the antecedent is false at all possible worlds, or some worldw at which both p and q are true, is closer to the actual world than a world at whichp and not-q is true. However, the problem is that, given (22) above, (8) is false.

Assume that the big conjunctive contingent fact (BCCF) just is the aggregate ofall contingently true propositions on the actual world conjoined into one large con-junction.13 Every world, were it actual, would have a respective BCCF. Let Aw0individuate w0’s BCCF, let q be Aw0, and let us also assume that w0 is the ac-tual world. The variable p will pick out ∼Aw1, where w1 is a metaphysically pos-sible world. Assume that (8) above is true. So if (Aw0 & ∼Aw1 & ♦Aw1), then

10 Pruss (2006, pp. 243–247).11 Where in premise (22) I use @ = the actual world; wp = p is true at a possible world w; and wp <@

w1∼p = a world w at which p is true, is closer to the actual world @, than w1, and let w1 be any world atwhich ∼p is true.12 Lewis (1973, p. 16); Sider (2010, p. 274); Bennett (2003, pp. 152–176); cf. Kvart (1986, p. 163).13 I find the set-theoretic objections to the coherence of something like the BCCF to be unconvincing. SeeWeaver (2009).

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[Aw1 � → (∼Aw1♦ → Aw0)]. By modus ponens, we can infer that the conse-quent (i.e., [Aw1 � → (∼Aw1♦ → Aw0)]) is true. Proposition (22) now suggests thefollowing:

(Consequence #1): [Aw1 & (∼Aw1♦→ Aw0)] <@ [Aw1 & ∼(∼Aw1♦ → Aw0)]However:

(Consequence #2): (∼Aw1♦ → Aw0)↔∼(∼Aw1� → ∼Aw0)

The proposition (∼Aw1 � → ∼Aw0) holds at w1 if and only if a (∼Aw1 & ∼Aw0)-world is closer to w1 than any (∼Aw1 & Aw0)-world is. Since it is quite obvious nowthat there is only one (∼Aw1& Aw0)-world, vi z. w0, and a (∼Aw1 & Aw0)-world justis a world different from w0 and w1, it follows that (∼Aw1 � → ∼Aw0) holds at w1if and only if no other world is closer to w1 than w0 is. This entails:

(Consequence #3): The closest world to any possible world w1 is w0, the actualworld.14

Instead of abandoning (8) above by deriving consequence #3, some logicians havetaken consequence #3 to be a reductio for Lewis’s semantics for counterfactuals.15

However, I think that Lewis’s analysis of the ‘would’-counterfactual is approximatelycorrect, the cost of which is consequence #3 (assuming that (8) is true), and that costis too high, especially when the counter-example to (8) in Sect. 2.2 is considered. Itherefore recommend abandoning (8).

2.2 A counter-example for Pruss’s Brouwer analog

The Brouwer Analog states that the conjunction of q and a contingent truth p, materi-ally implies the proposition that, were p false, it would be the case that, were p true,q might still be true, since things might be as they actually are. This was expressedformally by proposition (8) above. This premise is potentially problematic.16 Supposethat the truth of p was guaranteed by natural nomicity at the actual world. Giventhis assumption, had p been false, the laws would have been different than what theyactually are. It is very important, however, to say further that the laws would not nec-essarily be different in such a way that that difference would entail p’s falsehood. But

14 For a very similar argument (one that I am indebted to) showing this consequence of Lewis’s semantics,see Pruss (2006, p. 245)); cf. Pruss (2009a, p. 72).15 Though there are doubtless other problems with Lewis’s semantics, cf. Elga (2001) where it is arguedthat Lewis’s semantics for counterfactuals doesn’t yield the asymmetry of counterfactual dependence (i.e.,that later states of affairs or events depend counterfactually on earlier ones); cf. Rescher (2007, pp. 166–169) who attacks Lewis’s semantics on the grounds that its understanding of similarity relations is not welldefined; Loewer (2009) suggested in correspondence that an analysis of natural nomicity which understoodthat nomicity probabilistically would be problematic for Lewis’s semantics for counterfactuals. However, Iam in complete agreement with Timothy Williamson who admits that the “best developed formal semanticsof counterfactuals makes use of the apparatus of possible worlds or situations” (alluding of course to theLewis–Stalnaker approach). See Williamson (2007, p. 142. cf.).One logician has even attempted to provide a semantics for counterfactuals that is consistent with (8) above:cf. Pruss (2009b).16 Thanks to Ned Hall for correspondence that helped in developing this counter-example.

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now it looks like we have a counter-example to (8). Consider the following principlefor counterfactual conditionals:

(Nomicity Principle: NP): Necessarily, a counterfactual antecedent that is notcontrary to natural nomicity at the actual world should not take us to worlds atwhich the laws are different.17

With NP in hand, we can go on to understand the variables of (8) in the following way:let p be the proposition that, I am not moving faster than the speed of light, Ł will bea list of the laws of a p-world, and let q therefore be the proposition that the laws areŁ. Quite obviously, the laws at a p-world are akin to the laws at the actual world. Sonow it can be the case that, were I moving faster than the speed of light, then it wouldbe the case, that were I not moving faster than the speed of light, then it would still bethe case that the laws would not hold. Or we get:

∼p � → (p � →∼q) (23)

But [∼p� → (p� → ∼q)] is equivalent to [∼p � → ∼(p ♦ → q)]. And so now,given the contingent truth of p (me not moving faster than the speed of light), andthe truth of q (the laws are Ł), we get the claim that not-p leads to inconsistency.For, on the Brouwer Analog, we should have [∼p � → (p ♦ → q)], but insteadwe have derived [∼p � → ∼(p ♦ → q)]. And, according to Lewis’s semantics{[(p � → q)&(p� → ∼q)] ⇒ ∼♦p}. We can therefore infer ∼♦∼p. So now wehave [(q & p & ♦∼p) ⊃ ∼♦∼p], which further implies that the Brouwer Analogmaterially implies a contradiction. So, we should abandon (8) above.

The above argumentation would not be successful against the Pruss-argument ifnatural nomicity were understood in such a way that it coincided with metaphysi-cal necessity.18 Pruss is obliged to object to such a view of nomicity—in fact, hemust reject such a view since his ‘new cosmological argument’ will not run withoutsuch a rejection.19 So, he will have to look elsewhere for a proper response to thecounter-example on offer.

One might try to challenge the idea that natural nomicity guarantees the contin-gent truth of propositions, by denying that there are any such things as laws. Severalaccounts of nomicity recommend themselves in this respect: (a) (arguably) NancyCartwright’s theoretical realism, (b) Bas van Fraassen’s rejection of laws, and (c) the

17 This principle was recommended to me by Ned Hall. I should note two further things: (1) Laws can bedifferent in two ways (a) some law that actually obtains might not obtain, and (b) some law that does notobtain might obtain. NP is concerned with a difference of the former, i.e., (a), not (b). (2) Originally I thoughtthat David Lewis would have been rather at home with something like NP, though it is not altogether clearthat he would affirm NP, since he seems to think that small localized miracles are prices to pay for certaincontinuity across space-time cf. Lewis (1979, p. 473). However, he does say elsewhere that we should tryto avoid “even small, localized, simple violations of Law” (Ibid.: 472). I am therefore a bit unsure aboutwhether or not Lewis would have affirmed NP.18 This view of laws is defended by a gaggle of scholars: Smith (2001), Bird (2007), Swoyer (1982),Shoemaker (1998). Though I should add that for Shoemaker it is a contingent fact what laws obtain. Thanksto Barry Loewer for this correction.19 cf. Gale and Pruss (1999).

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Plantinga/Ratzsch view of laws.20 It is not at all clear that, on Cartwright’s view oflaws, one could not have the guaranteeing of the truth of a proposition by virtue ofwhat laws are like at the actual world. Cartwright’s phenomenological laws are truein nomological machines, but she does admit that they are sometimes true as found innature (as things in and of themselves per se).21 With respect to such naturally situatednomological machines, it does seem that phenomenological laws could guarantee thetruth of a proposition about the world. The van Fraassen, and the Plantinga/Ratzschpositions seem to me to be desperate since the latter entails occasionalism, and theformer depends too heavily upon the inference and identification problems. So, theredoes not appear to be any escape from the counter-example through appealing to theseexotic accounts of laws, and for those uncomfortable with nomic necessitarianism, anargument not dependent upon (8) will have to be proffered for my thesis.

3 The Lewisian argument

As was noted previously, Lewis’s semantics for counterfactuals does not fare wellwith premise (8) of the Pruss-argument. In this section, I present a modified versionof the argument, which gets us the same desired conclusion without invoking Pruss’sBrouwer-Analog. I call the argument “Lewisian” because it starts from the claim thatcounter-factual dependence is a necessary condition for causation, and because it isconsistent with Lewis’s semantics for counterfactuals.

Consider the following four principles:

(1) ∀x[Cx � → (∼Cx� → ∼Ox)]22 [Counter-Factual Dependence Principle](2) ∀x(�x ⊃ ♦Cx)23 [Weak Causal Principle](3) ∀x(�x ⇒ Ox) [Predicate Clarification](4) (�q & p & ♦∼p) ⊃ [∼p � → (p ♦ → q)] [Principle Alpha]24

The first principle states that, for any event x , if it were the case that x were caused,then it would be the case that if x were not caused, then it would be the case that x wouldnot have occurred. A consequence of this principle is that counterfactual dependence isa necessary condition for causation. This seems very plausible, a fortiori given Lewis’sanalysis of causation. I am well aware of the great plethora of objections to principlessuch as (1) (e.g., problems related to issues of early, late, and trumping preemption). I

20 Cartwright (1983, pp. 54–73) , cf. Hoefer (2008, p. 5ff), van Frassen (1990), Plantinga (2007, p. 132–133), cf. Foster (2007), Ratzsch (1987, pp. 383–402). The Plantinga/Ratzsch view takes laws to be truecounterfactuals of divine freedom.21 Cartwright (1999, pp. 49–59).22 The domain or universe of discourse is once again events. Let Cx mean that x has a cause. Let Ox meanthat x is such that it occurs.23 The designation �x will mean that x is a wholly contingent event. An event of S’s having P at ametaphysical index I is wholly contingent just in case S’s having P at I could have failed to obtain.24 This proposition is a schema of sorts.

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think that each of these would-be objections has corresponding responses to them.25

However, due to space constraints, I will be unable to reproduce those responses here,and can only refer the reader to the relevant literature.

With respect to premise (2), there are several ways we can motivate an affirma-tion. First, the adherent of the Lewisian version of the argument can simply take herbelief that (2) is true to be properly basic for her.26 If proper basicality is an intelligi-ble notion, then our adherent need only entertain would-be objections to (2) and notnecessarily provide independent non-epistemically circular propositional evidence for(2). Second, several philosophers have proffered rather plausible arguments for theclaim that all wholly contingent events are such that they could be caused.27

I have limited the admission here in (2) to wholly contingent events, because statingthat every event has a cause leads quickly to absurdity.28 For recall that, at the startof the paper (note one), I disclosed that I would be assuming that events are propertyexemplifications. Surely though, there is an event (on that analysis) that just is all ofreality, or at least the sum of all actual property exemplifications. Or, if we were toassume mereological universalism, we could posit that, the mereological sum S thatis all of reality, has the property of being such that it is all of reality (call this propertyP) at the actual world, is an event proper (call it E).29 Given the idea that all eventshave causes, E must have a cause. But, the causing of E must itself be incorporatedby S. It seems that we are caught in a vicious regress.

If we limit our claim to, all wholly contingent events have causes, then we canescape the reductio above. This is because S is not merely the sum of all whollycontingent events. We cannot coherently suppose that all of reality is an aggregate ofwholly contingent events. This is easy to show. S having the world-indexed propertybeing such that it exists at w, at the metaphysical index w, will not be wholly contin-gent.30 But S’s having that property at w, is a part of S. At every world, S has thisproperty at w. Since my analysis of a wholly contingent event demands that the eventof S’s having P at a metaphysical index I is wholly contingent just in case S’s havingP at I could have failed to obtain. S’s having the world-indexed property P at w isnot wholly contingent. Thus, S is not merely the sum of all wholly contingent events.

Suppose though, that you were to focus the objection in such a way that you main-tained that S should be restricted to the sum of all wholly contingent events. Is it not

25 See some of Lewis’s responses in Lewis (2004), cf. Paul (2009), and Ramachandran (2004), thoughRamachandran’s final analysis suggests that counter-factual dependence isn’t even a sufficient conditionfor causation (2004, p. 400).26 And here I’m assuming the following account of warrant and basicality: belief b has warrant for acognizer c if and only if c forms b with properly functioning cognitive faculties, faculties which functionin a congenial epistemic environment, an environment with a design plan, and an environment aimed attruth. Furthermore, there must be no successful defeaters for b. Now b will be properly basic for c when itis the case that c has not inferred b from propositional evidence, and when the experience of grasping theconceptual relations of the proposition b is about—just is the warrant-contributor for b. Plantinga (1993).27 Koons (2000, pp. 107–119), Pruss (2009b, pp. 67–70).28 I spell out what’s meant by a wholly contingent event in note 24 above.29 Van Cleve (2008, pp. 321–333), cf. Markosian (2008, p. 341) who seems to admit that mereologicaluniversalism is somewhat of a majority position among contemporary philosophers.30 Keeping in mind the Kimian analysis of events as some object S having P at an index I.

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the case that S’s having a cause is itself a wholly contingent event? If it is, then, if Cwere a cause of S (and call the event of some cause C causing S, E), then we wouldalso need a cause of E . As a consequence, it seems we are now stuck in the self-samevicious regress suggested above.

Following Robert Koons, the best way to respond to this objection is to note thatthere is no reason to think that E is a wholly contingent event. The event of the firstcause causing the cosmos, for example, would appear to be composed of two furtherevents: viz., the first cause on the one hand, and the cosmos on the other. The truth thatthe first caused the second, does not represent a third event E in addition to the firsttwo. Rather, each statement is about single-case causal connections which superveneupon the cause, the effect, and certain counterfactual dependence relations betweenthe cause and the effect. Therefore, the wholly contingent part of the causal nexuswhich has as its relata C and S, is simply S itself. As Koons notes, we are forced onlyto reaffirm that C caused S, and not that there is some further event E which requires acause.31 This response does assume that causal truths supervene on non-causal truths,such as counterfactual dependence relations (or stepwise counterfactual dependencerelations).32

There is a second response to the reductio argument on offer, which does not dependupon a reductionist counterfactual analysis of causation. Suppose that the cause C ,which brings about S, informs us about the causal activity of a god, or necessarybeing. If we agree with Aquinas and other medievals about this god being simple, thengod just is god’s activity, in which case god’s activity must be necessary itself, beingidentical to him. Thus, C’s bringing about S will not be a wholly contingent event thatis a part of the mereological sum that is S.33 It seems then, that the vicious regress isescaped.

We should therefore limit our claim about possible causation to wholly contingentevents. But is there an argument that all wholly contingent events could have causes?I have already alluded (cf. footnote 28) to the fact that some philosophers have prof-fered arguments in support of positive answers to this question. But one argument Ican quickly advance here suggests that support for the claim that all wholly contingentevents possibly have causes comes from empirical considerations. Every success inscience in reconstructing the causal antecedents of particular wholly contingent eventsprovides substantiation of (2). So there is inductive evidence for (2).34

Some libertarians about free will might object to (2). If the event in question is afree choice, then, by analysis of what a “free choice” is, it is metaphysically impossiblefor it to be caused.35 So how does the proponent of (2), even the one who claims that(2) is properly basic for her, respond to the libertarian?

Suppose, as Pruss does, that we could think of agents as the sorts of entities thatcould stand in causal relations. Assume that we have an agent a1 who performs an

31 Koons (2000, pp. 118–119).32 Paul (2009, p. 168).33 See on this Pruss (2009a, p. 76).34 See Koons (2000, pp. 111–112).35 Thanks to Tomis Kapitan for bringing this objection to my attention.

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action p freely (in the libertarian sense of the term). But now why can we not saythat a1 is the cause of p? Even if you thought that freedom is an intrinsic propertyof an action, we could still say that the state of affairs of a1 freely choosing p has acause, vi z. a1. We could also say that a1’s making the choice between p and q whilebeing impressed upon by reasons r1 − rn is the cause of p. So, if this understandingof agent causation is even possibly true, then it is possible that the relevant event ofa1’s freely choosing p has a cause, and the Lewisian argument can still run in a waythat is applicable to events possibly caused by agents.36

What about principle alpha? The principle says that, when q is necessarily true,and it is conjoined with a contingent truth p, such a conjunction materially impliesthat, were p false, then it would be the case, that were p true, q might still be true. The‘would’-counterfactual [∼p � → (p ♦ → q)], if true, is non-vacuously true, sincethe antecedent of (4) supposes that p is contingently true. Thus, if p were assumed tobe false, it is only at least false at one possible world and not necessarily false.37 Theproposition [∼p � → (p ♦ → q)] is true just in case the [∼p & (p ♦ → q)]-worldis closer to the actual world than the [∼p & ∼(p ♦ → q)]-world is. But, ∼(p ♦ → q)is true at no world, since it is quite easy to see how there is a p-world, viz. the actualworld (as the antecedent of (4) grants), that belongs to a sphere S in a system ofspheres centered on itself (call that system centered on the actual world $i, where i isthe actual world). Since q is a necessary truth, every sphere S in $i that contains atleast one p-world, will also contain at least one world where both p and q are true.Thus, ∼(p ♦ → q) is always false on Lewis’s semantics, given the antecedent of (4)above. So, the [∼p & (p ♦ → q)]-world is closer to the actual world than the [∼p and∼(p ♦ → q)]-world is. Therefore, [∼p � → (p ♦ → q)] will be true on Lewis’ssemantics for counterfactuals, on the assumption that (�q & p & ♦∼p).

Prior to exploring the Lewisian argument further, I will need to say more about myview of events. I construe events as property exemplifications, i.e., exemplificationsof properties by objects at an index. With Jaegwon Kim, the indices in question weretemporal indices. However, I affirm an inclusive disjunctive analysis, where eventsare exemplifications of properties by objects at an ontological index more broadly. Anontological index can include any of the metaphysical indices, particularly the indicesof space, time, and world.38 The exemplification of a property by an object at a worldis an event proper, irrespective of the fact that the object does not have a property at atemporal index.

If one is sufficiently realist in one’s ontology about propositions, then every worldwould have a corresponding BCCF.39 In fact, a world’s BCCF marks that world out asdistinct from other worlds. On the realist view of propositions I am assuming, propo-sitions can be rightfully understood as objects with properties. Moreover, propositionswould have world-indexed properties. For example, the property truth-in-w charac-

36 See Pruss (2009a)37 As a consequence, on Lewis’s semantics the proposition [∼p ♦ → (p ♦ → q)] will also be true. SeeLewis (1973, p. 21).38 Yagisawa (2010, p. 53).39 The BCCF on a world just is an aggregate of all contingently true propositions on that world. cf.Plantinga (2003, pp. 229–233) for an argument for realism about propositions.

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terizes all the propositions that are in fact true (where w is the actual world). Only theBCCF at the actual world will have the unique property of being such that it is actual,since the actual world’s BCCF has the unique property of being such that it is true atthe ontologically privileged world (on actualism).40 To illustrate this, one need onlygive attention to the fact that, had w been actual and w1 not, the essence of w1’s BCCFwould not have been exemplified. For some object x , the property of being such thatx is actual, entails the property of being such that x’s essence is exemplified, where aproperty p entails a property q, if it is not possible that p be exemplified by an objectthat lacks q. While the essence of the BCCF for w1exists, it remains unexemplified atw the actual world. The above follows only if essences are conceived of, as propertieshad by objects essentially and incommunicably.41

Give attention with the minds eye to the world at which nothing more than nec-essarily existent objects exist. Call that world w2.w2 has a respective BCCF, and w2is the only world at which its BCCF has the property of being such that its essenceis exemplified since the BCCF is a mereological sum of contingently true proposi-tions.42 By consequence, w2 it is the only world at which the essence of its BCCF isexemplified. In fact, no matter what world w − wn is actual, the event of that world’sBCCF having the property of being such that its essence is exemplified at that worldwill be a wholly contingent event. That event could have failed to obtain preciselybecause the BCCF in question could have failed to exemplify its essence.

One very interesting consequence of my argumentation is that, for any world w thatwe presume is actual, the wholly contingent event of that world’s BCCF having theproperty of being such that its essence is exemplified at w, occurs. We can thereforebegin an explication of the Lewisian argument with the following assumption:

(5): ∃x(��x & ∼Cx) [Assumption]

The ordinary reading of this premise on S5 quantified modal logic is that there isat least one event x such that x is essentially wholly contingent and x does not havea cause. I have argued that, no matter what world is actual, there is some event at thatworld that is wholly contingent. I have further argued that we can confidently affirmthe left conjunct of (5) with its quantifier, since at the actual world there is some eventthat is essentially wholly contingent (viz. (where w is the actual world), w’s BCCFhaving the property of being such that its essence is exemplified at w).43 We can nowinfer the following:

40 In fact, I will use the locutions, ‘being such that it is true at the ontologically privileged world’, and‘being such that it is actual’ interchangeably.41 This view of essences has been defended nicely in Plantinga (2003, pp. 111–120).42 There are contingently true propositions at worlds at which everything that exists is a necessarily existentbeing. For example, it will be contingently true that that maximal state of affairs obtained. That is to saysome other world could have been actual. Likewise, at that world, it will be contingently true that AlvinPlantinga does not exist.43 This event (call it E) is essentially such that it is wholly contingent in that this event only occurs at w,since w is the only world at which S can have the property of being such that its essence is exemplified.Therefore, E only exists at w. On actualism, a property p is essential to an object o, just in case at all worldsat which o exists, o never fails to have p. Since w is the only world at which E occurs, all the properties ithas at w will be essential to it. Therefore, its being such that it is wholly contingent will be essential to it.

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(6): (��ê & ∼Cê) [EI (5)](7): (�ê ⊃ ♦Cê): [UI (2)](8): �ê [Nec. Elim., Simp. (6)](9): ♦Cê: [MP (7), (8)](10): (��ê& ∼Cê & ♦Cê): [Conj. (6), (9)](11): [Cê � → (∼Cê ♦ → �ê)]: [DN, MP (3), (10)](12): [Cê � → (∼Cê ♦ → Oê)]: [(3) an events being wholly

contingent entails that it occurs](13): [Cê � → (∼Cê � → ∼Oê)] [UI (1)](14): [Cê � → ∼(∼Cê ♦ → Oê)] [from (13) by Lewis’s semantics

for CFs](15): ∼ ♦Cê [Lewis’s semantics for CFs44](16): (♦Cê & ∼ ♦Cê): [Conj. (9), (15)](17): ∃x(��x & ∼Cx) ⊃ (♦Cê & ∼ ♦Cê): [CP (5–16)](18): ∼∃x(��x & ∼Cx) [Reductio (17)]

And obviously, from (18), we can infer that there is no wholly contingent eventwhich occurs without a cause:

(19): ∼∃x(�x & ∼Cx) [Nec. Elim (18)]

3.1 The Lewisian argument and counterexamples

The reductio argument from Sect. 2.1 will not run per the Lewisian argument, sincethe Lewisian argument does not make use of the Brouwer Analog, but instead appro-priates principle alpha, which, as I have shown, is thoroughly consistent with Lewis’ssemantics for counterfactuals.45 Likewise, the counter-example of Sect. 2.2 will notbe applicable per the Lewisian argument, since that counter-example maintained thatq was the proposition that the natural laws are what they actually are. On non-neces-sitarian accounts of laws, this is merely contingently the case. Thus, in order for thereto be a strict implication relation between q, and p’s contingent truth, q must be acontingent truth. But, the Lewisian argument affirms principle alpha, and, as a conse-quence, suggests that q is a necessary truth.46 Therefore, the Lewisian version escapesthe reductio of Sect. 2.2.

4 Objections from quantum physics

I have argued that, it is a consequence of our best logic of counterfactuals, and maybeour best understanding of at least one necessary condition for causation that, all whollycontingent events have causes. Some scientists and philosophers, however, think thatthis conclusion faces opposition in the physics of quantum mechanics (QM). I wantto turn my attention to such opposition here in this section.

44 The axiom used here to make this inference is: [(p � → q) & (p � → ∼q)] ⇒ ∼p ♦.45 I explained just how it is consistent with those semantics.46 cf. Principle alpha above.

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Is the quantum state of a physical system at the end of a temporal interval, deter-mined by its quantum state at the beginning of the interval? Yes. The evolution ofthe state is governed by the famous Schrödinger equation, and this determines thequantum state at a later time from its form at the beginning of the interval. Supposedindeterminism is introduced at the stage of determining the values of the outcomes of ameasurement.47 Specifying the observable quantity we use the measurement relevantphysically possible outcomes for the measurement process is fixed (see footnote 47).But, the quantum state attributed to the system allows one to infer that a given one ofthese outcomes will arise with a certain probability. Knowing that a system has “beenprepared in a given manner at one time, and even knowing that the system has notbeen interfered with in a given time interval, will not generally let us predict that oneand only one value of an observable will be obtained, if that observable is measuredat the end of the time interval (see footnote 47).”

Although the quantum state does not fully determine which of the outcomes willobtain, the outcome that does obtain might be determined by some factor not taken intoaccount by the quantum state. Quantum states could be less than complete descriptionsof the world, and something outside the quantum state may play a role here. DavidBohm’s view of QM invoked an idea like this, and his ideas were later sharpened byJohn Bell.48 The Bohm–Bell interpretation uses the same empirical content that wesee in QM, and it also utilizes the same mathematical formalism peculiar to QM. Thedifference between it and other interpretations is its underlying metaphysic. Bohmand Bell assumed that every material particle has a perfectly determinate position.49

For Bohm and Bell, the motion of particles is completely deterministic. Probabilitieson this view of QM are merely epistemic. The so-called modal interpretations of QMare very similar in that they also (arguably) view the probabilities merely epistemi-cally.50 And, in the literature, (despite the brief attraction to von Neumann’s proof),most philosophers of science seem to be comfortable with the claim that there is noth-ing inconsistent in the postulation of deterministic models of quantum phenomena.Even the no hidden-variable argument of Kochen51, will not hold up against view-ing the nature of a measured quantity contextually. The Kochen proof attempted toshow that, the interrelationship among measured values predicted by quantum the-ory, are incompatible with any possibility of these values being fully determined byunderlying values of hidden parameters. Such a conclusion will be false, given theunderlying metaphysic of an interpretation akin to Bohm–Bell, or modal views ofquantum phenomena. So, quantum physics seems to provide my argumentation withno real defeaters, since the Bohm–Bell interpretation of the phenomena appear to beat least possibly true.52

47 Sklar (1992, p. 205).48 Bell (1987).49 Albert (1992, pp. 134–135).50 Bohm (1952), van Frassen (1991).51 Kochen (1985, pp. 1–20).52 cf. Salmon (1998, pp. 280–281).

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I should add that, even if Copenhagen interpretations of quantum phenomena werecorrect, I have not specified precisely what I mean by the Rxy relation with respectto the Pruss-argument. There is a significant group of philosophers of science andmetaphysicians who adhere to probabilistic analyses of causation.53 On such analy-ses, the presence of spatio-temporally continuous causal processes between events isnot required for real causal connectivity between those two self-same events. On theprobabilistic analyses, what is important to causal relations is the fact that changinga cause makes a difference to the relevant effect(s), and that such difference makingshows up in probabilistic dependences between cause and effect.54 This is particularlypertinent because the criticisms of the theories of Good, Suppes, and Reichenbach55

(adherents of probabilistic analyses) lean heavily upon the suggestion that causal rela-tions involve spatiotemporally continuous causal processes which join any two causalrelata in view together. The odd thing is that while folks like Wesley Salmon andothers affirm this, they admit that their demand for such connectivity is moot at thequantum level.56 But what good is our analysis of causation if it has counterexamplesat the quantum level?

If Salmon and others were to suggest that there just is no such thing as causation inquantum physics, then we will need an argument not dependent upon the stipulationthat causation requires spatial-temporal contiguity via causal processes between causeand effect. An appeal to the absence of causation in quantum physics by definitionalfiat, in no way shows that probabilistic analyses (or even counterfactual analyses ofindeterministic causation) are mistaken. For, if the non-classical correlations betweenobservables in quantum entanglement can be appropriately understood in terms ofone observable being counterfactually dependent upon another, or in terms of oneobservable making a difference to the other, and that difference showing up in proba-bilistic dependence, then it would seem we have causation at the quantum level with-out spatio-temporally continuous causal processes joining the two causally connectedevents or objects together.57

What I have suggested in the above discussion, is that if the counter-example ofSect. 2.2 can be overcome by advocating a necessitarian view of laws, and furthermore,if one could provide a semantics for counterfactuals that was consistent with Pruss’sBrouwer Analog58, probabilistic analyses of causation provide the proponent of thePruss-argument with good reasons for embracing even indeterministic interpretationsof QM without thereby jettisoning the conclusion of that argument. Moreover, with

53 Good (1961, 1962), Salmon (1998, pp. 208–232), Hall (2005, pp. 505–506).54 Williamson (2009, p. 187).55 Suppes (1984), Reichenbach (1946).56 cf. Salmon’s comments in Salmon (1998, pp. 280–281).57 One may still very well have present spatial-temporally continuous causal processes joining the imag-ined events or observables in question, even in the quantum entanglement case. For on something likethe Bohm-Bell interpretation of QM, a guide wave exists which connects the observables (the particles)such that the hidden variables are the particles themselves existing as functions of the guide wave. So evenif one thought there should be spatial-temporally continuous causal processes joining causally connectedevents, on Bohm’s interpretation of QM we have just such contiguity even when considering quantumentanglement.58 As perhaps what one sees in Pruss (2009b).

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respect to the Lewisian argument, one need only appeal to the possible truth of deter-ministic interpretations of quantum phenomena in order for the relevant conclusion tofollow.

5 Conclusion

Every event that occurs at the actual world that could be caused must be caused. Thatwas my audacious thesis. I have provided two deductive arguments for this thesis. Oneargument requires a non-standard semantics for counterfactuals, the other argumentdoes not, since it rests on weaker principles consistent with Lewis’s take on counter-factuals, as well as his analysis of causation. The Pruss-argument cannot be affirmedwithout embracing an exotic view of natural laws so as to escape an intuitively plau-sible counter-example to one of its premises. The Lewisian version of the argumentis therefore preferable. Quantum physics is nothing to worry about, since causationcould be probabilistic in nature, and Bohm–Bell interpretations (and perhaps modalinterpretations) of QM are at least possibly true.

Acknowledgment I would like to thank Valia Allori, Joshua Armstrong, Carl Gillett, Ned Hall, TomisKapitan, Timothy O’Connor, J. Brian Pitts, Alexander Pruss, Joshua Rasmussen, Michael Sweiger, andChristina Weaver for their comments on this paper.

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