Counterfactual conditionals describe what would happen if some event or condition or property that has not occurred in fact or belief had occurred.
Most contemporary studies make use of possible-world semantics to analyze counterfactuals but the project of making sense of suppositional and hypothetical events is not new. David Hume explicitly articulated an argument for causation in the very form that persists today. “We may define a cause to be an object followed by another, and where all the objects, similar to the first, are followed by objects similar to the second. Or, in other words, where, if the first object had not been, the second never had existed,” he wrote. Were it not for Hume's insight, it could be argued, there would be no coherent theory of counterfactuals. If we had made that argument, it could be the case that we would have been wrong.
The 'standard' explicatory apparatus of modern counterfactual logic owes mostly to David Lewis, whose exhaustive treatment of the subject, Counterfactuals, remains—despite substantial revision—the de facto account. However, it was the early efforts of Stalnaker, Mackie, and Ramsey that proved instrumental in the incipient development of the coherent truth-conditional semantical framework that permits the prevailing orthodoxy in logic where it remains a field of broad applicability and active interest. The power of counterfactuals is particularly seductive to Bayesians and decision theorists, for whom it accommodates quantified probability valuations, a purpose for which it is put to normative ends, and to philosophers of mind, for whom the salvation of mental causation is thought to reside in the generous logic of modality. Of course it is also a form of reasoning we use every day—an inescapable and constant presence in our minds.
The notion of causation in counterfactual conditions should be broadly understood as imposing relations of dependence, a concept that has proven immune to facile designation as either sufficient or necessary. Lewis himself mostly resisted these categorial strictures, though any robustly conceived counterfactual account vindicates both interpretations. It is sometimes argued that counterfactuals concede necessity only when forced to account for preemption (c.f. Ruben, 1981) but by Lewis's inverse formulation, ¬A → ¬B, that is, 'there would/might not be B if it were not for A,' causal necessity inheres in the logic itself (and is in fact explicit in the concentric-worlds graphs in Counterfactuals). The positive expression A → B, that is, 'if A were the case then B would/might be the case,' on the other hand, broadly evinces sufficiency. It is thus evident that both concepts are subsumed in counterfactuals; they do not, however, abide the seemingly analogous equivalency relations found in the indicative mood or in the logic of material implication. Further, the mores of formal semantic and syntactic structure in counterfactual logic are inchoate at best and so do not submit to the rigor found in many other logics. (Note that the capital letters in our symbolic notation denote events for which truth valuation is measured ontologically; semantically, then, it is sometimes appropriate to regard these variables as representing propositions as well.)
We will adopt Lewis's notation for counterfactual conditionals wherein the necessity operator '□' and the possibility operator '◇' are appended to and modify the modal force of the subjunctive conditional operator '→' and not to the antecedents or consequents themselves; this notation honors the precision of the model since notions of counterfactual similarity are not measured in a purely technical sense. To negate the consequent is to assert a denial of the conditional; it does not necessarily affirm the relata. The proposition, A □→ ¬B can be understood as 'if A were the case then B would not be. ' To negate the proposition is, in contrast, to deny its truth-value. We should interpret the proposition ¬(A ◇→ ¬B) as 'it is not true that if A were the case then B might not be.'
Bivalent considerations recommend that we understand that A, if B is not possible, necessitate the occurrence of something else—i.e., a non-B. It has to be the case then that the negation of this condition compel the occurrence of B. The proposition A □→ B is thus equivalent to ¬(A ◇→ ¬B). A ◇→ B is likewise equivalent to ¬(A □→ ¬B), which asserts that it is not true that A denies the possibility of B. Moreover, by this manner of reasoning we see that A ◇→ B ≡ ¬(A □→ B); so A □→ B ≡ ¬(A ◇→ B). This model (which preserves duality) follows from □A ≡ ¬□¬B. This is, however, an interpretation that invites the objection (among others) that it fails to disambiguate likelihood and possibility. One recourse we have to solve problems of this nature is to impose pragmatic considerations on the deductive criteria. However, the doxastic improvements on offer threaten to weaken the theory to a point of provisional fluff.
In any case, counterfactual conditionals are intelligible only insofar as it is possible to individuate the causal relata. Events that are causally consequential to and conditionally dependent upon their antecedents are obviously dependent nevertheless on innumerable peripheral conditions; absent any delimiting metrics of salience or relevance the domain of necessary causes is therefore constrained only by the temporally-receding light-cone of physical cotenability, and this constraint holds only in worlds contingently similar to the actual world. "Beyond this,” McGee argues, “to try to say which of the many selection functions that originate at the actual world is the actual selection function, we rely on pragmatic considerations in the form of personal probabilities."
In order to elucidate effectual causes, we must therefore exclude—or at least relegate to the contextual background—consequentially promiscuous causal conditions and events on grounds of relevance from the outset. Certainly it is true, for instance, that a forest fire counterfactually depends on the presence of oxygen, but if the fire were started by an arsonist's match we are led by intuition and causal parsimony to preferentially assign the causal responsibility (to say nothing of culpability) for the fire to the arsonist, because only the latter establishes causal necessity with specific efficacy for the putative effect.
We must note, however, that the causal relation between the arsonist and the fire is not necessarily symmetrical (although the evident disparity in any case is conditioned by the time and manner in which it is measured) but the salient point is that it does not disclose the flagrant causal asymmetry of a background condition like oxygen. After all, the presence of oxygen is necessary for the presence not only of the fire, but of the forest and the arsonist as well; the arsonist's causal influence, on the other hand, cannot reciprocate this broad causal context; it begins with the fire. This influence might spread, along with the fire, through space-time, but what it cannot do is exceed the causal constraints of its contextual conditions across possible worlds.
The valuations of counterfactual truth conditions are measured against a conceptual contrivance (we will ignore for now what its actual ontological status is) in which suppositional changes to the actual world are simulated as relevantly deviant possible worlds and graded by similarity to ours, in which the conceptual modifications accede to a quantifiably reified ontology for didactic or methodological purposes. David Lewis went for broke defending the claim that these other worlds are as real as you and I. This is not the mainstream view, but given the difficulty adducing evidence for any position in this debate, and because the utility of possible worlds semantics persists under any metaphysical judgment, it is a point that can be ignored.
The logical form of the counterfactual conditional proposition, A □→ B ("if A were the case, then B would be the case”) is given foundation by several considerations of dependence. Lewisian counterfactual dependencies are conjunctively expressed as, for instance, (Ax → Bx iff (A1 → B1) ∧ (A2 → B2)), etc. In terms of Lewis's weighted counterfactuals, the crucial evaluation is: A □→ B iff B is true in the A-world most like the actual world. Much of the discourse in counterfactual logic is preoccupied with making sense of this claim. What, for instance, should constitute this likeness—the ‘comparative similarity’? And to what extent can this be objectively determined?
A second, more fundamental notion of dependency, causal dependency, is also required, as we have seen. E depends on B iff B → E and ¬B → ¬E. We have also seen that the counterfactual dependency sequence is insufficiently precise. Hence, we should prefer to establish a causal sequence, such as BB iff (b, d, e) where (b → d) ∧ (d → e), and (b → e) iff BB entails (b. . .e). Salient causes are picked out from the background causal clutter; a given cause C is considered to be independent from A1, A2 iff C is truly invariant of Ax. (This may ultimately be a promise of futility however, because, as Leibniz observed long ago, "Since each thing influences all the others in such a way that, supposing that it were abandoned or changed, all things in the world would henceforward be different from the way they actually are.")
Lewis proposes a nomic dependency as well. This feature of his theory profiles too Humean for many philosophers who would prefer to defend a logical or metaphysical relation instead. Nevertheless, Cx depends, if we adopt this requirement, on Ax iff there are sets L [law] and F [fact] such that (L ∧ F → (Ax → Bx)) ∧ (¬F → (Ax → Bx)). In conventional causal formulations, events M and P (by convention, designating the mental and the physical, respectively) are not—by Kim’s exclusion principle—distinct. But this doesn't hold in the counterfactual mode, so we have: C → E iff (¬(C = E) ∧ (¬C → ¬E)), and C → E iff C1 → C2, etc. But should (¬(C1 = C2) ∧ (P → (E ∧ M)) → E or if P and M are distinct and (P → E) ∧ (M → E) then overdetermination threatens. While such a scenario could theoretically proceed from a premise conflict outside the strictures of counterfactual dependency, the independence principle—given the strong counterarguments that obtain in its absence—is the requisite contrivance in Lewis's theory—and necessarily entails this change. It also permits Lewis to revert—with minor modifications—to standard rule-weighting. For Kim's theory of mind, the independence theorem threatens to undermine the exclusion principle.
This is precisely what Barry Loewer alleges. Paul Raymont, however, dismisses the charge. "Given that strong supervenience relations hold across the nomologically possible worlds," he claims, "anything that has M in any such world also has a physical property (other than P) from the supervenience base of M, and this physical property will (for the reasons set forth in Kim’s exclusion argument) have at least as strong a claim to efficacy with respect to P* as M has." In either case the application of counterfactual semantics would still evince proximate causal relations. The causal candidates consist of statements to be evaluated by a similarity relation. For instance: Intention Ix → Action Ax vs. Physical-event Px → Action Ax. This pair can be helpfully broadened to betray the winner by contrastive distinction: Intention Ix → Action Ax and Intention ¬Ix □→ Action ¬Ax vs. Physical-event Px □→ Action Ax and Physical-event ¬Px □→ Action ¬Ax. Since physical-event ¬Px □→ Action ¬Ax is false, the intention relation should be selected in defiance of Kim's exclusion principle, and the preceding difficulty is made nugatory. However, it does not follow that Kim's exclusionary principle should be forfeited. It would, however, persist only as a contingent condition and not, as Kim claimed, an analytic truth. (Raymont, 2003).
╭─ ─ ─ ─ ▶─ ─ ─ ─╮ Ⓐ──▶ ───╮
Ⓑ ──▶◯Ⓐ─ ╮ / ⑊ /
╭◀─╯ \ / Ⓑ────▶◯Ⓒ●◀─╯
(2a) ●Ⓒ \ /
╰─▶Ⓓ◀─╯ (2b)
Event D in figure (2a) represents Daisy's possibly being killed by Carl (the diagram assumes Carl's success, depicted by the black arrow). But because Bob fails to prevent Anna from preventing Carl from killing Daisy (the white circle signifies failure to causally intervene) it seems reasonable to conclude that Bob, in some sense, caused Daisy to be killed. After all, were it the case that Lewis had not acted, Daisy—thanks to Anna—would (very probably) still be alive; thus, we must, in asserting that B → D express a sense of dependency that exceeds causal sufficiency. Since D is dependent on B, B satisfies conditional necessity for D. This is more clearly expressed by its inverse ¬B → ¬D. Hume conflates (and Lewis equivocates) the two formulae; both express the notion of changing a part of the world—a part that makes a difference—by supposition.
In other words, if Bob doesn't stop Anna then Carl doesn't kill Daisy. Bob's actions by themselves are of course insufficient to bring about D; the point here is only that so long as they are necessary then Carl's actions too are insufficient for D. We might suppose then that the concomitant counterfactual expression of necessity—that is, 'if Bob's action weren't the case then Carl's action would not be'—should assume the form ¬B □→ ¬D. But this isn't satisfactory, as the following example illustrates.
In Figure (2b) we suppose that C now stands for the killing of Carl, a fate both Bob and Anna wish to bring about. Bob's obligation to do so, however, is obviated by Anna's murderous initiative. His causal distance is indicated in figure (4b) by the white circle. But were it the case that she lost her resolve, Bob would have nevertheless taken action. Carl's demise was therefore ineluctable, and so because ¬(¬A ◇→ ¬C) ⇒ □(C) it cannot be the case that ◇(C). Furthermore, we can strengthen the case against Carl and for causal redundancy by supposing that, after Anna fired her gun, Bob went ahead and shot Carl anyway, or by supposing that he fired his gun after Anna fired hers, but before the bullet from her gun killed Carl. However, the actions of neither killer—in contradiction to the scenario above—are necessary; instead, each killer, per se, is seemingly sufficient for this fate. I say 'seemingly' because the apparent delimited sufficiency of this model is simply a function of diagrammatic parsimony. The world, on the other hand, provides no such categorical delineation, only innumerable entangled causes.
Counterfactual theories of causation must contend as well with the odd case in which the alleged counterfactual event is simply absent. Let us suppose that Bob—in his haste to prepare for a vacation—forgets to feed his fish before departing. Are we justified in concluding that his not doing something—his not feeding them—should be the cause for their subsequent demise? After all, it does seem as though, absent Bob's forgetfulness, his fish wouldn't have died. We might reason counterfactually therefore that had Bob not done nothing, the putative causal consequent would not have obtained, which we can logically state as ¬B □→ ¬E. But this ignores a fundamental problem: what exactly is the ontological character of B? The short list of possibilities comprises: (a) a non-actual possible event; (b) an actual displacing event; and finally, (c) a simple fact.
The likeliest candidate seems to be (a) simply because not feeding fish is not something that Bob or anyone else does—despite a misleading grammatical inflection that can seem to impart semantic weight—but rather something that one does not. The event in question then exists only as the fact (c) that Bob did something else. That something else is the displacing event (b) and it is has been argued that (a) simply is (b). But such an argument is tragically indistinct since we would not expect a counterfactual to disclose a symmetrically feasible form analogous to the hypothetical event in question.
Let us suppose nevertheless that Bob's packing lunch (or some sub-event within or duration thereof) should be regarded as the veridical cause of his not feeding his fish. We should thence claim that were it not for his packing he would have fed his fish. But why should we not suppose with the very same confidence that Bob—alleviated of his packing chores—would not have spent the time arranging his matchsticks? After all, such is an endeavor suggested by the very same authority of evidence. Furthermore, neither (a) nor (b) constitute anything which might be confused for a cause and all three possibilities are an affront to physics.
But why should we consider contra-causal outcomes solely in terms of Bob at all? It is understandable that we should have this doxastic bias; but the argument is counterfactual, and the event deviance offers no assurance of comparative modesty.
t1 t2 t3 t1 t2 t3
Ⓐ─/───/──▶Ⓒ Ⓐ──/▶Ⓑ──/▶Ⓒ──/▶Ⓓ──▶Ⓔ
\ / / / ⑊ / ↗/
╰──▶Ⓑ─────▶ Ⓕ──▶Ⓖ──▶○Ⓗ
(3a) (3b)
Suppose that as figure (3a) depicts there is an event A that is always followed by B and C, such that whenever event A occurs, B occurs and then C occurs. From this regularity it would appear that the regularity of this conjunction is eligible for the accusation of backtracking inasmuch as one could claim that C □→ B and suffer no fear of disconfirmation because, as depicted, such a condition would always obtain. Furthermore, absent any information to the contrary, C□→B would exhibit similar regularity. If we should prefer a forward-tracking interpretation—out of presumptive temporal bias, or to give account for B's occurrence, for instance—we might prefer instead, and again without countervailing evidence, that B □→ C. Andyet the backtracking condition persists here too, along with ordinary forward-tracking temporal progression. One could even infer a transitively causal time loop through B→A→C. It is true that, in terms of sufficiency and necessity, that A causally accounts for C such that ¬A□→ ¬C a condition that B cannot match. It stands to reason nevertheless that if B obtains whenever A occurs, and if B can in fact sustain its fraudulent causal credentials indefinitely, in virtue of what is A entitled to causal authenticity?
We might similarly argue on the basis of figure (3b), that: ¬C □→ ¬D and ¬C □→ ¬B. It is of course true, as in (a), that if it weren't for C, it wouldn't be for D; the claim that ¬C□→ ¬B in figure (b) exhibits a rather different relation: one that tracks back through time. The antecedent of ¬C □→ ¬B causally circumvents B's inhibitory authority over H; thus ¬B □→ (H ⩘ D). This in turn then implies H □→ D. But if H □→ D is true then ¬C □→ ¬D is not. However, from the vantage of t2, it is seemingly impossible that ¬C □→ ¬D could be wrong, since it is rather obviously true that B □→ ¬H. The problem seems rather to inhere in the backtracking claim ¬C □→ ¬B. If this modal proposition is invalid, it does not follow, however, that C □→ ¬B. The situation would be correctly described instead by the possibility operator, the notation of which is just C ◇→ ¬B. But C ◇→ ¬B precludes neither C ◇→ B nor ¬C ◇→ ¬B, the latter of which, predictably, admits the possibility of E's occurring in its absence. This is a troublesome logical exigency, since ¬C ◇→ D both denies ¬C □→ ¬B and affirms ¬C □→ B. Although ¬C □→ ¬B may seem unimportant compared to the more intuitive ¬B □→ ¬C or perhaps even dispensable, it is inescapably troublesome—whatever the temporal structure might be—that a case should eventuate wherein B occurs and C does not; conversely, under no circumstance could C occur without B's occurring. This, however, is precisely what is implied by ¬C ◇→ D.
It does not follow from these data however, that backtracking is possible or even meaningful; the likely culprit is the logical schema. Lewis was careful to avoid opening his theory to accusations of "back-tracking"; to that end, he prudently avoided temporally-asymmetric constructions. "We very easily slip back into our usual sort of counterfactual reasoning, and implicitly assume that facts about earlier times are counterfactually independent of facts about later times," he observed. Nonetheless, Lewis did account for the asymmetry of causality; these relations are handled by dependency hypotheses whose counterfactual constructions mediate agent-contingent state-responses. But Adam Elga discovered these machinations did not extend to thermodynamical processes. Although Lewis claimed "it is of the first importance to avoid big, widespread, diverse violations of actual law," his ambiguous assertion of possible world law-compliance is less than emphatic. "Careful attention to the dynamical properties of thermodynamically irreversible processes shows that in many ordinary cases, Lewis’s analysis fails to yield this asymmetry," Elga noted.
It is generally acknowledged that transitivity too creates problems for—or perhaps simply does not persist in—counterfactual conditionals at all. The archetypal specimen in the academic literature, however, is far from persuasive. By conventional reasoning it is true that 'if Hoover were a communist he would have been a traitor' and that 'if Hoover were born in Russia he would be a communist,' but that the consequent of the first supposition does not follow from the antecedent of the second, since the condition of being a Russian is seemingly insufficient to ensure one's being traitorous. But so too is being a Hoover; therefore whatever suffices to validate the first supposition in question might therefore give it transitive legitimacy. Nevertheless, given the depth and breadth of the salient cotenability domain, this example (which goes back to Stalnaker) is simply unconvincing.
Suppose instead that I perform an action A—flipping a light switch, say—and that in doing so, I illuminate the room. Suppose moreover (in accordance with the Davidsonian pedagogical tradition) that my flipping the light switch also scares off a prowler of whose presence I am unaware. We can interpret the causal character of this sequence of events thus:
A ────▶ B A ──▶B
╰──────▶ C \ ↘
╰────▶ C
(4a) (4b)
Figure (4a) adequately represents event C's dependency on action A. My flipping on the light switch is, after all, a necessary condition of the prowler's being alerted by the light. This condition only obtains, however, if my flipping the switch does in fact bring about B, the illumination of the room. So B is also a necessary condition of C, as depicted in figure (4b). It is rather obvious, however, that A is only a necessary cause of C if it is a necessary cause of B. Its necessity therefore suggests a transitive interpretation of the causal chain. Thus we can allow that (A → B → C) ⇒ A → C. If I were pressed to identify the reason for which I flipped the light switch, I might plausibly claim that I did A in order that B occur; that is, I flipped the switch in order to turn on the light in the room. This explanation is satisfactory in the sense that it accounts for A, but it cannot account for C, since this consequence obtained without either my knowledge or intention. It remains the case, however, that I could just as credibly identify as the reason that the prowler was scared off the fact that the room was illuminated. And so the reason the light went on, by this manner of reasoning, was because I flipped the light switch. But the reason I flipped the light switch is so that the light would turn on. What suffices to constitute a reason is thus vague:
The counterfactual interpretation of causality borrows the intuitive reasoning processes and linguistic structures by which we make sense of everyday occurrences. The concept of regret, for instance, is intelligible only in terms of counterfactual supposition. When multiple sufficient conditions preclude a particular desired outcome, it is common for us not only to rescind the blame we would otherwise attach to the cause of our initial disappointment, but to excuse from causal implication every such intervening event.
I might, for instance, be inclined to blame a friend for forgetting to bring the tickets to a ball game. However, should the game be subsequently canceled on account of rain, I might let him off the hook. But the latter event only modifies the causal character of the former by making it an unnecessary causal condition—not an insufficient causal condition—for my disappointment. I might go further, and say something like, "it wasn't meant to be." Whether this statement is uttered with the conviction of its idiomatic force matters little for we do not ordinarily consider events—before they occur—to be inevitable. But even after their occurrence we ordinarily consider events to be inevitable only if their happening were compelled by forces outside the domain of plausible human intervention.
Indeterminism complicates the picture. Suppose Bob and I are playing blackjack and that Bob wins a million dollars. If I had got Bob's hand, I say, I'd have won a million dollars. But this counterfactual is unacceptably vague; it does not account for the apparent ontologically displacement implied by such a case. We could of course simply suppose that Bob, for his part, had got my hand. Such a condition could have obtained, for instance, if the dealer had shuffled the deck differently, or if the deck had been cut between different cards. These contrivances don't strike us as deviantly implausible—at least not in the way as does, for example, the claim that I would have won a million dollars if only the game of blackjack had been formulated against the number twenty-two instead of twenty one. But deferring to the actions of the dealer isn't necessarily a safe bet; it could be the case that the dealer is a cheat. The safest bet, in any case, lies within us, since a volitional adjustment is conceptually simpler than any conceivable machination in the world.
We might suppose instead that, were it the case that Bob was seated in my chair and I in his that I would have won the million dollars, that I might have, indeed could have, taken that seat were it not for my late arrival, that I would have been on time were it not for the traffic accident on the expressway, and so on. Yet the notion that I could have been the difference-maker nevertheless persists. Since it seems that the relevant requisite amendments to the world are entirely within the domain of my own volition, and whatever causal ripples slip into the world at large do not in any case interfere with the putative consequences to which they are attached. The claim that I would have won a million dollars if I had taken Bob's seat is preferable to the claim that I would have won a million dollars if I were Bob simply because it is easier to sit in Bob's chair than it is to be Bob, a claim which lacks even the dignity to qualify for impossibility.
If the possibility represented by the branching histories at time t1 is indeed ontological then the forward-looking modality specifies much more than the promise of epistemic regularity. The agent could have done otherwise. Sometimes this claim is saddled with the stipulation that doing otherwise would necessarily present a counterfactual identical in intentional terms such that one's compliance with Moore's gambit would entail as well his violating the dictates of desire. There is here again the temptation to interpret counterfactual events as explanatory fictions. This is modally possible in the indicative mood; the agent's knowledge state for w, however, can be expressed subjunctively so that βₐ(w) ⊆ Kₐ(w) and w ∈ Kₐ(w). But there is no meaningful modal interpretation under the epistemic conditions prevailing and so (◇Aq → p) collapses to ◇(p → p) taking the counterfactual logic with it.
Under indeterminism, if we picked the numbers 1, 2, 3, and won a million dollars, we should then reason that if it were the case that we had picked the numbers 1, 3, 2 then although we might have nevertheless won, the chances of our having done so are close to zero, because whether we conclude that our respective numerical choices were or were not, in any sense, the same, we would have nevertheless crossed the threshold of causal deviation. Of course, this is the unconditional guarantee of indeterminism; there is simply no conceptually intelligible modal condition that would obviate this problem—not even the fraudulent counterfactual supposition that borrows its antecedent from reality. The unlikely possibility that this alternative should result again in my winning disallows the conclusion that 'things would have been different.'
It is not just that indeterminism offers no reason to believe that non-deterministic events are consistent with free will—though it does. It is instead that indeterminism is widely considered to be incompatible with free will entirely and to threaten a nonsensical fate for the world. That the world is not a chaotic jumble of events seems to be a very good reason to conclude that indeterminism is false. "The conception of indeterminism that now emerges," Richard Taylor wrote, "is not that of a free person, but of an erratic and jerking phantom, without any rhyme or reason at all." But Taylor's pessimism is hard to take seriously since his account disproves itself.
While probability theory can make sense of some of the practical problems of chance, it is nevertheless the case—as De Finetti unwaveringly insisted—that the notion of randomness itself is ultimately inexplicable in Bayesian terms. Wesley Salmon says that trying to attach probability to indeterministic chance "is as pointless as it is unjustified." Even Judea Pearl argues for nothing stronger than subjective degrees of belief. "Bayesian methods provide a formalism for reasoning about partial beliefs under conditions of uncertainty. In this formalism, propositions are given numerical parameters signifying the degree of belief accorded them under some body of knowledge, and the parameters are combined and manipulated according to the rules of probability theory," he noted, leaving plenty of room for indeterministic happenings.
Lewis showed that P(A ⩘ Oj) is not necessarily equal to P(Oj | A). This distinction evinces the limitations of evidentiary decision theories—where these two expressions are always equal. Lewis's argument, unsurprisingly, makes use of possible worlds and modal reasoning. However, explicit concepts of subjective conditionals—common in other CDTs are given to dependency hypotheses in Lewis's philosophy. Brad Armendt summarizes his method: "Lewis recommend[s] that the agent consider various sets of counterfactual conditionals describing possible causal patterns the world might have that are relevant to the actions and consequences in question. Which of these causal patterns obtains is taken to be outside the agent's control, and in both theories the agent is told to weight the values he gives to the possible consequences by his degrees of belief in the competing conjunctions of causal counterfactual conditionals." This is a glimpse at the tortuous logic required to establish causality. Most recent developments in causal decision theory (CDT) broadly adopt the closely-related methodologies of Fishburn, Lewis, Skyrms, Gibbard, and Harper. These CDTs have in common a framework based on the agent's set of beliefs that the world is in a state not caused by his or her actions. One elaboration of this strategy is Lewis's dependency hypothesis, defined as "a maximally specific proposition about how the things [the agent] cares about do and do not depend causally on his present actions. Or in other words, "If I could have, then I would have.”
