An early account (Chisholm 1946) of the meaning of counterfactual conditionals treats them as claims that the proposition, B, contributed by the consequent,¹⁹ follows from the proposition, A, contributed by the antecedent, in conjunction with a body of supplementary facts, C.

In other words, the bare counterfactual conditional A>B compresses a specific piece of reasoning that may rely on additional (factual) premises C and contain intermediate steps.²⁰

This inferential analysis makes the comparison with inferential will from the previous section straightforward, and for that reason we take it as our starting point. Though the “similarity” semantics of Stalnaker and Lewis is more popular, the inferential approach is largely intertranslatable with it,²¹ and the added complexity of the Stalnaker-Lewis model does not in this case confer any benefits.

The relevant body of supplementary facts, C, is left unspecified in (9). This may be seen as an attractive feature. Suppose it is Alma’s birthday. She’s turning 40, and the year is 2020. It seems that either counterfactual below is acceptable in the context:

(10)	a.	If Alma had been born in the year 2000, she would be turning 20.
	b.	If Alma had been born in the year 2000, it would now be 2040.

The antecedent is the same in both sentences, but the conclusions are incompatible: if Alma had been born in 2000 and it was now 2040, she would certainly not be turning 20. The analysis in (9) allows this by leaving C open; the antecedent is supplemented by a different fact in either case. In (10a), the hypothetical proposition that Alma was born in the year 2000 is supplemented by the true proposition that the current year is 2020. In (10b), it is supplemented instead by the true proposition that Alma is turning 40. If we supplement with one of these propositions, we must exclude the other (since taken together they are inconsistent with the antecedent), but it would seem that either choice is possible. And hence, perhaps, there is no absolute truth about “what would have been the case” had Alma been born in the year 2000.²²

It is possible to spell out the supplementary factual premises in the surrounding linguistic context. One way is to present the counterfactual as the conclusion of an argument:²³

(11)		It is now 2020, so
		a.	If Alma had been born in 2000, she would be turning 20. [True]
		b.	If Alma had been born in 2000, it would now be 2040. [False/#]
(12)	Alma is turning 40, so
	a.		If she had been born in 2000, she would be turning 20. [False/#]
	b.		If she had been born in 2000, it would now be 2040. [True]

In the discussion to follow, we spell out the supplementary premise whenever it helps drive home an empirical point. We assume that, in so doing, we are making explicit what would otherwise have to be guessed from the context (rather than introducing a substantially different piece of linguistic data).²⁴

As Goodman (1947) famously pointed out, there are cases where B follows from A and certain facts C (and A is contrary to fact) yet the would-counterfactual A>B is either false or infelicitous. Though this wasn’t yet apparent to Goodman, the cases turn out to be those where B is a cause or explanation either of A or a member of C. The excluded inferences are abductions (Bjorndahl and Snider 2015).

Here’s an example similar to Goodman’s. Suppose the lawn is wet and the only possible explanations are inclement weather or the sprinkler (which, it is important to note, operates on a timer and so is independent of the weather). In the event that the sprinkler has been off and rain is responsible for the wet lawn, we get the following judgments:

(13)	If it hadn’t rained, the grass would be dry. [True]
(14)	If it hadn’t rained, the sprinkler would have been on. [False/#]

From the counterfactual hypothesis (it hasn’t rained), supplemented by the fact that the sprinkler was off, we can derive the consequent of (13): the grass is dry. But note that from the same hypothesis, supplemented instead by the fact that the grass is wet, we can also derive the consequent of (14): the sprinkler was on.²⁵ The basic inferential analysis (9) therefore predicts that either conditional, with the right supplementation, will be true. But this prediction is incorrect: (14) cannot be heard as true. Indeed, if we try to make the supplementary premise explicit, the result sounds incoherent:

(15)	#If it hadn’t rained, the sprinkler would have been on, since the grass is wet.

The problem seems to be that the inference supporting the consequent of (14) is abductive: the conclusion (the sprinkler was on) provides the explanation for one of the premises (the grass is wet). This is not so for (13), where the inference runs from cause (no rain and no sprinkler) to effect (dry grass). In further support of this diagnosis, observe that (14) out of the blue suggests a mechanism that turns the sprinkler on after a dry spell. Left to fill in the details ourselves, we default to a non-abductive argument.

Anti-abductivity in would-counterfactuals parallels the case of inferential will, already discussed. Abduction to the consequent of the counterfactual is blocked—which is to say, to the prejacent of would. And would, of course, is the past tense of will. One is tempted, therefore, to see woll, the lexeme common to will and would (Abusch 1997: fn. 14), as the source of anti-abductivity in both cases.²⁶ There is correspondingly little temptation to attribute the constraint on would-counterfactuals to their counterfactual marking (such an account would not extend to will). An upshot of this is that the linguistic argument for the distinctive nature of counterfactual reasoning and modality is seriously compromised.

For if we return to Humberstone’s contrasting pairs of conditionals, meant to exemplify the difference between indicative and counterfactual, we find that anti-abductivity (in the (b) sentences) is sufficient to explain the relevant contrast.²⁷

The (accidental) generalization that everyone at the bus stop is wearing a hat is explained in part by the proposition that Pennie (who, by hypothesis, is at the stop) is wearing a hat (see (6) above). Thus, if that generalization is used as a premise in the derivation of that proposition, the derivation is an abduction. The same goes for the blocked premises in the remaining two (b) examples. Failing to detect (i.e., not tasting) the pepper that is (by hypothesis) in the tea would explain the absence of that flavour when the tea was sipped; and the kitten in question’s being concealed from view explains the fact that no kitten can now be seen, though one is in the room (by hypothesis).

A serious issue for the view that the (a) and (b) sentences above exemplify the difference between indicative and counterfactual is that the same contrast holds between indicatives: between those that do and those that do not contain (inferential) will—just as our account tracing anti-abductivity to the auxiliary (will/would) predicts (cf. Dudman 1988: 120).

(19)	a.	If Pennie is at that bus stop, she is wearing a hat.
	b.	If Pennie is at that bus stop, she will be wearing a hat.
(20)	a.	If there was pepper in the tea, I didn’t taste it.
	b.	If there was pepper in the tea, I won’t have tasted it.
(21)	a.	If there is a kitten in this room, it is concealed from view.
	b.	If there is a kitten in this room, it will be concealed from view.

Once again, the (b) sentences most naturally suggest some sort of propensity—for Pennie to wear a hat, for the speaker not to taste pepper, for kittens to conceal themselves from view. Inference from such a “law” to a particular instance is not abductive; the law explains the instance, rather than vice versa. What the indicative conditionals with will in the consequent do not seem to allow are, once again, inferences that conclude with explanations of particular premises: the headwear in evidence at the bus stop, an insipid sip of tea, a scene without kittens.

We take ourselves by now to have made a case for the first two items from our earlier list:

will and would apply the same constraint, namely anti-abductivity.

Further support for our account emerges from a consideration of counterfactual conditionals that allow abduction.

On our account, the anti-abductivity of would-counterfactuals is due to the choice of auxiliary in the consequent (would), rather than their counterfactuality. We are therefore in a position to explain another feature of the linguistic data: the existence of counterfactual conditionals—with a different auxiliary structure in the consequent—that permit abduction.

Consider the following variant of (13):

(22)	If it hadn’t rained, the sprinkler would have to have been on (since the grass is wet). [True]

This is not a would-counterfactual, but a ‘would have to’-counterfactual. Instead of the single auxiliary would, the consequent stacks two auxiliaries, would above have. This variant form allows abduction: whereas (13) was false or infelicitous, (22) is true.²⁸

Counterfactual forms²⁹ that permit abduction furnish counterexamples to the hypothesis that reasoning from counterfactual premises is inherently anti-abductive, and hence to the idea that the counterfactuality of (1b) is responsible for this restriction on its use. But while the ‘would have to’ form and its properties have long been observed, and even carefully documented,³⁰ the lesson has been resisted, and the importance of counterfactuality upheld, by various conservative tactics.

Broadly speaking, two lines of resistance are possible: (i) deny that ‘would have to’-counterfactuals are truly abductive, or (ii) deny that they are bona fide counterfactuals. A way to pursue the first approach is to dissect the double auxiliary into an ordinary (i.e., anti-abductive) would-counterfactual and a further embedded necessity claim (viz., that so-and-so has to be the case) in its consequent.³¹ There are various versions of the second approach; an extreme one is to claim that abductive counterfactuals are only true if reinterpreted as indicatives.³²

Schulz (2007: Ch. 5) offers a more nuanced position. She holds that every counterfactual sentence can be interpreted as either an “ontic” (i.e., metaphysical) or an epistemic modal, and while the epistemic reading allows abduction, the ontic does not. Though conceding that not all counterfactuals are anti-abductive, her position is similar to the second line of resistance above, in that abductive conditionals are consigned to a different—and peripheral—modal category. She is, in this way, able to maintain a connection between anti-abductivity and the primary sort of counterfactual modality: ontic/metaphysical.

There is no real evidence for such an ambiguity, however. If all counterfactuals have an epistemic reading, then it should be possible to hear would-counterfactuals with abductive support as true.³³ We ought to hear a true reading of (14), and, for that matter, this wonderful example of Arregui’s (2005: 91):

(23)	A driver stops her car. The bridge she is driving on is unfinished, and a gap yawns ahead.
	If the driver had kept going, the bridge would have been completed.

But no true reading can be heard.

We have limited ourselves, in the consideration of counterfactual forms, to an examination of English. Our finding is that, here at least, verbal auxiliaries make up a paradigm in which counterfactuality and anti-abductivity are independently marked (see Table 1).³⁴ Our hunch is that this will turn out to be the rule across languages. As Winans (2016) already showed, there are anti-abductive indicatives (counterparts of will) in other languages. We expect to find abduction-allowing counterfactuals (counterparts of ‘would have to’) in other languages too, even among those with very different ways of marking counterfactuality.³⁵ We are skeptical, then, of any close conceptual connection between anti-abductivity and counterfactuality, and doubt that the one can be derived from the other. It seems to us that permitting a hypothetical premise to be contrary to fact and blocking abductive inferential support are simply different jobs that languages don't need to bundle together.³⁶

The barbecue scenarios from §2 revolve around the following propositions: it’s Friday night (F), the neighbours are barbecuing (Q), there’s a smoky smell (S), the neighbours are drunk (D). The dependencies between them are represented in Figure 2.

Given the formulation of anti-abductivity in (24), we correctly predict that (25) is acceptable when its prejacent Q is inferred from F, but not when it is inferred from S; since, as may be read off Figure 2, Q≺S, Q≺F.

(25)	The neighbours will be barbecuing.

Moreover, we predict that (26) is acceptable when its prejacent is inferred from S (a common cause inference), since D≺S.

On the other hand, supposing must is not anti-abductive, we predict that (27) is felicitous even if the prejacent is inferred from S.

We propose the following schematic truth conditions for an utterance of ‘will B’ backed by the premise set C:⁴²

While clause (a) checks that the premises are all true, clause (b) models the indirect evidence requirement by ensuring that B is not a trivial consequence of C.⁴³ Clause (c) ensures that the prejacent follows once we add the dependencies in M.⁴⁴ Finally, clause (d) is the anti-abductivity constraint. Since we are ignoring anaphoricity,⁴⁵ the corresponding analysis of ‘must B’ simply drops (d):⁴⁶

On the basic inferential analysis, a conditional is the claim that the consequent follows from the premise set plus the antecedent. We arrive at the following by adding non-triviality and anti-abductivity clauses to that analysis (or, thinking compositionally, by adding the expected contribution of the if-clause to the earlier analysis of ‘will B (C)’).

Note that only an (un)negated atomic clause can occur as the prejacent of will, so the template above is not a general schema for the conditional.⁴⁷ The analysis is the same as (28) except that we have replaced ‘C’ with ‘A∪C’ in clauses (b)–(d).⁴⁸ Thus, for instance, clause (b) now requires that B is not a trivial consequence of C augmented by A.⁴⁹ We can get the truth conditions for ‘if A, must B (C)’ by dropping (d), as before.

Counterfactual marking, on our account, relaxes the requirement that A is epistemically possible. Since we make no attempt to represent this constraint in (30), the analysis applies unchanged to the schema ‘if A, would B (C)’, and applies minus clause (d) to ‘if A, would have to B (C)’.⁵⁰

Structural equation models are limited in two directions. They are limited vertically—they can only go back so far in the causal ancestry of a proposition—and horizontally—a proposition can only have so many parents (i.e., there is a finite limit on the complexity of the function determining its truth-value). In reality, the list of potential factors that immediately bear on a given proposition might be open ended.

Consider the equation we used to model the state of the lawn based on the weather and the sprinkler: W=R∨S

This is an obvious simplification. For one thing, rain doesn’t guarantee a wet lawn. Someone might have parked their car on it (C), or even set up a marquee to protect it from the elements (M ). We could add these conditions to the equation, but the result W=R∧¬C∨M∨S is not obviously an improvement. The longer formula still fails to capture the true relationship between the variables (the marquee could blow down, the car could leak fluid), and the concession to nuance doesn’t balance the downgrade in elegance.

Since we can’t represent the true situation in a finite formula, we follow Pearl in allocating an additional exogenous error variable UX to every formula ϕX. “Error” is understood to subsume any combination of additional factors (including marquees and the like) that would make the prediction of the original equation come out false.

The introduction of the error variable is not without consequences. For suppose we update the equation as follows⁵¹ W=R∨S∧¬UW and then use it to evaluate the counterfactual

Our account now predicts that (31) is only true if the premise set includes the claim that the situation is normal (i.e., UW=0). Without this further premise, we cannot infer W from the antecedent R. This prediction is not correct.⁵² For consider that UW is really the disjunction of every claim to the effect that a particular (inhibiting) error has occurred, and hence ¬UW spelled out in full is the claim that no marquee has been erected, no car is parked on the lawn, etc. Now suppose someone utters (31) without considering whether a car might be parked on the lawn. We do not judge the utterance false merely because the negation of this unlikely proposition did not occur in their reasoning.⁵³ Indeed, anyone who utters (31) must fail to consider some potential source of error, since sources of error can be invented without limit. The problem is a serious one: no one can utter (31) truly on our account as it stands.

But how should we amend it? Note that (31) is not automatically true either (Stone 1994: 184). Instead, its truth-value depends on the facts. Was a car in fact parked on the lawn or not? If one was, (31) is false; if no car was, and indeed nothing else was there to keep the rain off, then (31) is true.

The situation is different for factors that truly are important or salient. We can fault someone who says

(32)	It must have rained, since the grass is wet.

by pointing out that the wetness of the grass could have been due to the sprinkler instead. The speaker is not rationally exonerated if it turns out that, though they were not aware of it, the sprinkler was in fact turned off. Our analysis captures this by predicting that (32) is only true relative to a premise set that includes ¬S.⁵⁴

The distinction we are relying on between a relevant factor and an abnormal interference is reflected in the interpretation of weak necessity modals (i.e., should and ought).

We can follow (33b), but not (33a), with ‘but it’s not, because I covered it with a tarp.’ Weak modal claims can be true even if the error term is tripped (Stone 1994: 184). But even weak modals sound false if an important or salient variable is ignored:

(34)	The grass should be dry, since the sprinkler hasn’t been on.

That isn’t true: it might have rained!

We propose a revision, in light of the above, to clause (c) in our analysis of inferential modals. For strong modals, the prejacent is required to follow from the premise set along with the actual value of every error variable UX (for endogenous X). We update (30) accordingly:

(35)	If A, will B (C)’ is true in M,v iff
	a.	∀P∈C, vP=1
	b.	A∪C⊩B
	c.	A∪C∪UX=vUX:X∈End⊩MB
	d.	There is no X∈atomA∪C such that atomB≺MX

Weak necessity modals, by contrast, are true if the inference goes through on the assumption that no error has occurred (whether or not that assumption is correct). They therefore receive the analysis below:⁵⁵

(36)	‘should/ought B (C)’ is true in M,v iff
	a.	∀P∈C, vP=1
	b.	C⊩B
	c.	C∪UX=0:X∈End⊩MB
	d.	There is no X∈atom-(C) such that atomB≺MX

Interventionism (Pearl 1996; 2009) is a precise proposal about the semantics of counterfactuals with widespread currency in contemporary philosophical accounts of modality and psychological accounts of reasoning.⁵⁶ It shares with Bjorndahl and Snider’s (2015) anti-abductivity analysis the idea that the direction of explanatory dependency is crucial to the truth conditions of (would-) counterfactuals. However, the interventionist, not content with stipulating an embargo on abduction, provides a conceptual rationale against it—one that aligns counterfactual reasoning with experimental intervention.

Performing an experiment is a way of distinguishing causation from correlation. The experimenter brings about a certain state by intervening in the (normal) causal order and disrupting the relationship with that state’s (normal) causal antecedents. (For example, an experimental proband is selected at random to avoid the confounds of self-selection.) The causal consequences of the intervened-on state follow in the usual way, but intervention isolates these from the state’s usual causal ancestors, along with any other state that in the normal course of things it would be (merely) correlated with. In other words, while causal consequences may be inferred from an intervened-on state, other standard inferences, including—but not limited to—abduction to a causal ancestor, are blocked. If you intervene to make the lawn dry (with a blow dryer, say), you cannot turn around and claim that the weather must have been fine.

The anti-abductivity of would-counterfactuals can, therefore, be seen as the surface ripple of something deeper: reasoning based on (hypothetical) intervention. Interventionist protocol both constrains reasoning—in a manner that counterfactuals are supposed to emulate—and gets at what is really there: the underlying causal structure. By comparison, reasoning from what one has “passively” observed is unconstrained (abduction, for instance, is legitimate). But this is so precisely because the objective causal relationships have not been sorted from the myriad other sources of correlation—let alone from the “strange combinations of rare eventualities … that have nothing to do with general traits of influence and behavior” (Balke and Pearl 1994: 231).

Corresponding to the interventionist mode of reasoning is a modality structured by objective features of the world: necessity in virtue of causal and other forms of metaphysical dependency.⁵⁷ In assigning a profounder origin to the anti-abductivity of would-counterfactuals, interventionism also lends support to the objective (or metaphysical) characterization of their modality—in line with the received view that counterfactuals diverge from the epistemic modality of indicatives.

Interventionism, then, is a formidable alternative to the basic anti-abductivity account we have assumed; one that, moreover, supports the traditional view of the modal difference between counterfactuals and indicatives. We reject it, however, because it fails to provide an accurate characterization of the truth conditions of would-counterfactuals.⁵⁸

By lumping abduction in with the other inference patterns blocked by intervention, the interventionist ascribes a stronger constraint to (would-) counterfactuals than neat anti-abductivity. This is clear from the following informal first pass at the interventionist truth condition (Cf. Gibbard & Harper 1978; Pollock 1976: 42):

(37)	A>B is true iff either
	a.	A, when combined with the facts that do not depend on whether A, is sufficient to bring about B, or
	b.	B is true and does not depend on whether A.

Route (a) requires the consequent to occur downstream (dependency-wise) of the antecedent and any supplementary factual premises (this is the intended sense of “bringing about”).⁵⁹ This is a stronger requirement than anti-abductivity, which only says that the path to the consequent must not be upstream of any premise. This discrepancy will become important shortly.

The interventionist algorithm derives the truth condition above by an imagined procedure. We first wind the causal-explanatory history back, undoing everything that depends on the truth-value of the antecedent, A. Holding fixed the resulting causal-explanatory background (captured by the actual truth-values of the exogenous variables), we intervene on A, setting its value at “true.” The intervention is modelled by detaching A from its parents, so that its value can only propagate forward in the graph. Finally, we let a new history “play out” from the same causal-explanatory background—but now with A set to true—and see whether the truth of the consequent B follows.

This computation is defined below. In the following, MA is the model identical to M except that ϕA≡T, while v′~Endv means that v′ matches v except possibly in the assignment it makes to endogenous variables.

With the truth condition now in hand, we can present a counterexample to interventionism. We first observe that, intuitively speaking, a would-counterfactual A>B may be judged true though B follows from A by an inference that goes by way of a common cause. This holds in the case below, where we infer the bang from the flash via a (counterfactual) detonation (Hiddleston 2005):

(39)	A firework fails to detonate.If there had been a flash, there would have been a bang. [True]

Now a common cause inference is not an abduction B≺A, but it is not a purely forward inference either A≺B. It consists of a step back (to a causal ancestor) followed by a step forward (to another effect of the same cause). As such, it lands in the gap between the weaker anti-abductive constraint and the stronger interventionist one. While anti-abductivity predicts that (39) is true, interventionism predicts—incorrectly—that it is false. Intervening to make it the case that there was a flash deletes the connection to the parent node representing the detonation of the firework. This blocks any inference from the truth of the former that runs via the truth of the latter (see Figure 3).⁶⁰

So far in this paper we have avoided the term backtracking. The term was introduced (into the counterfactuals literature) to describe an inference that first went “backwards” and then “forwards.” Thus ‘backtracking counterfactual’ originally meant one that was true in virtue of a common cause inference (as Bennett [2003: 208] chronicles, this meaning was lost, and ‘backtracking’ came to mean simply ‘backwards’ or abductive). The original sense (or at least the underlying concept) merits a revival. As we have seen, it delineates a kind of counterexample to interventionism, and hence an important contour to the empirical landscape.⁶¹ ‘Backtracking’ should not be merged with ‘backwards’, precisely because the former is acceptable in would-counterfactuals (and with inferential will, §2), while the latter—abduction—is not (cf. Arregui 2005: 84–85; Bjorndahl & Snider 2005: 7–9).⁶²

Anti-abductivity provides an account of would-counterfactuals that is empirically superior to interventionism and related accounts from the causal modelling literature. But how does it compare to quite different approaches that distinguish good and bad examples of would-counterfactuals without restricting the causal direction of the inference? Kratzer 2012 and Ippolito 2016 are prominent accounts in this vein, and useful to consider, though their arguments do not in the end cast doubt on the importance of abduction to the understanding of would-counterfactuals.

Ippolito 2016 contains a useful reanalysis of certain examples that seem at first to call for a causal treatment. To begin with, consider Pavel Tichý’s (1976) case:

(40)	Invariably, if it is raining, Jones wears his hat. If it is not raining, Jones wears his hat at random. Today, it is raining and so Jones is wearing his hat.
	But even if it had not been raining, Jones would have been wearing his hat. [False]

Since the conditional is false, the fact that Jones is wearing his hat (in the actual world, where it is raining) must be off-limits as a supplementary premise. But why? Suppose we change the example slightly (Veltman 2005):

(41)	Every morning, Jones tosses a coin. If it is raining that day, he wears his hat, but if it is not then he only wears his hat if the coin came up heads. Today, the toss comes up heads, it is raining, and Jones is wearing his hat.
	But even if it had not been raining, Jones would have been wearing his hat. [True]

This time the inference goes through. Evidently, certain facts—in this case the fact that the coin came up heads—can occur as premises. Indeed, the judgments may be captured by a familiar rule: facts are admitted as long as they don’t depend (causally) on the antecedent (cf. (37a)). Jones’s hat-wearing depends on the weather; the result of a coin toss does not.

A different diagnosis competes with this causal one, however. In the false case (40), the consequent B can only be derived from itself or the negation of the antecedent ¬A. While both propositions are true in the scenario, admitting either into the premise set (alongside the antecedent A) ensures a trivial derivation of the consequent B. By contrast, the true case (41) is one where the consequent follows from a premise (the coin came up heads) that is logically independent of both clauses of the conditional. Both judgments could be explained, therefore, by a restriction to nontrivial inference (Ippolito 2016: 18). This explanation is especially attractive, given that such a restriction is independently motivated by the (indirect) evidential requirement of the modal.⁶³

Note further that in the scenario of (41) the following seems false:

(42)	If Smith had tossed the coin instead of Jones, it would still have landed heads. [False]

A causal dependency account like (37) predicts this only if it can be made out that the result of the toss “depends” (in the relevant sense) on the the person who makes it.⁶⁴ But now consider the following (cf. Ippolito 2016: 4):

(43)	Smith’s last counter is twelve squares away from home. Jones throws the dice and rolls a double six.
	If Smith had rolled the dice instead of Jones, he would have been home. [True]

The causal structure of (43) does not differ appreciably from that of (42), so it is hard to see how a causal theorist would account for the difference. On the nontriviality approach, however, the explanation is clear: while the consequent of (42) directly concerns the result of the toss (something that can only be trivially inferred from the facts of the case),⁶⁵ (43)’s consequent concerns a further ramification of this result (which may be nontrivially inferred from it).⁶⁶

As Ippolito is aware, nontriviality cannot account for every case that would seem to turn on causal features of the situation. In particular, it doesn’t suffice for a correct prediction (of falsity) in our central sort of case, exemplified by (14). For here the supplementary premise (the grass is wet) is logically independent of both conclusion (the sprinkler was on) and antecedent (it hasn’t rained).

Instead, for such cases, Ippolito calls upon a recent account by Kratzer (2012). Kratzer, as it happens, is convinced that causation is a red herring in (13)/(14), since she has found an analogous case where the causal direction is reversed (2012: 140). We set this case aside for the time being, and jump directly to her alternative proposal.

Instead of causal direction, Kratzer’s account is based on the asymmetry between situations that confirm a nonaccidental generalization and those that, while not upsetting the generalization, don’t confirm it either. (‘All ravens are black’ is supposed to be confirmed by the sighting of a black raven, but not by the sighting of a non-black thing that is also not a raven; the sighting of a non-black raven would of course falsify the generalization.) Notoriously, what counts as a confirming situation in this milieu depends on the wording of the generalization (‘All ravens are black’ is confirmed differently than the logically equivalent ‘All non-black things are non-ravens’). In particular, Kratzer’s account predicts (2012: 141–43) that if the generalization is worded as ‘whenever P, Q’ then a confirming situation is one where both P and Q are the case (as opposed to one where ¬Q and ¬P are the case).

Suppose the generalization operative across (13) and (14) is ‘Whenever it doesn’t rain and the sprinkler is off, the grass is dry.’ Hence a case where it doesn’t rain, the sprinkler is off and the grass is dry is a confirming instance of the generalization, while a case where it doesn’t rain, the sprinkler is on, and the grass is wet, is not. According to Kratzer’s (2012: 144) confirming proposition constraint, facts that yield confirming instances of nonaccidental generalizations are preferred when it comes to supplementing the antecedent of a counterfactual. This means that we will supplement the hypothesis that it hasn’t rained with the fact that the sprinkler is off (rather than the fact that the grass is wet), and hence that (13) is true, and (14) false.⁶⁷

Note that Kratzer’s account makes different predictions to one based on causal asymmetry. We can compare it to such an account by considering a minimal pair of conditionals that differ in causal direction (from antecedent to consequent), but not in their verifying situation.⁶⁸ Suppose Bob decides whether to go to the office holiday party based solely on whether Alice is going. If she is going, then he stays at home, but if she isn’t going, he goes. This year, Alice stays home, and so Bob goes.

We judge the first counterfactual true and the second false. On a causal account, this is easily explained: it is because Bob’s doings have no effect on Alice’s actions. Kratzer’s noncausal account, of course, cannot say this. It treats the rule as a causally neutral biconditional: A↔¬B.⁶⁹ Of course, by this rule, A follows from ¬B just as much as ¬B follows from A. Note that no supplementary premises are required for either inference, so Kratzer’s confirming proposition constraint (which constrains the supplementation of a counterfactual antecedent) doesn’t get traction here. But even if the constraint did apply, the verifying situation is the same for both sentences (i.e., one where Alice goes and Bob stays), so no matter the wording of the rule (whether it is ‘Alice goes if and only if Bob stays’, or ‘Alice stays if and only if Bob goes’), no difference in judgment would be predicted.⁷⁰

Kratzer (2012: 140) prefers a noncausal account of the difference between examples like (13) and (14) in part because she takes the following case to belong to the same camp (viz., “Goodman’s Puzzle”), and hence to warrant an explanation in kind.

(46)	Whenever the Royal Bavarian Flag is up and the lights are on at Leoni Castle, the King of Bavaria is home. As it happens, the lights are on, but the flag is down and the king is away.
	a.	If the flag had been up, the king would have been home. [True]
	b.	If the flag had been up, the lights would have been off. [False]

Our anti-abductive semantics for would-counterfactuals makes the wrong predictions for (46). When the lights are on and the flag is up, they are so, presumably, because the king is home. It follows that the inference from the flag hypothetically being up (and the lights being on) to the king being home is abductive—and so the account (incorrectly) predicts (46a) to be false or infelicitous. Furthermore, the inference supporting (46b) is not an abduction,⁷¹ and hence a constraint on abduction cannot explain its falsity.

The second disparity is easier to address than the first. We agree that (46b) is supported by a non-abductive inference, but propose that this inference is obscured by a more salient one supporting (46a).⁷² Note that once a since-clause specifying the appropriate factual premise is added, foregrounding the inference in question, the counterfactual sounds true:

(47)	If the flag had been up, the lights would have been off, since the king is away. [True]

Our proposal is similar to Kratzer’s own explanation of the initial falsity judgment for (46b). On her account, the operative generalization privileges a different argument, whose supplementary premise and conclusion are primed by the wording of the rule (see §5.2).⁷³

Our account of (46b) presupposes the (more salient) veracity of (46a). How then are we to account for this judgment? The first thing to note is that the flag and the lights function as a signal of the king’s presence at the castle. Indeed, this is critical to the way Kratzer’s case operates (cf. Schulz 2007: 89–90). If we remove this feature, the judgments revert to the pattern predicted by anti-abductivity:

(48)	The only person authorized to raise the flag at Leoni castle is the steward. So:
	a.	#If the flag had been up, the steward would have been home.
	b.	If the flag had been up, the steward would have to have been home. [True]

Now it is well known that evidential marking can be sensitive to the distinction between the content of a signal (or report) and the conclusion of an inference.⁷⁴ For instance, will and must, which we know to be compatible with inference, are both incompatible with reportage. Suppose you are some distance from the library and observe a sign on the door that reads ‘closed’, it would be moderately eccentric to then muse to yourself:⁷⁵

(49)	??The library will be closed.

This could be because the inference from the sign to the store’s being closed is abductive, but in that case we would expect the sentence to improve with must. However, neither is it quite right to say:

(50)	??The library must be closed.

The appropriate thing to say is just, ‘The library is closed’. What is more, changing the direction of causation doesn’t alter the judgments. Suppose the game ends when the referee blows her whistle (the signal, in this case, is the cause of what it signals). It is odd, once again, to utter either (51a) or (51b) upon hearing the whistle. Only (51c) is acceptable.

Note that will is fine if the inference is to other (non-signalled) consequences of the whistle blowing (e.g., ‘Her ears will be ringing’).

Kratzer’s King of Bavaria case would seem to belong to the same category. Suppose you are passing Leoni Castle and you notice the flag and lights (the official signs of the King’s installation there). The first two remarks (directed to yourself, or a blind companion) seem odd by comparison with the third:

(52)	a.	??The king will be home.
	b.	??The king must be home.
	c.	The king is home.

Note that (52a) is improved if the king’s presence is inferred instead from the fact that today is the national holiday that he traditionally spends at the castle.

What we propose, then, is that the justification for the prejacent of would as it occurs in (46a) is not inferential, but testimonial. It is the “report” made by the flag and the lights, rather than an inference to their likely cause, that secures the connection between antecedent and consequent. Of course, for this approach to predict the judgment that (46a) is good, it must be that would is compatible with reported prejacents. But this seems to be so, based on other judgments:⁷⁶

(53)	a.	If there had been a ‘closed’ sign on the door, the library would have been closed. [True]
	b.	If the referee had blown her whistle, the game would have been over. [True]

Note that the judgment in (53a) is surprising if the connection is inferential rather than reportative, since the inference in question is abductive. (Why is there a ‘closed’ sign on the door? Because the store is closed.)

Up to this point in the paper, we have dwelt on analogies between would and will. But here at the end we suggest a difference: will is anti-reportative, while would is not. Hopefully it is clear that this proposal is open to us; the mere fact that will and would share morphology does not mean they must restrict evidence in exactly the same way. Indeed, there is reason to expect just this difference. While there is an alternative to will (and must)—the (reportative) simple clause—in the non-counterfactual paradigm, there is no similar option in the consequent of a counterfactual.⁷⁷ As a result, if testimonial justifications belong in counterfactual structures, the task of admitting them will fall to a modal auxiliary, and would is a natural choice.