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Chapter2.tex
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%!TEX root=If.tex
\documentclass[If.tex]{subfiles}
\begin{document}
\chapter{Closeness}
\label{chap:cem}
\begin{comment}
\begin{itemize}
\item
Other arguments for CEM: Neg-raising and small differences.
(What were we thinking here? Maybe the idea is that the CEM-lover can explain why we are as careless as we are about the placement of ‘not’, whereas for strictists it should be analogous to the case of ‘have to’ where there is no temptation to conflate ‘don't have to’ with ‘have to not’. Not sure if there's really a whole separate argument here.)
\item
The super-contextualist response [response to what exactly?]. It has to posit lots of mid-sentence context-shifts in cases where it seems really implausible.
\item
Just a footnote on the terrible Reverse Sobel Sequence argument for strictism - appeal to Moss.
\item
The deepest considerations in favour of strictism come from the desire to have a uniform treatment of propositional conditionals and conditionals with adverbs of quantification. We'll get to this in a later chapter.
\item
Does Kratzer have any arguments that the hidden operator in indicatives is ‘must’-like? One could make an argument from the premise that we never see a \emph{lexicalised} modal that is epistemic but has the kind of ordering source feel that would be required by a CEM lover.
\end{itemize}
\end{comment}
\section{Three views}\label{why-closest}
Here, again, is our bare-bones theory of conditionals:
\begin{prop}
\litem[CLOSEST] \label{closest}
A conditional with antecedent $p$ and consequent $q$ is true iff either there is no accessible $p$-world, or the closest accessible $p$-world is a $q$-world.
\end{prop}
In saying this we are taking it for granted that where there are accessible $p$-worlds, there is a unique closest one. The aim of the present chapter is to say more about the relevant notion of closeness, and to motivate the claim that it obeys the required uniqueness assumption.
Some theories of conditionals use the ideology of worlds and closeness but make no assumption that there is always a unique closest accessible $p$-world when there is any accessible $p$-world. The most influential such theorist is Lewis \citeyearpar{LewisCounterfactuals}. Lewis's theory allows for two kinds of failures of uniqueness. First, there can be ties: there may be several maximally close accessible $p$-worlds (i.e.\ equally close $p$-worlds such that no $p$-worlds are closer than them). Second, there may be no maximally close accessible $p$-worlds: it could be that for every accessible $p$-world, there is a yet closer accessible $p$-world. Lewis's truth-conditions agree with ours in the case where there is a unique closest accessible $p$-world, but introduce a new element of universal quantification to deal with the other cases. When there are several maximally close accessible $p$-worlds, the conditional is true just in case all of them are $q$-worlds. When there are accessible $p$-worlds but no maximally close accessible $p$-worlds, the conditional is true just in case there is some accessible $p$-world such that every accessible $p$-world at least as close as it is a $q$-world. In fact the latter case also covers the case where there is one or more maximally close $p$-worlds; so Lewis's theory can be stated as follows:
\begin{prop}
\litem[LEWIS] \label{lewis}
A conditional with antecedent $p$ and consequent $q$ is true iff either there is no accessible $p$-world, or there is an accessible $p$-world such that every accessible $p$-world that is at least as close as that world is a $q$-world.%
\footnote{Lewis avoids the need to say ‘accessible’ all the time by setting things up in such a way any accessible world is closer than all inaccessible worlds (if there are any inaccessible worlds). But this is not essential to the aspect of Lewis's theory that we are presently concerned with.}
\end{prop}
Lewis himself only applied this schema to the analysis of counterfactual conditionals, but it is certainly worth exploring whether a theory with this shape could also work for indicatives.
Another important family of rival theories of conditionals makes use of the ideology of worlds and accessibility, but does not appeal to closeness at all in the specification of truth-conditions. On these ‘strict conditional’ accounts, the truth-conditions of conditionals are straightforward:
\begin{prop}
\litem[STRICT] \label{strict}
A conditional with antecedent $p$ and consequent $q$ is true iff every accessible $p$-world is a $q$-world.
\end{prop}
(One might be tempted to think that at the schematic level we are
presently working at, \ref{strict} is perfectly compatible with our account:
if we redefine ‘accessible world’ to mean what we previously meant by
‘closest accessible world’, won't \ref{strict} then be just a terminological
variant on \ref{closest}? No: in the expression ‘the closest accessible
$p$-world’, ‘closest’ is not playing the role of a monadic predicate of
worlds. Being a closest accessible $p$-world is not just being (a)
closest, (b) accessible, and (c) a $p$-world, just as the property of
being a shortest spy is not the conjunction of the property of being
shortest and the property of being a spy. Being a shortest spy is being
a spy who is at least as short as any spy; being a closest accessible
$p$-world is being a $p$-world that is at least as close as any accessible
$p$-world.)
Lewis's argument against \ref{strict} (for counterfactual conditionals) is
well-known and prima facie compelling. Lewis notes that \ref{strict} validates
\emph{Antecedent Strengthening}: whenever ‘If $P$, $Q$’ is true, ‘If $P$
and $R$, $Q$’ is true (for any $R$). But this rule does not look to be valid. For example, the inference
from
\begin{prop}
\nitem \label{dog}
If I bought my son a pet dog, he would be delighted
\end{prop}
to
\begin{prop}
\nitem \label{strangle}
If I bought my son a pet dog and strangled it, he would be delighted
\end{prop}
seems invalid. Or what comes to much the same thing: the conjunction of the first with the denial of the second seems perfectly consistent and felicitous. We note that exactly the same kind of argument can be given for indicative conditionals: for example, the inference from \ref{idog} to \ref{istrangle} seems just as terrible:
\begin{prop}
\nitem \label{idog}
If he bought his son a pet dog, his son was delighted
\nitem \label{istrangle}
If he bought his son a pet dog and strangled it, his son was delighted.
\end{prop}
To keep \ref{strict} going in the face of these prima facie compelling considerations against it, its proponents have posited widespread context-shift as regards what counts as accessible. Their thought (\emph{cite von Fintel}) is that in the context in which (1) is uttered truly, worlds where the speaker buys a pet dog and strangles it are inaccessible; but because of the presupposition of nonvacuity, uttering (2) triggers a new context in which at least one such world is accessible. The central mechanism that triggers contextual shifts is thus held to be that of presupposition accommodation. On this view, \emph{Antecedent Strengthening} is valid in the sense that it preserves truth when context is held fixed; but realistic cases where the putative counterexample sequences are uttered are not cases where context is held fixed.%
\footnote{Note that if one allowed context-shift to run completely rampant, one could even reconcile the truth-values delivered by \ref{strict} with those delivered by \ref{closest}. The idea would be that at the context where a conditional is uttered, the set of accessible worlds (in the sense relevant to \ref{strict}) are all and only those accessible worlds (in the sense relevant to \ref{closest}) that are at least as close as every accessible world where the antecedent is true. Unlike a version of \ref{strict} that tries to try its account of context-shift to the familiar phenomenon of presupposition accommodation, this kind of mad-dog contextualism seems to depart so much from the standard way of thinking about what it means for distinct utterances to belong to the same context that evaluating it would require getting a lot clearer about what theoretical role is being assigned to the new conception of ‘context’.}
Since the views about accessibility we developed in chapter 1 also involve quite an extensive amount of context-shift, including context-shift driven by presupposition accommodation, we are in no position to discount \ref{strict} simply on account of the contextualism it requires.%
%\footnote{*** One might think that the contextualist diagnosis of apparent failures of Antecedent Strengthening can't work, on the grounds that presupposition accomodation is a conversational phenomenon and we can consider the relevant arguments and evaluate their validity just in our own inner monologue, divorced from any conversational context. We think this is far too sanguine. \textbf{Say more here.}}
% \begin{itemize}
% \item
% *** Also mention Contraposition and Transitivity.
% \end{itemize}
One feature that distinguishes \ref{closest} from both \ref{strict} and \ref{lewis} is
that status of the principle of Conditional Excluded Middle:
\begin{prop}
\litem[CEM] \label{CEM}
Either if $P$, $Q$ or if $P$, not-$Q$.
\end{prop}
Given \ref{closest}, instances of this schema will be true independent of context.%
\footnote{At least if we continue take it for granted that every world is either a $Q$-world or a not-$Q$ world. Chapter 4 will consider whether we sometimes need to invoke “incomplete” worlds that do not conformt to this generalisation.}
By contrast, \ref{strict} obviously allows for interpretations of instances of CEM on which they fail to be true, because the set of accessible worlds includes both $P$-and-$Q$ worlds and $P$-and-not-$Q$ worlds. Likewise, \ref{lewis} allows for two kinds of failures of instances of CEM. In one kind of case, there are several $P$-worlds tied for maximal closeness, which differ with regard to the truth-value of $Q$; in the other kind of case, there are no maximally close $P$-worlds, and for every $P$-and-$Q$ world there is a closer $P$-and-not-$Q$ world, and for every $P$-and-not-$Q$ world there is a closer $P$-and-$Q$ world.
Proponents of \ref{strict} and \ref{lewis} are well aware of this feature of their views; indeed their views have been partly motivated by the desire for CEM to come out invalid. However, we think there are strong reasons to like CEM. We will consider several such reasons in the present section; in section 2 we will address some arguments against CEM.
\section{Chance and confidence-theoretic arguments for CEM}
\label{sect:probs}
The most central considerations for us turn on facts about the chances of conditionals, and the levels of confidence we should have in conditionals. We are going to look at some instances of CEM involving fair coins, which seem like good test cases: if there were a problem for CEM, one would expect it to show up for conditionals concerning the outcomes of fair coin tosses.
Let's begin with chance (understood as objective rather than epistemic). Consider one of the coins currently in your pocket. What's the chance that it would land Heads if it were tossed in the next minute? Around 50\%, surely (unless you are in the habit of carrying around trick coins). Similarly, the chance that it would fail to land Heads if it were tossed is around 50\%. But chance is a kind of probability, and it a basic theorem of the probability calculus that the sum of the probabilities of two propositions equals the sum of the probabilities of their disjunction and their conjunction. And in this case, the conjunction of the two conditionals---namely that if the coin were tossed it would land Heads, and if the coin were tossed it would fail to land Heads---is absurd, and deserves zero credence. So the chance of the disjunction of the conditionals---which is an instance of CEM---is roughly one. In fact, it would seem to be \emph{exactly} one, since any surprising factors that might elevate the chance of one disjunct above 50\% would presumably reduce the chance of the other disjunct below 50\% by the same amount. But given that the disjunction has chance one, it would be preposterous to deny that it is true.%
\footnote{We wouldn't quite want to assume that \emph{all} chance-one propositions are true: certain examples involving infinitely fine-grained outcomes make trouble for that simple generalisation. Similar points apply to arguments for the truth of a proposition from the premise that its epistemic chance is 1, or for the premise that we ought to assign it a credence of 1. But it seems hopeless to try to leverage these considerations into a defence of the denial of CEM.}
%*** Think somewhere about cases like ‘What is the chance that what this piece of paper says is true?’ when it's a counterfactual. It actually seems fine - and that's a good argument against NTV style theorists, Kratzerians, etc.
We can generate a similar pattern of judgments by considering the chances at a certain past time of counterfactuals concerning times in the future of that time. For example, if you didn't toss a certain coin yesterday, it seems that the chance two days ago that it would land Heads if it were tossed yesterday was in the region of 50\%, as was the chance two days ago that it would fail to land Heads if it were tossed yesterday, so again, the chance of the disjunction of these counterfactuals would seem to have been 1.
Another way to motivate CEM for counterfactuals is to consider the levels of confidence that seem to be appropriate. For example, concerning a certain untossed coin which you have no special reason to suspect of being a trick coin, it seems that you should be about 50\% confident that it would have landed Heads if you had tossed it, and about 50\% confident that it would have failed to land Heads if you had tossed it. After all, discovering that the coin and the table it is being tossed on are magnetised in such a way as to favour Heads looks like it should make you more confident that the coin would have landed Heads if you had tossed it. Discovering a setup of magnets that favours Tails would make you less confident. Without evidence for any such setups, it seems obvious that your credence should be middling. And for exactly the same reasons, it looks like you should be about 50\% confident that the coin would have failed to land Heads if you had tossed it. Since you should be certain that the conjunction of these counterfactuals is false, and since rational levels of credence conform to the probability calculus, your level of confidence in the disjunction---an instance of CEM---had better be about 1. Indeed, it seems that it should be exactly one, since any reasons for raising your credence in one of the disjuncts above 50\% seems like an equally good reason for lowering your credence in the other disjunct by the same amount.%
\footnote{A closely related argument for CEM appeals to judgments about epistemic probability rather than about appropriate levels of confidence, e.g.~that the epistemic probability that the coin would have landed Heads if it were tossed is about 50\%.}
These ways of assigning credences to counterfactuals can be further supported by considering the motivating role of counterfactuals in deliberation. Suppose that you face an uncomfortable choice between opening two boxes. You are 49\% confident that Box A contains a bomb primed to explode on opening the box, and 51\% confident that it contains nothing. You know that Box B contains a bomb linked up to a fair coin: if the box is opened, the coin will be tossed and the bomb will explode if it lands Heads. Obviously you should open Box A here. And the obvious explanation of this is that you are more confident that you would be killed if you opened Box B than that you would be killed if you opened Box A. But your credence that you would be killed if you opened Box A equals your credence that there is a bomb in Box A, namely 49\%. So your credence that you would be killed if you opened Box B, which is equal to your credence that the coin in Box B would land Heads if it were tossed, looks to be over 49\%--in fact, 50\%. Standard CEM-deniers, by contrast, will think that insofar as you are confident that you won't take Box B, and hence confident that the counterfactual ‘You would be killed if you opened Box B’ has a false antecedent, your credence in that counterfactual should be very low. They will need some other, more complicated and (we think) less natural way of relating credences in counterfactuals to good deliberation.
These natural confidence-theoretic judgments are far out of line with the sorts of credences that seem to be recommended by standard versions of \ref{lewis} and \ref{strict}. For Lewis, cases involving chancy processes like coin-tossings are a primary motivation for allowing ties in the closeness ordering; given his actual theory of closeness, when we know that a coin is fair and was not tossed, we can be quite confident that the set of maximally close tossing worlds contain a mix of Heads and Tails worlds, and hence quite confident that each conditional are false. (Even if we aren't sure whether the coin was tossed or not, there will be some significant portion of our probability space devoted to the hypothesis that it was not tossed and that both conditionals are false because of ties.) Similarly, extant versions of \ref{strict} tend to say things about accessibility that encourage the idea that the accessible worlds in this case include both Heads-landing and Tails-landing worlds, in which case both conditionals will come out false according to \ref{strict}. We will certainly get this result if we interpret \ref{strict} using anything like the conception of accessibility described in chapter 1. One our claims in that chapter is that the same notion of accessibility relevant to counterfactuals can also be picked up by certain modals like ‘has to’ and ‘might have’; and obviously, given that the coin might have been tossed, it might have landed Heads, and might have landed Tails, and didn't have to land Heads, and didn't have to land Tails.%
\footnote{Of course, proponents of \ref{strict} might attempt to preserve our motivating confidence judgments by introducing some devious new conception of accessibility under which, in the case of an untossed fair coin, the accessible worlds where the coin is tossed are all alike as regards how it lands, and our 50\% confidence reflects our uncertainty about which worlds are accessible. We will briefly consider this view later. ***One point to note: given the kind of contextual shiftiness that such a view would require, we can't be thinking that accessibility is ‘sticky’ in the way required by the von Fintel/Gillies account of Reverse Sobel Sequences. We also can't assimilate accessibility restriction to the broad model of quantifier domain restriction in general, where the stickiness phenomenon is clearly a real thing.}
We find a similar pattern of confidence-theoretic judgments when we turn from counterfactuals to indicative conditionals. Here the relevant confidence-theoretic judgments are ones we already discussed back in \autoref{sect:antimaterial}, and their force has been widely acknowledged. For example, if you are not sure whether a certain coin was tossed yesterday, and have no special evidence favouring the hypothesis that it was tossed and landed Heads over the hypothesis that it was tossed and landed Tails, it seems that you should be about 50\% confident that it landed Heads if it was tossed and about 50\% confident that it landed Tails if was tossed, and hence---since you should be about 0\% confident that it both landed Heads and landed Tails if it was tossed---you should be about 100\% confident that either it landed Heads if it was tossed, or it landed Tails if it was tossed. So the confidence-theoretic case for CEM looks as strong for indicatives as for counterfactuals.%
\footnote{We haven't included an argument based on the objective chances of indicative conditionals, because it's not so easy to find sentences where it's clear both the relevant notion of chance is objective and that the conditionals are genuine indicatives---recall that we have suspended judgment on the status of ‘does-will’ conditionals. One could however try to make something of examples like `There's a fifty-fifty chance that if he tosses this coin during the next hour, he wins a prize'.}
While many opponents of CEM seem by our lights to have let their theory ride roughshod over their natural confidence-assignments, some of them have shown enough awareness of the ordinary practice to want to explain away data of the sort we have been relying on. Here is Jonathan Bennett:
\begin{quote}
Admittedly, we often find it natural to say things like `There's only a small chance that if he had entered the lottery he would have won', and `It's 50\% likely that if he had tossed the coin it would have come down heads'. In remarks like these, the speaker means something of the form $A>(P(C)=n)$---if the antecedent were true, the consequent would have a certain probability; yet the sentence he utters means something of the form $P(A>C)=n$\ldots{}. When we use one to mean the other, we employ a usage that is idiomatic but not strictly correct. (Bennett \page{251}.)
\end{quote}
Bennett is not very clear about whether the relevant notion of chance here is epistemic or objective. \textbf{check} But whichever way we go, we don't think the diagnosis is very promising. First of all, Bennett seems to be accusing us of conflating things that we seem in fact to quite good at distinguishing. This is particularly clear if we replace likelihood claims with claims about particular people's degrees of confidence, as in
\begin{prop}
\nitem \label{confcf}
I am pretty confident that if this dice had been rolled without my knowing that it was rolled, it would have landed on some number other than 6.
\end{prop}
The analogue of Bennett's move in this case would be to say that \ref{confcf} is conflated with \ref{cfconf}:
\begin{prop}
\nitem \label{cfconf}
If the die had been rolled without my knowing that it was rolled, I
would have been pretty confident that it had landed on some number
other than 6.
\end{prop}
But it is hard to believe that we could conflate such obviously different claims. Moreover, even if such a conflation were plausible, there would be no prospect of using it to explain away our temptation to regard \ref{confcf} as true, since \ref{cfconf} is manifestly false.
Even confining our attention to claims of chance, as Bennett does, the “conflation” strategy strategy delivers terrible results. The problem is especially clear when the time-index of the chance ascription is made explicit:
\begin{prop}
\nitem \label{nowlikely}
It is likely right now that if Jim drank arsenic tomorrow, he would be dead by the weekend.
\end{prop}
Bennett's transformation would turn this into
\begin{prop}
\nitem \label{wouldbelikely}
If Jim drank arsenic tomorrow, it would be likely right now that he would be dead by the weekend.
\end{prop}
Assuming that it is not in fact likely right now that Jim will be dead by the weekend, \ref{wouldbelikely} seems false: it's not true that if he drank arsenic tomorrow, he would have all along been likely to have been dead by the weekend. And this is so whether the relevant notion of likelihood is understood as objective or epistemic. In neither case are the probabilities today plausibly regarded as counterfactually dependent on how things play out tomorrow.
A desparate fix for this problem would be to say that we conflate \ref{nowlikely} with some counterfactual concerning probabilities at later times, such as
\begin{prop}
\nitem \label{wouldbelikelythen}
If Jim drank arsenic tomorrow, it would be likely then that he would be dead by the weekend.
\end{prop}
Quite apart from the intrinsic implausibility of such a conflation, there are cases where a shift to narrow scope coupled with a rewriting of the time won't deliver acceptable truth conditions, no matter what time is used in the rewrite. Consider:
\begin{prop}
\nitem \label{leftpoison}
There is a 50\% chance if Jim drank poison tomorrow, he would drink poison with his left hand.
\end{prop}
On the natural way of filling out the story, if Jim drank poison tomorrow, it's not clear that there would any time at all such that the chance at that time of his drinking poison with his left hand was 50\%. Once the drinking happened, the chance would be one or zero; beforehand, probably, the chance would be much lower than 50\%, since there would still be a chance of Jim not drinking poison at all.%
\footnote{Perhaps given the continuity of actual-world physics we should admit that if Jim had ended up drinking it with his left hand, there would have been a moment when the chance of his doing so was 50\%, en route to 1 from its initial low value. But this does nothing to help explain the seeming assertability of the sentence, since for one thing few of us are apprised of such physics, and for another thing we could only have this reason for believing that there would have been such an instant if we believed that if Jim drank poison tomorrow he would in fact have done so with his left hand.}
Similar remarks apply to the epistemic interpretation of \ref{leftpoison}. Suppose the situation is one in which if he drank poison, it would be completely obvious to all relevant parties which hand was being used. \ref{leftpoison} still sounds fine in this case, although there is no reason to think if he drank poison, there would be any time at which the epistemic probability of him drinking with his left hand was 50\%.
% *** Insert here a cross-reference forward to future discussion of the Kratzerian treatment of the chance- and confidence-theoretic judgments.
\section{CEM and denying conditionals}
Another group of considerations in favour of CEM have to do with the interaction of negation and denials with conditionals. The validity of CEM is equivalent to that of the inference from ‘It is not the case that if P, Q’ to ‘If P, it is not the case that Q’. However such inferences are hard to evaluate directly: after all, explicit negations of the form ‘It is not the case that if P, Q’ and ‘It is false that if P, Q’ are not common in natural language, and so it would be unwise to place too much weight on any instincts regarding sentences of that form. But there are forms of denial that are much more natural. For example, we can consider negative answers to questions concerning a conditional:
\begin{prop}
\nitem \label{coindialogue}
\begin{prop}
\item
Would this coin have landed Heads if it had been tossed?
\item
--- No.
\end{prop}
\end{prop}
This answer is clearly quite inappropriate unless you have some very unusual knowledge about the characteristics of the coin or the tossing setup. Furthermore, in any setting where that answer is felicitous, one is also in a position to assert
\begin{prop}
\nitem \label{wouldtails}
If the coin had been tossed, it would have landed Tails
\end{prop}
assuming that one's insider information does not disrupt the standard assumption that if the coin had been tossed, it would have either landed Heads or Tails and not both. But the standard versions of \ref{strict} and \ref{lewis} just described seem to entail that merely knowing that the coin was fair would be enough to justify the negative answer to \ref{coindialogue}, and that the inference from this negative answer to \ref{wouldtails} is completely unjustified.
This kind of consideration also supports CEM in the case of indicatives. For example, the reasoning of the client in the following dialogue looks cogent:
\begin{prop}
\nitem \label{taxdialogue}
\emph{Client:} Will I have a big tax bill if I have won the lottery?
\emph{Accountant:} No.
\emph{Client:} Great: so if I have won the lottery my financial
troubles are completely over and done with.
\end{prop}
In saying ‘No’, the accountant is clearly committed to `You won't have a
big tax bill if you have won the lottery'.
It might be suggested that the inappropriateness of the negative answer to \ref{coindialogue} should be assimilated to the phenomenon of ‘Neg-raising’ that applies to words like ‘believe’ and ‘want’, wherein sentences in which negation takes wide scope over some other operator fail to entail, but seem in some other way to suggest, the truth of the corresponding sentences in which negation takes narrow scope. For example, saying `I don't believe it is raining outside', or answering ‘No’ to the question ‘Do you believe that it is raining outside?’, tends to suggest that you believe it isn't raining outside. Likewise `I don't want to go to London' suggests ‘I want not to go to London’. Could this mechanism be enough to explain the seeming goodness of the inference to \ref{wouldtails} from the negative answer to \ref{coindialogue}, and the infelicity of that negative answer? We doubt it. The suggestions associated with Neg-raising are easily cancelled: for example, if my answer to the question ‘Do you want to go to London?’ is `No, I don't care either way', no-one would be tempted to infer that I want not to go to London. By contrast, there is nothing which one could to ‘No’ in answering \ref{coindialogue} which would make it felicitous and block the inference to \ref{wouldtails}. For example, saying `No, it's a fair coin' does nothing to make the answer any better.
Some have argued that when we put focal stress on the word ‘would’ we get judgments more in line with those of CEM-deniers. Hajek (2007), for example, claims that \ref{WOULD} sounds true:
\begin{prop}
\nitem \label{WOULD}
It is not the case that the coin WOULD land tails if it were tossed.
\end{prop}
We have no clear judgment about what is going on in this sentence; we suspect that sentences generated by prefixing conditionals with ‘It is not the case that’ are so unnatural that our views about them are especially likely to be theory driven. One might try to avoid this by using the question-answer format. But it doesn't seem that focusing ‘would’ in \ref{coindialogue} makes the answer ‘No’ much more acceptable. (The most obvious reason for focusing this ‘would’ is to signal one's challenge to a previous assertion of ‘The coin would have landed heads if it had been tossed’: this context certainly does nothing to improve the negative answer.) Perhaps, however, we can find some more natural examples where focusing ‘would’ provides evidence against CEM, at least for whatever reading of the conditional is activated by such focus. Suppose for example that have a pile of coins, some normal, some double-headed and some double-tailed. It is not unnatural to say things like \ref{threekinds} in describing this situation
\begin{prop}
\nitem \label{threekinds}
There are three kinds of coins in this pile: those that WOULD land heads if tossed, those that WOULDN'T land heads if tossed, and the rest.
\end{prop}
And once you have got into this mood, you can for example, issue the instruction
\begin{prop}
\nitem \label{WOULDorder}
Just give me the coins that WOULD land heads if tossed
\end{prop}
where one expects obedience to consist in handing over the double-headed coins.
One point to make about these focus-based examples is that they carry across to ‘will’, including uses of ‘will’ that are not embedded in conditionals. Compare \ref{threekinds} and \ref{WOULDorder} with:
\begin{prop}
\nitem
There are three kinds of students: there are the ones that WILL pass, there are the one's that WON'T pass, and there are the students that might go either way.
\nitem
Don't give a student a lot of time unless they WILL pass
\end{prop}
This provides a reason for caution about the putative anti-CEM data from focus, since even those philosophers who reject CEM for counterfactuals will likely still want to preserve sentences like ‘Either you will pass this exam or you will not pass this exam’. Granted, there is the radical option of adopting a theory of ‘will’ according to which it involves universal quantification over a range of possible futures, and hence fails to commute with negation even when only one future time is relevant. But we suspect that the kind of focus-driven effect on display is something much more general, that does nothing special to illuminate the semantics of ‘would’ or ‘will’. For example, we are not seeing a big difference between the foregoing examples and the following:
\begin{prop}
\nitem \label{sick}
There are three kinds of patients in this ward: those that ARE sick, those that AREN'T sick, and those that might or might not be sick.
\nitem
Don't waste any more time doing tests on the patients who ARE sick.
\end{prop}
A tentative hypothesis: putting focal stress on bland small words like ‘are’ and ‘would’ can make the relevant clause behave as if it were prefixed by some epistemic operator like ‘It is known that\ldots{}’ or ‘It is knowable that\ldots{}’.%
%\footnote{***\textbf{Add a vagueness-theoretic note here? Some might suggest that the relevant operator is along the lines of ‘It is definitely the case that\ldots{}’. This might be OK for us, but examples like \ref{sick} suggest that that's not always what's going on.}}
\section{Other arguments for CEM}
Let us briefly report on some other arguments for CEM that can be found in the literature, having to do with a range of embedded constructions. The first argument, due to \textbf{Barker and von Fintel}, turns on the behaviour of ‘only if’. In a wide range of cases involving both indicatives and counterfactuals, ‘P only if Q’ seems to entail ‘If not Q, not P’. Given the standard account of ‘only’ (\textbf{cite Rooth etc.}), the semantic effect of adding ‘only’ to a sentence is to negate a certain range of relevant alternatives to that sentence, which differ with respect to the constituent acted on by ‘only’. Using this machinery it is easy to account for the validity of an inference from ‘P only if Q’ to ‘Not: P if not Q’: we need only say that in context, the relevant alternative to ‘if Q’ is ‘if not Q’. (It would also suffice if we posited a range of relevant alternatives whose disjunction is equivalent to ‘not Q’.) But without CEM, there is no way to bridge the gap from ‘Not: P if not Q’ to ‘If not Q, not P’.%
\footnote{Things get more complicated when the range of relevant alternatives to Q do not exhaust the not-Q possibilities. Suppose that we know that Jim was thinking about going to the museum today. We can say ‘He is having a good time only if he is getting one of the GUIDED TOURS around the museum’. This speech is fine even if we are open to the possibility that in fact, Jim didn't go to the museum and is having a good time in the pub. The standard semantics for ‘only’ predicts this, since it may be that in context, the only relevant alternative for the ‘only’ is tantamount to ‘Jim is in the museum but not getting a guided tour’. By contrast, assuming modus ponens, `If Jim is not getting one of the guided tours around the museum, he isn't having a good time' has to be false if Jim is having a good time in the pub, given that it has a true antecedent and a false consequent. So it seems that the inference from ‘P only if Q’ to ‘If not Q, not P’ can fail in cases where the relevant alternatives do not exhaust the accessible worlds. There are similar complications for the standard logic 101 idea that ‘Only Fs are Gs’ entails ‘All Non-Fs are not Gs’: ‘Only MALE ostriches eat meat’ does not unproblematically entail `Everything that isn't a male ostrich doesn't eat meat'. **However there is a complication here: with focus, ‘If Jim is not getting one of the GUIDED TOURS around the museum, he is not having a good time’ can sound alright even when we think he might be having a good time in the pub. {[}\emph{Can it?}{]} In context, the antecedent seems to be strengthened by the disjunction of the relevant alternatives. This kind of behaviour is of course familiar from unembedded utterance of negated sentences with focus\ldots{}\ldots{} Similarly, sentences like ‘Nothing other than a MALE ostrich eats meat’; `No-one who doesn't take a taxi to this museum enjoys the visit', etc. One approach to this data is to say ‘so much the worse for the reflexivity of accessibility’, i.e.~so much the worse for modus ponens - but we think this is too costly. What we prefer here is a mechanism of ‘embedded scalar implicature’. Just as ‘some’ in ‘Those who ate SOME of their pizza were less ill afterwards than those who ate ALL of it’ can be strengthened to ‘some but not all’, it seems we can strengthen `Those who didn't eat ALL of their pizza were still not as happy as those who didn't eat ANY of it' to `Those who didn't eat all, but did eat some of their pizza\ldots{}'. (\emph{Note that the whole point of these sentences is to contrast two groups among those to whom the relevant restrictor literally applies, so it's hopeless to think that the mechanism is that of domain restriction to exclude those who don't fall under the strengthened meaning.}) Similarly: `If you don't know ALL of your alphabet you will still get some credit for knowing some of it'.}
\begin{itemize}
\item
Mention here: you need CA (‘Disjunction’/‘Cases’) to get to ‘If not-$P$ then not-$Q$’ from the conjunction of ‘if $P_i$ then not-$Q$’ for each of a range of alternatives $P_i$ to $P$.
\end{itemize}
The second argument, due to von Fintel and Iatridou (2002), turns on a range of examples due to Higginbotham (???) in which ‘if’-clauses seem to occur in the scope of determiner quantifiers like ‘Every’ and ‘No’. Higginbotham's observation is that \ref{everygoof} is equivalent to \ref{nogoof}, assuming that failing can be equated with not passing:
\begin{prop}
\nitem \label{everygoof}
Every student will fail if he goofs off
\nitem \label{nogoof}
No student will pass if he goofs off
\end{prop}
Von Fintel and Iatridou observe that if we give these sentences the form they seem to have, then if we allow that there might be some students for whom neither ‘x will fail if x goofs off’ nor ‘x will pass if x goofs off’ are true, \ref{nogoof} will fail to entail \ref{everygoof}. Similarly, if we allow that there might be students relative to whom \emph{both} ‘x will fail if x goofs off’ and ‘x will pass if x goofs off’ are true---as we would if we were equating ‘if’ here with the material conditional---\ref{everygoof} won't entail \ref{nogoof}. What's needed to get the equivalence is thus the relevant instances of CEM---namely ‘Either x will fail if x goofs off, or x will pass if x goofs off’---plus the relevant instances of “Conditional Non-Contradiction”, namely ‘It is not the case that both x will fail if x goofs off and x will pass if x goofs off’. (The material conditional view secures the relevant instances of CEM but not CNC; on our favoured view, CEM is unproblematic; while every instance of CNC that doesn't involve violating the presupposition of non-vacuity will be true.)
Note that while Higginbotham's examples belong to the ‘does-will’ category of conditionals that we have been trying to avoid, exactly the same phenomenon arises with standard indicatives and counterfactuals:
\begin{prop}
\nitem
Every student would have failed if he had goofed off
\nitem
No student would have passed if he had goofed off
\nitem \label{everygoofpast}
Every student failed if he goofed off
\nitem \label{nogoofpast}
No student passed if he goofed off
\end{prop}
Against this diagnosis of the equivalences, someone might complain that there is a further set of equivalences that need explaining and are not explained by an account like ours, namely the equivalence of some of the above sentences with sentences that don't involve conditionals at all. For example, \ref{everygoof} and \ref{nogoof} have been claimed to be equivalent, respectively, to \ref{everygoofwho} and \ref{nogoofwho}:
\begin{prop}
\nitem \label{everygoofwho}
Every student who goofs off will fail
\nitem \label{nogoofwho}
No student who goofs off will pass
\end{prop}
Similarly, \ref{everygoofpast} and \ref{nogoofpast} are arguably equivalent to \ref{everygoofpastwho} and \ref{nogoofpastwho}:
\begin{prop}
\nitem \label{everygoofpastwho}
Every student who goofed off failed
\nitem \label{nogoofpastwho}
No student who goofed off passed.
\end{prop}
Our account does not seem to predict these equivalences. For example, we would analyse \ref{everygoofpast} as ‘Every student is such that either there is no accessible world where he goofed off, or the closest accessible world where he goofed off is one where he failed’, which is strictly stronger than \ref{everygoofpastwho} (given that the actual world is accessible and closer than all other worlds). Such considerations might push one towards the view that---despite surface appearances---the relevant sentences don't have a logical form whereby a wide-scope quantifier controls a variable within the scope of a conditional, in which case it would be hard to learn anything about the logical behaviour of conditionals from these sentences. However such accounts are quite problematic---see von Fintel and Iatridou (???).
Kratzer (???) suggests a way of recovering the equivalence of the Higginbotham sentences with conditional-free sentences like \ref{everygoofwho}--\ref{nogoofpastwho} while retaining their apparent logical forms. Her idea is that in the quantifiers in \ref{everygoofpast} and \ref{nogoofpast} are by default read as implicitly restricted, so that the sentences are tantamount to:
\begin{prop}
\nitem \label{everygoofrestricted}
Every student who goofed off failed if he goofed off
\nitem \label{nogoofrestricted}
No student who goofed off passed if he goofed off
\end{prop}
The idea that the ‘if’-clause plays a dual role: as well as its standard semantic role, it also serves as a signal that helps to resolve the context-sensitivity of quantifier domain restriction. Given that the conditional validates modus ponens and and-to-if, these sentences are logically equivalent to \ref{everygoofpastwho}
and \ref{nogoofpastwho} respectively.%
\footnote{Kratzer also makes the further suggestion that the relevant accessibility relation for the conditionals in \ref{everygoofpast} and \ref{nogoofpast} is the trivial one in which every world is accessible only to itself, so that the conditionals become equivalent to material conditionals. But her way of recovering the equivalences does not depend on this suggestion: all that matters is that the conditional is one that respects modus ponens and and-to-if.}
This proposal also disrupts the use of the Higginbotham data to argue for CEM, since insofar as the domains are restricted in line with \ref{everygoofrestricted} and \ref{nogoofrestricted}, the only substitution instances of the conditionals we will ever need to consider will be ones with true antecedents, leaving us free to reject CEM for conditionals with false antecedents.
Kratzer in fact applies the domain restriction technique only to the past-tense sentences \ref{everygoofpast} and \ref{nogoofpast}. Her view of the original future-tense sentences \ref{everygoof} and \ref{nogoof} is different: she thinks that the most natural reading of these sentences is one on which they are not equivalent to the conditional-free \ref{everygoofwho} and \ref{nogoofwho}, and do not involve the restriction of the domain to students who will goof off. In support of this, she argues that \ref{everygoof} is false in a case where where one the relevant students is a ``teacher's pet'' who will pass come what may but doesn't in fact goof off, even though \ref{everygoofwho} could still be true. She concludes that the argument for CEM is sound for the ‘will’ conditionals, and indeed presents a CEM-friendly semantics for those conditionals.
We agree with Kratzer about the non-equivalence of \ref{everygoof}/\ref{nogoof} with \ref{everygoofwho}/\ref{nogoofwho}; moreover, we think that there are analogous worries about the claimed equivalence of \ref{everygoofpast}/\ref{nogoofpast} with \ref{everygoofpastwho}/\ref{nogoofpastwho}. Suppose that two busloads of students went to take an exam. The excellent students went in one bus, while the struggling students went in the other. You suspect that the bus containing the struggling students did not make it to the exam on time, since you know that that bus got held up in bad traffic. You thus suspect that only excellent students took the exam. In this setting, you have good reason to suspect that \ref{everytestwho} is true:
\begin{prop}
\nitem \label{everytestwho}
Every student who took the exam passed
\end{prop}
However you have far less reason to suspect that \ref{everytest} is true:
\begin{prop}
\nitem \label{everytest}
Every student passed if he or she took the exam
\end{prop}
This difference in your levels of confidence strongly suggests that \ref{everytest} and \ref{everytestwho} aren't being interpreted as equivalent. (We are open to there being contexts where sentences like \ref{everytest} are restricted in the way Kratzer suggests; but as the above example illustrates we doubt that there is any general pressure to do so.) So we think that there is still a strong argument for the validity of CEM both for the ‘will’ conditionals and for the past-tense indicatives.
We agree that if one simply looked at \ref{everytestwho} and \ref{everytest} in abstraction from any particular narrative there is some temptation to think that they are equivalent. One possible diagnosis as follows: first, \ref{everytest} straightforwardly entails \ref{everytestwho} by modus ponens. And while the inference in the other direction from \ref{everytestwho} to \ref{everytest} is not valid, it is \emph{quasi-valid} in the sense explained in section ???: the result of strengthening the premise by adding ‘must’ entails the conclusion. The argument thus has the same status as the arguments from ‘P or Q’ to ‘If not P, Q’ and the argument from ‘P or Q and not-P’ to ‘Must Q’. As we saw in section ???, arguments with this status tend to induce positive feelings akin to those produced by actually valid arguments; thus it is no surprise that there is we get a feeling of equivalence when we are confronted with such pairs as \ref{everytestwho} and \ref{everytest}.%
\footnote{\textbf{However, this diagnosis does not apply to ‘Most students who took the exam passed’ and ‘Most students passed if they took the exam’. Come back to this after we have written the chapter dealing with ‘usually if’ and so forth.}}
Some authors who reject the claim that instances of CEM are invariably true have suggested that they nevertheless have a different positive status, namely that of being \emph{true whenever their presuppositions are satisfied}. The idea is that ‘If $P$, $Q$’ presupposes what the corresponding instance of CEM, namely ‘Either if $P$, $Q$ or if $P$, not $Q$’, expresses. Most proponents of this idea have been advocates of \ref{strict}, so that for them, the content of this presupposition is that all accessible $P$-worlds are alike with respect to whether or not they are $Q$-worlds: the presupposition is thus in their hands a “presupposition of homogeneity”. A favoured analogy is with plural definites it's suggested that ‘The philosophers left the room’ presupposes that either all or none of the (relevant) philosophers left the room, so that ‘Either the philosophers left the room or the philosophers failed to leave the room’ is true whenever its presuppositions are satisfied.%
\footnote{Citations: \cite{vonFintelNLSECD,vonFintelCDC}, Klinedinst, Kriz.}% *** We'd better here have a discussion of the presuppositions of disjunction: the only cases that matter for us are relatively uncontoversial, namely when both have the same presupposition, or when the second disjunct entails the presupposition of the first. In each case it is uncontroversial that the relevant presupposition carries forward to the disjunction. }
This move gives instances of CEM the status of “Strawson validity”: true on any uniform interpretation on which everything they presuppose is true.
As noted in \autoref{sect:quasivalidity}, Strawson-validity confers nothing like the same kind of intuitive security as validity proper. For example, the following sentences are Strawson valid given standard tenets about the presuppositions of ‘the’ and ‘stopped’:
\begin{prop}
\nitem
\begin{prop}
\aitem
The elephant wearing a hat in my bedroom is wearing a hat.
\aitem
Either John has stopped shouting, or John is shouting right now.
\aitem
Either I regret eating the moon, or I ate the moon and do not regret it.
% \aitem
% She is female.
\end{prop}
\end{prop}
It is thus important not to assimilate the kind of thorough embrace of CEM that we have been advocating to the more lukewarm endorsement that goes along with Strawson-validity.%
\footnote{The contrast between Strawson-validity and the stronger form of validity tends to be obscured in theories according to which presupposition failure is held to make for truth value gaps. If one holds this, then not even the law of non-contradiction is sacrosanct, since it will have instances that fail to be true because of presupposition-failure, such as ‘It is not the case that (John regrets eating the moon and it is not the case that John regrets eating the moon)’. Within this framework, then, one might think that Strawson-validity was the strongest status that could reasonably be claimed for any schema. However, even within this framework, one can still draw a contrast between schemas which are guaranteed to be true whenever non-gappy expressions are substituted for the schematic letters and other schemas whose Strawson-validity is due to the presupposition-theoretic properties of the non-schematic expressions in the schema.}
Nevertheless, we agree that there are some cases where a diagnosis of Strawson-validity provides the best all-things-considered explanation of the positive felt status of some schema. Indeed, we have provided such a diagnosis in the case of the following good-looking schema:
\begin{prop}
\litem[CNC]
Not ((if $P$, $Q$) and (if $P$, not $Q$))
\litem[CNC*]
Not (if $P$, $Q$ and not $Q$)
\end{prop}
On our account, these is false when there are no accessible $P$-worlds; however, in such cases all the ingredient conditionals---and hence also their conjunctions and negations---will suffer from presupposition failure thanks to the presupposition of non-vacuity.% discussed in \autoref{sect:nonvacuity}.%
%\footnote{As the fact that we are willing to speak of false sentences with false presuppositions indicates, we are not working with a gap-theoretic approach to presupposition; however our remarks could easily be adapted to such a setting.}
The thesis that CEM is Strawson-valid gives proponents of \ref{strict} or \ref{lewis} a story about the goodness of some of the inferences that feature in certain of our subsidiary arguments against those views. For example, the fact that answering ‘No’ to the question ‘Would this coin have landed Tails if it had been tossed?’ is inappropriate if one takes the coin to be fair (and not to have been tossed) could be accounted for by saying that such an answer would signal aquiescence to the presupposition of the question (that either the coin would have landed Tails if it had been tossed or the coin would have failed to land tails if it had been tossed), a presupposition which is false if the coin is fair and untossed. However, the Strawson-validity of CEM does not even begin to make sense of the facts about chances and credences that drive our central argument against \ref{strict} and \ref{lewis}. For example, on those approaches, ‘I am pretty confident that if he had rolled the die it would have come up between 1 and 5’, as uttered in a setting where one is pretty confident that the die is fair, will be like ‘I am pretty confident that the elephant in the room is more than five feet tall’, as uttered in a setting where one is pretty confident that there is no elephant in the room.
We can also probe more directly the question whether conditionals carry the presuppositions that would be required for CEM to be Strawson-valid given \ref{strict} or \ref{lewis}. In general, when a sentence $P$ carries a presupposition, ‘$S$ doesn't know whether $P$’ carries the same presupposition: for example, ‘Bert don't know whether Alice stopped smoking’ presupposes that Alice used to smoke.%
\footnote{It also seems to presuppose that Bert knows that Alice used to smoke, though that won't be important here.}
Given this presuppositional profile, then, the Strawson-validity approach ought to predict that that claims of the form ‘S doesn't know whether the coin would have landed Heads if tossed’ would be infelicitous when the coin in question is known to be fair, or indeed even when it is not known not to be fair. In fact, however, such claims seem completely fine under such circumstances. In our view, such attributions of ignorance are acceptable even in the case of a coin that's known to be fair; but their felicity may be even more evident when fairness is only of the epistemic possibilities. Suppose for example that Sally picked her coin from the bucket with a mix of fair and double-Headed coins---‘Sally doesn't know whether her coin would have landed Heads if tossed’ seems like an excellent description of this situation. Or consider a self-attribution of ignorance like ‘I don't know whether he would have said yes if she had asked’. Here it is fine to follow up with ‘He is so unpredictable: he might have said yes and he might not have’. There is a contrast here between conditionals and the case of the plural definites that forms the inspiration for the view: ‘I don't know whether the philosophers left the room’ does still seem to carry some suggestion that the philosophers moved or stayed as a bloc. Note also that these tests for presuppositionality give a much more favourable verdict in the case of CNC and CNC*: ‘I don't know whether cannons would both work and not work if Aristotelian physics were true’ is quite bizarre.%
\footnote{And following up this speech with ‘That's because Aristotelian physics is impossible’ would be completely bizarre, although the presuppositional view of CEM suggests that this should be the analogue of the ‘He is so unpredictable’ followup mentioned above.}
\begin{itemize}
\item
Of course one can consider hybrid views: for example, Kratzer (???Conditionals) develops a view that conforms to \ref{strict} for indicatives and \ref{lewis} for counterfactuals.
\item
Like \ref{closest}, \ref{lewis} and \ref{strict} can be fleshed out in different ways,
according to one's favoured gloss on accessibility, worlds, and (in
the case of \ref{lewis}) closeness.
\item
Add: pointing out some nice logical features of CEM, e.g.~the way that
it makes various things entail CSO and one another.
\end{itemize}
\section{Closeness and similarity} \label{objections-to-cem}
In the literature---especially thanks to the influence of \citet{LewisCounterfactuals}---closeness is standardly understood as some kind of similarity. For one world to be closer than another is for the former to be more similar to the actual world than the latter in the relevant respects. What the relevant respects are and what weight they carry is supposed to be up for grabs: Lewis emphasises that the respects and the weighting might not be recoverable simply from general pre-theoretic ideas about overall similarity between worlds. Lewis also thinks that there is plenty of context-sensitivity as regards respects and weightings. Nevertheless, the use of the word ‘similar’ isn't meant to be utterly divorced from its home in ordinary language. There are various structural expectations which are triggered by the ideology of similarity which are preserved on Lewis's view: the kind of relation we are supposed to be thinking about is one specifiable by a scale that comes in degrees, where the degrees in question are determined by some aggregation procedure whose inputs are degrees of resemblance in a range of specified respects, where resemblance comparisons in the particular respects are supposed to be much more straightforward.%
\footnote{As Goodman (???) points out, some of the structural expectations invoked by Lewis's use of the word ‘similar’ are actually ones which he disavows in his most careful moments in a way that many expositors have overlooked, including Lewis himself in some of his less careful moments. In particular, if one pronounces the semantically relevant three-place relation as ‘$w_2$ is more similar to $w_1$ than $w_3$ is’, one would expect that the following pattern can never arise:
\begin{itemize}
\item
$w_2$ is more similar to $w_1$ than
$w_3$ is
\item
$w_3$ is more similar to $w_2$ than
$w_1$ is
\item
$w_1$ is more similar to $w_3$ than
$w_2$ is
\end{itemize}
For, letting $|ww'|$ be the degree of similarity between the worlds $w$ and $w'$, these claims would seem respectively to entail the jointly unsatisfiable $|w_1w_2|<|w_1w_3|$, $|w_2w_3|<|w_1w_2|$, and $|w_1w_3|<|w_2w_3|$. But in fact, Lewis is open to the idea that the relevant three-place relation does contain triples with this cyclical structure, and indeed, as Goodman argues, such cases can arguably be constructed for the particular similarity relation described in \cite{LewisCDTA}. The key to resolving the mystery is that for Lewis, the key three-place similarity-theoretic relation should really be pronounced something like ‘$w_2$ is more similar to $w_1$ than $w_3$ is in the respects that matter at $w_1$’; differences in which respects “matter” at different worlds can then create the surprising cycles.}
And even this very schematic notion of similarity would lead one to expect there to be many cases where two worlds are equally similar to a third world. After all, the relations of resemblance in particular respects that enter into the final aggregation procedure are generally things that do allow for ties: for example, two worlds could be equally similar to a third in respect of \emph{mass} even when one was greater than it in mass and the other less. Moreover, if the aggregation procedure ever allows tradeoffs between different respects of resemblance without one trumping the other, one would expect there to be cases where the tradeoff results in an exact balance. In view of the richness of the space of possible worlds, it is thus hard to see how anything we could think of as a relation of overall resemblance could fail to generate many ties.
If the closeness relation is understood as one of similarity, there is also considerable pressure to allow for failures of the Limit Assumption. Certainly for particular respects of resemblance where comparisons are straightforward, there can be propositions such that for any world where that proposition is true, there is another world where that proposition is true that resembles that actual world more in the relevant respect. Take total mass as the relevant respect, and consider the set of worlds whose mass is greater than $m_1$, where $m_1$ is greater than the mass of the actual world. Clearly for any world in this set, there is another world that resembles actuality more in respect of total mass, e.g.~one whose total mass is halfway between $m_1$ and that world's mass. And given standard assumptions about the continuity of various fundamental magnitudes it is hard to see how this sort of phenomenon could fail to be replicated at the level of overall similarity. Thus, even prior to its details being filled in, a similarity-driven conception of closeness strongly suggests both kinds of Lewisian counterexamples to CEM.
Our response is to completely jettison the similarity-driven conception of closeness. We have various reasons for going this route. Two of them are broadly logical. First, as discussed in \autoref{sect:cf}, infinite agglomeration is just as compelling as finite agglomeration, but as we have just seen, identifying the closeness ordering with a similarity ordering makes the Limit Assumption and thus infinite agglomeration look untenable. Second, as discussed in the previous section, we think there are good reasons to accept CEM, which is again threatened by identifying the closeness ordering with a similarity ordering.
Even if one accepted these logical principles, one might think it was an overreaction to abandon wholesale the connection between similarity and closeness: one might might propose a picture where similarity facts constrain but do not determine closeness facts, or more generally, a picture where there is a general tendency for the similarity and closeness orderings to line up. (If we only had to deal with the problem of ties, we could entertain the obvious constraint that whenever one world is is more similar to actuality than another it is closer, treating the closeness ordering as a mere tie-breaking refinement of the similarity ordering---cf.~\cite{StalnakerDCEM}. It is far less clear how we could understand the constraining role of similarity if we think that the closeness ordering does, while the similarity ordering does not, respect the Limit Assumption.)
But what really pushes us to reject even this moderate attitude towards the relevance of similarity to closeness is the the pattern of confidence judgments we find for both counterfactuals and indicatives. As a warmup, let us begin with a familiar kind of concern. Suppose first that we tossed a fair coin ten times yesterday and saw that it landed Heads on four of those times. Consider the counterfactual
\begin{prop}
\nitem \label{ninecoins}
If the coin had landed Heads nine times, it would have landed Heads all ten times.
\end{prop}
Our judgment is that the right way to assign confidence to the proposition expressed by \ref{ninecoins} is to consult what you know about the conditional chances: of the $2^{10}$ equiprobable ways the coins could have landed, ten involve exactly nine Heads outcomes while one involves ten Heads outcomes, so your credence in the proposition should be $1/11$. But if the kind of similarity that constrains closeness is understood in a pre-theoretically natural way, one would expect that any world where the coin comes up Heads ten times will be less similar to actuality than at least one world where it comes up Heads only nine times, so that \ref{ninecoins} would get a vanishingly low credence. For essentially similar reasons there will be a mismatch in the case where you don't know how the coin landed. So long as the coin is fair we think the appropriate credence is still $1/11$; but the similarity-constrained approach threatens to entail that the only way \ref{ninecoins} could be true would be for ten Heads to in fact have been tossed, a hypothesis to which we assign a much lower credence (namely $1/2^{10}$). (Unless the coin actually landed Heads every time, then for every ten-Heads world we should be able to find a nine-Heads world that is more similar to actuality.)
The general problem is that once we connect closeness to similarity---even in the more modest, constraining way---we will be left with odd confidence distributions. In particular, when there are various ways of a thing happening some of which are more similar to actuality than others, the similarity will push us to be unreasonably confident that if that thing had happened it would have happened in one of the more similar ways. Essentially this objection was made in an early review of Lewis's \emph{Counterfactuals} by Fine (???): Fine considered the counterfactual ‘If Nixon had pressed the button there would have been a nuclear holocaust’, and objected that Lewis's theory yields to the problematic judgment that this is false, since worlds where the button is pressed and some subsequent misfire blocks the expected global nuclear war are more similar overall to the actual, nuclear-war-free world. In response, Lewis makes the point that the relevant similarity metric does not have to coincide with the one elicited by pre-theoretic similarity judgements about worlds, and denies that the relevant peaceful worlds are any more similar to actuality than the war worlds on his \emph{intended} similarity metric.%
\footnote{Lewis's final similarity ranking is intended not merely to prevent the peace worlds from counting as more similar to actuality than the war worlds, but further to get some of war worlds to be more similar than any of the peace worlds. We have further concerns about whether his official specification actually achieves this goal. Lewis's trick for promoting the war-worlds above peace worlds---assuming determinism---is to say that all of the latter contain at least two small ‘miracles’, i.e.~localised exceptions to the actual laws, whereas some of the former contain only one small miracle. We are not sure why Lewis is so confident that a delicately adjusted small miracle couldn't do the job of simultaneously ensuring a button-pushing and its failure. For example, Nixon could trip and fall in such a way that his finger hits the button just after his teeth sever the wire connecting it to the nuclear arsenal. (The problem is even more obvious in a case where there are various ways of pressing the button, a few of which are ineffective.) One could however tinker further to address this worry, e.g.~by saying that the aforementioned tripping-and-falling small miracle is more “remarkable” than certain miracles that lead to war, and for this reason makes for more dissimilarity to the actual world. (This suggestion would be in the spirit of some of Lewis's own ideas about the generalisation of the account to the case of indeterminism which we will discuss below.) By contrast, the problem for Lewis's account which we focus on in the main text is not one that could plausibly be evaded by small adjustments to the similarity ranking.}
But structurally the same problem recurs if we look at the respects of similarity that really do matter according to Lewis. For Lewis, the most important consideration (besides the avoidance of large, widespread exceptions to the actual laws of nature) is the extent of the spatiotemporal region of perfect match. When the region of perfect match between $w_3$ and $w_1$ is a proper part of the region of perfect match between $w_2$ and $w_1$, and neither $w_2$ nor $w_3$ contains ‘big miracles’ with respect to the laws of $w_1$, $w_2$ is more similar to $w_1$ than $w_3$ is. If closeness is constrained by a similarity ranking that works in this way, the upshot is that we can be confident, generally speaking, that if things had gone otherwise than they actually go, the departure would have been later rather than earlier (since the later the departure, the larger the initial chunk of perfectly matching spacetime). But this still leads to problems in the coin example. Suppose that the coin landed Tails the first time it was tossed. Then on Lewis's account some worlds where it lands Heads exactly nine times---namely, those which match actuality throughout a period including the first toss---are more similar to actuality than any world where it lands Heads all ten times. So if we know that it landed Tails the first time, we should be practically certain that \ref{ninecoins} is false. Moreover, if we don't know anything about how it landed but know that it was fair, given that we should still be at least 50\% confident that it landed Tails the first time, the only way we could generate the intuitively correct credence of 1/11 in \ref{ninecoins} would be to assign \ref{ninecoins} a credence of 2/11 conditional on the hypothesis that the coin landed Heads the first time, which seems completely crazy. (After all, conditional on the hypothesis that the coin landed Heads the first time, we are certain that \ref{ninecoins} is true just in case the coin would have landed Heads on all of the final nine flips if it had landed Heads on at least eight of those flips. Intuitively, we ought to assign the latter proposition a credence of 1/10, both unconditionally and conditional on the hypothesis that the coin landed Heads the first time.)
The problem here is essentially the same as the problem of Pollock's Coat which we discussed in chapter 1. There, we saw that a similarity metric that favours late divergence will license us to be very confident in the truth of
\begin{prop}
\nitem \label{coatagain}
If my coat had been stolen last year it would have been stolen on
December 31st.
\end{prop}
when we know that our coat was not in fact stolen last year. And for reasons similar to those discussed above, an approach that licenses such confidence will also generate unreasonable-looking credence profiles in propositions such as \ref{coatagain} in cases where we are uncertain whether our coat was stolen. In fact, we think that the appropriate way to assign credence to the proposition expressed by \ref{coatagain} on its most obvious interpretation is to set this credence equal to our expectation for the conditional chance of the coat being stolen on December 31st conditional on its being stolen some time during the year.%
\footnote{Calculating this number is not straightforward given that the coat being stolen on one day requires it not to have been stolen earlier. In the case where you have no relevant discriminating evidence it will be less than 1/365 but certainly not zero.}
Another respect of similarity that proponents of similarity constraints on closeness have actually regarded as important is similarity with respect of the \emph{absence of remarkableness}, particularly in respect of the outcomes of chance processes. The task these theorists set themselves is to craft a notion of similarity on which, assuming that a certain plate is not in fact dropped, some worlds where it is dropped and hits the floor are more similar to the actual world than any world where it is dropped and quantum-tunnels right through the floor. On the (realistic) assumption that such events of quantum-tunnelling always have some miniscule but nonzero chance of happening, the idea that the disruption of \emph{laws} is a count against closeness does not achieve the desired result, since no actual laws need be disrupted by either kind of world. Clearly the worlds in question need not differ with respect to the size of the region of exact match, so Lewis's idea about the primary importance of exact match doesn't help either. Saying that \emph{low-probability} events count against similarity is a non-starter: whatever happens, the precise trajectory of the plate will be a low-probability event, maybe even a zero-probability event. Lewis's suggestion is that among the various low-chance events we should distinguish a subclass of “remarkable” ones, and to count only these as making for dissimilarity. There are various ways of making this suggestion more precise. One possibility worth mentioning is to understand the remarkable outcomes within a given partition of possible outcomes to be those low-probability outcomes that are much more \emph{natural} than most of the low-probability outcomes in the partition (using something like the notion of relative naturalness for properties developed in \cite{LewisNWTU}). For example, the property of being a sequence of ten coin-tosses all of which land Heads looks considerably more natural than the property of being a sequence of ten coin-tosses which land in the pattern HTTHTHHTHH.%
\footnote{Of course both of these properties are a long way from being perfectly natural; but this need not disrupt the judgment of comparative naturalness.}
Similarly, consider any specification of a way for the plate to hit the floor and break that is detailed for the probability of the plate doing \emph{that} to be roughly as low as the probability of it quantum-tunnelling: plausibly, each of the aforementioned specifications defines a property far less natural than the property of being a quantum-tunnelling through a potential barrier of such-and-such strength.%
\footnote{The naturalness-theoretic gloss on “remarkableness” brings it close to the notion of “atypicality” invoked by Williams (???), drawing on earlier work by Elga (???). The notion of atypicality invoked by these authors is rooted in a mathematical characterisation of the contrast between “random” and “non-random” infinite sequences of coin-tosses due to Gaifman and Snir (???). Williams hopes that Gaifman and Snir's insight can be generalised to finite sequences, and even to particular localised events, and uses expressions like ‘simplicity’ in characterising this generalisation. But Williams mainly focuses on sequences of coin-tosses; it is not straightforward to extract a particular treatment of the quantum-tunnelling case from his remarks.}
Any attempt to specify a similarity relation that turns on considerations of remarkableness will have to reckon with two facts: (a) Assuming that the actual world is quite extensive, we can be very confident that many remarkable events actually occur; and (b) if things had gone differently in some specified way, it would still have been very likely for the ensuing history of the world to contain many remarkable events that don't actually occur. There is plenty to say here (see Hawthorne ???), but we think the best bet for the remarkableness lover is to add a dose of contextualism to the story. For example, we could allow context to contribute a “reference property”, such as the property of being a flip of a certain coin by a certain person during a certain period, or a pattern of motions of a certain plate during a certain period. The extension of this property gives us a reference class which varies from world to world, and may be sometimes empty. We could then articulate a remarkableness-driven respect of similarity such as the following: $w_2$ is more similar to $w_1$ than $w_3$ is iff the reference class at $w_3$ has more remarkable properties than the reference class at $w_1$ or the reference class at $w_2$.%
\footnote{Note that this toy theory counts $w_2$ as more similar to $w_1$ than $w_3$ in the relevant respect even if the reference class has ten remarkable properties at $w_1$, eleven at $w_3$, and only one at $w_2$. A more mechanical similarity measure based on counting remarkable properties would not give this result. But the failure to demote $w_2$ seems desirable. Suppose that in the actual world, I threw ten plates at one wall and they all quantum-tunneled. We don't want to be able to say ‘If I had thrown the plates at the opposite wall, then at least one of them would have quantum-tunnelled’.}
(Of course a final story will have to say something about how this respect of similarity is to be aggregated with others.)
The structural problem we have identified for similarity-based constraints on closeness recurs for the remarkableness-based similarity measures. The basic problem, as always, is that similarity constraints mandate giving more credence to counterfactuals whose consequents characterise non-actual outcomes that are more similar to actuality, in cases where considerations of chance push in a different direction. Suppose for example that a fair coin landing heads or tails a hundred times in a row count as remarkable outcomes, while a certain specific sequence S of heads and tails outcomes counts as unremarkable. In the actual world we know that the coin was never tossed. Before the coin-tossing was called off, Fred bet that the coin was going to either land all Heads, all Tails, or in sequence S, and George bet that it was going to land in sequence S. Given a remarkableness-based similarity constraint, we should then be able to know that worlds where the coin is tossed a hundred times and the sequence of outcomes is S are closer than worlds with a hundred heads or with a hundred tails.%
\footnote{Of course what matters according to similarity-lovers is overall similarity, not just the particular remarkableness-based respect of similarity that we are currently considering. But in this case, none of the other respects of similarity that have been thought to play a role---perfect match, approximate match, lack of miracles, typicality of the world as a whole, and so on---do anything to favour the all-heads or all-tails worlds over S-worlds, so the S-worlds will plausibly be counted as more similar overall as well as in respect of remarkableness.}
And we should thus be in a position to assign very high credence to the proposition expressed by \ref{hundred}:
\begin{prop}
\nitem \label{hundred}
If Fred had won his bet, George would have won his bet too.
\end{prop}
But intuitively this proposition deserves a credence equal to the conditional chance of George's winning given Fred's winning, namely 1/3.%
\footnote{Could we avoid the undesirable result that (37) is true by appeal to some further contextualist trick, according to which (37) for some reason evokes a context in which the all-heads and all-tails outcomes don't count as remarkable in the way that detracts from their similarity? In principle yes, but it's hard to see how such an account would go. For example, saying that the mere \emph{salience} of a particular low-probability outcome makes that outcome stop counting as remarkable will disrupt the proposed explanation of the truth of `If the plate had been dropped it wouldn't have quantum-tunnelled through the floor'.}
% \begin{itemize}
% \item
% *** Add note about how the case relates to the arguments in Hawthorne 2003 and Williams's responses to that. (That paper primarily uses assertion rather than credence.)
% \end{itemize}
The problem with similarity-constraints on closeness is thus quite general. But without a general theory of closeness that guarantees that the worlds where the plate hits the floor and breaks are closer than worlds where it quantum-tunnels through the floor, how are we to explain the assertability of \ref{break} and \ref{notunnel}?
\begin{prop}
\nitem \label{chancycfs}
\begin{prop}
\aitem \label{break}
If the plate had been dropped, it would have broken
\aitem \label{notunnel}
If the plate had been dropped, it would not have gone right through the floor
\end{prop}
\end{prop}
The first point to note about these sentences is that, if we form degrees of confidence to counterfactuals in the manner endorsed in \autoref{sect:probs}, we should have a high degree of confidence in the propositions expressed by \ref{break} and \ref{notunnel}, assuming that we know that the antecedents are false and that the conditional objective chances of the antecedents on the consequents were high.%
\footnote{Even if we don't know that the antecedent is false, the counterfactuals will still deserve high credence so long as we don't have evidence relevant to their consequents that is “inadmissible” with respect to the relevant time.}
Of course, high credence isn't in general sufficient to explain assertability---consider ‘This is a losing lottery ticket’. Perhaps assertability requires knowledge; in any case, the results of prefixing ‘I know that’ to \chisholm{chancycfs}{a-b} are also assertible. How, without some general analysis of closeness in terms of similarity, or some analysis of accessibility that guarantees that there aren't any accessible worlds where the plate is dropped and tunnels through the floor, could we explain our ability to know that the closest accessible world where the plate is dropped isn't a tunnelling world? A good starting point for thinking about this question is the assertability, despite small objective chance of error, of simple claims about the future: for example, our background knowledge about quantum tunnelling does not block us from asserting ‘This plate will soon be broken’, or from self-ascribing knowledge that the plate will soon be broken. It is not an easy task to come up with a workable theory that pinpoints the relevant difference between future lack of quantum tunnelling and future lottery-losing. (For some ideas about this, including ideas according to which our reactions betray deep-seated errors, see
\cite{HawthorneKL}.)
%\footnote{Our talk of objective chance here need not be understood as building in a commitment to indeterminism: for accounts of chance-talk that make non-trivial chance-ascriptions compatible with determinism, see ***. Nor do the epistemological issues turn essentially on the assumption of indeterminism: even if we knew that determinism was true, ‘I will lose the lottery’ would still be unassertable (in ordinary circumstances), and it would still be hard to find an explanation of this unassertability that didn't overgeneralise to all manner of ordinary assertions about the future.}
We can remain neutral about this here: the main suggestion we want to make is that one should approach the knowledge and assertability of counterfactuals in the same way as the knowledge and assertability of claims about the future. Whatever explains the unassertability of ‘I will lose the lottery’ will also explain the unassertability of ‘If I had bought a ticket in that lottery I would have lost’; whatever explains why the unassertability of the first claim doesn't carry over to all sorts of other claims about the future which have small chances of being false will explain why the unassertability of the second claim doesn't carry over to all sorts of other counterfactuals whose consequents have a small chance of being false conditional on their antecedents.%
\footnote{Here we are in agreement with \citet{MossSCSH}.}
Of course whatever defensive manoeuvres we make, we should concede that in the future case, if you are unlucky and the plate does in fact tunnel through the floor, then the earlier self-ascription of knowledge was false after all; similarly, if you are unlucky and the closest accessible world where the plate falls is one where it tunnels, your self-ascription of knowledge that the plate would break if it fell is false.%
\footnote{In the case of future-tense claims, the prima facie attractive idea that knowledge is closed under many-premise deductions leads to further puzzles: by performing a conjunction introduction based on many intuitively knowable high-chance premises, one can come to know a conclusion that one knows to have very low chance. This is both odd in its own right, and makes further puzzles for the project of formulating an account of the rational connection between a proposition $p$ and the proposition that $p$ has a certain objective chance. On one approach, one's knowledge of $p$'s low chance means that one \emph{ought} to have a low degree of confidence in $p$ (despite the fact that one in fact knows $p$ and thus presumably \emph{in fact} is quite confident in it). On a different approach, one ought to have a high degree of confidence in $p$ (despite the fact that one knows its chance is low). In the terminology of Lewis's Principal Principle \citep{LewisSGOC}, the latter option will involve claiming that one's evidence at the time in question is “inadmissible” at that time. All of this structure carries over to the case of counterfactuals. Given multi-premise closure and (finite) agglomeration for counterfactuals, we will sometimes be in a position to know the proposition that if $p$ were true $q$ would be true despite also knowing that its current objective chance is low. Possible reactions include giving up multi-premise closure; tolerating the idea that one can know things to which one ought to have assigned low confidence; and tolerating the idea that we should sometimes be confident in something whose current chance we know to be know.}
The challenge of connecting up facts about the appropriate degrees of confidence in conditionals with facts about assertability and knowledge arises for indicatives as well as for counterfactuals. However, for indicatives, there is more scope for explaining assertability by appeal to the accessibility parameter. Suppose that we're in a context where it takes for a world to be accessible is that be compatible with my knowledge, and that I know that the plate won't be dropped without breaking. Then ‘If the plate is going to be dropped, it is going to break’ is guaranteed to be \emph{true}. And if I \emph{know} I know that the plate won't be dropped without breaking, I can know the proposition expressed in context by the conditional without having to draw at all on knowledge of closeness. On our view this kind of explanation is often apt, but is not the only path to knowledge of indicative conditionals: one can also draw on whatever resources one used to make room for knowledge of counterfactuals like \chisholm{chancycfs}{a-b}.%
\footnote{By contrast, it is much harder to see how a proponent of \ref{strict} can make room for knowledge of indicative conditionals in cases where iterated knowledge is unavailable.}
\section{Metaphysical worries}
We have rejected similarity-based analyses of closeness. And as will be becoming increasingly clear to the reader, we are not going to put anything their place: we will be “treating the concept of closeness as primitive”. At least, we will not be offering the kind of analysis of closeness that would allow us to break out of the circle of concepts that includes closeness as well as conditionals: on our account, so long as we are in a context where both $w_1$ and $w_2$ are accessible, ‘$w_1$ is closer than $w_2$’ is equivalent to ‘If one of $w_1$ or $w_2$ were actualised, $w_1$ would be actualised’ and to ‘If one of $w_1$ or $w_2$ is actualised, $w_1$ is actualised’. We are a little wary of calling this an ‘analysis’, in part because we are a bit unclear about what it takes for something to count as an analysis. Claims of analysis are sometimes understood to carry implications about what grounds what, or what is more metaphysically fundamental or natural than what, or what is more conceptually basic or “explanatorily prior” to what: we are not endorsing any such claim or priority for closeness over conditionals.
So, in a sense we are less ambitious in our goals than many other theorists of conditionals have been. Nevertheless, the claims we are making are by no means trivial - indeed as we have seen they are inconsistent with the views of many other writers, so we feel no need to defend the substantiveness or interest of the views we are putting forward.
“Treating closeness as primitive” is merely refraining from offering any definitions or equivalences that break out of the circle just noted. It is not any kind of claim about closeness. We are thus not making any claim to the effect that closeness is fundamental or perfectly natural or unanalysable or anything else of the sort. Nevertheless, the structural claims we are making about closeness invite a certain hard-to-pin-down metaphysical worry. Take, for example, the claim that a certain untossed coin is either such that some world where it lands Heads is closer than any world where it lands Tails, or such that some world where it lands Tails is closer than any world where it lands Heads. What, people want to know, could conceivably \emph{make it be the case} that one rather than the other of these two possibilities obtains? In what could the difference between the untossed toins that would have landed Heads if tossed and the ones that would have landed Tails if tossed conceivably \emph{consist}?
% The worry is an old one...
% \emph{mention the tradition of thinking that middle knowledge is weird}
\begin{comment}
\begin{itemize}
\item
Is the appeal to vagueness enough to defuse all the worries?
\begin{itemize}
\item
No. On some views of vagueness, we shouldn't have the relevant
intermediate credences when we are thinking that they are vague.\\
\item
Further worry: if there is all this vagueness it looks to infect
even the very good conditionals which we seemingly can know and
assert (in ways analogous to how we can know and assert some
things about the future that have high chances short of 1.)
\end{itemize}
\item
Mention Jeremy's argument from CEM to non-physicalism etc. (Perhaps
just briefly discuss that here and really address it in the concluding chapter. )
\item
Duality arguments against CEM.
\end{itemize}
\end{comment}
\end{document}