Abstract: The standard account of counterfactuals that most philosophers endorse—Lewis's 'Analysis 1' — is wrong. The correct theory is one invented by Jonathan Bennett in 1984 which he called 'The Simple Theory'. Bennett later argued himself out of that theory and went on to champion the standard account. But those arguments fail. The Simple Theory has been right all along.
Let us, following Lewis, help ourselves to the idea that some possible worlds more or less overlap other worlds in matters of particular fact and say that the extent to which this is true measures how much one world "resembles" another. We'll also help ourselves to the idea that two worlds might resemble each other at one time but not others.
Let us, following Bennett, call worlds with exactly the same natural laws as our world "legal" worlds. Call our world, "a". For the sake of simplicity we will assume, in this post, that the laws of a are deterministic.
Laws of nature are unrestrictedly universal so that if any world has a's laws at any time it has them at every time. And yet there is a way in which a possible world can at least appear to only partially overlap ours with respect to laws: it seems logically possible that a world might, for some period of time, behave as if it obeyed exactly the same laws as our world but then suddenly, at time T, begin behaving in ways that were inconsistent with our laws. In such a case let us say that this world "behaves legally until time T". Likewise we can imagine a world which, up until a time T, behaved in ways inconsistent with the laws of a but, after T, suddenly began behaving as if it were a legal world. Let us say, of such a world that it "behaves legally after time T".
Finally let us help ourselves to the idea that some statements are about some times and not others. So we will say
Obama became president.
is about something that happened in 2008 and
Obama was reelected.
is about something that happened in 2012. We'll write 'TP' to indicate time the proposition p is about.
Our topic is counterfactual statements of the form:
If ANTECEDENT had been the case then CONSEQUENT would have been the case.
which we'll write:
To keep things simple, we will only deal with cases where A is false at a but nomologically possible: that is, where A it is true in at least one legal world. We'll call worlds where A is true, 'A-worlds'.
In these terms we can frame six different analyses of counterfactuals:
THE FIXED PRESENT THEORY
A > C iff at any possible world that is exactly like a at TA except that A, it follows from the laws of nature that C
THE FIXED LAWS THEORY
A > C iff C is true at the legal A-worlds which most resemble a.
THE FIXED PAST THEORY
A > C iff C is true at A-worlds exactly like a prior to TA and which behave legally after TA.
THE SMALL MIRACLES THEORY
A > C iff C is true at the A-worlds which exactly resemble a until some time TM prior to TA; behave legally after TM; and resemble a as much as possible at TM.
THE LEGALIZED MIRACLES THEORY
A > C iff C is true at the legal A-worlds which exactly resemble the closest SMALL MIRACLES worlds at TM.
THE SIMPLE THEORY
A > C iff C is true at the legal A-worlds that most resemble a at TA.
The Untenable Present
The FIXED PRESENT theory deserves to be be first on our list because it marks the start of modern thinking about counterfactuals. It makes its debut in Nelson Goodman’s 1947, “The Problem of Counterfactual Conditionals”.
To see how it it is supposed to work suppose that these are the facts: Here is a match. It is well made and unbroken. There is oxygen in the room. The match is not wet. There is no wind. There is a good, dry striking surface close to hand. The match is not struck but surely:
(1) If the match had been struck, it would have lit.
The FIXED PRESENT theory tells us that to see why this is true, all we need to do is to describe a world exactly like the actually world except that the match is struck. When we conjoin its being struck with its being being dry, the presence of oxygen … &c. with the laws of nature it will surely follow, as a matter of logic, that the match will light.
But there is a problem because all we have said thus far leaves some inconvenient facts about the world unchanged. For example, it is a fact that at no time, in the actual world, does the temperature of the match exceed room temperature. If we leave that fact unchanged then the laws of nature will tell us that the match does not light, since the laws require that no match lights at room temperature. Worse yet, since the laws of nature tell us that no struck match remains at room temperature, we can conclude that the match was not struck, or was not dry or…&c.
We get contradiction because the world we get when we have the match struck but leave its temperature unchanged is nomologically impossible. It is a possible world, but not a legal world.
What has gone wrong is obvious: if the match had been struck it wouldn’t have been at room temperature. As Goodman put it “The match remains at room temperature” is not “jointly tenable” or “cotenable” with “The match is struck, the match is dry, there is oxygen present … &c.”.
“A is cotentable with S if it is not the case that S would not be true if A were” 💬
To get round this problem it seems we will have to stipulate that in evaluating A > C we must consider a world different from the actual world not just in respect of A, but also different in any respect which is not cotenable with A. But adding this proviso renders our theory circular because, of course, cotenablity is a counterfactual notion: A is cotenable with B provided that B would still be true if A were true.
To repair the FIXED PRESENT theory would require an account of cotenability that did not rely on counterfactuals. Goodman called this “the cotenability problem”. It was, he said, the problem with counterfactual conditionals and closed his famous essay by confessing that he had no solution to it.
That failed start should have been the end of FIXED PRESENT theory. But in the kind institutional amnesia so characteristic of modern philosophy (there are so many new papers, who can read the old stuff!) the theory keeps turning up like a bad penny. Hall and Paul adopt it as their working theory of counterfactuals in their “User’s Guide”:
To evaluate “if C had not occurred, then E would not have occurred” we construct a counterfactual state of the world at time t as much like the actual state of the world as possible save for the fact that C does not occur. Think of taking the actual time-t state of the world, and wringing carefully localized changes on it just sufficient to make it the case that C does not occur. 💬
This can’t work unless we understand that in “wringing our carefully localized changes” we are careful to change the things that would be changed if C.
Hall and Paul credit this account of counterfactuals to Maudlin . Maudlin at least flirts with the problem of cotenability. His example is:
If the bomb dropped on Hiroshima had contained titanium instead of uranium it would not have exploded.
All we need to do to see that this is true, he says, is to take the state of the world shortly after the bomb was dropped “save that the physical magnitudes are changed in this way:uranium is replaced with titanium in the bomb”, and then “allow the laws to operate”.
But wouldn’t certain other things have to be different if there were titanium in the bomb? Maudlin’s answer:
Thus the ‘if the bomb dropped on Hiroshima had contained titanium instead of uranium…’ directs us to a moment shortly before the atomic explosion and instructs us to alter the description of the state of the world so that titanium replaces uranium in the bomb. And there is a tacit ceteris paribus condition: leave everything else the same. Don’t fool with the position of the plane or the wiring of the bomb or the monetary system of France or anything else. Similarly, if I command you to take the roast beef in to the guests you have not carried out the command if you step on the roast beef first, and if you murder one of the guests in the process you did not do so on my instructions. Of course, one cannot just change the uranium into titanium and leave everything else the same. The total amount of titanium in the universe will be changed, as will the ratio of titanium to steel in the bomb. So the clarity of the instruction, what counts as fulfilling it and what not, depends on the clarity of the ceteris paribus clause, and this is a function of the context. 💬
But the problem is not with obeying a ceteris paribus clause to “leave everything” the same. The problem is that not everything can be left the same. Some things would have to be different if it were titanium not uranium: the mass of the bomb for example. If it weren’t, any number of conservation laws would be violated and “allowing the laws to operate” will lead straight to contradiction. To know “what to fool with” we need to know what facts would be different if the antecedent of his counterfactual were true. Mauldin says we would figure this out from the “context”. Doubtless we would, but in this Mauldin’s theory would be no help.
The problem with the method of FIXED PRESENT is that gives us illegal worlds. Why not then just stipulate that relevant worlds must preserve the laws of the actual world? This brings us to FIXED LAWS.
Hooray for Small Miracles!
Everyone agrees that the FIXED LAWS theory is wrong. To see why, take our mundane counterfactual:
(1) If the match had been struck it would have lit.
said on an occasion when the match was not struck. According to FIXED LAWS, to decide if this is true we would have to consider (entirely) legal worlds in which the match was struck; ask which of them most resembles the actual world and then determine if the match lights in those worlds.
Now we can be confident that there are some legal worlds where the match is stuck insofar as we are confident that it is physically possible for the match to be struck; to say that it is "physically possible" just is to assert that it happens at some legal world. But to be confident on this score is not the same thing as being confident about what such a world might be like. If the laws of a are deterministic then a world in which the match is struck will have to have had a different past from a's, different for all eternity at every moment. Those differences will in turn will make for other differences— in some worlds small differences, in some worlds large; differences that will extend far beyond the lighting or not lighting of the match. None of us can be sure what even one such world be like, let alone compare alternatives for overall — that is over all time and space — resemblance to a.
The problem with FIXED LAWS, then, is it would make it impossible for us to determine if any counterfactuals were true or false.
Contemplating the unknowable eternal pasts of the closest FIXED LAW worlds, we might well think we should stick with the past we know (or, if we don't know, we can at least point at). So what if we forget about preserving the laws and focus on preserving the actual past? Thus, the FIXED PAST theory.
On the FIXED PAST account we treat the closest worlds as ones which have exactly the same past as a right up to the time of A. This will make the truth of A a miracle according to a's laws, but we will require that these worlds behave legally after TA.
The merit of FIXED PAST as compared to FIXED LAWS is that does give us a viable way of determining if counterfactuals are true or false. Unfortunately it gives us the wrong answers. Consider this example.
(2) If President Obama were in North Korea right now he would have no secret service protection.
This is risibly false, said at any time in the Obama presidency, but it comes out true on the FIXED PAST account. Even if Obama and his retinue are now safely ensconced in the White House, the closest worlds where the antecedent is true but which have exactly our actual past are worlds in which Obama is instantaneously transported to North Korea. And (2) will be true according to those worlds since worlds at which Obama is teleported alone and his retinue remains in Washington will more resemble the actual world than ones in which they are teleported with him. Counterexamples like this to FIXED PAST can be multiplied indefinitely.
What is most obviously wrong with a world like this—the reason we don't count it as relevantly close to ours— is the sudden jump that takes Obama instantly from the USA to the DPRK. The problem we, may think, is not just that the jump would be a miracle— we have already seen problems that can arise from trying to avoid miracles— but that it seems altogether too big and sudden a miracle.
We think (2) is false because we know that the president never goes anywhere without secret service protection and would not make a state visit like this without many months of prior planning leading to a long flight on Air Force One. A world where that happens would have to have a different past than ours — at least over the last several months— and if it would have taken miracle to set this alternate course of events in motion it could at least have been a small miracle—say a change of someone's mind some months ago— rather than the big discontinuity required if we kept the past entirely fixed.
Thus: the SMALL MIRACLES theory.
In this setting SMALL MIRACLES appears as a kind of compromise between FIXING the past and FIXING the laws. SMALL MIRACLES, like FIXED LAWS, requires that A come about legally and, like FIXED PAST, it prizes similarity in matters of past particular fact. But SMALL MIRACLES allows for miracles and past differences prior to TA provided they are small miraculous divergences just big enough so that A comes about lawfully.
Here is Lewis's canonical formulation of SMALL MIRACLES in "Counterfactual Dependence and Times Arrow" (CD&TA):
"Analysis 1. Consider a counterfactual "If it were that A, then it would be that C" where A is entirely about affairs in a stretch of time TA. Consider all those possible worlds w such that: (1) A is true at w; (2) w is exactly like our actual world at all times before a transition period beginning shortly before TA; (3) w conforms to the actual laws of nature at all times after TA; and (4) during TA and the preceding transition period, w differs no more from our actual world than it must to permit A to hold. The counterfactual is true if and only if C holds at every such world w." 💬
Which he elsewhere glosses in CD&TA as:
"Roughly, a counterfactual is true if every world that makes the antecedent true without a gratuitous departure from actuality is a world that also makes the consequent true. " 💬
And here is Jonathan Bennett's version in "A Philosophical Guide to Conditionals ":
"... the closest A-worlds exactly resemble a until shortly before TA, diverge from a in an inconspicuous manner that may be a small miracle, and conform strictly to causal laws of a thereafter." 💬
Lewis is properly credited with inventing SMALL MIRACLES but it was never quite his official position. He certainly thought that SMALL MIRACLES gave mostly the right answers about which counterfactuals were true or false but he thought he could get those same results with a different account: the one he gives in CD&TA. The merit of the CD&TA account was supposed to be that it doesn't assume — as the SMALL MIRACLES model does— that past resemblance is more important than future resemblance. Lewis wanted such a time-neutral account for all sorts of interesting reasons but, alas, there are good reasons to think the CD&TA account doesn't work: there are counter-examples; they are moreover counterexamples where the CD&TA account gets things wrong and where the SMALL MIRACLES gets them right.
So SMALL MIRACLES became and remains the standard theory. "Standard" both in the sense that it is the model most philosophers assume when they want to put the logic of counterfactuals to work on topics like causation, the laws of nature or free will and also in the sense that it is generally accepted by philosophers that it accurately describes how non-philosophers ("folks", as the professionals call them) go about deciding which counterfactuals are true.
Troubles with Small Miracles
But if SMALL MIRACLES is a compromise between FIXED PAST and FIXED LAWS it is, I think, an uneasy compromise. It seems to get the right results, but why and how it does so is not so clear.