Counterfactuals: The Short Course

with Kadri Vihvelin

Consider this possible world. 

w1

It’s a world in which signals appear at the left at time t1 and emerge on the right at t2. The dark circles indicate the presence of a signal; an empty circle, the absence.

w1  is a very simple world. It is governed by one simple law.

(L1)    It is nomologically necessary that: (A and B) ≡ C

We can describe its workings with a simple truth table using T or F to indicate the presence or absence of a signal.

 

 The rows represent all the combinations of events that the Law (L1) permits. They don't represent all logically possible worlds, only the nomologically possible worlds given (L1).

If you think there is something fishy about circuit diagrams.  note that we can build a world which is exactly like w1 for all our purposes out of  billiard balls:

 

A Billiard Ball AND Gate

 

 The simplicity of L1 makes the nomological logic of w1 easy. For example, from L1 we can infer derivative laws like:

(L1')    It is nomologically necessary that: ~A ⊃ ~C

The counterfactual logic of w1 is also simple. Consider, for example, the question "What would have happened at w1 if ~A (that is, if there had been no signal on A)?”  The obvious answer is that in that case there would have been no signal at C:

~A > ~C

At first glance, it might seem that we can read the truth of this counterfactual directly off L1 by way of L1'.  But counterfactual reasoning is not that simple. To see this, think of one of the possible worlds where A and C are both false:

w3

Ask yourself: "What would happen at w3 if A were true?"  The correct answer seems to be that, in this situation, if A were true, then C would also be true.

A > C

After all, since B is already true at w3, all that is required to make C true, according to L1, is that A be true. And yet that conclusion is not entailed by L1 all by itself. In fact L1 entails:

It is not nomologically necessary that if A ⊃ C

The moral is that to decide if a counterfactual is true we must pay attention not only to the laws but also to the facts on the ground. A counterfactual can be true at one world but false at another. Thus in this situation:

w4

it is not true that A > C.  In this world B is false and just making A true would not, according to L1, be sufficient to make it the case that C. So:

A>C is true at w3

A>C is false at w4

This relativity of counterfactual truth to worlds makes the logic of counterfactuals more complicated than the strict conditionals we study in introductory logic classes. So complex that it was only in the 1970's that philosophers — the seminal figures here are David Lewis and Robert Stalnaker — got a grip on it.

Lewis and Stalnaker's central insight was that the truth of counterfactuals turns on relations of similarity between worlds. We say that a counterfactual of the form:

If ANTECEDENT were the case, then CONSEQUENT would be the case.

is true at a world if the worlds most similar to it at which ANTECEDENT is true are worlds at which CONSEQUENT is also true. *

Thus, to return to our example,  w1:

w1

and w2:

w2

are the nomologically possible worlds where the antecedent, A, is true:

But w2 is dissimilar to w3 in a way in which w1 is not.  At w2, B is false whereas at both w1 and w3, B is true. This makes w1 more similar to w3 than any other world at which A is true. And because C is true at this closest world, A > C is true at w3.

Pretty much everyone who thinks about counterfactuals agrees that this is a fruitful way to understand their logic, but it still leaves open issues. These centrally devolve around the question of how to measure similarity. w1-w4 are so simple that they don't leave much room for debate: we can count similarities by counting signals. But we do not have to make things much more complicated for deeper questions to emerge.

Consider this world.

w1’

The new branch is captured by the additional law:

(L2) It is nomologically necessary that: C ≡ D

So the nomologically possible worlds look like this.

Now consider the question: "At w1', what would have happened if C had been false?"

The obvious answer seems to be that if C had been false then D would have to be false given (L2).  Moreover, if C had been false then either A or B would have to have been false since (L1) entails:

(L1')  It is nomologically necessary that: ~C ≡ ~( A & B)

Which means that by the measure of counting signals there is a tie among ~C worlds closest to w1'. A tie between

w2’

and

w3’

We capture ties like this in our ordinary subjunctive speech by talking about the way things "might have been". We say that if C were false at w1' then either A or B might have been false.

But already our similarity measure might seem to fall short of capturing everything we want to say. Ordinarily we would say that if ~C, then A might be false or B might be false or both.  But the simple signal counting method excludes that third option, w4'

w4’

because w4’ is more dissimilar to w1' at t1 than either w2' or w3'. 

This seems odd.  Our simple counting method seems to require us to say that if C were not true then one of A or B would be false but not both.

Here it is natural to look to David Lewis for guidance. But when we do, we get a surprise.

On Lewis's account, none of w2', w3' or w4' is the right answer to the question “What if ~C?”.  According to Lewis the most similar world to w1' at which C is false is:

w*

Now you might well protest that w* is not a nomologically possible world given L1 and L2.  But Lewis thinks that is beside the point.  Lewis says that when we think counterfactually we are not obliged to confine ourselves to nomologically possible worlds.  We are allowed to think of worlds that violate the laws, worlds that contain “miracles”.

On Lewis's account, when we compare worlds for similarity:

1. It is of the first importance to avoid big, widespread, diverse violations of law.
2. It is of the second importance to maximize the spatiotemporal region throughout which perfect match of particular fact prevails.
3. It is of the third importance to avoid even small, localized, simple violations of law.
4. It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly.
   

At w*, C is false even though both A and B are true. This is a violation of L1. A miracle. But the required miracle does not seem "widespread or diverse" in the sense prohibited by Lewis’s requirement (1).  A single "small and localized" failure in the AND gate or in any one of several connections would do the job. In the billiard ball example: a tiny diversion of the course of a single ball at any point.

w2’,  w3’ and w4’ involve no violations of law, but Lewis ranks that consideration, (3), as less important than (2): maximizing the extent of spatiotemporal overlap between worlds. In that respect, w* beats w2’-w4’ hands down.

In treating w* as closest, Lewis denies the backtracking counterfactual.

~C > ~(A & B)

This is "backtracking" because its antecedent is about a time later than its consequent. This is not an anomalous result for Lewis's theory. It was Lewis’s official position that counterfactual backtracking was always a bad business and that most backtrackers are false.  This prohibition against backtracking was centrally important to his metaphysics and is widely accepted even by metaphysicians who disagree with Lewis on other points.

And yet in the context of our simple worlds the refusal to backtrack is simply absurd. Any electrician who reasoned like this would soon be out of a job. In these simple worlds, if C were false then either A or B or both would have to have been different.

Let us hasten to say that we  have not come this far just to offer a contrived counter-example to Lewis.  One can imagine any number of arguments to the effect that Lewis is not committed to this absurd result in these cases or that the results are not absurd in these cases or that these examples are anyway not relevant to "real world" counterfactuals.  We will not join those arguments here.

Tomkow has already devoted a lengthy post to arguing against Lewis's theory. There he argued that the correct account is Jonathan Bennett's 1984 "Simple Theory".

Bennett's Simple Theory doesn't trade in miracles: It requires that we find the closest worlds among nomologically possible alternatives. It holds that a counterfactual is true at a world if its consequent is true at the nomologically possible worlds most similar to it at the time of the antecedent.

To see how that plays out in the case of w1': Note that all the nomologically possible alternatives to W1' where C is false are equally similar and equally dissimilar to w1' at the time of C, t2.  All that happens, or fails to happen, at any of these worlds at t2 is C or ~C.  On Bennett's theory, any similarities or differences among these worlds at any time before or after t2 are irrelevant to closeness. Only similarity at the time of the antecedent matters. The upshot is that Bennett's account will treat w2', and w3' and w4' as equally similar to w1'. Thus Bennett allows us to say, as we wanted to say, that if C had been false at w1' then either ~A or ~B or both ~A and ~B.

These simple cases illustrate the merits of Bennett’s theory over Lewis's. Though simple, we think them profoundly important. 

Lewis was a great philosopher, but his theory of counterfactuals was a disastrous misstep.  Lewis’s miracles theory disconnects counterfactual possibility from nomological possiblity at the most fundamental level. It requires us to regard counterfactual reasoning as inherently different from scientific reasoning, to treat metaphysics as disjoint from physics. 

Bennett’s theory can mend this divide because it describes direct connections between laws and counterfactuals. In forthcoming posts we will try to show that these connections have much to tell us about the nature of causation, events, and the arrow of time.