Counterfactuals
Counterfactuals are conditionals in which the
antecedent is presumed to be false. We use
them constantly in practical and causal
reasoning.
Indicative:
 If Brandon goes to graduate school, he
remains a pauper.
 If Brandon does not take logic, then he
remains a cipher.
Strict Modal:
 It is necessarily the case that if Brandon
goes to graduate school, then he remains a
pauper.
 It is necessarily the case that if Brandon
does not take logic, then he remains a
cipher.
Counterfactual:
If Brandon were to go to graduate school, then
he would remain a pauper.
If Brandon had not taken logic, then he would
have remained a cipher.
Traditional skepticism about counterfactuals.
Philosophers have always been suspicious of
counterfactuals, and some have even suggested
that they are completely unverifiable, hence (on
intuitionistic grounds) neither true nor false or
(on verificationist grounds) meaningless.
Consider:
 If we hadn’t both been in the bar that night,
we would never have met.
 If we had never met, then we would never
have gotten married.
 If you hadn’t hid the keys, I would have
driven home drunk.
 If you would have just told me the answers, I
would have passed the test.
These are all counterfactuals. They are pretty
obviously meaningful, and they vary in terms of
there verifiability. Some philosophers think that
because, by definition, the antecedent condition
of a counterfactual is false, that it is impossible
to know whether it is true or false. But this
view combines the worst elements of extreme
empiricism and extreme rationalism.
The proper analysis of causal relations is still
disputed, but it is widely agreed that causal
claims require counterfactual analysis.
The weird logic of counterfactuals
1..Counterfactuals are not material conditionals.
If they were, they would always be true. Why?
2. Counterfactuals are not strict conditionals
because the following inference pattern holds for
strict conditionals but not counterfactuals.
pq
 (p & r)  q
For example:
 If you were to ask me to marry you, I would
say yes.
 Therefore: If you were to ask me to marry
you and have sex with a poodle in front of
my mother I would say yes.
3. Countefactuals aren’t transitive
While this might seem valid
 If we hadn’t both been in the bar that night,
we would never have met.
 If we had never met, then we would never
have gotten married.
 Therefore, If we hadn’t both been in the bar
that night, we would never have gotten
married.
This is pretty clearly invalid
 If Barry hadn’t taken steroids, he still would
have been one of the great baseball players
of all time.
 If Barry had never played sports as a child,
he never would have taken steroids.
.
 Therefore, if Barry had never played sports
as a child, he still would have been one of
the great baseball players of all time.
The meaning of counterfactuals
First, we introduce a sweet new connective: 
A  B reads as “ If it were the case that A,
then it would be the case that B”
The basis of this connective is that that
counterfactuals clearly are modal, but they are
nevertheless not strict conditionals. This fact is
most easily appreciated if we think about really
strong claims like:
 If you were to buy a lottery ticket, then you
would be a dollar poorer.
 If you were to go to school naked, people
would notice.
If we interpret these as strict conditionals, then
we have to say they are false, since there are
possible worlds in which you win the lottery and
there is a possible world in which you go to
school naked and no one notices.
But both of these worlds are what we would
intuitively call highly improbable worlds. The
counterfactual way of expressing this idea is to
talk about the proximity of worlds. When we
say:
 If you were to go to school naked, people
would notice.
we are saying
 In every world close to the actual world, if
you go to school naked, then people will
notice.
The concept of proximity or “closeness”
obviously requires some analysis, but this is the
basic idea.
So, what a counterfactual claim really seems to
say is not the strict modal claim:
 A B is true in a world w iff B is true in
every possible world in which A is true
but rather something like this (p.401):
 A B is true in a world w iff B is true in all
the worlds in which A is true that are closest
to w.
If we define an A-world as a world in which A is
true, we can say this a little bit more intuitively
as:
 A B is true in a world w iff B is true in all
the A-worlds closest to w.
Relative strength of the conditionals:
A B is stronger than the material conditional
A B , because it applies to other possible
worlds, not just the actual world. But it is weaker
than strict conditional A  B because it is not
true in every possible world.
So: (A  B) implies (AB) implies (A B)
Truth Trees for Counterfactuals
This is an extension of S5 we can call System C.
So, we subsume all the rules for S5, and adopt a
few more for the .
First, the Counterfactual rule, which works just
like  and .
Counterfactuals
√(A  B)
A
B
A new rule, Counterfactuals* applies to
situations where we know A to be true. This
mimics E. (Note this rule does not exist in our
truth tree method for any  or . A must be
given independently. It does not amount to
decomposing
A B into A & B.)
Counterfactuals*
√(A  B)
A
B
Counterfactual Negated requires a world shift
line and the idea of the “closest A world”
articulated in the semantics. This rule interprets
the negated counterfactual as saying “There is a
closest A-world in which A is true and B is false.”
Counterfactual Negated
√(A  B)
A
B
A
To understand what sort of formulas survive
world-shift lines we have to introduce the new
technical term “tantamount” which weakens
slightly the idea of logical equivalence to
subsume counterfactuals. (p. 404)
We can say that A is counterfactually equivalent
to B just in case both of the following are true:
 A B
 B A
Both of these conditions will hold when A is
identical to B or when AB is true, but not, of
course, when only A  B is true.
So, the answer to the question of what survives
world-shift lines is this:
We can use counterfactual information A
B across a world shift line marked with C
iff
 A is tantamount to C; or
 A is tantamount to a conjunction of live
formulas containing C below the worldshift line.
This last condition is important because it allows
us to deal with conjunctive counterfactuals as
follows.
√ (p r)
* q
(p &q)  r
p
r
q
(p&q)
p

r
q


p
Examples from book. (p. 405)
Recall from our English language examples
above that both transitivity and strengthening the
antecedent fail for counterfactuals. System C
preserves this sweetly.
p  q
q r
√ (p r)
p
r
q
P
But here is actually a nice variant on transitivity
that does work:
p  q
(p&q) r
√(p r)
p
r
q
r

P
Strengthening the antecedent fails as follows
p  q
((p & r) q)
p&r
q
p
r
P&R
To complete System C we need to modify our
possibility rule from S5 slightly so that the world
shift line can be crossed by counterfactuals
when necessary.
A
A
A
All this does is to restrict the inference from “A is
true in some possible world” to “A is true in the
closest A world”. Bonevac gives a traditional
example on p. 406, but it can be shown to be
important in practical causal reasoning contexts
as well.
If I were to fall in love and get married, I would
be happy.
If I were to fall in love and get married, I would
be miserable.
Therefore, if I were to fall in love, it would not be
possible for me to marry.
(p & q)  r
(p & q)  r
 ((p  q)
Unfortunately, to understand the rest of the
Counterfactual chapter we have to return to the
Necessity chapter and learn some modal
deduction. Fortunately, this isn’t terribly difficult,
as much of it’s rationale mirrors the rational for
modal truth trees.
Counterfactual Deduction
Counterfactual deduction is an extension of
modal deduction, and since we only used the
tree method when we studied necessity, we
need to trot back and pick up the modal
deduction rules. Most of them are analogous to
some tree rule, so this isn’t too hard.
Necessity Exploitation is an unrestricted.
Necessity Exploitation
n.
n +p
A
A E, n.
The modal proof rule is essentially the analog of
world travel. To show that A you simply show
that A, where the only formulas above the Show
line available are modally closed.
Modal Proof
Show A

n.
n+p
A
Possibility Introduction is unrestricted.
Possibility Introduction
n.
n. + p
A
A
I,n
Possibility Exploitation
The basic idea here is that if we know A is
possible, we know from the semantics that A is
true in some world. Hence, if A strictly implies
some formula B, we know that B is true in some
world. But if B is modally closed, then we know
that B is true in all possible worlds. So,
possibility exploitation works like this
Possibility Exploitation
n. A
m. A  B
p. B
E n,m
B must be modally closed
Strict Conditional Exploitation works just like
modus ponens.
Strict Conditional Expl.
n.
m.
p.
A B
A
B E, n,m
Modal proof for strict conditionals is just
conditional proof with the restriction that all
formulas above the show line used in the
subproof be modally closed.
Modal proof for strict conditionals
Show A B

A
AP
.
.
.
B
n.
Finally, we have the expected rules for strict
biconditionals.
Strict Biconditional Exploitation
n.
m.
p.
A B
A (or B)
B (or A)  E, n,m
Strict Biconditional
Introduction.
n.
m.
A B
B A
Examples:
1. p
A
2. Show p
3. Show: p  p

4. p
AP
5. p
E 1,3
1.
2.
3.
4.
5.
(p  q)
A
Show (p  q)
Show (p q)  (p q)

p q
AP
pq
E 1,3
1.
2.
3.
4.
p  q
Show p  q

p
q
A
AP
E 1,3
Counterfactual Deduction adopts all of the
above rules, and then the following, which are all
just counterfactual versions of modal rules.
Counterfactual Exploitation
n.
m.
p.
A B
A
.
B E, n,m
Counterfactual proof is like Strict Conditional
proof, and restricted similarly to the restrictions
on crossing counterfactual world lines in the
truth tree method.
Counterfactual Proof
n.
Show A  B

A
A  P
.
.
.
B
As with strict conditional proof, all modally
closed formulas are available within a
counterfactual proof. Further, counterfactuals
are available if the antecedent to the
counterfactual is tantamount (in the technical
sense described in the truth tree method) to (a)
the antecedent of the conditional being used for
conditional proof or (b) the conjunction of the
antecedent of the conditional and other
information already available in the sub proof.
Strict and Counterfactual Conditionals. The
following rule just follows from the relative
strength of strict vs. counterfactual conditionals.
Notice it only goes one way.
Strict and Counterfactual
Conditionals
n. A  B
n +p. A B , n
Possibility exploitation* This rule is the exact
analogue of the modal rule.
Possibility Exploitation *
n. A
m. A B
p. B
E*, n
B must be modally closed
Example:
1. p  q
A
2. q  r
A
3. q  p
A
4. Show p r

5. p
AP
6. q
E, 1,5
Counterfactual Denial
Things get philosophically interesting when we
try to deal with the denial of the counterfactual.
 ( A B)
Counterfactual:
 If you had told me to scram, I would have.
Counterfactual Denial:
 No, it’s not the case that if I had told you
scram, you would have.
What, exactly does this mean? Here are two
alternative interpretations.
Interpretation 1: Robert Stalnaker
 If I had told you to scram, you wouldn’t have.
Interpretation 2: David Lewis
 If I had told you to scram, you might not
have.
Reasonable people can differ on the proper
interpretation of counterfactual denial. The crux
of the matter is this:
 How many closest A worlds are there?
To see why, recall that
 A B is true in w just in case B is true in
every A world closest to w.
So, how many closest A worlds are there?
Specifically, is there only one closest A world.
Or can there more than one?
Stalnaker’s interpretation: One closest A world
Stalnaker’s interpretation makes sense if we
assume there is only one closest A world.
If there is only one, then to say
 It is not the case that if I had told you to
scram you would have.
is to say:
 In the one closest A world if I had told you to
scram, you would not have.
Hence, Stalnaker’s rule for counterfactual denial
is
CS Counterfactual Denial
(A B)
==========
A B
For Stalnaker, then, the question immediately
arises: How do you capture the meaning of the
alternative interpretation; i.e.
 If I had told you to scram, you might not
have.
The answer is that we introduce yet another
connective:
 AB
This is read as:
 “If it were the case that A, then it might be
the case that B”.
So, “If I had told you to scram, you might not
have” would get translated as
 A  B
What does this actually mean, though?
The answer is given by the following rule:
CS Might Counterfactual.
(A B)
==========
( A B)
So, to summarize, in Stalnkaker’s system.
“It’s not the case that if I had told you to scram
you would have” = (A B)
“ If I had told you to scram, you might not have”
= (A B) = (A B)
Lewis’ Interpretation: Multiple closest A worlds.
For Lewis, there can be multiple A world’s all
equidistant from some world w.
Hence, to say that
 It is not the case that if I had told you to
scram, you would have.
is to say that
 In at least one of the A worlds closest to w, if
I had told you to scram, you wouldn’t have.
This, of course, translates directly into: A 
B.
But, clearly Lewis can not accept Stalnaker’s
definition of A B. If he did, then he would
have to engage in radical reinterpretation of
counterfactuals proper and assign a different
interpretation to  (A  B).
Instead, then, Lewis adopts a different rule. For
Lewis:
 If I had told you to scram, you might not
have.
translates into counterfactual denial:
 It’s not the case that if I had told you to
scram, you would have.
So Lewis’s rule for  is
CL Might Counterfactual
(A  B)
==========
( A  B)
To capture the non-negated version of our
example:
 If you I had told you to scram you might
have. = A  B
means
 It’s not true that if I had told you to scram,
you wouldn’t have. = (A B)
These different rules create different conditions
for world travel, and hence for what can and can
not be proved within the respective systems.
Bonevac gives you a few of these differences.
At this point in the semester, we are likely to find
these arguments to be pretty arcane. But if you
allow yourself to recall that counterfactuals are
the basis of causal reasoning, then there is
actually quite a bit at stake with respect to the
proper interpretation of scientific claims.