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 .
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.)
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.”
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 AB 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 world-shift line.
This last condition is important because it allows us to deal with conjunctive counterfactuals as follows.
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.
But here is actually a nice variant on transitivity that does work:
Strengthening the antecedent fails as follows
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.
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.
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.
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.
Possibility Introduction is unrestricted.
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
Strict Conditional Exploitation works just like modus ponens.
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.
Finally, we have the expected rules for strict biconditionals.
Examples:
Counterfactual Deduction adopts all of the above rules, and then the following, which are all just counterfactual versions of modal rules.
Counterfactual proof is like Strict Conditional proof, and restricted similarly to the restrictions on crossing counterfactual world lines in the truth tree method.
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.
Possibility exploitation* This rule is the exact analogue of the modal rule.
Example:
Counterfactual Denial
Things get philosophically interesting when we try to deal with the denial of the counterfactual.
· Ø( 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
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:
· Aà®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 à® ØB
What does this actually mean, though?
The answer is given by the following rule:
So, to summarize, in Stalnkaker’s system.
“It’s not the case that if I had told you to scram you would have” = (A ®ØB)
“ If I had told you to scram, you might not have” = (A à®ØB) = à(A ®ØB)
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 à® ØB.
But, clearly Lewis can not accept Stalnaker’s definition of A à®B. If he did, then he would have to engage in radical reinterpretation of counterfactuals proper and assign a different interpretation to à (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 à® is
To capture the non-negated version of our example:
· If you I had told you to scram you might have. = A à® B