lundi, février 06, 2006

Propaedeutics: Counterfactuals

Some properties of counterfactuals


This note is intended to cover questions some readers have about some of my comments on works on counterfactual conditionals. In the following I repeat some of the things I have already said, and the examples I give are found in the literature.

Can counterfactuals be explained temporally, via concepts like time-line/ history? To answer this question, one has firstly to define what one means by counterfactuals and which properties characterise them.
Counterfactuals ought not to be confused with contrapositions. Contraposition is a kind of transposition in traditional logic. Thus, although in the literature it is common to find sentences like (3) and (4) as examples of counterfactuals, they are actually contrapositions:
(1) If Jerry is a cat then he eats fish.

(2) If you must to go to Lisbon, you take train A.


(3) If Jerry does not eat fish, then he is not a cat.

(4) If you do not take train A, you cannot go to Lisbon.

Counterfactuals are conditionals of a different nature. First of all they state things that are contrary to the facts. If a conditional states something that is in accordance with the facts, then the statement is not counterfactual. Counterfactual statements serve as kind of explanations and, perhaps, are closely related to reduction to the absurd demonstrations. Furthermore, Counterfactuals display some key properties of counterfactuals, whereby some modes of inference do not apply:
Failure of the Hypothetical Syllogism (Transitivity)

(5) a. If Hoover had been a Communist, he would have been a traitor.

b. If Hoover had been born in Russia, he would have been a Communist.

*⇒ c. If Hoover had been born in Russia, he would have been a traitor.

Failure of Contraposition

(6) (Even) if Goethe hadn’t died in 1832, he would still be dead now.

*⇒ b. If Goethe was alive now, he would have died in 1832.

Failure of Strengthening the Antecedent

(7) a. If kangaroos had no tails, they would topple over.

*⇒ b. If kangaroos had no tails but used crutches, they would topple over.

Failure of Inversion with equal values (T&T or F&F)

(8) a. If kangaroos had no tails, they would topple over. (F→F)

*⇒ b. If kangaroos had tails, they would not topple over. (nonsense) (T→T)

(9) a. If Goethe did not die in 1960, he would still be dead now. (T→T)

*⇒ b. If Goethe died in 1960, he would still be alive now. (nonsense) (F→F)

Failure of Modus Ponens

(10) If high interest rates were a remedy to control inflation, piglets could fly.

*⇒ a. High interest rates are a remedy to control inflation.
b. Piglets can fly.

These properties can be explained in terms of different histories, i.e., alternate time-lines. (A particular history h can in principle be identified with a possible world w, but we shall set aside the discussion on whether we have really distinct concepts or just different terminologies to old ideas.) First assume a set of all possible events E. If we impose an ordering upon the elements of E, then we derive a set of possible histories or time-lines. Similarly, if we take a finite subset of E, let us say E1= {e1…en} and we impose an ordering upon the elements of E´, then each different order of events will be a different possible time-line or history.
Now assume that we are in a particular time-line h*, which is history as it has been (our time-line). We know or imagine that History could have been otherwise, the several imaginable possibilities being the alternate time-lines. In our time-line the allies have won World War II, China has annexed Tibet, Armstrong was the first man to walk on the Moon and the United States lost Vietnam War. Each of these points in our time-line is crucial. History would have been very different if, for instance, the Nazi-Fascists regimes had won World War II, the Tibet had annexed China, Gagarin was the first man to walk on the Moon and the United States won the Vietnam War. We can pick up each of these crucial points in our time-line and relate it to alternate time-lines via branching nodes [1]. Accordingly, reasoning on abstract grounds, there was previously a point in the time-lime where one of the possible outcomes of World War II would be inserted. If we entertain two possibilities, i.e., either the Allies would win or the Nazi-Fascists regime would win, then we have a point where the line splits into two: we are in the branch of the time-line where the first possibility was the event that succeeded the preceding historical events. In the alternate time-line, the other possible took place. Thus, considering the other paring of events, i.e., the annexation of China by Tibet versus the annexation of Tibet by China, etc, we get more branches of the time-line (or time-tree, if you like).
In this sense, a counterfactual conditional is a statement about sequences of events that could have happened, but did not. This is precisely why they state things contrary to the facts and cannot be subject to modus ponens.
In other words, counterfactuals may be characterised as statements about time branching, about the alternate time-lines. If someone says one of the sentences below, he will be talking about different sequences of events:
(11) If you stroke these matches, they would have lit.

(12) If you stroke these matches and soaked them in the water, they would not have lit.

By changing the events that are included in the line or by changing their orders, the results affect the ways we interpret the sentences and so make certain modes of inference unavailable.
Let us examine the examples above. In (11) the conditional construction accesses a line with two ordered events:
Event A: the matches are striken;

Event B: the matches lit.

Event B is the outcome. In (12) another event C is interposed between events A and B, but then B is no longer accessible: event C causes the line to split and creates a binary branching, leading to another outcome ¬B. So the addition of the clause and soaked them in the water does not strength the antecedent, it changes the time-line that is accessed, it creates a space-time branching.
Examples like those in (5) seem trickier, but they are not: although (5a) and (5b) share one common event, the one in which Hoover would have been a Communist, they access two different alternate time-lines, with two different orderings. (5a) branches from some point in our actual history after the event of Hoover being born in the US. While (5b) departs from a branching exactly at instant of Hoover’s birth: in the actual time-line Hoover’s place of birth is the US, in the alternate time-line his place of birth is Russia and the counterfactual accesses the latter.

To be continued…