Counterfactual conditional

''This article is about the grammatical subject. For the type of history, see Counterfactual history. For the type of literature, see Alternate history.''

A counterfactual conditional, subjunctive conditional, or remote conditional, is a conditional (or "if-then") statement indicating what would be the case if its antecedent were true. This is to be contrasted with an indicative conditional, which indicates what is (in fact) the case if its antecedent is (in fact) true.

An example
The difference between indicative and counterfactual conditionals can be illustrated with a pair of examples:
 * 1) If Oswald did not shoot Kennedy, then someone else did.
 * 2) If Oswald had not shot Kennedy, then someone else would have.

The first sentence is an indicative conditional that is intuitively true. The second is a counterfactual conditional that is intuitively false (or at least not obviously true).

A counterfactual connective
In order to distinguish counterfactual conditionals from material conditionals, a new logical connective '>' is defined, where A > B can be interpreted as "If it were the case that A, then it would be the case that B."

The truth value of a material conditional, A → B, is determined by the truth values of A and B. This is not so for the counterfactual conditional A > B, for there are different situations agreeing on the truth values of A and B but which yield different evaluations of A > B. For example, if Keith is in Germany, the following two conditionals have both a false antecedent and a false consequent:


 * 1) if Keith were in Mexico then he would be in Africa.
 * 2) if Keith were in Mexico then he would be in North America.

Indeed, if Keith is in Germany, then all three conditions "Keith is in Mexico", "Keith is in Africa", and "Keith is in North America" are false. However, (1) is obviously false, while (2) is true as Mexico is part of North America.

Possible world semantics
Philosophers such as David Lewis and Robert Stalnaker modeled counterfactuals using the possible world semantics of modal logic. The semantics of a conditional A > B are given by some function on the relative closeness of worlds where A is true and B is true, on the one hand, and worlds where A is true but B is not, on the other.

On Lewis's account, closeness is analysed in terms of overall similarity, a real relation of which we are supposed to have some intuitive grasp. A > C is (a) vacuously true if there are no worlds where A is true (for example, if A is logically impossible); (b) non-vacuously true if, among the worlds where A is true, some worlds where C is true are closer than any world where C is not true; or (c) false otherwise.

Consider an example:
 * If I had eaten more at breakfast, I would not have been hungry at 11am.

On Lewis's account, the truth of this statement consists in the fact that, among possible worlds where I ate more for breakfast, there is at least one world where I am not hungry at 11am and which is more similar to our world than any world where I ate more for breakfast but am still hungry at 11am.

Sometimes people condense this slightly difficult mode of expression and simply say that, at the closest world where A is the case, C is the case. So, at the closest world where I eat more breakfast, I don't feel hungry at 11am. This mode of expression embodies the Limit Assumption, which is just the assumption, made of a given counterfactual, that there is a single closest world where the antecedent is true. Although it is commonly made, and can be useful for exposition, the Limit Assumption will often be strictly false. For example, consider whether there is a closest world where my coffee cup is to the left of its actual position. On the face of it, it seems not; for in principle, there might be an infinite series of worlds, each with my coffee cup a smaller fraction of an inch to the left of its actual position. (See Lewis 1973: 20.)

Ramsey
Counterfactual conditionals may also be evaluated using the so-called Ramsey test: A > B holds if and only if the addition of A to the current body of knowledge has B as a consequence. This condition relates counterfactual conditionals to belief revision, as the evaluation of A > B can be done by first revising the current knowledge with A and then checking whether B is true in what results. Revising is easy when A is consistent with the current beliefs, but can be hard otherwise. Every semantics for belief revision can be used for evaluating conditional statements. Conversely, every method for evaluating conditionals can be seen as a way for performing revision.

Ginsberg
Ginsberg (1986) has proposed a semantics for conditionals which assumes that the current beliefs form a set of propositional formulae, considering the maximal sets of these formulae that are consistent with A, and adding A to each. The rationale is that each of these maximal sets represents a possible state of belief in which A is true that is as similar as possible to the original one. The conditional statement A > B therefore holds if and only B is true in all such sets.

Within empirical testing
The counterfactual conditional is the basis of experimental methods for establishing causlity in medicine, natural and social sciences, e.g., whether taking antibiotics helps cure bacterial infection. For every individual, u, there is a function that specifies the state of u's infection under two hypothetical conditions: had u taken antibiotic and had u not taken antibiotic. Only one of these states can be observed, since the other one is literally "counter factual." The overall effect of antibiotic on infection is defined as the difference between these two states, averaged over the entire population. If the treatment and control groups are selected at random, the effect of antibiotic can be estimated by comparing the rates of recovery in the two groups.

Pearl
The tight connection between causal and counterfactual relations has prompted Pearl (2000) to reject both the possible world semantics and those of Ramsey and Ginsberg. The latters were rejected because causal information cannot be encoded as a set of beliefs, and the former because it is difficult to fine-tune Lewis's similarity measure to match causal intuition. Pearl defines counterfactuals directly in terms of a "structural equation model" -- a set of equations, in which each variable is assigned a value that is an explicit function of other variables in the system. Given such a model, the sentence "Y would be y had X been x" (formally, X = x > Y = y ) is defined as the assertion: If we replace the equation currently determining X with a constant X = x, and solve the set of equations for variable Y, the solution obtained will be Y = y. This definition has been shown to be compatible with the axioms of possible world semantics and forms the basis for causal inference in medicine, natural and social sciences, since each structural equation in those domains corresponds to a familiar causal mechanism that can be meaningfully reasoned about by investigators.