Counterfactual conditional
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A counterfactual conditional, or subjunctive conditional, is a conditional statement aimed at capturing the meaning of if-then statements in natural languages. Differently from logical conditional statements, a counterfactual conditional may be false even if its antecedent is false.
The meaning of if-then statements in a natural language is not always correctly formalized by logical conditionals. In particular, logical conditionals are always true whenever their antecedent is false, while an if-then statement in a natural language can be false in that case. For example, the statement “If Keith is in Mexico, then Keith is in Africa” would typically be considered false. However, the corresponding logical conditional is true if Keith is currently not in Mexico. In other words, if the statements “Keith is in Mexico” and “Keith is in Africa” are formalized by the propositions <math>m</math> and <math>a</math>, respectively, one would not expect the first to imply the second. Nevertheless, if <math>m</math> is currently false, then <math>m \rightarrow a</math> is true in propositional logic.
In order to distinguish counterfactual conditionals from logical conditionals, a symbol <math>></math> is defined, so that <math>A>B</math> means “if <math>A</math>, then <math>B</math>”.
The semantic of the counterfactual conditional <math>A>B</math> cannot be defined in terms of the truth value of the conditions <math>A</math> and <math>B</math>, as it is done for the logical conditional. Indeed, there are different situations agreeing on the truth values of <math>A</math> and <math>B</math> but yet expected to give different evaluation of <math>A > B</math>. For example, “if Keith is in Mexico then Keith is in Africa“ is expected to be false even if Keith is in Germany, which implies that both the antecedent and the consequent of this conditional are false. However, the statement “if Keith is in Mexico then Keith is in America” is true in the same condition in which Keith is in Germany. Therefore, both the antecedent and the consequent of these conditionals are false in both cases. However, the conditional statement itself is false in one case and true in the other case.
Philosophers such as David Lewis and Robert Stalnaker modeled counterfactuals using the possible world semantics of modal logic. The semantic of a conditional <math>A>B</math> is based on considering the most likely situations in which <math>A</math> is true, and checking whether <math>B</math> is true in all of them. Formally:
- <math>A > B</math> is true at a world <math>w</math> if, in all the worlds closest to <math>w</math> where <math>A</math> is true, <math>B</math> is also true.
For example:
- If the braves had won, Keaton would've eaten his hat.
To evaluate this statement, consider a possible world where the braves did win, and imagine that this world is otherwise as similar to the actual world as possible (so, for example, it is not a world ruled by Nazis). Then ask whether, in such a world, Keaton proceeded to eat his hat.
Counterfactual conditionals can be evaluated using the Ramsey test: <math>A>B</math> holds if and only if the addition of <math>A</math> to the current body of knowledge has <math>B</math> as a consequence. This condition relates counterfactual conditionals to belief revision, as the evaluation of <math>A>B</math> can be done by first revising the current knowledge with <math>A</math> and then checking whether <math>B</math> is true in what results. Revising is easy when <math>A</math> is consistent with the current beliefs, but can be hard otherwise. Every semantics for belief revision can be used for evaluating conditional statements. Vice versa, every method for evaluating conditionals can be seen as a way for performing revision.
A semantics for conditionals has been proposed by Ginsberg assuming that the current beliefs form a set of propositional formulae, considering the maximal sets of these formulae that are consistent with <math>A</math>, and adding <math>A</math> to each. The rationale is that each of these maximal sets represents a possible state of beliefs in which <math>A</math> is true that is as similar as possible to the original one. The conditional statement <math>A>B</math> therefore holds if and only <math>B</math> is true in all these worlds.
Note
The semantics of logical conditionals is defined so that <math>A \rightarrow B</math> is equivalent to <math>\neg A \vee B</math> because this is the only semantics based on truth values that ensures that <math>\{A, A \rightarrow B\}</math> entails <math>B</math> and <math>\{\neg A, A \rightarrow B\}</math> does not affect the truth value of <math>B</math>. Logical conditionals model rules of inferences like if <math>A</math> is true then <math>B</math> must be true as well, that are supposed to be trivially satisfied whenever their antecedent is false.
See also
References
- J. Bennett (2003). A Philosophical Guide to Conditionals, Oxford University Pres.
- D. Bonevac (2003). Deduction, Introductory Symbolic Logic, 2nd edition, Blackwell Publishers.
- M. L. Ginsberg (1986). Conterfactuals. Artificial Intelligence, 30:35-79.
- D. Lewis (1973). Counterfactuals, Blackwell Publishers.
