A
proposition whose
truth-value depends on a state of affairs that is not actually the case. For example:
Now if we take the '
if' to be the logical operator of
material implication (
->), then if we have not(A), it follows for any
B that A ->
B ("A implies B", or, "if A then B") is true, because the
truth table for the
implication operator, ->, is:
A B A->B ("If A then B")
--+----+--------------------
F F T
F T T
T F F
T T T
It is only false when A is true and
B is false.
So, given that monkeys do not eat soy, the statements
- If monkeys ate soy they would have superpowers.
- If monkeys ate soy then all custard would be blue.
must both be evaluated as 'true'.
It has been observed that most, if not all, scientific theories can be expressed in the form of counterfactual conditionals, and it has even been speculated that the counterfactual conditional is the basic propositional form of a scientific theory. Therefore, attention has been paid to improving on the boolean model of implication, which, as seen above, does not cope terribly well with distinguishing between plausible and implausible counterfactual conditionals.
Saul Kripke's Possible world theory is one such attempt, where, roughly, the truth-values of propositions are calculated according to which maximal consistent sets of propositions (or 'possible worlds') they appear in. We might say that if, for any possible world where monkeys ate soy, all custard was indeed blue, then the statement is true, otherwise not. This is intended to produce a viable logic that can express causality, belief, and so on, by helping to distinguish such absurdities from true facts, such as 'if monkeys ate soy, they would have superpowers'.