Impossible World - Applications - Counternecessary Statements

Counternecessary Statements

A counternecessary statement is a counterfactual conditional whose antecedent is not merely false, but necessarily so (or whose consequent is necessarily true).

For the sake of argument, assume that either (or both) of the following are the case:

1. Intuitionism is false.

2. The law of excluded middle is true.

Presumably each of these statements is such that if it is true (false), then it is necessarily true (false).

Thus one (or both) of the following is being assumed:

1′. Intuitionism is false at every possible world.

2′. The law of excluded middle is true at every possible world.

Now consider the following:

3. If intuitionism is true, then the law of excluded middle holds.

This is intuitively false, as one of the fundamental tenets of intuitionism is precisely that the LEM does not hold. Suppose this statement is cashed out as:

3′. Every possible world at which intuitionism is true is a possible world at which the law of excluded middle holds true.

This holds vacuously, given either (1′) or (2′).

Now suppose impossible worlds are considered in addition to possible ones. It is compatible with (1′) that there are impossible worlds at which intuitionism is true, and with (2′) that there are impossible worlds at which the LEM is false. This yields the interpretation:

3*. Every (possible or impossible) world at which intuitionism is true is a (possible or impossible) world at which the law of excluded middle holds.

This does not seem to be the case, for intuitively there are impossible worlds at which intuitionism is true and the law of excluded middle does not hold.