Impossible world

Non-normal modal logics

Non-normal worlds were introduced by Saul Kripke in 1965 as a purely technical device to provide semantics for modal logics weaker than the system K — in particular, modal logics that reject the rule of necessitation:

⊢ A ⇒   ⊢ ◻ A .

Such logics are typically referred to as "non-normal." Under the standard interpretation of modal vocabulary in Kripke semantics, we have ⊢ A if and only if in each model, A holds in all worlds. To construct a model in which A holds in all worlds but ◻ A does not, we need either to interpret ◻ in a non-standard manner (that is, we do not just consider the truth of A in every accessible world), or we reinterpret the condition for being valid. This latter choice is what Kripke does. We single out a class of worlds as normal, and we take validity to be truth in every normal world in a model. in this way we may construct a model in which A is true in every normal world, but in which ◻ A is not. We need only ensure that this world (at which ◻ A fails) have an accessible world which is not normal. Here, A can fail, and hence, at our original world, ◻ A fails to be necessary, despite being a truth of the logic.

These non-normal worlds are impossible in the sense that they are not constrained by what is true according to the logic. From the fact that ⊢ A , it does not follow that A holds in a non-normal world.

For more discussion of the interpretation of the language of modal logic in models with worlds, see the entries on modal logic and on Kripke semantics.

Avoiding Curry's Paradox

Curry's paradox is a serious problem for logicians who are interested in developing formal languages that are "semantically closed" (i.e. that can express their own semantics). The paradox relies on the seemingly obvious principle of contraction:

( A → ( A → B ) ) → ( A → B ) .

There are ways of utilizing non-normal worlds in a semantical system that invalidate contraction. Moreover, these methods can be given a reasonable philosophical justification by construing non-normal worlds as worlds at which "the laws of logic fail."

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.