Impossible world

In philosophical logic, the concept of an impossible world (sometimes non-normal world) is used to model certain phenomena that cannot be adequately handled using ordinary possible worlds. An impossible world, w, is the same sort of thing as a possible world (whatever that may be), except that it is in some sense "impossible." Depending on the context, this may mean that some contradictions are true at w, that the normal laws of logic fail to hold at w, or both.

Applications

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:

$\vdash A\Rightarrow \ \vdash \Box A$.

Such logics are typically referred to as "non-normal." For more on this, see the article on modal logic.

Avoiding Curry's Paradox

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

$(A\rightarrow B)\rightarrow (A\rightarrow (A\rightarrow B))$.

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.

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.

Resources

• Kripke, Saul. 1965. Semantical analysis of modal logic, II: non-normal modal propositional calculi. In J.W. Addison, L. Henkin, and A. Tarski, eds., The Theory of Models. Amsterdam: North Holland.
• Priest, Graham (ed.). 1997. Notre Dame Journal of Formal Logic 38, no. 4. (Special issue on impossible worlds.) Table of contents
• Priest, Graham. 2001. An Introduction to Non-Classical Logic. Cambridge: Cambridge University Press.

External links

• Edward N. Zalta, A classically-based theory of impossible worlds (PDF)