Depending on the context, this may mean that some contradictions, statements of the form
, or that the normal laws of logic, metaphysics, and mathematics, fail to hold at
Impossible worlds are controversial objects in philosophy, logic, and semantics.
They have been around since the advent of possible world semantics for modal logic, as well as world based semantics for non-classical logics, but have yet to find the ubiquitous acceptance, that their possible counterparts have found in all walks of philosophy.
Possible worlds are often regarded with suspicion, which is why their proponents have struggled to find arguments in their favor.
Many philosophers, following Willard Van Orman Quine,[5] hold that quantification entails ontological commitments, in this case, a commitment to the existence of possible worlds.
Quine himself restricted his method to scientific theories, but others have applied it also to natural language, for example, Amie L. Thomasson in her paper entitled Ontology Made Easy.
[6] The strength of the argument from ways depends on these assumptions and may be challenged by casting doubt on the quantifier-method of ontology or on the reliability of natural language as a guide to ontology.
A similar argument can be used to justify the thesis that there are impossible worlds,[3] for example: The problem for the defender of possible worlds is that language is ambiguous concerning the meaning of (a): does it mean that this is a way how things couldn't be or that this is not a way how things could be.
[2] It is open to critics of impossible worlds to assert the latter option, which would invalidate the argument.
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: Such logics are typically referred to as "non-normal."
Under the standard interpretation of modal vocabulary in Kripke semantics, we have
These non-normal worlds are impossible in the sense that they are not constrained by what is true according to the logic.
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.
The paradox relies on the seemingly obvious principle of contraction: There are ways of using non-normal worlds in a semantical system that invalidate contraction.
This yields the interpretation: 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.