Logical necessity and beyond: are there impossible worlds?
Position: Hyperintensional sententialism
The concept of impossible worlds is a common one in philosophy. Of course, "impossible" is not precise enough; what precedes has taught us that we should ask "relative to which modality?" As we’ve seen, for instance, there are worlds that are metaphysically possible but physically impossible. But the term "impossible world" has generally been used with the idea of metaphysical impossibility in mind. Even within the metaphysically impossible, there are still distinctions to make, for logical possibility encompasses metaphysical possibility (because the laws of logic are prior).
Let’s first consider the metaphysically impossible, but logically possible. We already know how to represent such worlds: via the same construction as above, but with closure under the logical entailment \(\vdash^{log{}}\) (if we attempted to use metaphysical consequence instead, some contradiction would have us land on the universal object that encodes everything). Here are some situations which would fall in this category:
An object that is both abstract and concrete (this violates the definition of these concepts).
A concrete object that encodes a property (this violates one of the metaphysical axioms).
A property that doesn’t have a Form (this goes against the comprehension principle).
Two existing objects whose mereological fusion does not exist (this goes against the rules of our mereology).
An intentional mental state that’s independent of any physical object (in contradiction with our buildup of mental states).
A version of you that wasn’t born from the same parents (this one requires more development: by using the term "you" in our language, I implicitly refer to a human being. As we’ve seen, in our metaphysical construction, that is an ordinary object which is individuated by some criteria. In the case of human beings, that normally involves some core genetic identity: Nixon could still have been Nixon if he had never been President, but he couldn’t have been Nixon if his mother hadn’t been Hannah Milhous. Therefore for you to have other parents than yours is not logically impossible at all, but metaphysically impossible).
Logically impossible worlds are beyond any form of necessity. They can be represented as state of affairs without any rule of closure. Basically, "anything-goes" world. Situations can include:
An object that exemplifies \(P\) and not-\(P\) (which is against the propositional logic that our ontology is built on. That can be a truth however in the theory of paraconsistent logic, which plays the role of an impossible world where statements like "\(P\) and non-\(P\)" can be true at that world).
\(1=0\) (at the most basic level, these are just some inkblots on a page. But implicitly, when humans use these symbols, they refer to some version of arithmetic that’s consistent with Peano Number Theory, so we’re really talking here about \(0{_\mathfrak{pnt}}\) and \(1_{\mathfrak{pnt}}\). We know that equality to be false in Peano arithmetic, which is closed under logical consequence; therefore it can only be true in a logically impossible world).
An object in Euclidean space that is both round and square.
All in all, the Zalta-inspired formalism we’ve described above has proved its worth once again. We use the term of sententialism to characterize the present section in order to distinguish more sharply between metaphysically possible and impossible worlds. Although the abstractionist formalism is similar on either side, it is not the same to define objects in the context of our metaphysical theory that enforce the laws of metaphysic in their internal semantics, or to define objects that are metaphysically inconsistent, and thus live in contradiction with the language they’re couched in. The latter are best characterized as sentences that we can talk or think about, but hardly deserve the world moniker.