Modality
Our construction of the world is progressing nicely, with the nature of ordinary objects and properties elucidated. Complex sentences involving these elements seemingly make up most of our thoughts and discourse: "I am thirsty", "Buck the dog has a tail", etc. All those sentences participate in a local state of affairs, but there is an understanding that the situations pertaining to our direct environment are part of larger and larger states of affairs, up to a maximal one, "the world".
Yet, this is far from covering the whole extent of possible discourse. Consider:
"I should have taken an umbrella this morning."
"If Nadal had become a pro soccer player, Federer may have won 29 majors."
"I didn’t think this could ever happen."
None of those describe a state of affairs in the actual world. Instead they relate to possible situations. Modality is the conventional term to encompass notions of possibility and necessity. What is true in no possible world is impossible; what is true in at least one possible world is possible; what is true in every possible world is necessary; what is not necessary is contingent.
The metaphysical nature of possible worlds has already been pondered above (in the "abstract world" section). But prior to taking a metaphysical stance, we need to distinguish between all the different types of modality. Consider now these sentences:
It is necessary for the President of the United States to be at least 35 years of age.
It is impossible to travel at more than 300,000 km/s
If \(A\rightarrow B\) and \(A\) is false, it is still possible that \(B\) be true.
They are all couched in the language of modality, yet they clearly appeal to very different flavors. Ultimately, in a formally rigorous framework, modal logic should be reducible to the understanding of a set of fundamental laws: something is possible if it wouldn’t violate any of those laws. Or, equivalently: a possible world is a way in which our world could have been if things had been different, while still being subject to the fundamental laws.
Our three modal sentences above are true by virtue of, respectively, the U.S. Constitution, Einstein’s general relativity, and classical logic. Let’s outline the following varieties of modality, and the corresponding laws:
Physical necessity: what is necessary under the exact physical laws of our Universe.
Nomic necessity: what is necessary under "some" physical laws.
Metaphysical necessity: what is necessary under the laws of metaphysics.
Logical necessity: what is necessary under the laws of logic.
There is no reason why the metaphysical nature of the possible worlds implied by these modalities should be the same. You may remember that we observed how most metaphysical conceptions of possible worlds simply fall under "concretism" or "abstractionism": either there are other universes, metaphysically indistinguishable from the "actual" one (whose only differentiating factor is that I live in it), or possible worlds are just another abstraction. As I’ve outlined, the two approachs, save for the ultimate ontological truth, are conceptually equivalent for the most part. Concretists typically favor their approach because modal statements stem from a simple fact of the matter: something is possible because it is really true somewhere. Abstractionists dispense with concrete worlds but are left to contend with defining what’s possible. The systematic approach to building a formalized ontology that we’re engaging in right now gives us the best of both worlds: we don’t need concrete worlds to guide us because a clear set of laws would be enough to build the modal abstraction of possible worlds, if that was our only option.