Start with the idea of *grades of necessity*. At the bottom, say[note 1], lie ordinary empirical claims like that I am typing now, which have no necessity. Higher up lie basic structural claims about the world, such as that, say, there are four dimensions and that there is matter. Perhaps higher, or at the same level, there are nomic claims, like that opposite charges attract. Higher than that lie metaphysical necessities, like that nothing is its own cause or that water is partly composed of hydrogen atoms. Perhaps even higher than that lie definitional necessities, and higher than that the theorems of first order logic. This gives us a relation: *p*<*q* if and only if *p* is less necessary than *q*.

Let → indicate *subjunctive* conditionals. Thus "*p*→*q*" says that were it that *p*, it would be that *q*. Let ⊃ be the *material* conditional. Thus "*p*⊃*q* basically says that *p* is false or *q* is true or both. Then, the following seems plausible:

- If ~
*p*<(*p*⊃*q*), then*p*→*q*.

Suppose it's a law of nature that dropped objects fall. Then the material conditional that *if this glass is dropped, then it falls* is nomic and hence more necessary than the claim that this glass is not dropped, and the subjunctive holds: were the glass dropped, it would fall.

Moreover, the subjunctives that (1) can yield hold non-trivially, if there are grades of necessity beyond metaphysical necessity (on my view, those are somewhat gerrymandered necessities), and this yields non-trivial *per impossibile* conditionals. Let *p* be the proposition that water is H_{3}O, and let *q* be the proposition that a water molecule has four atoms. Then ~*p*<(*p*⊃*q*), because *p*⊃*q* is a definitional truth while ~*p* is a merely metaphysical necessity. Hence were *p* to hold, *q* would hold: were water to be H_{3}O, a water molecule would have four atoms.

I wonder if the left-hand-side of (1) is necessary for the *non-trivial* holding of its right-hand-side.

## 5 comments:

If we stipulate that every falsehood is less necessary than every truth, then we get the rule that if p and q are true, then p→q.

~p < p ⊃ q only if p -> q. Let p be I do not jump from the building. Let q be I fly. It is not less necessary that I do not jump from the building than that I jump from the building only if I fly. They're at most equally necessary (though I admit I don't understand how propositions that are not necessary at all are more or less necessary than others, but let's skip that). But it might well be true that were I to jump from the building, I would fly. Otherwise, I simply would not jump.

"But it might well be true that were I to jump from the building, I would fly. Otherwise, I simply would not jump."

Surely the right thing to say is: "Were I to jump, I would fall. That's why I'm not going to jump."

However, if some one is a pigeon and this is known to be 100% true that he/she is definately a pigeon, and if that some one has been on the building waaaaaay too long then they should jump. :-)

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