The Sense of Contingency

Bill has a very interesting post just out that prompts some thoughts on modality.  After giving a couple of examples to motivate Hume's (and his own) conviction that anything can consistently be thought of as nonexistent, he says,
I now define the sense of contingency as the sense that everything is thinkable without logical contradiction as nonexistent. I claim that this sense is essential to the type of mind we have. I also claim that the sense of contingency does not entail that everything is modally contingent, i.e., existent in some but not all metaphysically (broadly logically) possible worlds. So from the mere fact that I can think the nonexistence of God without logical contradiction, it does not follow that God is a contingent being. I further claim that we have a hard-to-resist tendency to conflate illicitly the sense of contingency (precisely as I have just defined it) with genuine modal contingency.
I am not sure there is any such thing as 'genuine modal contingency'.  I suspect that our sense of the necessary derives from logical inference: accepting a and a→b we are compelled to accept b.  But this makes necessitation a relative concept: p is necessary relative to some set of propositions Γ iff Γ⊢p.  Perhaps when we say that necessarily p, some set Γ of propositions such that Γ⊢p is implicit.   Thus for narrow logical necessity Γ merely contains the tautologies; for broad logical necessity Γ is augmented with propositions explicating the sense of predicate symbols (Bachelor(x)→Male(x)∧~Married(x), for example); for nomological necessity Γ is augmented with yet further propositions.  On this interpretation our conviction that possibly p arises from our inability to derive ~p from our assumptions Γ: we can say that p is not ruled out by Γ.

If this is right then God's necessity arises from the proof of his existence (for example, the prime mover argument) relative to some assumptions Γ we deem unassailable.   But then something strange occurs:  we can't, contrary to Bill's claim, without contradiction think that God does not exist.  Whatever it is that Bill is thinking without contradiction as non-existent it can't be that God whose existence follows from the primer mover argument; it must be something else. 

Bill also claims that necessary abstracta such as the number seven can also be thought of as non-existent, without contradiction.  Well, I have a much clearer idea of what seven is than of what God is.  It's the successor of six.  If I'm to give up seven I have to give up six, and if I have to give up six I must give up five, and you can see where this is going, as Nick Knowles says.  Eventually I'm reduced to abandoning unity and indeed nullity, which rather undermines a logical language ascribing properties to individuals.  So  I claim, contra Bill, that I can't, without contradiction, think of seven as non-existent.  If you agree with this I say that this is evidence that the sense of contingency/necessity and 'genuine modal contingency/necessity' do not come apart in the way that Bill suggests they do.   And this in turn is evidence for the thesis that modality derives from inference.

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