We may say that something is brute—in the sense of “brute” as it occurs in “brute fact”—if there is no reason, in any relevant sense of “reason”, why it obtains or occurs. If something is brute, it is inexplicable: there is no reason why it is the way it is. Many theses in philosophy seem to hinge on the existence of brute necessities: necessities which, if they obtain, obtain for no reason in any relevant sense of “reason”. Consider logically brute necessities. For our purposes, we can take these to be necessities which, though they obtain, cannot be demonstrated to obtain by classical logic. Some examples may be necessary existents—including God, on many conceptions—as well as essential-but-unshared properties, non-analytic necessities or “necessary connections”, and undoubtedly many more. If there are any logically brute necessities they are logically arbitrary. For example, it might very well be true that God necessarily exists, but a system of classical logic could never have “God exists” as a theorem; as far as logic is concerned the truth value of “God exists” is just as arbitrary as “there are eight planets in the solar system”. Even though the former might be metaphysically necessary and the latter not, logic is blind to this distinction.
Are there any cogent arguments for or against the existence of logically brute necessities? Let us assume for the moment there are no logically brute necessities. In that case, it must be logically demonstrable that there are none; that is, it must be logically demonstrable that no logically indemonstrable proposition is necessarily true. This is so because if there were no logically brute necessities, and this itself was logically indemonstrable, we would have a necessary truth—that there are no logically brute necessities—that was logically indemonstrable, giving us at least one example of a logically brute necessity. So if there are no logically brute necessities, it logically demonstrable that there are none. What would such a demonstration look like? How could someone logically prove all necessity is logical necessity without implicitly or explicitly defining it to be so? If they didn’t define it to be so in their proof, aren’t they appealing to the very sort of metaphysical necessity they claim to reject? How else could one prove the two notions coincide? I won’t venture to say I can see a priori there is no such proof, because I can’t, but nevertheless I remain skeptical. If you think there is such a proof, I’m open to it: All I ask is that you show me.
Finally, I’d like to close with a potential example of a logically brute necessity. Consider:
(1) “(1) is necessarily true.”
Assuming (1) expresses a proposition and bivalence holds for propositions, what (1) expresses is either true or false. If true it is necessarily true, and if false it is necessarily false. Yet there is a plain sense in which (1)’s truth value, though necessary, is utterly arbitrary. There is no logical proof of its truth or falsity to be had. (Once again, I’m open to the idea there’s a logical proof if you think you have one.) One might even imagine that there are many such propositions, each asserting its own necessary truth, some being necessarily true while others are necessarily false, each as a matter of brute fact. Of course, (1) bears dangerous affinities to the Liar and related paradoxes, and it remains unclear how a solution to them would affect (1). All the same, it gestures in the right direction.
That’s enough from me. What do you think? Any comments, questions, or criticisms are welcome. ^_^
 I am assuming here something along the lines of S5, that modal matters are themselves necessary: It couldn’t be the case, for example, that something might have been necessarily true even though it in fact isn’t. So if there are no logically brute necessities, it is logically impossible for there to be any, otherwise it would be logically possible for there to be some, and thus we would have something which could have been necessary even though in fact it wasn’t.
 This requires the same assumption as footnote 1. Assuming S5 is the correct modal logic, (1)’s modal status is itself necessary, irrespective of whether it is necessarily true or necessarily false. It just can’t be true that, though a proposition is necessarily true, it might not have been necessarily true.