“Logic has it’s own peculiar truth and value.” – Augusta Ada King, Countess of Lovelace

Way back in 2000 JC Beall and Greg Restall (B&R hereafter) wrote:

There are too many modal logics to hold each of them as the logic of broad metaphysical necessity. So given a particular interpretation of each of the symbols in our formalism (including consequence) we can admit that there is a great deal of scope for rivalry. For the propositional modal logic of necessary truth preservation, we think that a logic somewhere between S4 and S5 is a candidate for getting things right. Anything else gets it wrong when it comes to metaphysical necessity. There is scope for rivalry and disagreement, when the meaning of the basic lexicon is settled [2000, 489].

And only slightly less way back in 2004 I wrote:

Here is the main source of my puzzlement. Beall and Restall claim that neither classical nor relevant logic gets the logic of the sentential connectives wrong, but that S4 gets the logic of metaphysical necessity wrong. What, one wonders, is the relevant difference? [2004,413]

Now I don’t know if B&R have changed their minds about this: to the best of my knowledge* they haven’t revisited the issue, certainly it doesn’t appear in the little book. Maybe I should take silence as consent?

But I haven’t stopped being puzzled. Further in 2009 an honours student of mine, Julia Zochodne, wrote a thesis which in part address the question of what it is to be a logic. And after this I found myself thinking about this issue again. Since I am a write in order to think sort of person, I thought I would think about it publicly, here.

So a preliminary to start, with anything of real interest postponed until next time. Our topic, as described by B&R is the logic of broad metaphysical necessity. What is metaphysical necessity? Logicians sometimes use the phrase ‘metaphysical modal logic’ to pick out the logic of necessity and possibility, in contrast with deontic modal logic (obligation and permissibility) or epistemic modal logic (knowledge**), etc. Another contrast B&R might have in mind here might be with epistemic necessity (what is necessary given everything that I know) or natural necessity (what is necessary given scientific laws). But in those two cases it is prima facie reasonable to think that what we have is metaphysical necessity plus some extra axioms in the non-modal part of the logic (the known or laws), so it’s not immediately clear to me that these forms of necessity would require different logics in the pluralist sense. A final contrast they might have in mind is with pure logical necessity. It is, it is sometimes claimed, logically possible that something is red without being coloured, but (so the claim continues) it is not metaphysically possible that something is red without being coloured. Now the truth of that depends on some substantive issues surrounding the question ‘What is a logic?’ which is on the agenda for this series of posts very very soon. But it is, I have come to suspect, more central to what B&R had in mind in that comment in their 2000 paper than I (or other commentators) had thought in 2004.

*They are both rather prolific, and I admittedly have spent most of the intervening six years worrying about either empty names or illocutionary silencing, so I may have missed something.
**Modal logic is probably a terrible way to treat epistemic logic for anyone other than an ideal knower.