One of the foundational ideas behind philosophical logic is that when you say something, that has further implications beyond the single thing you said. Like, if I think ‘every single frog is green’ and ‘Fran is a frog’, then I am committed to thinking that Fran is green. I don't have to have actually thought to myself or said out loud that Fran is green—I'm just required to believe that Fran is green, given that I thought the first two things, and if I fail to believe that, I've made some kind of mistake. Like I haven't thought through all the consequences of my beliefs.

Modal logic studies how we reason about obligation and permission. For example, f I think that Bob is obligated to visit his parents for the holidays, it follows from that that he isn't permitted not to visit his parents for the holidays. (The term for this in philosophical logic is that obligation and permission are duals.) There are lots of inference patterns that pop up, some of them familiar and some of them surprising, the moment you start thinking about how the notions of ‘obligated to’ or ‘permitted to’ interact with notions like ‘if/then’ or ‘and’.

Free choice permission is a funny case where it feels like out in the wild, you would have to draw a certain conclusion from something you said, but our best formal, mathematical theory of obligation and permission tells us that you aren't allowed to draw that conclusion. So although the theory gets most other things impressively right, it seems to get this one thing wrong.

Here's the example. Imagine you're a customer at a cafe and a waiter says to you, ‘Since you ordered our prix fixe lunch menu option, you may have coffee or tea’. Translated into the terminology of obligation and permission, we could think of what the waiter said as ‘it is permissible for you to have either coffee or tea’. And there seems to be no way the waiter could think that and not thereby also be committed to thinking it is permissible for you to have coffee. If you're allowed to have either coffee or tea, then surely you're thereby allowed to have coffee. Right?

The problem is that the best available formal mathematization of how reasoning about obligation and permission works (believe it or not, this is given the humorous-sounding name normal modal logic) predicts that you are not allowed to draw that conclusion. So since it seems obvious that any rational person would draw that conclusion, but our theory predicts that you aren't allowed to draw it, that means the theory has a problem. The trouble is that revising the theory so as to correctly make that prediction is quite technically difficult, because most of the obvious things you might do to have it make that prediction have the side effect of breaking other aspects of it that work well.

In this episode, Melissa Fusco sketches out a highly original and ambitious approach to the puzzle, using a more sophisticated framework called two-dimensional modal logic. Two-dimensional modal logic is based on a subtle but interesting distinction between a statement that's automatically true the moment you start thinking about it, and a statement that is necessarily true, no matter what. It may sound a bit counterintuitive, but just wait till you hear the examples that Fusco gives! Trust me—her idea about how you can use that distinction to explain what's happening in the waiter example is super cool.

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