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Episode post here. Thanks to Caroline Wall for the transcription!

Matt Teichman:
Hello, and welcome to Elucidations. I’m Matt Teichman, and with me today is Melissa Fusco, Assistant Professor of Philosophy at Columbia University. And she is here to discuss free choice permission. Melissa Fusco, welcome.

Melissa Fusco:
Hi. Thanks for having me.

Matt Teichman:
So, free choice permission is a really interesting linguistic and logical puzzle that first arose in the ‘70s, and which people have been discussing especially recently. In the last 20 years, there’s been more and more interest in it. What exactly is the puzzle of free choice permission?

Melissa Fusco:
Well, the puzzle of free choice permission is an unexpected inference that people make on the basis of sentences that involve a weak or diamond-type modal, like ‘may’, or ‘might’, or ‘can’ in English.

Matt Teichman:
Or, like, ‘possibly’?

Melissa Fusco:
‘Possibly’, yeah—disjunction under the scope of that modal. Or there’s some controversy about whether it needs to be under the scope.

Matt Teichman:
So like, when you’re using ‘might’ and ‘or’ at the same time.

Melissa Fusco:
Yeah. So I’ll use an example with ‘may’, because that’s permission. (One of the original titles of the puzzle is ‘the puzzle of free choice permission’.) If I say, ‘You may have coffee or tea’, what I say seems to entail that you may have coffee and you may have tea. Not necessarily both, but that you have a choice—so both disjuncts are individually permissible. Sometimes it’s a little hard to see why that inference is puzzling.

Matt Teichman:
Yeah, it seems natural, right?

Melissa Fusco:
It seems very natural. So part of what comes out of studying modal logic, or classical logic and then modal logic, is that certain inferences that we make all the time become more surprising than they would have seemed at first, because they are not predicted to be valid on the semantics that you learn from those textbooks or in those courses. So I think, actually, one of the best ways of understanding why, with free choice permission, a positive entailment—in other words, something that does seem to be entailed by something that you said—why that’s mysterious comes from looking at the negative half of that. Which is that certain inferences don’t seem valid that are predicted to be valid on the classical semantics. So those are connected.

The best way to start, I think, is with a different puzzle—Ross’s puzzle. So that’s also got a lot of attention in pretty recent literature. Ross’s puzzle involves the ‘box’ companion of the ‘diamond’ that’s in a free choice inference. So ‘box’ is usually translated in English as ‘must’ or ‘ought’. So that’s a necessity rather than a possibility operator.

Matt Teichman:
And when we say ‘box’ and ‘diamond’, we’re talking about symbols you would write in a formal language, sort of of the type—

Melissa Fusco:
That’s right. In a modal logic, where the box is some form of necessity and the diamond is some form of possibility, such that modal logic might be described as the formal study of possibility and necessity. It’s not, at a first pass, a study of the way that we talk about those things. Just like classical logic, which involves truth-conditional disjunction, conjunction, negation, and a material conditional—truth-conditional ‘if-then’—that’s not, first and foremost, an empirical theory about how we talk with ‘if-then’, for example, or with negation. But obviously, they’re connected. If we would like to understand what follows from what we say, then we would be interested in valid inference patterns that may come from logicizing the connectives that we, in fact, use.

So the same idea, with the same caveats, applies to modal logic. If modal logic is the study of necessity and possibility, it seems like that should be informed by—and in turn, inform—how we actually talk about necessity and possibility, how we talk about what we might or must or ought to do, maybe in part because how else would you study necessity and possibility? It’s not like possibility is a thing you can go dig out of the ground. You don’t figure out what’s possible by ordinary empirical methods. It seems like we just look at the way that we talk and reason about those things.

So here is Ross’s puzzle. Ross’s puzzle is of the form, ‘You ought to do P. Therefore, you ought to do P or Q’. So that, if you wrote it in modal logic, would be: $$\Box P$$ Therefore: $$\Box (P \vee Q)$$

So that’s predicted, in a normal modal logic, to be a valid inference.

Matt Teichman:
What would be a good example of something where the first thing is true and the second thing is true—like, an English example.

Melissa Fusco:
Well, actually, I’m not sure that there are that many. If you just talk about something that you ought to do—‘I ought to pay my taxes’—I suppose you could say, then, ‘I ought to pay my taxes or Q’, where Q is something else. ‘I ought to pay my taxes, so I ought to pay my taxes or go on vacation’. But aha—actually, it doesn’t feel like that follows, right?

Matt Teichman:
It’s weird. It feels like going on vacation is irrelevant. What does that have to do with your obligation to pay your taxes?

Melissa Fusco:
Right, and you can get examples of that form of reasoning that seem worse. They’re not merely irrelevant; they just seem false. They just seem wrong. So one of the original examples was, ‘I ought to post the letter. Therefore, I ought to post the letter or burn it’. ‘I ought to pay my taxes, so I ought to pay my taxes or flee the country without paying my taxes’.

Matt Teichman:
Yeah. That seems nuts, right? Because if you burn the letter, you obviously haven’t fulfilled your original obligation, the one where you—originally, we thought you were just obligated to send the letter. But if you burn it, you can’t send it. So it seems like that really conflicts with the first thing.

Melissa Fusco:
Right, right. So think of this as the failure of ‘or introduction’ in the scope of ‘ought’. From the fact that you’re mailing the letter, it follows that you are mailing or burning it, at least, again, on a classical semantics for ‘or’. Whenever P is true, then P or Q is true. But that pattern, that you can always introduce an extra disjunct—even if it’s irrelevant, even if it wouldn’t be as nice as the first disjunct—but it would still be true to say it. If it’s true that you’re posting the letter, then it’s true that you’re posting the letter or burning it. If Mike is a duck, then if it’s true that Mike is a duck, then it’s true that Mike is a duck or an insect or whatever.

Matt Teichman:
Yeah, uninterestingly true. He’s one of those two things—namely, a duck. So therefore, he’s also a duck or an insect, although we know he’s not an insect.

Melissa Fusco:
That’s right. So in modal logic, both the box and the diamond are predicted to be what we call upward entailing. So that means that they preserve the direction of consequence in their scope. So if ‘P or Q’ follows from P, then ‘ought (P or Q)’ should follow from ‘ought P’. And if ‘P or Q’ follows from P, then ‘possibly’ or ‘may’ or ‘might’—however you want to say it—‘P or Q’ will follow from ‘may’ or ‘might’ P.

And that does come from the way that we ordinarily analyze possibility and necessity. The idea is when we talk in terms of unmodalized Ps and Qs—which, of course, we don’t actually do—when we talk about states of affairs, like Mike being a duck or posting a letter, we’re talking about what goes on in the actual world. When we talk about what’s possible, we talk about what’s going on at some possible world, not necessarily the actual one. And when we talk about what’s necessary or what ought to be the case, we’re talking about what goes on in every possible world. Not every in the unrestricted sense, sometimes in various restricted senses.

So for example, ‘ought P’ doesn’t mean that P is actually going on. But it does mean that P is going on at every world which is in some sense good, every world where the rules are obeyed, every world that is morally ideal, every world, maybe, which is as ideal as a world could be (subject to various constraints of consistency, if you have conflicts between your rules). So if the inference from P to ‘P or Q’ is good at the actual world, then it should be good at an arbitrary world. So that’s, in some sense why, if P entails ‘P or Q’, that entailment should be preserved, even when you shift off-world. So it should be true under the scope of a box for necessity, and it should be valid, rather, under the scope of a diamond for possibility. And yet, it seems that we have counterexamples to this—again, because if you ought to post the letter, just because you ought to post the letter, it doesn’t follow that you ought to post it or burn it.

Now, here’s the connection to free choice permission, the way that I would frame it. Why can’t you conclude—I mean, again, this seems like a question so obvious that it’s not worth raising, but, to take a naive approach—why can’t you conclude from the premise that you ought to do P, that you ought to do P or Q? Well, because the statement that you ought to do P or Q is stronger than your premise. It has entailments—it would license inferences that you’re not allowed to conclude on the basis of the premise that you ought to do P. So in particular, it seems to carry the felt entailment to the permissability of each disjunct embedded under the operator. So the reason you can’t conclude, from ‘you ought to post the letter’, ‘you ought to post the letter or burn it’, is because burning is impermissible. So you isolate a strong entailment property of the putative conclusion of the thing that is not, in fact, the conclusion of your inference.

It seems like free choice permission gives us a window onto what that is—why it’s stronger. So the idea is if you know that you ought to do P or Q, then you can conclude that it’s permissible to do P and it’s permissible to do Q, which is exactly what you’d conclude if you knew that it was permissible to do P or Q. So the idea is that ‘or’ embedded under the box (‘ought’) is like ‘or’ embedded under the diamond (‘may’, in this case, for deontic modals). And it licenses the permissability of each disjunct. And the question is: why? So from the unremarkable fact that if I say, ‘You may have coffee or tea’, you can conclude that you may have coffee and you may have tea, that can explain what’s wrong with other inferences, like Ross’s puzzle.

Matt Teichman:
Nice. So we’re killing two birds with one stone, ideally, if we can get both of these things under control. What would be an example of free choice permission with ‘ought’ rather than ‘may’?

Melissa Fusco:
Well, free choice permission is usually—by its nature, it involves a modal that describes possibility. But let me just give you—I think this would work in the same way—let me give you a Ross’s puzzle-type example with permission. So the original Ross’s puzzle is, ‘You ought to post the letter; therefore, you ought to post it or burn it’, except that ‘therefore’ can easily be challenged. It doesn’t seem like the inference is valid. Likewise, I can say, ‘You may post the letter; therefore, you may post it or burn it’. That inference sounds just as bad as the original.

So I think the data—what’s a little bit funny about the original way that the literature developed is: free choice permission is this puzzle where you get something stronger than you expect. Ross’s puzzle is that you get something weaker than you expect. Call the weaker-than-expected phenomenon a negative datum, a failure of entailment. Call the stronger-than-expected datum a positive entailment, something stronger than you expected. But each can be transformed into the other. That’s why they’re related. So again, from the negative datum that I can not conclude ‘I ought to post or burn’ from ‘I ought to post’, I can just transform that, in the case of the diamond, to ‘I have permission to post’ doesn’t entail ‘I have permission to post or burn’.

Matt Teichman:
Yeah. Okay. Interesting. So it seems like the conflict here is between the intuition that from a certain point of view, this inference pattern we’re calling disjunction introduction, where I say, ‘Mike is a duck; therefore, Mike is either duck or an insect’—that seems like that’s a robust inference pattern. Because it’s impossible for Mike to be a duck and not be either a duck or an insect. It seems like that follows. But then, if we hold onto that reasoning principle, and we think that’s a good inference pattern, we get both of these undesirable Ross and free choice results. The Ross result is, ‘I ought to mail the letter; therefore, I ought to mail the letter or burn it’, which seems completely wonky.

Melissa Fusco:
Suppose I said, ‘You may have coffee or tea. I don’t know which’. Okay, now that I’ve said, ‘I don’t know which’, it no longer seems like you can choose. It seems like you have to go check.

Matt Teichman:
Like I’m getting something out of a box with no label on it. I don’t know what it is, but you can have one of these boxes. I don’t know what’s in it.

Melissa Fusco:
Yeah. So ‘I don’t know which’-type riders on free choice permission are sometimes interpreted as canceling the free choice effect. Now, if it follows by logic, then it doesn’t seem like you can cancel. Using the term ‘cancellation’ typically appeals to pragmatic effects. So to use an example, you went on a blind date, and I ask you the next day, ‘How did your date go?’ and you say, ‘The restaurant was nice’.

Matt Teichman:
Ouch.

Melissa Fusco:
There, it seems like you are telling me—but you’re telling me in an indirect way, you’re telling me in a pragmatic enrichment type of way—that the date did not go well. You’re not actually saying that, right? I’m just in a position to conclude that if it had gone well, you would have told me so. And so since you didn’t tell me so, I can conclude that it didn’t go well. You’re just too nice to say so, or something. However, that kind of inference—the inference from you didn’t tell me it went well, so considering that you changed the subject and so on, I should conclude that it went poorly—that’s cancelable. I can run the dialogue again. ‘How does your date go?’ You say, ‘The restaurant was nice’. And then, you say, ‘The date also went well. I don’t mean to say the date didn’t go well. I just mean, wow, I can’t stop thinking about the pasta I had at the restaurant’.

Matt Teichman:
Right. You might try to explain it away, like, ‘Oh, I guess that was kind of misleading. Actually, I was just thinking about the restaurant. But the date was also pretty good’. You’re not committed to thinking—it’s not logically required—

Melissa Fusco:
Exactly. You’re not committed to it. It doesn’t follow as a matter of logic. Whereas if I had said, ‘How did the date go?’ and you had replied, ‘I have never been on a date that went well’—well, okay. Again, you didn’t exactly answer the question, in the sense that you said not quite the yes or the no that I expected. But you said something that entails that—

Matt Teichman:
Yeah, you can’t be like, ‘I’ve never been on a good date, and this one was great!’ That makes no sense.

Melissa Fusco:
Yeah, exactly. So from the fact that you said something of the form ‘all Fs are G’—all dates went badly for you—I can conclude that the particular one you went on last night didn’t go well. And again, I conclude that by logic. So no cancellation, it seems, is possible. You can’t go: ‘I have never gone on a date that went well, but I don’t mean to say that the one I went on last night didn’t go well’. That doesn’t seem to make sense.

So if I can cancel the free choice effect, then it looks like it’s not semantic. It looks like it’s not logical, in some sense. I think that that impression is misleading. I think it actually is likely to be an entailment, or anyway, I think that observation doesn’t show that it fails to be an entailment.

Matt Teichman:
It does seem weird to go: ‘Would you like some coffee or tea? You may have either coffee or tea, but you can’t have tea’. That does feel a little weird to me.

Melissa Fusco:
Right. So it seems like ‘I don’t know which’-cancellation for the free choice effect does something very special. My line on this, for what it’s worth, is: I think it’s a scope distinction. So I think the best thing to predict, or the best view that’s the most flexible with regard to when people actually make these inferences and when they don’t, has to do with a distinction of logical form. So at logical form, either the modal operator—the diamond—will have scope over the negation, or vice versa. It seems like we get free choice readings on one of those scopal disambiguations and not on the other.

So here’s the problem with ‘I don’t know which’-cancellation. Again, the overall dialectic being that saying, ‘You may have coffee or tea. I don’t know which’—that ‘I don’t know which’ rider seems to cancel the free choice effect. So it seems to show that it’s pragmatic and not semantic. It’s an issue for the theory of cooperative communication and not for a theory of logic to deal with. ‘I don’t know which’ riders tend to systematically cause people to reinterpret scope. So I’ll give you an unrelated example, an example that doesn’t involve modal operators. And it involves disjunction.

So suppose I say—let’s see. We’re talking about philosophers, but I want to talk about a philosopher who I may or may not have read a lot of books by. And you are not ideally situated, maybe, to figure out whether I’m an expert on this author. So let’s take Wittgenstein, right? Wittgenstein is a philosopher of language, in at least a salient respect, but not particularly close to the areas in philosophy of language that I work on. So I say: ‘I haven’t read a book by Wittgenstein’.

Matt Teichman:
Ooh!

Melissa Fusco:
Okay. So there’s two scope-taking elements in that sentence, at least. One is the not—an ‘I haven’t’—and the other is ‘a book’, the existential ‘a book’.

Matt Teichman:
Yeah.

Melissa Fusco:
So when I say, ‘I haven’t read a book by Wittgenstein’, I could mean two things. And logical form will disambiguate which. This is the true order—not the surface order, but the true, logical ordering.

Matt Teichman:
So whether ‘a’ comes first or ‘not’ comes first.

Melissa Fusco:
Excellent. So I could mean—reading one—there isn’t a single book by Wittgenstein that I’ve read, right? That is one reading of ‘I haven’t read a book by Wittgenstein’. The other, which you might expect more if you thought I was closer to being an expert on Wittgenstein, is just that there is a single book by Wittgenstein—this is the existential having wide scope over the negation—there is a single book by Wittgenstein such that I haven’t read it. Right? So ah, Wittgenstein wrote n books, and I’ve read n - 1 of them. So there’s a book by Wittgenstein, the n-th book, that I haven’t read.

Matt Teichman:
There’s a new compilation of outtakes from the Nachlass that just came out, and you haven’t read that yet.

Melissa Fusco:
Right, exactly. So which do I mean? Well, typically you don’t know, in the sense that surface form underspecifies logical form. You have to take your cues from context. Or you could ask for a clarification, where I could follow it up with something that makes it obvious. For example, I could say, ‘I haven’t read a book by Wittgenstein. It’s the Investigations’, right? Okay, that’s a bit unlikely, since typically, if you have read any book by Wittgenstein, it’s the Investigations. But what was the book that you used as an example?

Matt Teichman:
Oh, I was just imagining Jimi Hendrix-style, the way they’re dredging up old recordings every year, pretending it’s a new album, and releasing it—that Wittgenstein people might do that with Wittgenstein’s old notes.

Melissa Fusco:
Okay. Let’s say that they dig up something, and it’s—there’s the Brown and the Blue Books. And so let’s say they dig up—

Matt Teichman:
Oh my god, the Purple Book!

Melissa Fusco:
A Purple Book–exactly. So I can say, ‘I haven’t read a book by Wittgenstein. It’s the Purple Book’. Now, the point of that just is in that dialogue, because of the second sentence I said, because I said, ‘It’s the Purple Book’. It’s the Purple Book that I have not read. Then you would interpret the scope of ‘a book’ in the original sentence, ‘I haven’t read a book by Wittgenstein’, as scoping wide over the negation. Otherwise, apropos of nothing, again, not knowing that much about me, but maybe not thinking that I work very much on Wittgenstein, and I just say, ‘I haven’t read a book by Wittgenstein’, you would conclude that I haven’t read a single book by Wittgenstein—that I have no copies of Wittgenstein books.

Matt Teichman:
Yeah. This is really interesting, right? Because a lot of the things we say are what’s called semantically ambiguous, which means they have two different literal meanings they could possibly have. And then, we’ve got to figure out, in context, which of these two literal meanings was the one the person intended. But even when there are two literal meanings the person intended, often, one of them is generally more likely the one that, all things being equal, the person probably meant. It’s like we need inside information about the person to know that. There’s a weighting, as it were, of the two possible readings.

Melissa Fusco:
Right. So let me just make a point about the ‘I don’t know which; cancellation, or the putative ‘I don’t know which’ cancellation of the free choice effect. ‘I don’t know which’ tends to make existentials and disjunctions read as scoped wide. So I’m going to illustrate that with the sentence, ‘I haven’t read a book by Wittgenstein’. So if I say, ‘I haven’t read a book by Wittgenstein’, suppose, again, that you don’t think that I know very much about Wittgenstein. So I mean the ordinary thing, the thing I would probably mean if I said, ‘I haven’t read a book by Norman Mailer’, ‘I haven’t read a book by John Lennon’, or whathaveyou. It means there isn’t a single book that I’ve read by that author.

So I say, ‘I haven’t read a book by Wittgenstein. Guess which’. Okay. Now, even if originally you were tempted by the first reading, the ‘not a single book’ reading, when I say ‘guess which’, it seems like now I’m asking you to name a particular book by Wittgenstein that I haven’t read. And that question, my prompting you to guess which—that command, or question–only makes sense if the existential scopes above the negation.

Matt Teichman:
Yeah, so you’ve completely disambiguated it with that follow-up.

Melissa Fusco:
Right, so I’ve changed the most salient logical form of what we could call the antecedent, the sentence that precedes the ‘which’—the thing that’s, ‘I don’t know which’ or ‘Guess which’. So I think that’s what’s going on in the case of free choice permission, as well. There’s two possibilities for the logical form of the sentence, ‘You may have coffee or tea’. One is: there is a single proposition, the disjunction ‘you have coffee or you have tea’, which scopes under the possibility modal that’s picked out by ‘may’. The other is that it’s a wide scope disjunction. This is like the wide scope existential with the book by Wittgenstein. So it’s a disjunction of two possibility statements that have been agglomerated to make the surface form shorter. When I say, ‘You may have coffee or tea, but I don’t know which’, then the ‘I don’t know which’ raises a certain logical form to salience. And that’s not the logical form on which free choice seems to obtain.

So again, that’s just, big picture, an argument that the cancellation data is not as good as it may seem. That’s controversial. I made some controversial assumptions about syntax, about the different ways that surface form can interact with logical form. I made an assumption about the robustness of logical form—

Matt Teichman:
I’m outraged by all this, absolutely.

Melissa Fusco:
—that there’s always a fact of the matter about whether the things that you say have one logical form or another, right? You could be a skeptic about all of those things. But philosophers of language tend to work with a strong, robust assumption of logical form that goes all the way back to Russell. If you think about Bertrand Russell’s examples, he has those funny examples where he’s bringing out different scope interpretations of logical form, and then pointing out that certain of them are invisible to us because they would be silly, or because they would be communicatively uncooperative to assert. So one of his jokes involves the yacht being longer than it is, right?

Matt Teichman:
I love the yacht example.

Melissa Fusco:
So somebody says to another person, ‘I thought your yacht was longer than it is’. And—I don’t know, these are two aristocrats sniping at one another about the length of the yachts they can afford, I suppose. And so, the second person, whose yacht has just been denigrated, says, ‘Well, no, in fact, my yacht is not longer than it is’. Right? So what’s going on there, when I say, ‘I thought your yacht was longer than it is’—not that I’ve ever had occasion to say that about anyone’s yacht, actually—

Matt Teichman:
‘I thought your desk was longer than it is’, maybe, is what it would be in our case.

Melissa Fusco:
Yeah, ‘I thought your desk was longer than it is’, ‘I thought you were taller than you are’. One logical form is that you thought that my desk’s length exceeded itself, which is impossible. No length can exceed itself. No matter how long anything is, it can’t be longer than it is.

Matt Teichman:
Yeah.

Melissa Fusco:
But that’s the ridiculous reading, right? That’s one disambiguation at the level of form that we tend not to hear. We would only go back and pick it out if we’re trying to be testy and uncooperative in conversation. Which is, of course, what is happening with the yacht example. That’s why I can snipe at you by saying, well, no, it’s not, in fact, longer than it is. I’m deliberately misinterpreting you. But what makes that misinterpretation possible is that at surface form, the claim ‘I thought your yacht was longer than it is’ actually does admit of at least two logical forms. So it doesn’t just explain how we understand sentences, but how, under pressure in conversation, we could go back and reinterpret those very same things.

So that’s a little bit, I think, what’s happening with ‘you may have coffee or tea, but I don’t know which’. What’s happening is: right after I finish saying, ‘You may have coffee or tea’, you are tempted to make the free choice inference. And then I say something further, which causes you to reinterpret the logical form of the first thing that I said. And now you don’t think that it follows. But it doesn’t show that the thing you originally thought I meant didn’t have that entailment profile. It just shows that now, under pressure, you’ve reinterpreted it to be something that doesn’t have that entailment profile.

Matt Teichman:
Yeah. They’re trying to make you look foolish by obviously incorrectly assuming that you meant something that makes no sense: the thing is longer than itself.

Melissa Fusco:
Right. So to bring it back to free choice permission, it’s this unexpected modal entailment pattern. It’s formulated in a very austere language, right? The very austere language just involves modal operators and disjunction. So, this is within the language of classical propositional logic, a very simple, well-studied language. And that’s a non-classical entailment pattern that I argue—but there’s a lot of arguments that need to be made here, arguments about cancellation arguments, about how Gricean reasoning works, and so on—that is not best explained by assuming that the inference is merely pragmatic. If it’s semantic, then we need a new modal logic, or we need a new way of understanding the way that classical modal logic interacts with or models features of natural language. At a bird’s eye view, that’s the hard and radical way to go: to say, let’s go back to modal logic and try and reformulate it. But it’s also a fun project, even if it’s an extremely broad one.

Matt Teichman:
So what do you think is the best way to reconcile these conflicting intuitions? On the one hand, we think, ‘P is true; therefore, either P or Q has to be true’. That seems like a good inference pattern. But ‘you ought to mail the letter; therefore, you ought to either mail the letter or burn it’ seems like a clearly bad inference pattern. And then, these other free choice inference patterns are intuitively good, but they’re predicted to be bad by the rules of classical propositional logic and modal logic. How can we square all these conflicting intuitions, and have our cake and eat it too?

Melissa Fusco:
Yeah. So the pattern that I would like to assimilate, for example, Ross’s puzzle, to—the failure of ‘or introduction’ in the scope of a modal—has to do with a tradition that is pretty old in both philosophy of language and in modal logic: two-dimensional modal logic. So this goes back to Kripke’s Naming and Necessity. In the modal logic literature, it goes back to the 1970s: work by Segerberg and by Kamp. And, the idea is that there’s basically two forms of consequence that are particularly salient to us, in—either you could say in natural language, or just as reasoners. And they involve the so-called ‘two dimensions’.

So, one way that Kripke cashed this out in Naming and Necessity is to reflect on the difference between what is a priori and what is necessary. So he argued that there were necessary a posteriori truths, things that were necessary even though we weren’t in a position to know that they were. And that there’s a contingent a priori—things that are a priori in the sense that we know them on the basis of reflection alone. We know them prior to experience—that’s the original meaning of a priori—and yet they’re contingent. They could have been otherwise. So divorcing the a priori from the necessary corresponds roughly, in Naming and Necessity, to divorcing epistemic and metaphysical necessity.

So there are these examples—and this is along the lines of what two-dimensional semantics tries to formalize—between sentences that we know a priori—they’re logically valid. In some sense, we know them prior to knowing anything about the world. But they do not express necessities. They’re not true at every possible world. So maybe the simplest example of that—and this is a dialectical tool for formulating two-dimensional semantics—is the claim that—in my semantics, I call it bowtie. $$\bowtie$$ That’s just so I can have a primitive symbol for it. It’s the bowtie.

Matt Teichman:
It’s pretty.

Melissa Fusco:
Bowtie is the claim that everything is the way it actually is. So it involves the word ‘actually’, which is a very central expression to two-dimensional semantics. Some people—short story—think that two-dimensional semantics is the project of axiomatizing an actuality operator. But instead I will use this primitive statement, which is a whole sentence. It’s not just an operator like ‘actually’. And it’s the claim that everything is the way it actually is—or everything is the way it in fact is. You don’t really need to use the word ‘actually’. Now, take the claim that everything is the way it actually is.

Matt Teichman:
It kind of feels necessarily true, or something, doesn’t it?

Melissa Fusco:
It feels like it’s sort of got to be true—

Matt Teichman:
How could that be false?

Melissa Fusco:
—it couldn’t be false. It seems to have a particular privileged modal status. Suppose I blindfolded you and took you to a new location. So there’s many things concretely about the world, especially where you are in the world, that you do not know, in the intuitive sense. You don’t know what color the walls are. You don’t know how high the ceiling is. You don’t know what the weather is like outside, because you can’t see outside the windows, because you can’t see anything at all. Yet, here is something you know, a priori in the sense that before you get any experience about your surroundings, at least in the limited a priori sense of that, having cut off as much of your experience as I can, you still know that everything is the way it actually is. So in that sense, it seems like it’s a priori. You know that everything is the way it actually is, even though you don’t know what particular ways those are.

However, is it necessary that everything is the way it actually is? Typically, again, in a two-dimensional framework, you will say certainly not. It’s certainly not necessary that everything is the way that it actually is. Things could have been otherwise. That’s the very start—that’s the germ of modal logic—to be able to reason not just truth-functionally about how the world is, but how the world might have been. And ‘how the world is’ categorically doesn’t exhaust ‘how the world might have been’.

Matt Teichman:
Like, it’s certainly conceivable that I could have walked in and taken your chair instead of my chair, and we would have been sitting in opposite chairs.

Melissa Fusco:
Yeah, and then things would be different than the way they actually are. So everything would not be the way it actually is. If you had worn a different shirt today—maybe you’re blindfolded, and you forgot which shirt you’re wearing. And yet, you know if you had worn a different shirt, it would not be the case that everything is the way it actually is. Something would have been different.

So why go through all this? The idea with the contingent a priori, for example, is that you introduce modal operators, the modal operator of necessity. But then, you say, look. There are certain truths, like bowtie—again, which is just a simple sentence that has the conceptual status of the claim that everything is the way it actually is—bowtie is always true. So in that sense, it’s a truth of logic. You can conclude it on the basis of nothing. And yet, even though, in that sense, it’s a truth of logic or two-dimensional logic, you cannot necessitate it. It doesn’t follow that it’s necessary that everything is the way it actually is, of course. Because if you concluded that, you’d get modal collapse, and it would be useless to introduce your boxes and diamonds into the logic in the first place.

Matt Teichman:
And is there’s the shirt reasoning, too, which we had earlier.

Melissa Fusco:
Right. So, now I want to turn that observation about bowtie into a claim about inferences. So I’m going to isolate two forms of good inference. Those are going to be two notions of consequence that will be formalized: one, the so-called diagonal consequence in two-dimensions, and one, the so-called horizontal consequence. So it does seem like one way of saying that bowtie is always true is that you can make the following inference. From anything at all, conclude bowtie. Bowtie requires no premises. It is a conclusion that requires no premises.

So typically, when you formalize a modal logic, you have this rule that all the truths of logic, like ‘P or not P’, or ‘if P, then P’—those can be written down with no premises. They’re axioms, and they require no further justification. But there’s this other rule, that anything you can write down with no premises—anything that is just justified as an axiom ex nihilo—you can put a necessity operator in front of, because all the truths of logic are necessary. Now, in a two-dimensional semantics, that form of reasoning will fail. Because you can write down things like bowtie, ‘everything is the way it actually is’, and you can’t necessitate it. Because from the fact that you know it apropos of nothing, it doesn’t follow that it’s necessary.

Now, in my semantics, disjunction introduction, the inference from P to ‘P or Q’, has the same status as the inference from nothing to bowtie. It’s a good inference, in the sense that whenever you know P, you’re in a position to conclude ‘P or Q’. But it doesn’t mean that it’s necessary. It’s not necessarily true that ‘P or Q’ follows from P. So in other words, you can’t extend that form of reasoning under the box. And you can’t extend that reasoning under the diamond; the diamond is just the dual of the box.

So, from the fact that P is true, you can conclude ‘P or Q’ is true, only if you’re reasoning about the actual world. Just like if you’re reasoning about the actual world, you can conclude that everything is the way it actually is, but you can’t export that reasoning off-world. So it’s not the case that at an arbitrary world that is not necessarily the actual world, everything is the way it actually is. And from the fact that P is true at an arbitrary world, you can’t conclude that P or Q is true at an arbitrary world.

Now, rather than talking about arbitrary worlds, I’m going to talk about arbitrary deontically ideal worlds. So now, the box is ‘ought’. So from the fact that ought P is true, meaning ‘P is true at an arbitrary good world’—deontically ideal world—it doesn’t follow that ‘P or Q’ is true at an arbitrary deontically good world. Bringing it back to posting and burning, just because you post the letter at all deontically ideal worlds, it doesn’t follow that you posted or burned it at all deontically ideal worlds.

So what I’m extracting from two-dimensional semantics is this idea that disjunction introduction, and more generally, all classical tautologies that involve disjunction—so all statements of the law of the excluded middle, for example, ‘P or not P’, ‘phi or not phi’, those all involve ‘or’—that those are not necessary truths. Rather, they have a different privileged status, which logic in two dimensions can help us see. They have the status of the contingent a priori. They’re truths that you know prior to experience. They’re truths, maybe, of meaning itself, of the meaning of the logical connective ‘or’. But from that, it doesn’t follow that they’re necessary. And there’s actually a well worked-out framework that just hasn’t been applied to reasoning in deontic logic that gives us a guide to how that could be.

Matt Teichman:
Melissa Fusco, thanks so much for joining us.

Melissa Fusco:
Thanks for having me.

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