 Hi, I'm Peter Adamson and you're listening to the History of Philosophy podcast, brought to you with the support of the King's College London Philosophy Department and the LMU in Munich. Today's episode will be an episode about dynamic logic in medieval obligations with Sarah Uckleman, who is lecturer in philosophy at the University of Durham. Hi, Sarah. Hello. Thanks for coming on the podcast. Thank you for inviting me. You've published a lot about this genre of logical work or logical game, we might say, called obligations. And we'll get on to talking about that. But first, I wanted to ask you about your approach to this material, because you've argued in these publications that obligations show us an example of what's sometimes called dynamic or interactive logic already in the medieval period, which would be very exciting if it were true, if you're right about this. Because modern day logicians, I mean, really modern day, like this is something people are interested in right now, are interested in this phenomenon of dynamic logic. Can you explain what dynamic logic is and what it's being opposed to, like maybe static logic? Yes. So the idea is that generally, in logic, you have a set of propositions, and they each have a truth value, and this truth value is fixed. And so you can ask of a certain set of propositions, what do these propositions entail? And this is also going to be a fixed notion, because nothing is changing about the truth values of the propositions. Dynamic logic brings in a way of dealing with change. So what could be true now might be false tomorrow. Today it is Sunday, tomorrow it will rain. And so dynamic logic brings in a way for us to deal with these changes in the world, in our knowledge, in our belief. You can set up dynamics to happen at many different levels. But essentially, the idea is that you are no longer working from a fixed starting point, but your starting point might change, perhaps in the process of your reasoning. But why does that make a difference for the actual logic? I mean, we already had the idea in normal logic that things are either true or false. Why can't I just say, well, this proposition is true now. So the syllogism that uses this proposition as a premise is true now, or it's sound now. Tomorrow, if the proposition is false, then the same syllogism will be false. But that doesn't really seem to make any difference of logic. It's just which arguments come out true and which ones come out false. In that situation, perhaps. But suppose that you are dealing with a much more continuous set of reasoning, where I am in the process of reasoning, say, about what I should do. Today it is sunny, so I don't need to bring my umbrella with me. But I need to think ahead, well, what happens if tomorrow it is raining? Then I need to bring my umbrella. Or in another kind of context where you will see this sort of thing is, how about if I get new information? So I don't know if it's sunny or if it's raining. And so I will reason in one way, but then new information might come in that says, oh, in fact, it is raining right now. This is going to change the way I reason about what I do in the future. So as I said, there are many different kind of contexts in which dynamics can come up. It can either be facts about the world changing, or it can be our knowledge of facts about the world that changes. Okay. And why are modern day logicians interested in this then? Is it because it's just a better way of modeling the way that we reason in real life? So they're interested in it for, certainly that's one reason, but also because when you have a dynamic reasoning situation, it's also likely to be an interactive or another way to put it, a multi-agent reasoning setting, where you've got many different people with different knowledge, different actions, they say different things. This changes the knowledge that they have. So I tell one person something, he now knows something that he didn't know before. He knows that fact, but he also knows that I knew that fact. And so you can do much more complex modeling of kind of everyday situations because everyday reasoning doesn't involve simply sitting there doing syllogisms in your armchair. It involves talking to people, getting information from them, reasoning on the basis of that information, arguing with them, trying to get them to be convinced of your position. And these are all things that we do in everyday life. So that is one reason why logicians are interested in them. Some logicians are also interested in them from the computational side of things. Computers. Computers need to be able to reason sometimes from inconsistent or incomplete information. They will often have kind of new information coming in, say if a program feeds back the result of its computation to another program. And so you can use these sorts of developments in dynamic logic either for looking at kind of how people reason and interact, but also how machines reason and interact. Everybody likes reasoning about machines. Right, yeah, absolutely everybody. It is at least true that nobody could be listening to this podcast without machines. So if only for that reason we should support this kind of application of logic. Indeed. Does this sort of thing then actually get into what you might think of as the formal language or the symbolic structure of logic? I mean, are you saying that dynamic situations can't just be modeled with the standard apparatus of traditional logic? Because I was thinking that, you know, if again, I might have something like hypothetical reasoning. So if A, then B, and I need to be open to the thought that A might at first be true and then be false. So my computer, let's say, might need to know to keep checking whether or not A is true, because if it goes from being true to false, then it should stop inferring B. Right. But that's not a new logical device, is it? No, but what you can think of dynamic logic, it's not meant to replace static approaches, but to extend them. And to give us more tools, more refined means of modeling, you can make things more precise by adding in multiple people, by adding in changes over time. It just gives you a more accurate picture of how things are actually working. So that's kind of one advantage of this sort of approach. The other advantage is, is that if you can make this formal and precise, then you could actually feed it into a computer and tell the computer how it is that it is supposed to reason in these particular ways. Humans are surprisingly good at doing this without reflection. We can reason about what other people know on the basis of things that we hear them say, or things that we've told them. We have a lot of practice with reasoning about interactive settings, even if we don't ever make this articulated to ourselves. One thing that dynamic logic can do is make these things more explicit. And by making them more explicit, it helps us to recognize what we are doing when, and can therefore lead to making fewer mistakes, because you are better able to recognize what it is that's going on instead of doing this in a sort of non-reflective fashion. It's not only about teaching computers to think like humans, it's actually maybe about teaching humans to think like better humans? Yes. Okay. Turning then to the medievals, you're going to try to connect this idea of dynamic logic to this genre of logical writing called obligations. And I tried to explain this in the last episodes, what obligations were, and maybe why the medievals might be doing them. But you're the expert, so you're going to tell us what the real answer is. What were obligations? So obligations were a special type of disputation that arose. The first explicit treatises that we have on the subject are from the beginning of the 13th century, but you can find discussions of them throughout the 13th century and into the 14th century and even later. They are at root a type of formalized logical disputation that have two players, an opponent and a respondent, and a couple of very simple rules that the opponent and the respondent has to follow. Now, these rules differ from author to author. There were many people writing on these treatises. They were a part of the undergraduate logical curricula, so if you wrote a textbook, you would be writing on obligations. And so everybody had their own kind of idiosyncratic views. Basically, the most simple version of the most straightforward type, it's a type called positio for positing or simply putting forward a thesis, is that the opponent puts forward a statement initially, which is either true or false, and if it's not inconsistent, the respondent should admit it. That's the base condition for the game to start. Once that has happened, the opponent will continue to put forward further propositions to which the respondent is able to make one of three responses. He can concede the proposition, he can deny it, or he can be doubtful about it. Which of these actions he does is governed by certain rules. He is obligated by these rules to perform these certain actions, and that's where you get the name obligations for these types of disputations. Sentences that the opponent puts forward are divided into two types, those that are relevant and those that are irrelevant. The relevant ones are either logically entailed by what you've already conceded or logically inconsistent with what you've already conceded. The ones which are entailed, you should concede. The ones which are inconsistent, you should deny. The other sentences, the ones which are neither logically entailed by nor logically inconsistent, they are irrelevant, and you can respond to them, you can concede them if they are known to be true, deny them if they are known to be false, and then use the third option to be doubtful or to remain agnostic if you don't know. Okay, let me see if I understand that. Let's try this with an example. Let's say that I'm the respondent and you're the opponent. You're the opponent, so we start out by you telling me to concede something, and you tell me to concede that Socrates is a human, and I should concede this, in fact I have to by the rules, because it's not contradictory. Whereas if you ask me to concede Socrates is a human but not alive, I should deny that. So that's how things start. And then the next thing you do is you pose another proposition to me, and if it's something that's logically entailed by Socrates is human, for example, Socrates is an animal, I should say yes. If it's something that's refuted or inconsistent, Socrates is a donkey, then I should deny. And if it's irrelevant, like Socrates is in Athens, then I should go ahead and concede it. If you know that he's in Athens. I should only concede it if it's true. If it's known to be true. And if it's not, then I should doubt it. Yes. Then you can say, I don't know. And the rules say that I can't just sort of deny, deny, deny. If you say, I can't avoid being caught in a logical paradox or puzzle by just refusing to admit anything. That's right. Because as soon as you deny something that follows from something that you've already conceded, so for example, you've conceded that Socrates is human. If you then deny that Socrates is an animal, then I in my righteous Latin tone will come and say, ergo male, you have done badly, because you haven't followed the rules. Meaning I lose effectively. It does sound like it's really a game and there's a winner and a loser. The opponent wins or the respondent wins. And there's also a mechanism by which time is called, right? So basically the idea is that if the respondent can survive for long enough without making a mistake, then he wins. Right. And if the respondent at some point does make a mistake, he doesn't follow the rules properly, then the opponent wins. Right. And would they stop then? I mean, if the respondent loses, they'd stop. As soon as the respondent has made a mistake, then they will stop and they will say, you've made a mistake. Here's why. Here's where, here's what you should have done instead. And let's go through and try it again. So in this respect, the disputations have a very strong pedagogical component in that as soon as an error has been made, things stop and you try to analyze exactly where the error arose to make sure that that doesn't happen again in the future. I see. So this actually implies that the opponent is the teacher and the respondent is the student. Was that actually usually the case or do we not know? We don't know. One of the things that is kind of confusing about these disputations is we have a lot of theoretical treatises about them and no concrete evidence for their actual occurrence. We don't have anybody writing in their school diary. And today we spent the morning doing obligational disputations. But the general sense is that the opponent is going to be the teacher, the one who is trying to train the students in a particular tactic, and the respondent then is the one who is being tested, the student. Okay. Well, this raises the issue of why they were doing it, assuming they were doing it. And I think part of the answer must have been that it's fun. Yeah. Right? At least it was fun for them. I'm not sure whether the listener thinks it sounds like fun, but you could imagine it's sort of like a word game or a logic puzzle. So it's maybe like medieval Sudoku or something. But in addition to being fun, since it was being done in this medieval university setting, we would tend to assume, especially since it also has this pedagogical flavor, that they were trying to achieve something. And here, I guess the two obvious options are that it's for training the students' minds, so it's to maybe sharpen their wits or make them better at logic. That would be maybe a less exciting reason from the point of view of the history of philosophy, although not totally unexciting. But the more exciting reason might be that they thought they could actually discover things about logic. Do we think that that's true? Or do we just not know? We don't really know. And in fact, one of the big open questions about medieval logic is what was the point of these disputations? The first treatises that we have from the beginning of the 13th century don't touch on the question at all. It's almost as if they spring fully formed into the logical curriculum, that we are codifying something that everybody knows about and everybody has done, so we're just putting down into writing the rules and some useful examples. A number of approaches have been advanced. A number of modern commentators have tried to say that these represent an early attempt at axiomatic reasoning, or that they were systems for counterfactual reasoning, or for belief revision, or for thought experiments, or that this is the forerunner of the modern thesis defense. There are many different accounts that are actually advanced. I fall onto the camp of it's actually a bit of both, in that it has a pedagogical aspect, but it was also used for more theoretical purposes. On the side in favor of it being a pedagogical tool is the fact that this was a part of the undergraduate curriculum. These are found in textbooks that were used for training students. They provide a very good method by which you can learn to recognize logical inferences. When does something in fact entail something else? They're also very good on the mental side of things in helping you remember what you've already conceded. Now in a university setting that is based a lot on actual disputations for examinations and also just for public display, it's very important that you remember what you've said previously so that you don't end up tripping yourself up. A number of years back I was lucky enough to have a group of artificial intelligence students who actually took the rules for the basic form of obligations and created a program that would be the opponent's to a human person's respondent. Unfortunately it's not available on the website anymore, but during the period when it was I would regularly go and play these disputations because it is fun, it's interesting, it's a way to kind of exercise your logical brain, and I found that in general I was actually pretty bad. We could get about 12 or 15 steps in before I made a mistake, before I just couldn't keep everything in my head at the same time. And if you're going to do these on a regular basis, once a week, once a day, in a very varied setting, my feeling is that you could start very quickly to be able to keep a lot more in your head at once. So people who underestimate the pedagogical aspects of these I think are really missing out on part of the story. But there's another view about the theoretical purpose of these that's been advanced by Peter King, which I find very persuasive. And he wants to argue that what these disputations are training is not, they're not actual disputations, they're not about any substantive topic, but they are training you about how disputations can go. So the word that he uses is that they provide a meta-methodology for disputing. And some of the examples that show up in some of the treatises really display this nicely, because the propositions that are involved are not things like Socrates is human, Socrates is an animal, but things like that Socrates is an animal should be conceded. And so you get statements about the rules of the disputation occurring in the disputation itself. So you concede something like, the next thing I'm going to concede is true. Yes. Without even knowing what it was. Yes. And a common kind of paradoxical initial statement that is often put forward is, should you admit the positum is false? And That's basically a liar paradox. Exactly. Should I admit that the thing that I'm admitting is false? Yes. Right. And presumably the answer is no, or the only way to figure out the answer is to figure out the liar paradox. Yes. Okay. Right. Maybe that's a partial answer to the next thing I was going to ask, and then we really will get back to this dynamic logic issue, which is how this logic game relates to other logic games we know about from the history of philosophy. And for me, both because it's historically connected and because I happen to know about it and it's famous, the thing that leads to my mind is Aristotle's Topics. Aristotle's Topics is one of his works that the medieval at least would have considered to be a logical work. And it's a work in which he describes dialectical argument games and talks about the rules and the strategies for winning at these games. And again, it looks like there's two people, they're trying to win against each other by winning an argument. On the other hand, the thing you just mentioned, this thing about sort of second order propositions that are about whether first order propositions are true, that doesn't strike me as a very prominent feature of Aristotle's Topics. That's not very Aristotelian. And in fact, it seems to me that Aristotle's examples usually make it sound like there's content, there's philosophical content. So is one difference between what Aristotle was doing and what these medieval's are doing, that the medieval's were directing their attention more to the actual methodology of argument rather than just using argument in real disputation? Absolutely. So it's interesting that you mention Aristotle's Topics because some of the treatises that we have when they're kind of setting up the introduction, why it is that we do these disputations, why it is that I'm writing a treatise on this topic, will actually say something that sounds very much like what Aristotle says in Book 8, Chapter 4 of the Topics, where he's describing the point of disputations as taking something and seeing what follows from it. That if you have something that's possible, nothing impossible should follow from it. In that chapter, Aristotle makes a distinction between the dialectical, didactic, and heuristic disputations. The dialectical being two people working cooperatively to try to find the truth of the matter, the didactic being the teacher leading the student to a particular matter, and then the heuristic ones, or the sophistical ones, essentially arguing for the sake of winning. The rules there are a lot more flexible. So in that sense, there are certainly affinities between the obligational disputations and the disputations in the Topics. The, in particular, dialectical disputations can always be rewritten into a question-and-answerer format. So the questioner, say the opponent, puts forward questions and then the answerer is then going to say either yes or no, true or false, concede or deny to each of these. But the dialectical disputations of Aristotle only admit yes and no answers. So one way that the obligations differ from that is that we have a third option of I don't know. Doubt plays a role in these disputations that doesn't, where it doesn't have a role in Aristotle. Another is that while it may seem more on the lines of the dialectical disputations and that they're question-and-answer, yes and no, maybe with a third option of I don't know added, they aren't exactly cooperative. It's not that we have the opponent and the respondent working in tandem trying to find the root of the matter, but instead you have an opponent who is actively trying to trip up the respondent to make him act not in accord with the rules. And then this goes back to the point that you mentioned about the lack of substantive issues that are being discussed. The Aristotelian disputations are always about some particular thesis, the truth of which actually matters. But if you look at all of the, if you look at the obligational disputations, far from starting from some substantive thesis, the truth of which matters, they generally start from a thesis which is known to be false. For example, Socrates is a donkey. This is something that we would admit in the context of an obligational disputation because it's not inconsistent, but you certainly wouldn't be arguing for an Aristotelian disputation. In fact, the opponent might even deliberately start with that because it's probably a little bit harder to track your reasoning if you started with something false than if you started with something true. Exactly. Because actually if you're the respondent and all you have to do is be consistent with the truth, you just keep saying true things, you'll always be consistent. Exactly. And this is one of the particular things about this variant of positio in that you're required to concede the initial statement if it's inconsistent, but the game is only difficult if the statement is in fact false. Actually, one thing that strikes me about all this is that in antiquity there was a polemic, not involving Aristotle himself, but involving Aristotelians who criticized the Stoics for being interested in logic for logic's sake. And they said things like, oh, they're interested in these kinds of inferences like if A therefore A. And that's stupid. That's not even really part of logic because you can't use it. It's not truth producing or it doesn't help us expand our knowledge, maybe this kind of inference. And I find it very striking that the medieval heirs of the Aristotelian logicians were very interested in logic for its own sake, and maybe this obligations game is a manifestation of that interest. Now, let's finally get on to our central question, which is why we might think that obligations do manifest this thing that you're calling dynamic logic. So in order to do that, I need to kind of give some more details and a couple of examples. The canonical version that I've been talking about, as I said, there's a number of different variants. Every author had their own tweaks to the rules, the little things you can change. But the canonical variant is found in the works of Walter Burley. His treatise on obligations is not the first one, but it does it's one of the most comprehensive and it gives a very nice kind of canonical treatment of it. One of the things that is relevant, pardon the pun, in his notion of obligations is his definition of relevance. What is relevant is determined on the basis of what is logically following from or logically inconsistent with everything that you have conceded. And this also includes the negations of things that you've denied. That's conceding a negation is the same as denying the unnegated form. So relevance is something that is going to change at each step of a disputation. So I'd like to give just a couple of examples to illustrate this. The first is a very simple one that has a very nice, very nice conclusion. Let us assume that it is not now snowing. Thankfully, even though it's wintertime, we can say that this is true, but it's not inconsistent. It could be snowing. So I pause it to you. It is snowing. What do you do? I concede. Good, because it's not inconsistent. Next, I pause it to you. Either it is not snowing or you are a bishop. Okay, I concede because it's possible that I'm a bishop. You concede because it's not relevant. Okay. The disjunction, it is not snowing or you are a bishop doesn't follow from the statement it is snowing. Right. Okay. So it's irrelevant, but it happens to be true because it is not snowing. Okay. Now I pause it to you that you are a bishop. Ah, okay. And so now I have to try to remember what I've already conceded. Okay, what should I do? You should concede because from conceding that it is snowing and either it is not snowing or you are a bishop, it logically follows that you are a bishop. Great. Okay. So there you are. That is a very happy outcome. You can find versions of this example where the conclusion is you are an ass, but this just goes with the medieval desire of proving that everybody is a donkey. Even bishops. Even bishops. But suppose that we have the same starting point. We start with the sentence, it is snowing. You can see that. But then at my second step, I say you are a bishop. What will you do then? Then I doubt it? You'll deny it. I deny it. Because it doesn't logically follow from it is snowing. Because on his rule I only concede I will deny anything that doesn't follow from. It doesn't follow from and you know it is false. I see. Okay. So you deny that you are a bishop. Okay. Now I put forward either it is not snowing or you are a bishop. And presumably I should deny that too. Yeah. Because I know I'm not a bishop. It's false and I've already denied that it's snowing. Yes. You've already conceded that it is snowing so that is the same as denying that it is not snowing. Okay. Right. So these examples illustrate two things. One is that relevance is not simply a matter of the relationship between sentences, but it's a matter of which order you come across them in the disputation. Because in the first example, by the time that we got to the sentence, you are a bishop, it had become relevant. It was a logical consequence of the things that you'd already conceded or denied. Because it was introduced in a disjunctive proposition that I conceded. Yes. But in the second example, when I throw that forward as your second statement. I just denied it because it's false. Right. Okay. I see. So the reason that this approach to obligations is can be seen as dynamic is that at every counts as relevant changes. At every stage, you need to stop and recalculate. Okay. I may have possibly added in new information. Now I have to see what follows from what I've got. And at every stage, anytime that you concede something that is irrelevant or deny something that is irrelevant, this gives you more fodder for your logical gristmill. You now have more things that you could conclude from. So at every stage, you're going to have to stop and ask yourself, what is now relevant? Because this is going to change. And the change is what makes it dynamic. You can't sit down kind of in advance, calculate it all out, and then just kind of reason by rote. And would all of the authors who wrote about obligations agree with this idea that you need to bear in mind that the truth value of a proposition might depend on where it comes in the sequence? Absolutely not. In fact, this precise example of Burley's that we looked at, where I proved that you were a bishop, this is one that caused a lot of problems for certain authors, one in particular being Richard Swinesed, who was working in Oxford roughly the same time, about five to ten years later, I think. And he took strong objection to Burley's account of the rules whereby the order of the propositions mattered, whereby something could be irrelevant at one stage and relevant at another, and whereby you could essentially prove any contingent proposition. So the proof that I gave that you are a bishop, I could substitute any sentence in for that that I wanted, and exactly the same proof would work. Because as long as you offer me a disjunction and the second member of the disjunction isn't impossible, then I have to let it in. So you can immediately get in. But why is that so bad? I mean, it seems like that's in the spirit of the game, that you should be inviting me to acquire these commitments to contingently false propositions, and that you have to maneuver me into admitting them, but that's okay because I'm allowed to say false things, I'm just not allowed to contradict myself. Right. So you, I think, would be perfectly happy to continue playing by Burley's rules. Swain said wasn't. He didn't like the lack of systematicity that it gave. He thought that in this context, you could then just kind of prove too much. I see. So he took Burley's rules and changed them in one small but very significant way. He redefined what it meant to be relevant. In Burley's rules, remember, a proposition is relevant if it is a logical consequence of or logically contradicts what you have conceded or the negations of what you've denied so far. And that's why you have to kind of check at every stage whether new things have become relevant. For Swain said, something is relevant if it logically follows from or logically contradicts the initial statement. That's it. It doesn't matter what else you go on to concede or deny in the course of the disputation. Relevance will never change because the initial statement never changes. So actually, this is a much easier game, right? I just have to check whatever you offer me, whether it's consistent with the initial proposition. Yeah. So it seems like even a child could play this quite easily now. Yes. Okay. So this is actually one of my personal complaints against Swain said, is that he took what is a fairly tricky and fairly interesting and difficult game and turned it into child's play. Right. Because then it is merely a matter of looking at the logical relationships between two propositions, the one you started with and the one that has been put forward. And that is generally not going to be difficult to calculate. Right. So then it isn't as much fun. Yeah. Actually, there are even more complicated versions of obligations, which maybe are so complicated that we won't be able to talk about them without the assistance of a chalkboard. But I did want to ask you about one other variant. So far, we've been talking about this kind of game, which is called Positio, where you just begin by asserting a proposition. Yeah. But there's another kind of... There's several kinds of obligation games, and there's another one which is called the Dubitatio, which, as the name implies, must involve doubting something. Yes. How does this game work, and why might it be relevant to what we've been talking about? So Dubitatio is very interesting. If you go back to the three answers that the respondent can make, he can either concede a proposition, he can deny a proposition, or he can doubt it. In Positio, you generally start with a false proposition, which the respondent is obliged to concede, so long as it's not contradictory. There's a variant which is symmetric to it, Depositio, in which a true proposition is put forward, but the respondent is required to deny it. So you can see that that's going to work in basically a completely symmetric way. That's the same game, because as you said... It is. ...denying something is the same as... ...conceiting its negation. But there's the third response. It could be that some particular proposition that you know, either you know it to be true or you know it to be false, is put forward, and then you are required to doubt it. So I doubt that Socrates is human, or I doubt that Socrates is a donkey? Yes. Even though, in actual fact, you know that Socrates is human, and you know that Socrates is not a donkey. And why is that more interesting? Or why is that interesting at all? Yes. So many of the medieval logicians didn't actually seem to recognize that this was interesting. Just as Depositio can be reduced to Positio, they thought that Dubitatio could also be reduced to Positio, and therefore a lot of the medieval treatises don't even discuss it or discuss it in very little detail. However, Nicholas of Paris, writing in the middle of the 13th century, shows why Dubitatio is actually more than just a variation on Positio, and why it should be interesting to modern logicians and modern philosophers. And it's because it's working at a higher level. Depositio and Positio are at the level of truth. You take something that is true or false, you concede or deny it. If you take something that is known to be true or known to be false, and then pretend that you doubt it, this becomes much more difficult. And one of the ways that this is illustrated is by looking at the rules that are given. The rules for Positio are deterministic. For every proposition that is put forward, it is either relevant or irrelevant, and whichever one it is, there is a uniquely applicable rule that tells you what you need to do. There's never any choice. There's always a right answer. Exactly. In Dubitatio, there is always going to be a right answer, but there could be more than one. And to give an example of one of the rules, if you doubt some particular proposition, and it implies another one, then the rule is that you shouldn't deny the proposition that is entailed. So if I've doubted Socrates as a human, and then you ask, what about Socrates as an animal? You shouldn't deny it. I shouldn't deny it, but I could doubt or assent to it. Yes, and you're given the choice. So immediately there, this shows from the formal point of view that it can't be reduced, because it is non-deterministic in nature, and Positio is deterministic. These two are incommensurable. You can't reduce Dubitatio to Positio. And in fact, in Dubitatio, the respondent could use strategy, because he might think, oh, I think I'll assent to this, because it will make it easier as I go forward. Right. Then I have more things that are relevant, because I have brought in kind of new information. It gives me more to work with. Yeah. Yeah. Okay, that's really interesting. This way of thinking about obligations as having to do with dynamic logic, this is clearly an attempt on your part to show why obligations are interesting from the point of view of a modern-day logician. Do you think that that's the only way to show that they're relevant from this point of view? Or what do you think of other attempts that have been made to sort of bring obligations to the attention of modern-day logicians? There aren't that many. There are mostly kind of historians of philosophy, historians of medieval logic, who have looked at these and who have been unsure about what they have been for. I've mentioned people who have attempted to give accounts of them as counterfactual reasoning or as belief revision or as axiomatic systems. And for the most part, there's no agreement on any of these interpretations. And for any interpretation that you get, you can come up with some reason why it doesn't appear to work very well. In particular, the account of them as counterfactual reasoning. If you look at the people who have advanced this as an interpretation of obligations, what you will often find is them saying something along the lines of, if obligations are a theory of counterfactual reasoning, they're not a very good one. They don't give us the answers that we expect. They seem to get things wrong. They don't seem to be engaging with the right kinds of things. And so you might want to say, well, perhaps it's not just that they got counterfactual reasoning wrong, but that they weren't trying to do counterfactual reasoning at all. And this is something that I have recently argued with respect to Positio, because most people who advanced the counterfactual account are looking at Positio as the main, most common, most prominent kind of canonical version. This is not to say that I don't think counterfactual reasoning plays a role at all in obligations, because I think it does, just not in the version called Positio. It seems like it obviously involves counterfactual reasoning, because I start by supposing that something false is true, and then I think about what would follow from it. So isn't it counterfactual automatically? This is very much along the lines of what the modern commentators who advanced this view are saying, but if you then actually kind of see what sort of counterfactual claims you sent to and not, it doesn't work very well. If we still have time, then I'd like to give an example that actually shows how Positio is not counterfactual, but how another variant, one that is even less commonly talked about than Dubitatio, does seem to show some sort of counterfactual leanings. This variant is called variously, Sit Verum, Let it be True, or Re Veritas, The Truth of Things. And in this version of the game, the opponent starts off by saying, Let it be true that... And then this is before the game starts, this is not the initial statement that is to be conceded, it is sort of scene setting. So, let it be true that Antichrist exists, to take another common medieval example that you get. Let me first give an example following this of Positio where we don't have that sort of assumption, to show how the argumentation that's going on there is not counterfactual. Now I can posit to you that Antichrist exists, and this is not inconsistent, it's false. But you would concede it. Then I say Antichrist is colored, you admit that because it follows from the fact that he exists, that he has to have a color. I say that Antichrist is white. Now from the fact that he exists and he is colored, it doesn't follow that he is white. It's therefore not relevant, and we have to look to the actual world. In the actual world, Antichrist does not exist. He has no color. So he's not white. He's not white. So I should deny it, not doubt it. Now suppose that we say, exactly, suppose that we are actually doing re veritas, the truth of things is that Antichrist does exist. So now I posit that Antichrist exists, you concede it, I say he is colored, you concede it, I say he is white, and now you doubt it. Because now I'm thinking about this counterfactual world in which case, in which he might be white because he's really colored. Right. He might be white, he might be black, but you don't know which. But he exists, and therefore you are doubtful about it, as opposed to the case in the non-counterfactual situation where you look at the actual world, he doesn't exist, and you just deny it. So the difference is that in Positio, the actual world continues to be our standard for truth, whereas in Sidverum, actually you enter this counterfactual world. And that's what you look to for the evaluation of irrelevant claims. Right, okay, great. Yeah. Well, that was far from irrelevant. And that pretty much wraps up my look at logic in the 14th century, at least for the time being. I'm going to be moving on from there to look at the contributions of 14th century thinkers in mathematics and science. For now, I'll thank Sarah Uckleman very much for coming on the podcast. Oh, thank you for inviting me. This was lots of fun. And you are of course obligated to join me next time as I move on to later medieval mathematics and science here on The History of Philosophy Without Any Gaps. Thank you.