 Our first speaker for the afternoon session is Samuel Fletcher, who will speak about supervolumism and determinacy and the completeness of quantum mechanics. Thank you. So as you see from the title slide, this is a project with my colleague David Taylor at the University of Minnesota. The main topic that we're interested in here is quantum indeterminacy. So this is putatively a type of unsettledness in the world that quantum mechanics describes. And the kind of typical example that people use to invoke, there are many types of examples but a simple example involves a superposition state of an electron. An electron, it's like a little bit analogously to a tiny spinning object. And relative to a particular orientation, you can describe it as being in a state of, it's spin orientation being clockwise or counterclockwise. And we can describe those as being spin up or spin down. The idea here is you're using a right hand rule to determine which direction is. And there are, it seems, states for the electron where it's neither in one nor in the other. There's a kind of indeterminacy about what the direction of the spin is. And the idea here is that this sort of indeterminacy is not supposed to be merely representational. It's not as it were a defective precision in our language, right? It's not a defect in the predication of spin to the object. It's an indeterminacy or unsettledness in the world itself. Now, philosophers are interested in getting general accounts of what this sort of unsettledness could be. What this sort of metaphysical indeterminacy could be. And there's one account that has been very popular and influential amongst metaphysicians. This account adopts its basic ideas from an account of indeterminacy of language. And one of the main going accounts for that is known as supervaluationism. So the idea is take that account of indeterminacy in the context of language and just import it over to metaphysics. Maybe you paint over a couple of things and basically you have the same vehicle that you can use. Now, here's how the things get painted over, right? So on a general framework here for understanding indeterminacy on this account, it's metaphysically indeterminate whether P is a proposition predicating of some thing or other property or some relations or something like that. If and only if P is true in all actualizations of reality. So this is a little bit like necessity formally. In actualization, we're going to talk about it a little bit in more in a second. It's a little bit like a precisification of reality. It's like another possible world where the properties that things have or the predications that things have are less determinate, more precise. And it's metaphysically indeterminate whether P just in case P is true in some actualization of reality and false in another. So formally, this is a little bit like contingency modally. So on this account, indeterminacy or determinacy should be understood in analogy with necessity, indeterminacy in analogy with contingency. Of course, there are different accounts of what exactly an actualization is supposed to be. We're going to be mostly focusing on ersatz version of this where an actualization is a possible world that doesn't determinately misrepresent reality. And here we just understand possible worlds as classically complete representations of the way reality could be or have them. So here on this ersatz version, worlds are abstract and ersatz. You can contrast that with the realist version, which takes worlds to be concrete entities. Everything that we say can kind of like apply to the realist version. You just have to add in extra commitments. But we're going to be mostly focusing in the argument for the ersatz version. Now one of the attractions of this view is that it models metaphysical indeterminacy within classical logic. So the attraction is, in other words, you've got a metaphysical theory that doesn't require you revising your logical resources in order to be able to express it. Okay. Now what David and I are interested in is we're interested in the question of whether quantum theory, the sort of metaphysical indeterminacy we putatively find there, can actually be accurately modeled by this account of metaphysical indeterminacy. So in the question from science to metaphysics, the idea is what sorts of constraints does quantum mechanics put on the adequacy of this metaphysical theory of indeterminacy? Now about 10 or 12 years ago, Barnes and Williams and some other people had as well describe this account, supervaluationism as a general model for metaphysical indeterminacy. If that's right, then it should be able to model the quantum things. But around the same time, several folks argued that supervaluationism can't succeed. So there's some, for people who are like really into this debate, there's some influential articles by Scow and Darby around the same time. And they follow from their arguments perceived by invoking some very interesting but technical results from the foundations of quantum theory, known as the Coaches-Spectre theorem. A bit later, there are folks who tried to respond to these criticisms to try to save supervaluationism by tweaking certain assumptions. We don't really need to go into what the tweaks are. The basic thing is that you make some changes to some of the underlying structures that you use to describe this position. Instead of using complete worlds or using complete worlds, it doesn't really matter for our purposes. This is setting up the dialectic here, right? So a few people have done this. Darby and Pick up a few years ago and more recently, Marianne at all. David Taylor and I argued fairly recently that we don't think that supervaluationism can be salvaged. And we also think that the invocation of these kind of technical results, the Coaches-Spectre theorem, are not really needed. The Darby and Pick up and the Marianne at all are kind of designed to try to get around those arguments anyway. What we think all that's needed is a particular type of assumption about how quantum mechanics represents properties of physical systems that's known in the literature as the eigenstate eigenvalue link. So what I'm going to try to do today is give a more general argument against supervaluationism. Let me just describe what the eigenstate eigenvalue link is first. This is the central idea. A quantum system has a value v of a property. So if it's a categorical property, then the value is just like yes or no, it has it or it doesn't have it. If it's a quantitative property, it might be a particular numerical value of the property like how much charge it has or something like that. And that's going to be represented by an operator, a self-adjoint operator. So it has that property if and only if the quantum state is an eigenstate of the operator where the eigenvalue for that eigenstate is the value v. So that's the basic statement about how the formalism of quantum mechanics in terms of self-adjoint operators and eigenstates and eigenvalues connects with how properties are represented in it. So what we're going to do is we're going to kind of present a version of the argument against supervaluationism. What we think is stronger, that is it uses fewer assumptions, it has wider scope. It's more general, it applies to more versions of supervaluationism, including the ones that were designed to get around the arguments that invoke the Cauchy-Spectre theorem. And it's more refined in the sense that we make much more precise what the underlying assumptions are. The sense in which it's a version is that it argues for the same conclusion, but the internal workings of the argument that we think are actually quite different. We're going to call this the completion argument. A secondary goal is that I want to clarify the role that the eigenstate eigenvalue link plays in this argument and show that the success of this argument, surprisingly, and which hasn't been recognized so far, hinges around the disagreement regarding the completeness or incompleteness of quantum mechanics. So drawing that connection is a kind of secondary goal that I'll end up with. Okay, so this is the completion argument. Let me start on the, I guess it's for you, the left-hand side. I cannot do mirror symmetry in my head, I'm sorry. That starts with a little bit of the notation of the argument. We've tried to formalize this argument so that if you disagree with something, you can point to the thing that we assert that you disagree with. Okay, so let's start with the notation. The arrow up is just saying that our system, say, our electron is in a state which has spin up, or excuse me, it has the property of being spin up in some particular direction. Sigma i is the power of the operator representing that property. So I remember, I was telling you before that there's a representation relation between self-adjoint operators and properties according to yield. That's what the operator is in the case of spin. Now, delta and nabla are respectively determinacy and indeterminacy operators in our little toy language here. It's just a propositional language, but basically delta out in front means it is determinant that, and nabla out in front means it is indeterminate that. And we're going to let Q be the proposition that our system, S, is in an eigenstate of a particular pilot operator, the one representing spin line. Okay, our argument has four assumptions. Here's the first assumption that's labeled as A. If it's determinant, if it's determinant that the system is spin up in the x-direction, then it is determinant that it is spin up in the x-direction. By the way, I'm going to give the argument first and then I'm going to circle back to defend each of these assumptions. So the first thing is just to try to walk you through why the argument is a valid argument, and then I'm going to try to defend the soundness of the argument. Okay, so the second assumption is that if the system has the property of being spin up in the y-direction, then it is in an eigenstate of the Pali-Y operator. The third assumption, C, is that if the system, the electron, is in an eigenstate of the Pali-Y operator, then it's not the case that it determinately has the property of spin up in the x-direction. For those of you who know about quantum mechanics, just think about like the relation between an x and y is like a 45 degree rotation in the two-dimensional state space. If you're not into quantum mechanics, believe me. The fourth assumption is that if it's determinant that the system is spin up in the x-direction, then it's indeterminate whether it is spin up in the y-direction. There's two super-valuationist assumptions. These are going to come from the super-valuationism that we're critiquing. The first is that if it's indeterminate that P, so P is any proposition here, then there's an actualization, right? Remembering actualization is a world that doesn't determinately misrepresent reality, then there's an actualization at which P is true. Second, if it's determinant that P, then there is no actualization at which not P is true. So those are the assumptions. Here's the argument. Let's start with D. We assume D because that's an assumption. Now we apply E to that and it follows from that that at the world W, that's the one that is an actualization, right? So remember we're referencing E here, that spin up y is true. It had the electron in this actualization has spin up y for some actualization W. It then follows that from B that Q is true in that actualization. From C, it then follows that not determinately spin up x is true at that actualization. And then from F, here we're using modus tolens, it's not the case that it is determinant that it is spin up in the x direction. Then from modus tolens in A, we get the conclusion that it's not the case that it is determinately spin up in the x direction. And you can see that that's a contradiction. So with these background assumptions and these minimal super valuations commitments, we get inconsistency. Now if you're into like paraconsistent logic, you might be like, that's great. But remember one of the motivations for super valuation is the retention of classical logic. So this is really a problem for them. Okay, so now I'm going to try to defend each of these assumptions. So E and F are going to follow straightforwardly from the commitments, the super valuationist commitments that we made. They are, as it were, fragments of these by conditionals. The reason why we use these fragments of these by conditionals is that some of the people we referenced in the beginning who were trying to save super valuationism weakened the super valuationist commitment so that they don't have the complete strength of both directions of the by conditional. But all of those except E and F. So that's why we're weakening these strong commitments to E and F. Okay, so what is the defense of A? All we really need here is that you accept that there is some case of determinacy that is itself determinate. Maybe you don't think that that's the case for spin up in the x direction. But suppose you think that it's the case for some other property. We're just going to substitute in that property in for the argument. Okay. In other words, you just have to accept, you just have to assume that not all determinate property instantiations are indeterminate. Not all of the being determined are impermits. Could you say that again? Sorry, that might have been a bit less significant. Okay, all right, sounds good. Sounds good as you like. Okay, so we have a defense of E and F and A now. Now let's look at D. We claim that D is just part of the phenomenon of quantum indeterminacy that we started with at the very beginning. This is just an expression of the starting point for everyone involved in this debate. Without D, we don't have any quantum indeterminacy for spin. Okay, so now that leaves C and D. For C, we claim that this is just going to follow from the eigenstate eigenvalue link that we started with. It's a particular direction of the eigenstate eigenvalue link. So let me just elaborate a little bit on that. So there's a standard formulation. This is the one that I gave before. Sorry. There is a certain ambiguity here. The ambiguity here has to do with how the standard formulation is exactly put together. It has to do with what a determinant value represents. Okay, so let me walk you through what the ambiguity is. So in analyzing the ambiguity, I'm going to make a couple of presumptions. One is that I'm going to assume that what quantum physicists often refer to as observables are referred to properties. That the operators, like the polyoperator that I talked about, that I mentioned before, these represent determinable properties. And then the eigenvalues, these are the little v's, like the specific value of a physical quantity. These represent determinant properties, determinants of the determinable. Here I'm using a bit of metaphysics here. You might be familiar with this idea already from really pedestrian cases. The determinant distinction, excuse me, involves more or less specific ways of having a type of property. So for instance, something, an ordinary object can have a color. I wish I had a red thing. Oh yeah, yeah, yeah, right on. Thanks a lot. Okay, I'm going to assume that we all agree that this has a color. You don't have to give an analysis of whether it's a primary secondary property or anything like that. It's just, it's colored. Right, one specific way of being colored is it for it to be red. So color is a determinable property. The more specific instantiation with is being red. And red also can be a determinable that has more specific ways. So I wouldn't say that this is a crimson. I would say maybe it's more of a salmon. What would you call it? Burgundy. Sure, sure, but it's a more specific type of red, right? Okay, so these different types of color properties stand in relations of more or less specificity. We're assuming that the operators are the less specific. The eigenvalues represent the more specific. So the operator might represent having spin in some direction or other. The eigenvalues might be having spin up in this direction. Thank you very much. Okay, so now there is this question about when the standard formulation of eel says has a determinant and value of a property. This is ambiguous between these two different precisifications that we'll call strong eel and wheat eel. So in the strong version, determinately is just added for emphasis. It doesn't actually play any logical role in the statement. According to strong eel, the system instantiates the property, if and only if it is in an eigenstate of the operator that is representing, whose eigenvalue represents the property. So in this case, the property is the determinant of the determinants, the more specific thing. So on strong eel, the property instantiation doesn't involve any special reference to determinacy or indeterminacy operators. On the weak version, the connection between the formalism of quantum theory and properties is only through a determinacy operator. So on the weak version, states that determinately, system s, it instantiates p, if and only if f, excuse me, s is in an eigenstate of the operator whose eigenvalue represents that property p. The reason why it's weaker is that it applies to fewer cases. So it's making a connection between the formalism and properties in fewer cases. Now if you have a logic of determinacy where p and determinately p are extensionally equivalent, then strong and weak eel end up being equivalent with one another. But we're not going to assume that you have a logic of determinacy in which these are extensionally equivalent. Data and I actually think that they are extensionally equivalent, but we're not going to assume that in this argument. So in fact, c follows already from weak eel. So even if you think that no, no, no, eel, it's very specific. It only tells us about when systems have properties determinately, where there's a real determinacy operator that appears. We only need that to get c. It's just one direction of c. Okay, so the only thing left to defend is b. So we do claim that b follows from strong eel. Now if you're already sympathetic to strong eel, then the game is over basically, right? So we're going to proceed under the assumption that our interlocutor is someone who only assumes a weak eel. In which case they may object to assumption b and try to retain super-valuationism for quantum theory on those grounds. We're going to argue that strong eel follows from other assumptions that we've already made, plus an assumption about the completeness of quantum mechanics. Now the completeness of quantum mechanics is a controversial assumption we recognize, but we do think though that in these sorts of discussions which are kind of like in the background within a kind of like broadly realist interpretations of quantum mechanics, that that is a going assumption that people make in realist interpretations of quantum mechanics. But it is an assumption that we want to point out. Okay, so here are the three assumptions that we think that we need in order to prove strong eel. We need weak eel. We need an assumption that we call strong completeness. This is basically the idea that for any property that a quantum system can have, whether the quantum system has that property supervenes on the quantum state. So in other words, the quantum state is a complete description of the properties of the system. That's what strong completeness is. And a further assumption, thank you, of non-contradiction, which we take it to be an assumption, an ongoing assumption of the commitment to classical logic. Okay. So is that five minutes including discussion? No, no, no, without discussion. So it's still 15 minutes in total. Okay, great, thank you. Because if it were that, then I would skip some things. Okay, so, but I think I am going to skip a little bit. So this slide where I talk about some of the reasoning that's involved in the argument, it is more complex technically than the other slides, but I'm going to give you a kind of qualitative description about some of the essential steps that they're making it. I'm going to do this by going up to the screen and pointing, because there's a lot of things here. So in quantum mechanics, the systems that we're looking at, they have a state space which is two-dimensional. So each square here represents a state space of the system. One of the features of quantum theory is that states are going to be represented by, there's different ways of doing it, but in this case, effectively lines in this space. The ideal is going to fix the extension of these two properties here. Determinately spin up y and, and, and, and, and determinately spin down y. And we can pick, pick up, pick a way to represent things. So this is the horizontal line, and this is the vertical line. There are other features in quantum theory which require that these two lines have to be at right angles to one another. It will follow from a bunch of the other assumptions that there are only a finite number of options of the states that represent the extension of spin up y. So these are the quantum states that represent this quantum system having this property. And the white portions in each of the squares are different candidates for that. The reason why there's a finite number of candidates follows from the other assumptions. And I haven't explained that argument to you, but I'm happy to do it in the Q&A if you're interested in that. Two other assumptions follow. One is that the extension of spin up y has to include the extension of determinately spin up y. There's a technical argument for this, but the intuitive argument is that determinately spin up y is just one way to be spin up y at least. So the extension has to include the horizontal line. Conversely, it can't include the vertical line because no way of being determinately spin down y is a way of being spin up. And there's a more technical argument for that, but that's the basic idea. If we allow for that, we exclude this one, this one, this one, this one, this one, and this one. So six of them get excluded by that. The only one that's not excluded is this one here, this square. By parity of argument, though, we get an argument that if this were the extension of spin up in the y direction, then this one would have to be spin down in the y direction by parity of argument. And if that's the case, there would be states where they have both of the property of being spin up and not spin up. And that violates non-contradiction. So this one is ruled out. This is the only one that's left, which means that the extension of spin up y is the same as the extension of determinately spin up y. And remember, I showed you before that weak yield and strong yield are equivalent if determinately p and p are equivalent. So what this technical argument shows is that at least in this specific case, this is true. And so we get strong yield from yield. Finally, okay, the last thing which I promised is that we use this assumption of strong completeness. This assumption basically says that the properties of the quantum system supervene on the quantum state. The quantum state is a complete description of the properties. There is a way of rejecting our argument by rejecting strong completeness. Here's how you might do it. You might say, ah, quantum mechanics is complete, but not with respect to all properties, just with respect to the determinant properties. In that case, our argument doesn't go through. But we think that weak completeness in the background of this kind of realist, the realist commitments of this debate is really pretty untenable. The reason why that's so is that it makes it so that quantum mechanics, the formalism of quantum mechanics basically has nothing to say about the non-determinant properties. They float free of the formalism completely. It seems to us that this is a renouncement of the realist commitments of these sorts of positions. And for this reason, we take ourselves to be given a redactio of super-evaluationism as an adequate account of metaphysical indeterminacy. Thanks for your attention. I'm looking forward to your questions. This is very clear talk. We have nine minutes for questions. You can't get there. Sorry for interrupting you, just that the script forgets everything. Yeah, yeah, yeah. So your argument also assumes the classic model framework of metastal modality, right? Which is in model logic, it's S5, right? Ah! And so... No, I don't think so. Okay, right, so, but at least four is assumed, because if something is determinate, then you have to have that it is determinate, that it is determinate, right? So you might think, let me put it this way, we don't require in our argument that that hold for every proposition. Okay, cool. Ah, okay, okay, right. We only require that it hold for at least one. For one position. Yeah, and so you can have a weaker system. Indeed, we, in our opinion, so the super valuationist want to think about determinacy in analogy with elethic modal logic. David and I think that it's completely different type. We actually have a paper on the logic of this. I could tell you about later on, but we don't make any assumptions about that. So this is the reason why I say, oh, we don't make that assumptions. We just need determinate, determinacy for one proposition. Then if you need to have one property for which that's true, right? And then you have to point out why for that property it is true, and then you have to find a particular property. We just emphasize that, right? Because it's not obvious that there is a property for which that's right logically, that it is determinate, that it is determinate if it is determinate. So you have to find one property which is such that for the property that's true logically, right? Yeah, so we take that just to be an assumption of the argument. So if you want to be a super valuationist and you want to reject that, that there's not even any one for which that holds, then that's a way of resisting our argument. However, we're not aware of any super valuationist account that can accommodate that. Last thing to say, I don't know about super valuationist account, but in the philosophical literature there are arguments against horror that is used in the meta-technical modality, right? Okay, yeah. And there are arguments against horror, but there are also ways to keep horror by rejecting these arguments and then thinking about Williamson, Jessica Williamson. So she talks about relativized meta-technical modality and to relativize the truth of a modality playing a word into a context. Maybe in that case, if there's some death framework, there are arguments too. It might be possible, although maybe we have to talk about the details. So I really like this. I'm a little worried I'm missing a bit of the dialectic, so that's quite this question. That might become obvious in my question. So when I look at strongholds like this, I really associate that with a rejection of strongholds with bogeyans. Yeah. Right? Yes. And in the formulas for bogeyman mechanics, there's no indeterminacy whatsoever. Exactly. So there's this broader context in which you might say, look, the only way to save super valuationism is rejecting this thing that moves you toward a theory aperture. Is there any quantum indeterminacy anywhere? Yeah, yeah. I agree with that. I think that I started the discussion in a particular place where I've assumed that everyone involved in the dialectic agrees that there is quantum indeterminacy, and so then wants to give an account of it. And so for folks who have an interpretation of quantum mechanics, for which that's not the case, then this is all by the by. So I think we're actually on the same page. Well, I would just encourage you to. So I think at the end, once you've said all this, you might also just emphasize that there is an existing interpretation out there that's already, I mean, everybody is trying really hard to hang on to completeness. So if that's what you had to give up, it's not just that, for instance, like, oh, truth is floating free or you've got a distinction without a difference. It's like you've given out the game that you were playing with respect to the space of possibilities for interpretations of quantum theory. So I think that is all in favor of you having made a great point. Great. Thanks very much. Could you come back to the main arguments with all the assumptions? Yeah. Yeah, this one. Just to be clear, when you talk about the delta and delta and you say we are fixing the determinacy of the operator and the physical quantity you are assuming, is it fixing it in all the actualization or just one because I was confused about that part? Ah, yeah. So we intentionally, so we intend this to be a syntactic argument. What I mean by this is that we're not assuming any specific semantics for the determinacy operators and that's quite intentional. The reason why it's intentional is that different super valuations disagree about what the semantics for these are and so it's too messy for us to try to do this by giving a semantic argument. David and I actually like semantic arguments better. We think they're more philosophically transparent but we resorted to this syntactic strategy here to be acumenical with regards to the targets of our argument. I was asking this because in your completion argument couldn't you say that the delta operator shouldn't be used in the general sense but specified to one application? So you say instead of saying delta ty you say delta ty in some application and then you couldn't just use the delta in general. So instead of using delta you say delta index the acquisition. That's exactly what it was. Because your assumption, G, I guess it's saying that in general if ty is fixed then tx is not fixed and vice versa but you could just say no quantum mechanics is just saying that if it's fixed in some actualization then the other one is not determined in this actualization. That was going to be my question too. Let me make sure that I got it correctly. So you're focusing on D, right? I'm saying instead of using delta you should use delta index of the actualization in this world. Oh, okay, good, good, good, good, good, good. Good. I take the point and I think in the completion argument that's kind of implicit already because these statements are always going to be evaluated at some worlds and I take that what we need to do is we need to make that explicit. But if you were to find the argument with that remark you would say so delta index y prime of tx implies the nabla of ty at y prime but then when you say, then ty is true at some w it's a normal actualization, not the same. And there is no contradiction. Yeah, then because you would say that delta tx is true at actually w prime and then the negation is true. But then it's not really a contradiction. Yeah. We may have to talk about this because I'm not tracking all the aspects but I do take it that D is supposed to express in some world or other the phenomena of indeterminacy. And so if there's only supposed to be a connection between determinacy in one world with indeterminacy in another then I don't think that the changing thing is adequately expressing what quantum indeterminacy is supposed to be. However, I grant that it deserves to be thought about more clearly and so I'm going to, let's follow up later offline. I'm afraid we have to end the discussion here and we have to go through our stuff.