Quantum Logic & Truth Functionality (contd.)

0:00 This is a kind of structure, the NOT-A or the NOT-A and B are kinds of structures that appear if you want to describe both modularity in different terms. So if you are in a NOT-Modular lattice, it's pretty safe. It's much easier to speak about implications and things like this. So you can define it as an extra structure. If you're in a classical setting, if you're in a classical, your initial question was about classicologic. My initial question was not about classicologic. My initial question is that you have a language, the proposition language, which has three primitives. It's a churnia, negation, distribution, and junction. And then, of course, you have an invading into a Yes, it's described on the screen and that is in the pre-test. And then at the end, of course, we are describing, and then we go on, on page 4, but then at the end, we'll have a discussion about what is the resulting logic that you obtain and it has adequate, is it logical or is it not? I mean, as you pointed out, it has very good properties. Yes, but of course one of the questions some time I asked, well if we have imputation initially in the formula, and if the law of barbarism by them expands, then of course you'll have instances of imputation like A implies A, which do not hurt. And that probably is going to come sometimes. I mean, Feferman, logicians, will say, well, if you don't have a right notion of implication, that's not a logic. But of course it doesn't. This is a kind of meta-theological notion. Yes, yes, that's right. These are going to be weaker projects than classical logic, and again, one can debate whether they are strong enough to be our logic. Then one has different criteria, as I say, in English, what counts as a logic, what doesn't, and there has been a debate precisely about these logics, whether they are logic or not, and this idea of whether they allow for an implication connected or not has been a part of this debate, and this is what I was telling here was a short answer.

2:30 Yes, one can define an implication collective in certain cases, which does not behave exactly as the classical one, but it behaves more like a modal implication. and one can argue that, yeah, that's the kind of behavior one should expect from something like that. There is a chapter on this in Belchamete and Festivale, on implication, with all the results. I don't, I am, I apologize, I don't want to interrupt you. No, no, no. But it is beside me, in the light of such a context, in such discussion, that I asked the question, which is meant to be quite neutral in a sense. That's good, because this also comes between the two parts of the talk. Okay, so, second part about true functionality. So, we've seen that in any non-distributive ladder, No sir, there are propositions that are not always true or false, that is, given any non-distributive lattice, there are true valuations which are not total homomorphisms. And, ok, in specific case of Hilbert lattices this is in fact true of all propositions except 0 and 1, Only two propositions in a Hilbert lattice which are always true or false, and they're always true or false, are a tribulate false and a tribulate true proposition. But, if we look at other non-distributive logics, now whether or not they have a very connection with physics, If we look at non-distributive logics based on the class of all ortho-antises or the class

5:00 of all orthomodulises, we can have propositions other than a tribulary false or a tribulary which are true or false under all truth valuations. And although truth valuations in general are partial homomorphisms, certain non-tribular propositions are going to be true or false under all such partial homomorphisms. OK. These propositions, which are called classical propositions, can be characterized as follows. We define the center of the lattice as the set of all propositions which are compatible with all propositions in the lattice. So the compatibility of A and B is defined as the sub-wattles generated by A and B is distributed over. And, again, the center of the lattice is going to be the set of propositions which are true or false under all truth valuations. So, and the proof of this theorem, this follows from Proposition 1, which we have on page 5. Namely, okay, classical proposition, no, proof it. Classical proposition C is contained with A, John, A, neg compatible for any set A, because it's compatible with all propositions in the lactose cell. So, if we pick a proof variation and we consider, and for A we insert the set, if V of all proof propositions and for that proof variation, if sub V is an ultra-filter,

7:30 So, if something join, if something name-compatible, which is the entire form, which contains any, which contains all classical prepositions, is going to be contained itself in Fb join Fb name. So any classical preposition is I of two or fours. So, the center of our lattice is the set of prepositions for which divalence holds. all propositions which are true or false. And four attitudes which are not biblical attitudes, one can have biblical attitudes which have very rich centuries, not just a view of propositions in one proposition. And, well, if a classical proposition is two or four, it's trivial to show that the connectives behave too frontal on these classical propositions. One line just as in the case of classical logic. So A of B is 1, even if B of A is 1, or B of B is 1. If A and B are classical, they're both true or false, and then we just look at the different possible cases, And as in the classical case, the true functionality of this junction comes out the same with the conjunction. Now, what does this mean? So, in general, in orthologic and in orthomodular logic, there is no valid inference from A or B to A or B. However, if we choose an interpretation of the logic onto some non-distributive lattice

10:00 with a non-trivial centre, there will be formulas which are interpreted as a or b. The formula a or b which is interpreted as the disjunction of two classical propositions, and Because in this case, whatever truth valuations we then choose to apply to these interpreted formulas, the disjunction will be true if and only if and all of the disjuncts are true. interpretation, then the connectives do behave functionally. What's happening is that the The inference from A, B to A or B is not a logically valid inference, but once we've chosen the meaning of the propositions, and we can use meaning, well, we can think of the meaning as these subsets of this given set, concrete means for the proposition, or we can think of meaning as, in fact, the interrelation of the various propositions in the lattice. The fact that the proposition is classical So all means that in lattice theoretic terms, all sublattices which can be generated taking this proposition and any other proposition of the lattice are going to be distributive.

12:30 So if one has an original meaning as relations between propositions, whatever they are contentional meaning, That modifies as meaning. So depending on the meaning of the prepositions involved, once we've chosen an interpretation, we can operate with these classical prepositions using classical logic. So these are inferences based on concepts rather than inferences based on form. So it's a kind of content-based logic equation. But if we want to justify why in a given context that we seem to be able to apply class for logic, although the logic that we are considering is non-distributive logic, this is the way forward. And so, what seems to me to be the moral of this, or the possible application of this, and that's where I want to get the discussion going and where I want to get the feedback, is if one has a debate about revisionism in logic, or when the logic is a priori or not, then one might be in a situation that some trading logician proposes that true logic should be non-distributive, and then a classical logician comes back saying, come on, you can't

15:00 I seriously argue for non-missuitive quadric being the true logic, because you've just used class-quadric to make the argument. And the Australian tradition has now the scope to answer that, well, I've been using these inferences, but don't mistake them for logical inferences. I've been able to use them because of the specific meaning of the sentences I've been using in this meta-logical argument. And, well, I mean now he or she has to fill in the details, why meta-ortical propositions should be classical propositions. but in principle, that's a possibility that could be exploited. Similarly, if the debate is about a priorism in logic, If somebody comes with arguments about why classical logic is a priori, possibly an advocate of a non-distributive logic could try and pull the carpet underneath these arguments saying no, what you've been showing with these arguments is only that certain kinds of propositions are classical. They don't go as far as establishing that classical logic is a true logic, no, they only work for certain kinds of propositions, namely the classical ones.

17:30 I'm just sketching the things at a very abstract level, but I'm just guessing that these are possible moves in these kinds of debates, if we're considering the alternative between a classical logic on the one hand and a non-distributive logic on the other. If we wanted to make these considerations a bit more concrete, then now we can indeed go back to the case of of the debate in Martin Roderick and Putnam, because the obvious criticism which has been leveled against Putnam's claims of the 60s is very similar to what I've just sketched. that, namely, it's not at all obvious that if the true, well, it's up to Putnam to show that if the true logic is indeed quantum logic, or how if the true logic is quantum logic, We appear to be able to use classical logic in everyday situations or when describing quantum mechanical measurements and so on. It's a version of the quantum mechanical measurement problem, how to get the appearance of a classical world, the fact that measuring apparatus remains classical, even if it's influenced stochastically by interaction of a quantum system, why does this domain appear to obey the laws of classical physics and a-for-fiori or classical logic, if the world is really quantum mechanical and the true logic is really quantum law. And, okay, I mean, I don't want to go into the details of that debate, but one can imagine

20:00 at least a possible world in which proof theory is not standard quantum mechanics, but some generalized quantum mechanics based on the appropriate phenomenon algebra with lattices or projections with the appropriate sensors so that there are lots of classical propositions around Yes, yes, yes, that's it. You can imagine that kind of position, so that as a matter of physics, because of the structure of the lattices or propositions involved, measuring operations and so on, is described by classical propositions. and if the physics were like that, then Putnam could have a reply to that particular objection. Whether his other arguments they took through is a different question, but this is the analog, more concrete analog of the debate between the classical logician and the non-discutive logician. I have a failure in mind. So, really this is all I want to say. As I said, these are very old results from quantum logic, but I think they can be relevant to the base about the pedagogics a priori, and what is the true logic, and I'd be interested to hear your views as to how relevant these results might actually be to these debates.

22:30 Thank you very much. Well, good. Well, one thing is, speaking about centers, no one really works with an algebra, with a non-trial center, otherwise apart from the luck. what happens is that either you're interested in the structure well either your center is trivial it's real physics when the center is trivial if your algebra has a non-triple center which happens over here now in Bonoimann algebras what you do your step number next is factorized by the center so you get a factor so you work in a factor very rarely, well you start by just having some algebra, but very rarely people keep looking at the center and at the non-center, at the non-trivial part, at the same time. This just doesn't happen a lot. So it seems that all the beautiful results, almost all the beautiful results, are never sort of cross-center results. They're either about the non-trivial part, and then everybody forgets the center, or they somehow look at non-trivial centers and things like this, or whatever. So what you're saying is certainly, well, it does sound like Belovkin, because Belovkin's more or less the only guy in foundations of physics, in the algebraic foundations of physics who talks about algebras with non-trivial centers, which doesn't really happen a lot. Well, let me just distinguish the physics on one side and the logic on the other side. In the physics, yes, you're perfectly right at certain. Mostly when we talk about quantum field theory,

25:00 Yes, we didn't have a centre, but these are the global observables that you would need to be able to measure over the whole of spacetime in order to get a grasp on these observables. They are not the ones you deal with in all your realistic situations. So, yeah, you're perfectly right there. I'm just wondering whether in logic there might be, or whether there is, or there might be a debate on the distributivity or not of the logic, Somebody comes up with reasons for giving up bivalence, which have nothing to do with quantum mechanics. Is giving up bivalence a live topic of debate in philosophical logic? I don't know, but if it is, then I guess these considerations are relevant to these discussions, and if it isn't, why isn't it? Is it because of the immediate objection? Is it because of this kind of objection? If you give up bivalence, what are you doing writing papers? You're using classic logic all the time. And, okay, if that's the reason why there is an enliven debate about bivalence, then maybe that's the wrong reason, because these results show that, well, one can, at least in principle, there's an avenue which one could take to reply to this kind of objection. Then, of course, there are many discussions on binomies in the article. For instance, one discussion goes by that, that you have the core, the core, let's say,

27:30 a set of sentences in the language, and therefore sentences that behaves classical. And then there is another set of sentences in the language, because it is of a certain form, so they're actually can't distinguish between the form of one and the other, and these other ones are the ones which are sort of undisputed, in every single situation, but for the formal ones, which belong rather to the core, as we say, then they have classical, you know, and then they actually have the properties that you had mentioned, I mean, that goes on a conjunction, because that's what you, or the negation is So there's no doubt that there is an interest in the subject and there are debates around it. But now why there is a set of sentences which behaves always classical and there is another set which does not. then why does it, does the later one be, why better as they are, because then, then this converts to some considerations to do with some kind of partial information or some notion of truth which is more robust than the notion of classical truth, which then, which then does itself that, or causes that not enough information can be instructed in order to But I have still concerned about how that connects to the discussion of the end of the nucleus. I have a question. It's not a technical question to your logic. quantum logic, but Robert's question about the feedback in quantum physics, in quantum physics, about Putnam's statement that quantum logic can unite standard paradoxes, like a measurement problem in quantum

30:00 physics. What's your opinion? You say that it depends on the interpretation that you can position. Well, Putnam made a different series of claims. One set of claims was, quantum mechanics shows that two-logic is quantum logic and logic is empirical. And the other set of claims was quantum logic solves the paradoxes of quantum mechanics. Now, I think the latter claim is wrong, and that's the consensus. If you really take on the logic, what you get is reformulation of the same paradoxes, but the way Putnam was trying to resolve them is fallacious. What he was doing was he was playing with the semantics. Somehow he was slipping in classical semantics. You know, if the cat is alive or dead, quantum mechanically, ah, okay, so the cat must be alive or the cat must be dead. But that's precisely not how the semantics of quantum logic works. If you really want to make quantum logic work formally, you have to give up bivalence. So, those arguments by quantum were technically wrong. Then he has the other set of claims saying that, well, yes, we should revise logic because of quantum mechanics. And again, the problem with that is how do you explain that classical logic is valid in the everyday world, and that is a version of the measurement problem.

32:30 And there are approaches to the measurement problem, and there are very different approaches in variables, manuals, collapse theories, and in order to assess, begin to assess what has claimed, we have to distinguish what we think is the answer to the major problem, or we can distinguish the cases. If we think that in variables theory is correct, then solution to division and problem is, well, there are these hidden variables, and the result is either this or that, because the valid division variable is this or that, and we have the experimental context, et cetera, et cetera, and in that case the answer is classical logic is valid at the everyday level, because it's valid at the level of the hidden variables, and we don't need a review of logic. I think the answer in the case of many worlds is the the closest one can get to maintaining the upholding of these claims, because if the mini-world is really correct, then there's a sense in which the classicality genuinely emerges from the quantumness at the fundamental level, There is a sense in which the classical logic emerges from the quantum logic, and one could make the case, yes, all that is really fundamental is quantum logic, but because of the independence of the different components, things start behaving classically. Anyway, it's probably a tricky kind of argument, but I think that's the closest one can get, keeping to standard quantum mechanics, and not playing with the phenomenon algebra, keeping to standard quantum mechanics, to a position that's sympathetic to those things.

35:00 But certainly, whatever one does, it depends very much on one's opinions about the mainland problem, because that's the big problem for the public. It's not through the quantum logic. You just pray for an answer to the mainland problem. So the big question is how do you explain that classical water is valid in the everyday realm if quantum water is proven. I can actually give it to you right now. That's it. That's all that you have any flambines for. And actually the first half of Brita. I've actually got some more of, I've got his name, Mitsuz. Mr. Card is on another take if you want it. But the whole of flambines is on there. Yeah, both sides, so it was excellent. And if you do want any of the others, let me know. Thanks again, thanks again for everything you did. By the way, that last one you gave me, the one that is still inside, there was a full poem. The solution to the hold problem was simply that you had to press the off switch for about three seconds and then it cut the hold and then cut out. But I've just tried it and there are 33 tracks on there. I haven't listened to them of course. because I need headphones to listen to it. Maybe there's random, it seems to be recorded. Random recording, but... No, because I remember that... Well, it kept saying cold every time you pressed it, did it? Well, I'll listen to it. It may, may, may not. I'll try a few minutes, but then... But then it actually says 161 minutes. Is that cool? Yeah. It'll be quite interesting to listen to it and see if there is anything. It'd be great if Charles Parsons is on there, but I'm not counting my children.

37:30 But you say you can get a hold of a copy of his text anyway. It happened in Beijing, I think it was, yes, in Beijing. I put the device in my backpack, and then somehow... Yes, unfortunately, that was always a good idea to get the batteries out. I should have mentioned that. Well, you actually did that. But it's easy to go. And then I discovered that. And out of curiosity, I listened to it. Sasaki hook... So it just recorded... It must have just recorded the jogging of your backpack. It must have sounded good. But in the case... I erased most of it by recording on the top. Yeah, that wouldn't be a problem. But as I say, it has recorded something on that. And it shows it. We recorded 33 tracks in 161 minutes. So, you know, I'll let you know. But if not, then whatever happens, please send me Charles Parsons' text. That would be great. Yeah. Right. Are we going to take a drink or a coffee? Yeah, well, I suggest two things. It's possible that you can order it from EDAH. And if there's a problem, I can scan it in or make a photocopy of it. Yeah, but that's fine. But someone incorporated his results into some groups. I'm sure, yeah. Oh, this is before . HA1 is the right hand at the same time. Yeah, I mean, this is where I learned all these results. And I think he must be one of the plus. No, because if he has results for composite systems and algebra, he would not do star algebra. This is interesting. Oh, you look at our premise here, according to the title. He has. Yes, he has. That's the big problem. Yeah, so, so...

40:00 Yes, I should... Sorry to talk about it, you're not talking about Primoz, are you? No, but this guy, the student of Primoz... Because you know Primoz wrote this big little... No, Primoz wrote Primoz, but I've never heard about this Primoz. And he's supposed to have, according to the title, the thesis that Primoz wrote, So, this guy is supposed to have produced results on composite systems and double star algebras. Which is a big topic, so I want to see the thesis. I knew a guy who, this was about 10 years ago now, who was trying to redo Prima's book in a character theoretic setting. trying to find the right category for the representation of companies that are algebras. But he didn't really get very fine. I have to say, he got some... Right, so there's a huge problem about getting the correct definition at the centre of the algebras in the category, because it messes up all sorts of conditions that you want on It does seem to be there's actually a glimpse now by some of the constructions that they do in bi-categories. Because there's been a huge amount of progress in bi-categories in the last 10 years. And somebody should go back and look at Primus's stuff on that one here. But it would have to be a very good alphabet, indeed. A very good alphabet for students also. They're not always the same gift. It's a very interesting program. I think Fremas, in some respects, was way before he started. He was a strong predictor algebraist. Yeah, I have read a lot of his writing. It's difficult to read, but what I have read, I just found it. He obviously had this, you know, kind of contextual, quantum logic there, which came out with water. He was actually a theoretical chemist. He was actually a special theoretical chemist. But I didn't realize he had a cube. What was the name of the guy? Raggio. Raggio. Raggio has reasoned above him. What nationality would he be? Argentinian. Argentinian? I'd like to find out about that. That sounds very interesting. I'd like to find out more. You should ask the Argentinian on there. Yeah. Well, you can. I mean, he may have his pieces on the website.

42:30 If you Google him, he has a position somewhere in his face. Let me put this stuff around. What else? Thank you very much. Well, I look forward to seeing October the 12th. I'll probably see you before that. Sure, sure, sure. Yeah, I'll certainly see you around before that, so that's great. Thanks. Okay. I guess you probably want to get straight off to Jenny's... Actually, I need to talk to you. Oh, okay, right. Well, look, I'll say goodbye then. Thanks for everything. Okay, goodbye Michael. Thanks for everything. I'll email you with the reactions on the trustees issue. You're definitely one. I'll give you a few days to recover from the jetpack, obviously. Probably next weekend or I'll get back and have a safe journey. Thanks for everything.