Quantum Logic & Truth Functionality

0:00 Does everyone have a handout? So, for once, I'm not giving a talk about foundations of quantum mechanics. There will be very little, or almost no, quantum mechanics in this talk. The subject matter is inspired by quantum mechanics. We'll be talking about non-distributive logics, and these are the kind of wonderful structures that have been inspired by quantum mechanics, and such as come up in quantum logic literature. And I want to describe a few actually well-known and old results in content logic and point out that they have potential implications for debates in philosophical logic. And in fact, I'm particularly interested in hearing feedback from people who know more about these debates than I do, and see if, if in fact these results from quantum logic can really be interesting to current debates. So, the talk has two parts, first our sketch for non-distributed logics and in particular

2:30 We focus on the so-called orthologic and the so-called orthomodular logic, which are standard logics that arise in the fontomodic literature, explaining that indeed they give up bivalence, but nevertheless, on one hand, well-behaved logical systems. And then in the second part, I'll explain how bioavailants can be regained. So, depending on the meaning of the propositions involved, one has propositions, one can have propositions that are indeed true or false. And for these propositions, one can de facto apply classical logic, although these inferences which one can make an ostrich-speaking logical, because they're not formal inferences. They do depend on the classicality of the propositions involved. And, yes, as I said, these are all very old results, but I'm interested to see whether and how relevant it might be to debates that people take place outside the field of quantum logic. So, okay, just as a few definitions are needed if you are familiar with the theory of, say, Okay, this is all trivial. If not, they are used to all plot, no need to know this in great detail. So, partial ordered set, bounded, meaning it is the greatest element and the least element in the set. A lattice, A, in French, is a partial ordered set in which every pair of elements has a supremum,

5:00 as a least upper bound and an infinite number, greatest lower bound, and in a boundless lattice one can, if one can introduce the notion of a complement, not A, not necessarily unique, That's an element such that for any A, A or not A is 1, and A and not A is 0. And then one has an over complemented lattice if the complement obeys double negation and and a smaller than b implies, not b smaller than a. Now, a classical example of a complemented lattice is a Boolean algebra. That's a distributive lattice. That's when the distributive rule is satisfied. and there's a representation theorem that every distributive lattice can be represented as the lattice of subsets of some given sets. But, okay, in quantum logic, specifically in lattices that fail to be distributive, and instead satisfy, perhaps, some other conditions. So we define compatibility of two elements, A and B, an orthocomplemented lattice, if the lattice, the orthocomplemented sub-lattice generated by these two elements is distributive. And now one defines the condition of orthomodularity to be number three, if A is smaller than B, then B will be incompatible. So the comparable elements are compatible, that defines orthomodularity. and distributivity means that all elements are compatible.

7:30 And, okay, we'll be looking specifically, usually, at orthomodular lattices, or orthocomplemented lattices. There's also another class of lattices which is of special interest in quantum logic, but it appears that it's not definable in purely lattice-theoretic terms. These are the lattices, so-called Hilbert lattices, which are lattices of curved subspaces of Hilbert space. These are the two ones that have a sort of profile in quantum physics, as opposed to the abstract lattice theory, which can use us to talk about these things in logical terms. Okay, so what are non-distributive logics? To get to the non-distributive logics, let's look at the usual classical logic in a slightly different light. The usual notion of logical volility or classical notion of logical volility is most often defined using the notion of truth valuations, truth values, directly on the sentential formulas, which are glattis, ortholattis homomorphisms from this set of formulas onto the special Dugan algebra with two elements, V and I. That's perfectly standard. But one can also We also introduce an interpretation of classical logic in a two-step way. First we map the sentential formulas onto elements of some concrete distributive lattice, say, subset of some set. So in a sense we are giving meaning to the formulas, and then

10:00 we introduce the true fluctuations, that we say which formulas are now supposed to be true, and then we can equivalently characterize the logical validity of a formula if a formula turns out to be true under all possible proof valuations on all possible such interpretations. Whatever the specific meaning of the formula, and whatever the true variations on the interpretive formulas which we define, turns out to be true, that's a logical truth. And if you want, that's something similar to what one does in predicate logic, where interpretations that start from a given set with a subset of time and predicted and so on and so forth and then the logical validity of a formula defined in terms of truth under all such interpretations. OK. By the way, in the literature there's also different terminology. One talks about semi-interpretations as the mappings from the sentential formulas to a concrete Boolean algebra and interpretations as that whole thing if you also have the truth valuations I'm using terminology of interpretation is choosing some some elements of some concrete food in algebra as representing this integer formula and I'll use truth valuation for the assigned proof values to these interpretive form. Okay, now, if one does this, now one has the freedom to try to generalise

12:30 non-distributive lattices. The idea is, okay, let's imagine two elements of a non-distributive and then let's try and find true progression of non-discriminative lattices. Some specific class of non-discriminative lattices, say, orthomodular lattices, and then if the construction goes through, that's a way to define the orthomodular logic. or if it can be done for Hilbert matrices, that's Hilbert-logic, logic-form, natural-form mechanical system. But prima facie seems that this practically cannot work because truth valuations, in general, are not dependable on non-distributive lattices. So, in particular, if we look at the case of Hilbert Lattices, it was shown by John Bell, John S. Bell, the physicist, John Hill Bell's tradition, that the only plantices of closed subspaces, of Hilbert spaces, that admit total homomorphisms onto the Boolean algebra 0-1 are one-dimensional, and, in fact, the proof is very, very easy. let's take a two-dimensional space, and let's assume there is a total homomorphism from this two-dimensional space, or the lattice of subspaces of this two-dimensional space on to 0,1, well, there will be some,

15:00 well, a space that's more than one-dimensional, so there will be a two-dimensional subspace which is given the truth value 1 by this total homomorphism, and then we take a two-dimensional plane, this represents a true proposition, and then you choose two pairs of orthogonal one-dimensional subspaces in this plane, And because we have a total homomorphism, all of these subspaces aren't given truth values by the truth valuation, and since they are orthogonal, okay, one has to be given 1 and and in the other pair, okay, one is going to be zero and one is going to be given one. And now, if we take the infinimum of the two vectors that are given the root value one, R, call this P, call this R, we'll have the B of P and R as D1, we'll call this homomorphism, but because we have chosen these vectors to be non-polinear, they're not parallel or anti-parallel, So, the intersection of these two subspaces is the zero subspace, but the zero subspace must have the truth value zero, and that's a contradiction. And this is possible in any vector space that is more than one-dimensional. So, certainly for Hilbert lattices, so if we're interested in characterising the logic of Hilbert lattices, we can't do that with truth valuations.

17:30 There's a more complicated proof for orthomodular lattices, also in the case of orthomodular lattices. because the only orthomodular lattices that admit total homomorphism are distributive ones, results by Jauch and Pierron, also from the 60s, 63 or 64 I think. So, we can't define truth valuations in the obvious way, but there is a way of defining truth valuations, and, well, I suppose there are two ways anyway. You know, you could give up the homomorphism condition and then you get something strange in one sense, something like that, or you give up the idea that the homomorphism is total. So you give up bivalence. Every formula, every sentence, every interpretive sentence is true or false. And this is what is done in order to construct non-distributed logic. In a sense, this is required if we want to keep the homomorphism condition in the definition of a true valuation, then we have to give up bivalence. This is where all the questions of the philosophic logic come in, but let's just look at the formalism of how these partial homomorphisms work. So now there's a little technical bit which allows us then to improve various results. So one defines a filter of a complemented lattice as a non-empty set which does not contain

20:00 the zero element is closed under conjunctions and, well, is filtered, as one says. If an element A is in the set, any larger element is also in the set. And an ultra filter is a maximal filter, a filter that cannot be properly extended, cannot be contained in a strictly larger filter, and filters can always be made maximal or extended to an ultra filter. And And then if we define, if we take a filter and then take the set of all the complements of the filter, the so-called dual filter, we get a set of this form F or F-NA. This is an auto-sublastic of L, and we can define a partial homomorphism onto 0,1 just by saying all elements in the filter are false, and all elements in the dual filter are false. And this partial homomorphism will be maximal, even though the filter is an alpha filter. And conversely, if we start off with a maximal partial homomorphism, from the lattice to of all true propositions is an ultra-filter. So the idea is to define proof valuations as maximal partial enomorphisms, and they are in one-to-one correspondence with ultra-filters on the lactose. Okay, so we've given up bivalence because these group variations are not total

22:30 humoral prisms, but we are trying to keep as much as we can, and they're maximal. We're taking the largest possible sets of propositions to be true or false. And also, a proposition which is logically weaker than a true proposition is always going to be true because of the filtering property of a filter. And, okay, we can prove two results, one of which we'll use later on, crucially. So let's define this set A-compatible, and for any subset of a lattice, let's define A-compatible as the set of all elements which are compatible with all elements of the set, formula 14 here. In general, A-compatible is not a subset of A, and A is not a subset of A-compatible, but we have proposition 1, if F is an ultra-filter, then if we take the orthosupple lattice, F join F negative, and if we take the set of all the elements that are compatible with all of the elements of that, that is contained in the orthodox of lattice. So that means that for any truth valuation on the lattice, propositions that are compatible with all propositions which are true or false are themselves true or false. And from this, one can, for instance, show that on a distributed lattice, any particle homomorphism can be extended to a total homomorphism. take a particle homomorphism, extend it to a maximal one, so the set of true propositions

25:00 is an alpha filter. By the proposition, the set of all propositions which are compatible with A4F-nag is already contained in this set of propositions without proof of false, but because the lattice is distributive, this set is minimal lattice. And the intuition is that if A is compatible with all propositions of true or false, we can give it an appropriate truth value without spoiling the homomorphism. And indeed, in general, this is not always true. That's proposition 2. If we have two incompatible propositions in an ortho-lattice, then there always is a true valuation, that is, a maximal partial homomorphism onto 0-1, such that one proposition is true and the other one is neither true nor false. Since this is already a maximal partial homomorphism, it cannot be extended further to make the other preposition either true or false. So, indeed, even if we have these strong results, the group of lattices for the modular lattices that are the only ones that allow for total truth valuations that are distributive, So even if you don't have this strong kind of strong result for general upper complemented lattice, it is true. So even if there may be non-distributive lattices that allow for the existence of total homomorphisms, not all partial homomorphisms on a general non-distributive lattice can be extended to total ones. which cannot be extended to triple ones.

27:30 OK. So, now, with this notion of proof ovations as partial homomorphisms onto, or maximum, partial homomorphisms onto we can play the same game we did with distributive lattices and total truth valuations and define logical validity and logical consequence using non-distributive lattices as our structures interpreting the essential formulas. So a formula will be a logical proof if it is true under all proof valuations in all interpretations of a given potential formula. And one can define different types of non-distributive logic by choosing specific classes of non-distributive lattices which one uses to interpret the sentences, so if one takes all the Huber classes, one could get the logic all of the mechanics, So to speak, the Huber lattices are supposed to be the lattices characterizing polymechanical systems, but it is not known whether the resulting logic is axiomatizable. Instead, if one chooses larger classes of not just to be the lattices or the modular lattices for complemented lattices, one does get logics which are axiomatisable, and one can tell that corresponding logical systems are solid and complete, and that these different logics are in fact distinct logics, and that certain logics are very complicated, in the stronger

30:00 of the logics are not the truth of the weaker logics. And probably the most interesting of these logics and the most famous in the cosmological, the classic cosmological literature is automodular logic, because it allows for defining conditionals, polynomial connectives that support motor components, and under certain natural conditions one can in fact pick out a unique such conditional and the resulting logic resembles in certain ways of certain systems of modal logic. This is, if one talks about quantum logic, this is sort of the classic structure which quantum logicians have studied that are special in the in the 1970s and later on people have been realized projects based on those sets or from all your poses lattices is a visit is a there's a there's a reasonably large industry of of quantum logic, but this is the classic example of a quantum logic and I should say I should think it has all the hallmarks of a logic, all the formal properties which we we might want to refy our logic, an aximatisable, the system is sound and complete, a condition of, and so on. So it's, one gets a logic in which bivalence is given up, not every sentence is true or false, under all possible food regulations, but it is a serviceable water, satisfies lots of formal criteria for what a water could be, of course.

32:30 Then one gets into the debates about what is logic, and tie in with the debates in the foundations of quantum mechanics. So, famously Putnam in the late 60s, this and that, shows that logic is empirical, that the true logic is quantum logic rather than classical logic, and if one leaves just the safe realm of followers, and that starts to evolve and say, we should revise logic and adopt a logic that gives up my valence, then of course the debate becomes much more heated and I want to get back to this kind of debate, but yes. One very small thing. Disjunction is defined by negation and one of the connectives. Yes, that's the usual. Okay. All right. So. It depends on what comes first. The usual way to define the structure, the first thing of that is for the join and meet operations. And then you have negation if you have it, or sometimes you don't have it. If you have a pool of modular or whatever thing, then, of course, yes, you have it. No, I just might want to say. Oh, I mean, the definition of those laws, I mean, that's like it. The point I want to say is that if you have this junction, the indication we find this, let's say, A implies B is negation E or B, then we adjust it. Okay, no, no, that's not, it's not just junction and conjunction, it's meet and join. It's not logical, or what? Yes, all right, but I'm just thinking about the alternative authorization of classical Boolean logic.

35:00 Oh, okay. If it's classical, yes. Have I said something really stupid? Well, we are in the case of certain things. Yeah, you can do it with validation and a filter, it's not a problem. I'm just, sorry, you have page 3, you define a method, and then, of course, from formulas onto the boolean algebra, and I just, the formulas include implications, and the implication is defined by a negation, no, no, no, at this stage we haven't defined implication, We're just looking at, we just have, you could define a connective, not A or B, if you want, or whatever you like, and then we can study whether it has the properties of our implication. but we've got formulas, the usual formal language we have, and or, and not, and variables, and then, okay. I was going to say, you can still test to see whether it satisfies the rule for being a De Morgan Monoid. Yeah, even though you haven't, am I right? Still, yes, it's still demoralized. I'm just looking at it purely formally. So to that extent you have formally defined a kind of application operation. Why have I missed the point? It is classical on page three, so I mean, implication is just another way of saying a bunch of negations. It's just that if you do it for the non-distributive projects, if you take not A or B, that thing doesn't behave as not A or B behaves in the...