CEFISES Seminar: Lorenzo Rossi, “Supervaluational truth and quantifiers”

Yes, microphone is working. >> Amazing. Thanks everyone. Thanks Lorenzo for being here. Um, welcome to our last seminar of seminar. Last of all seminar. And today we welcome Lorenzo Rossi who's going to give a talk um about supervational truth and quantifier. Lorenzo, thanks again and the floor is yours. >> Thank you. Thank you so much Lundine and Peter for having me here. It's great pleasure and um so this is a joint paper with from the University of Bristol and it's very much work in progress. I think it's actually two papers rolled into one. So um you will help me to disentangle these two things and uh really thanks everyone for being here and for having me. Okay. So the plan let me bit more light. Yeah. The plan of this um of this talk is to extend a well-known framework to deal with indeterminacy which is a supervaluational framework to quantifier semantics and by this I mean to deal with generalized quantifiers so in general quantifiers that cannot be defined in first order logic such as finitely many or most okay and there's many sources of indeterminacy here in this talk I will just focus on three which I think lend themselves uh rather naturally to be dealt with within the supervaluational framework. Um I'm sure there's possibly many more and here they are presupposition failure vagueness and various phenomena connected with semantic paradoxes. So we have foundedness uh um paradoxicality ungroundedness and and such. Okay. So as I said the goal is to extend a supervaluation semantics which isn't quite a textbook supervaluation semantics is a tad different hopefully a bit more general to um generalized party fires because we wanted to explore the supervaluation and treatment of these three things so super valuation treatment subition failure vagueness and paradoxes now um I have been working uh mostly on theories of truth and paradoxes for a long time but I've uh forced myself to have exactly one slide on semantic paradoxes which isn't even about paradoxes about the truth predicate. So it'll be one slide and uh of course there is an appendix with the other slides. So if you have questions you're welcome to to ask. Um so the result as I said is actually not one semantics but two and they are competing because I think in the end there's a genuine trade-off between them but in both of them the interata concerning these three sources of indeterminacy can be arguably uh met and there is a substantial fragment of GQT which is uh my abbreviation for or or you know the abbreviation that I use for generalized uh quantifier theory. So there is a substantial chunk of the theory of generalized quantifier is recovered and I'm going to characterize precisely as best I can this fragment. Okay. So here's the outline. There's a lot of slides but don't worry the talk is much shorter and and here have an appendex uh a couple of you three appendices for for details. So let me start from um the setup which is a roughly forgian picture. I don't claim to be philologically accurate. I have no clue whether Fria would agree but this is frig inspired and the frigy inspired picture is the distinction between concepts and higher order concept. So in this distinction a concept not to be confused with the psychological notion of concept is a function from individuals to truth values. Okay and a higher order concept is the same pretty much but one level up. So rather than a function from individuals to truth value, you have a function from concepts to truth values and we are located at that level because quantifiers are high order concept. So to give you an example, consider the quantifier every. The quantifier every is the function that takes two inputs, two concepts, let's say dog and friendly, and outputs the value true. If whenever dog applies to an individual, so does friendly. So this shows that every dog is friendly is true. So you you feed it two inputs and whenever something is a dog but that individual is also friendly if it's true then the quantifier is also quantified sentence is also true. Okay, so of course this is the classical picture and in the classical picture whenever you have a concept you have two possibilities either the concept applies to an individual or it's negation does right so this is exclusive and exhaustive however if you have indeterminacy around it's not obviously the case I mean it might still be depending on your treatment of indeterminacy but at least it's possible that this is no longer warranted and in general there might be a partial which here I take it to be synonymous with tri ED interpretation of the language whereby a concept can be a partial function right so you have that C applies to an individual that not C applies to an individual or that neither um happens however and this is the usual you know run-of-the-mill supervaluation or motivation if you say that you're content with a single partial interpretation whether for any given concept C either C applies or not C applies or there is a gap right it means that you have a very precise way of drawing the line right between these three cases between C not C and the gap but it might be that there isn't a preferred way if you take a vague predicate for example it's the usual motivation there's many possible ways of drawing the line and you might want to respect some clear cases but the borderline cases admit many different uh ways of drawing the line so to the superval to the supervaluationist's eyes this suggests considering not just one interpretation many all those are compatible with your existing semantic information. Maybe you have some information you want to preserve that but then after that you can processify in however many ways you like. So the idea here is of course you combine these two strategies trivalent semantics to begin with and supervaluations later. Okay. Now the usual picture is that or one usual picture is that in a supervaluation semantic you start from a trivalent interpretation and then you make precifications which are bvealent themselves right so they are classical here we are going to relax the assumption our precifications may be trivalent themselves maybe are more informative than the one you had to begin with maybe the gap is actually smaller in the sense of set theoretical inclusion but they are allowed to be trivalent they needn't be they can be classical so if you like the classical approach it's perfectly compatible with what I'm going to say. So the issue of course is that as soon as you apply this picture to the level one concept, it also applies to level two higher order concepts. Quantifiers can be partial too, right? Every dog is friendly is true in a given interpretation. If as I said, whatever dog applies to individual D, so does friendly. But um super valuation of truth requires considering all compatible interpretations, right? Because you don't have just the friendly individuals and the dog individuals in your domain. you have many possible extension on extension gap. So every dog is friendly must hold in every partial interpretation compatible with the given one. Again if you are more classically minded just remove the water partial here and you have a standard super motivational picture. Uh so classically every dog is friendly is false if it isn't true. So if there is one individual in your domain uh uh which is a dog but not friendly. But of course with partial concepts you can have interpretations in which dog is not a subset of friendly so they do not stand in the subset relation but there isn't a counter example there isn't a counterwitness right simply because the counter witnesses only belong to the gap so no individual is both dog and not friend um so this suggests that we also need if we are going to give superv valuational truth conditions for quantifiers separate falsity conditions this is also not a surprise for many of these nclassical semantics you need truth conditions and falsity conditions. Okay, so this is the setup. Now before going to the specifics of the semantics and I'm also going perhaps in the interest of the discussion to skip some of the details because they're not so relevant for for the big picture. I would like to offer you a couple of possible semantic judgments concerning quantified sentences and inferences in the case of indeterminacy. These are quite controversial. So I've had many different responses to this example. Some people think they inferences go through or judgments are true or maybe they don't. But the answers to these judgments will help me to adjudicate between the two competing models that I'm going to produce in the end. Okay. So one thing that is sometimes discussed in um the context of indeterminacy coming from presupposition failure is um failure of contrast in a quantified setting. Okay. Um consider the first sentence. Everyone who quit smoking was told by their doctor to quit. Uh we have a presuppositional verb quit. So it assumes that the person was smoking on a very standard trivalent semantics for presuppositional verbs. Quit smoking is true if the person was smoking and then doesn't smoke is false if the person was smoking and then still smokes and is interminate if the person was never smoking. Okay, so this seems like a true sentence. But this sentence seems not true in the sense that well everyone who was not told by their doctor to quit smoking did not quit. I am a counterex. I never smoked. So I was never told by my doctor to quit smoking because my doctor never needed to tell me to quit smoking because they never smoked in the first place. So that's true. But it's false that I didn't quit if you adopt this particular semantics for quitting, right? Because did not quit is indeterminate. It's not true. You might think that this is a good country example. You might think this is not. As I said, people have a lot of different reactions to that. But this is one possible motivation. Failure of contropositions coming from presupposition failure to have this sort of indeterminacy behavior in quantified inferences. Um, another one, this is probably less strong is motivation. Category mistakes also motivating failure of controposition. Every expensive thing is on earth seems about right. Um, but if you say everything not on earth is not expensive, depending on what you mean by not expensive, you might think that this is false because things items that are not on planet earth are not for sale. So expensive and not expensive don't apply. Now the classically minded people will say wait not expensive is the complement of expensive. It doesn't mean cheap. Other people will say not expensive means cheap because there is a presupposition that you're only talking about items for sale. So depending on which of the two views you adopt, you might like or not like adding proof there. Um there's also a failure of the proposition for quantified claims that has to do with failures of the propositions in conditionals is doesn't revolve around the form of indeterminacy that uh we have here. But as if you're familiar with these counter examples to classical inference in conditionals, you might see them with every. For example, every person who orders a drink uh doesn't order wine. That might well be true in a given scenario. There are people who order a drink and none of them goes for wine. But if you say every person who orders wine doesn't order a drink, that's obviously false because wine is a drink. Okay. Okay. So the first semantics we are going I'm going to propose actually doesn't validate controposition. So if you want these counter examples to be counter examples, then maybe the first semantics is the one for you. Um other kind of linguistic phenomena that you might want to look at are cases of homogeneity in definite plurals. Um suppose that uh there are four kids, two of them are asleep and two of them are awake and consider the first sentence. The children are asleep. True or false? Yes. >> People tend to hesitate. Um how about the children are not asleep? the first one were false then the second one might be true but is it true two of them are asleep and two of them are not asleep >> also false that's one judgment that that that you find in the literature and so the idea is that here people tend to hesitate and say well you know we have exact uh an exact uh symmetry between the two cases so neither of them are acceptable how about this is optional but how about the children are either asleep or not asleep There's two readings, right? You can read this in a distributed manner. So for each given kid, that kid is either asleep or not asleep. Or you can read that in a collective manner so that all the children are asleep or all the children are not asleep, which would be false. So depending on your judgment, these are cases that are discussed in the literature of indeterminacy quantifiers. I should mention the definite article is a quantifier. is treated as a generalized quantifier in in generalized quantifier theory. So this is why this is quantification of indeterminacy and then my favorite example anti-restricts. So context uh my colleague and friend Mateo is in charge of interviewing the foreign applicants to our MA program. This is true and we have no Farohies applicants. Also I think it's extremely rare to have a Faroh applicant because there's so few of them. So sentence one Faroh applicants are interviewed by Mateo. Well, if you this is a bare plural, right? So also a quantifier. If you work in generalized quantifier theory, it everything start looking like a quantifier. So many things are quantifiers and that's a bare plural. So it is and that's an empty restrictor, right? Because there are no Farohies applicant. And since this would go with the semantics of the subset, the empty set is a subset of everything. So it would be true classically, but it would also be true that Faroh applicants are not interviewed by So if you adopt the classical semantics, you can replace far applicant with every far applicant if you like. They're both true. And that seems weird. I mean, in particular, it seems that since Mateo is in charge of interviewing all the foreign applicants, if we have Far applicants, well, the first will always be true because in case we have them, you know, they're interviewed by him and the second one turns out to be false. So these are all like non-classical features of inferences involving quantifiers that sort of motivate a supervaluation of treatment because all these inferences uh turn out to be um treated non-class in the semantics that I'm going to propose in particular the semantics I'm going to propose. It's possible that this is true and this is false. Okay. So these are motivating examples to get a sense of the kind of linguistic phenomena that um as I say motivate our work in the first place. And now let me go to the formal semantics. Um I'm going to skip over some details because there's a lot of uh ugly formulations that we haven't been able yet to make uh more elegant but I'll give the um the let's say the informal descriptions. So what is an interpretation for us? We start from an empty domain and then we take any function that behaves just like an interpretation in first order logic. So no fun business. uh the interpretation of a term is an individual. The interpretation of the function is a function. The only thing to uh bear in mind is that the interpretation of a predicate is a pair because it's an extension anti-extension pair. So we leave the possibility for there to be a gap. Okay. Um okay. So this is the first member of the gap. That's the second of the pair. That's the second member. Importantly, these interpretations are partial. So we don't assume that if you take the union of extension anti-extension you end up with a domain. Okay. Now for introducing gradually the supervaluational component suppose you have two of these interpretations right. So here is one interpretation what kind of object it is and suppose you have two. We say that they stand in the ordering relation. um we write that I uh is before J just in case basically the order the inclusion in the positive and negative uh clause um is preserved okay so basically J agrees with I okay on the positive cases and agrees with I on the negative cases okay so you can think of it as a precification right we just don't assume that precifications are classical but other than that it's just super valuation Okay. Now once we have this at our disposal, we view the space of precification as if it were a modal structure. Okay. So you can think of that as a frame essentially. So you have an empty domain. You have um um a set of interpretations and your set of interpretations don't disagree on uh the individual constants. So you might think as a constant domain first order model uh semantics and then you have an accessibility relation that basically at the very least tracks the extendability relation the personification relation. So and you can view the objects in our mode of space as interpretations. So a world a world can be an interpretation and the accessibility relation between worlds is extending or precifying. Okay. So uh obviously this is reflexive because you extend or precify yourself trivally and it's transitive because if you have one interpretation that extends um a first one and a third that extends a second and the third one extends the first one. Okay. So questionification why is it not just this relation small article then but but the age which because we wanted um to leave some space for just taking a sub uh rel having the accessibility relation in each model um uh changing from this but it's completely unnecessary in fact so you nothing bad happens if you identify age and the lesser than relation. So you can >> it would be interesting to to see if there are cases that >> yes there there are cases where this is required by some applications to the logic but not quite to the quantifier cases. So not quite to the cases that I'm going to consider. >> But if so you're right and for the particular purposes of today they actually can be safely identified. >> I see. >> Yeah. >> Okay. Um this is super valuational structure and now remember that we want to talk about quantifiers. So we need to go to the right level um of generality or sorry of um concepts of concepts and in particular we want to single out the domain of quantification. So suppose that we have a super valuational structure and let's consider the set of all the functions from the interpretations okay because remember it's a concept of a concept uh to the uh the power set. You can also view it by its characteristic function to true and false doesn't really matter. And what is a quantifier in our case? Well, is a subset of this particular um of this particular uh space. So here we have the space of all the functions that go into the power set of the domain to the power of n to the cartisian power of n. And here we have all these functions. Okay, it doesn't matter what the R is here. And a quantifier, okay, would be a subset of the cartision product of all these domains. This is only because we are dealing with functions rather than with subsets. So if you have in mind let's say the classical generalized quantifier picture where a quantifier is a set of sets. Okay, for example, let's take the type one one quantifier uh every classically. Every is the set of all pairs AB such that A is a subset of B. Right? So it's the extension of the subset relation. In generalized quantifier theories, uh quantifiers have an extension. They're not seen categoratic expressions like in task semantics. And here you have that the relation in question is a subset relation. Okay? and you take all the pairs such that a is a subset of B. So every contains the pair dog friendly cat shy whatever you want. Okay. Uh here however we need to deal with functions and so for us quantifiers are uh pairs of function fg. Okay, such that whenever you apply the function to an individual, the result has the property in question if and only if uh the second uh function applied to an individual has the property in question. Okay, so that's the forgiven picture that we were starting from. So remember the example every dog is friendly. So whenever you apply the first function here that is the function associated with the concept dog here friendly and then you pursue collusion. uh this is not this is a presentational difference that is essential to this way of spelling out things. However, in the end we are also going to work extensionally with these functions. So you can think of them as sets basically. Okay. And as I already anticipated we are going to consider the so-called type one one quantifiers. The type one one quantifiers are those exactly like every whose extension is a set of pairs. Okay, the type one quantifiers would be those whose extension is a set of sets like everything, something, infinitely many things and so on. You see the pattern here that there is a quasi logical word thing that does the job. Essentially everything, okay, is the set of all sets that coincide with the domain. There's a singleton of the domain. The extension of everything is the set of all of something sorry is the set of all sets that are non- empty right because something is a is true if the things that are a are the non- empty a set that is non empty okay so basically um some thing is the collection of a such that a is not empty but if I take sum. This is the type one quantifier and this is the type 1 one quantifier. This is the collection of pairs AB such that A and B have a non-MPT intersection. So if I say some dog is friendly um well that is the the pair dog friendly belongs to extension of some because the intersection of dog and friendly is not empty. So you take all of them. Okay. and we are going to focus on the ones whose extension is a pair. The reason why we focus on them is that linguistically they're very prominent. They correspond to the syntactic category of determiners. So you can also view that something is a special case of sum where you just fix one of the two arguments to be the whole domain the same for everything and so on. So those are the ones that receive a lot of attention in generalized quantifier here. other quantifiers do too. And in fact, an arbitrary a generalized quantifier is an object of of type nk dot dot nj. So it can be 1 2 7 3 blah blah blah for a finite taco that is as long as you like. But it's very difficult to find an object in natural language that corresponds to that particular quantifier type. Whereas type one quantifiers are extremely common. So they're very linguistically prominent. Okay. And I'm going to impose a further restriction on the quantifiers that we are going to consider which is the so-called conservativity requirement. A quantifier is conservative if that happens. A pair fg belongs to the quantifier just in case the pair f belongs to a quantifier. You already have some examples on the whiteboard. Every is conservative. Some is conservative. So every bike is red is felt to be equivalent to every bike is a red bike or um some um person has an umbrella is felt to be equivalent to some person is a person with an umbrella. So if you repeat the first argument the restrictor in the second argument the scope you don't change anything. Basically conservativity was a property of quantifiers generalized quantifiers that was isolated in the 80s um by a series of simultaneous work but in particular there is a very influential paper by baris and cooper who argue that conservativity is a so-called semantic universal which by analogy with syntactic universal is something that if it is true is true of the extensions of the quantifiers so of everything that interprets a determinary quantifier. ire in natural language and this is not my field I don't do comparative linguistics but I know that the semantic universal for uh conservativity has been tested very extensively across languages that have nothing to do with each other and has been found to be extremely robust. So there is a very strong empirical evidence that um natural language expressions which deter which denote quantifiers denote conservative quantifiers. uh exceptions have been proposed in the literature. If you want we can discuss one that is especially uh discussed in the literature but the consensus amongst people working on the linguistic side of this seems to be that natural language determiners u determine uh conservative quantifiers. So this is one reason to focus on on these conservative quantifiers. Um so as I said we focus on conservative quantifiers for their linguistic prominence and also the conservativity universal this thesis that is u that I just tried to summarized. Now we want to give a super valuational semantics for all these guys for all these conservative type one one quantifiers after having motivated why focusing on them. Now uh how many of these conservative quantifiers exist? Well a lot. Uh obviously infinitely many. If your domain is infinite, if your domain is finite, u I think is uh there is a paper of vanam where he shows that there's two to the 3 to the n conservative quantifiers type one for n the size of your domain. So there's a lot of them in general and also we are interested in the general case where your domain is allowed to be infinite. Um so we need a way of constructing this conservative quantifiers because I cannot give you a semantic theory with infinitely many clauses that wouldn't be a manageable theory for us to work with. Fortunately there is a beautiful result from the six from the 80s again uh by Kenan Davi that provides a simple and elegant characterization of the space of conservative type 111 quantifiers. So essentially they show that every type one quantifier can be defined by applying boolean operations so intersection union and complement to just three kinds of basic type one one quantifier. So you take three type one1 quantifiers and you give the semantics for them and then you close your semantics under these booted operations and you have all of them. Okay. All right. So the three type one one quantifiers we need are every so the subset relation basically um cardality quantifiers at least lambda many okay so of course this is definable uh if lambda is finite if lambda is infinite it's not definable so for every lambda up to the cardality of your domain you need to have the card the quantifier at least lambda many okay u and the so-called basic possessive quantifiers like John's okay like the ones you find in this genative uh constructions um this is also something that is interpreted as a quantifier if you have them then basically thanks to the keyni theorem uh you can define all the type one one conservative quantifiers now so the very first step to recover uh an amount of generalized quantifier theory in our framework which is now exactly the amount consisting of conservative type one one quantifier we need a version of the the Keenan Stabby theorem in the framework that we're working on. Okay, which is you know this super valuational non-classical functionbased uh framework. Um fortunately that's not complicated because the original there is a way of proving the theorem that uh makes it very abstract and purely algebraic and so you can reproduce it in whatever framework satisfies the basic constraints. So suppose you have a supervaluation structure and your set the basic quantifier will be the set including the denotation of every at least lambda many for every lambda up to the cardality of the domain and the possessives. Now I apologize because these look ugly but bear in mind the much nicer looking every that I've written on the whiteboard. So the semantics for every is that one. Now it bears a superficial resemblance to that but the spirit is the same. So rather than having all these sets or the pairs AB such that A is a subset of B we move to functions but functions are supervaluated. So we take all the functions FG okay that lives in your structure such that for every interpretation that extends the given one okay so you're given an interpretation J the sets that are outputed by applying F to J is a subset of the set as is outputed by applying G to J prime. So remember every dog is friendly. It has to be that for all the interpretation if this outputs a dog then the if this outputs a set of individuals that set of individual has to be a subset of this across all the space of the precifications. So remember this is a local structure you live here. This is your interpretation J and you have a bunch uh of I don't know accessible um other worlds which are precising all the way up. So here you have the subset relation here I have a subset relation here I have the substitation and so on. So it's not static, it's super valuated. Okay. So you have the denotation of every and uh I'm not going to comment extensively on the other ones because the trick is always the same. It's just that the relation that is preserved might not be the superet the subset relation might be another relation r. Okay. So for at least you have that uh the things that are f and g okay have a cardality that is at least lambda for all the interpretations that precisify the given one. And finally um um um possessives here in generalized quantifier theory when you deal with a possessive you typically take an individual John the an individual denoted by a term and you associate with it a relation which is the relation you know just a set of individuals that is are connected with John take John's cats for example and when you um take the intersection with the first argument that has to be um a subset than the second one. So, John's cats are shy, right? Is true if and only if the things that are John's and cats are a subset of the individuals um which are shy. Okay? And this again has to be preserved all the way up. So that relation there can be complicated doesn't have to be as simple as the subset transition. Now um I should put a disclaimer in English at least and other languages as well. Not every genative construction works like that. So John's cats are shy. It works like that. If you want to make it work for John's accountant lives far away, it doesn't quite work like that. But here we don't need it to be precisely tracking this gentle constructions. We need it mostly for technical purposes. Okay. We need that uh thing here to generate a filter in in the algebra. Okay. So now that we have that, let's define operations between functions. We define them in the obvious pointwise way. So let's define algebraic operations. This is a meet and that's a complement. The joint is defined uh in the usual classical way. So meet is defined pointwise. So for all J's it has to be the case that you take the uh intersection and you take the complement. Let's take a generalized uh version of the meat. So the infinitary version and then essentially what you have is this the set D here. Okay. So the uh quantifiers that you're going to define is the smallest set of quantifiers in my sense. So this set of pairs of function that includes the basic ones, the ones I just showed and is close under complement and meat. So this ugly formulation here is just closure under complement and meat. that whenever you have a lot of other quantifiers the meat is in there and we have a lot of other quantifi and when you have a quantifier in D then also the complement is in D I apologize for the set theoryic notation is is actually quite cumbersome sometimes in GQT you find the lambda notation which in this kind of case would be less cumbersome in other cases it would be more cumbersome so I go with the set theoretic notation okay what's the keen study theorem uh well this is adapted to supervaluational structure of course we were talking about models in the first order logic sense but you can lift it up to the supervaluational structures for every supervaluational structure the conservative type one modifiers are exactly the elements in D I'm not going to give you the proof because it would be long one direction is straightforward because for that uh direction here you just have to check that all the quantifiers in there are conservative and that's just a check of three quantifiers and that the closure preserves conservativity for the other one is more laborious because you have to reconstruct the set of all the quant of all the conservative quantifiers starting from D. And here we have adopted a a proof strategy that is in another paper that I have with Michael Glansberg where um basically which is not that different from the original proofs. It's is more algebraic but um the spirit is the same. You first show that the target set is a complete atomic boolean algebra and as soon as you know that the algebra is atomic as soon as you characterize the atoms you can characterize every element in the algebra by taking the boolean operations. So the goal is now characterizing the atoms. Um and the atoms themselves can be characterized as boolean combinations of non-atomic quantifiers that are in the conservative set. Now as you know the atoms of boolean algebra are inclusion small elements of the algebra. So they sit at the bottom of the algebra. Then you go up and you take all the other elements. But here we start from element in the conservative set that are big elements in the algebra. You go down with boolean operations. You characterize the atoms and then you go back up. You do this thing. Um and um yeah so basically at the end you can define everything from the quantifiers in your basic set. Okay good. Now that we have this, we can finally use the Kenan study theorem as a definability theorem so that we have an an object a linguistic object for which to give the supervaluation semantics and then we test how the supervaluation semantics fares with respect to desirable inferences involving quantifiers. Now we need now to formulate a language where these quantifiers live and then we can give truth conditions for these quantifiers. Now the problem is that the language is not going to be very pretty. The reason is that we have seen that we have to have these three quantifiers right we have to have um every at least not many and basic possessives and they can be taken as basic constructors of the syntax rather than having uh for all and there exists you take a binary variable binding operator which is every that's customarily done in gt at least lamb of any and the possessives but the other quantifiers are defined by taking the boolean operations and The complement is easy, but the other boolean operation is an infinitary operation. So you need an infinitary language, a language that allows for big conjunctions of a possible infinite size. Now you might say, well, I don't like infinitary languages. I want my conjunction to be finite. That's perfectly fine. But it means that you're restricting yourself to a finite domain. If you restrict yourself to a finite domain, then you don't need to construct quantifiers using a definitary conjunction. Um there there is um you know an unpleasant consequence of that is that some quantified sentences come out strange like you know in every finite domain it is true that finitely many prime finitely many numbers are prime and it's false that finitely many natural numbers infinitely many natural numbers are prime and the fact that there are infinitely many uh prime numbers is a basic theorem of arithmetic. So you would want that to be true for that to be reconstructed with the infinitely many quantifiers in this case you need an infinitary domain. So if you don't talk about math and mathematically oriented quantifiers it's perfectly fine and just imagine that whenever I write big conjunction there is a standard conjunction but if you also want that then you need the infinite one. Okay. So the vocabulary contains these um u these guys every and least lambda many which I abbreviate as a lambda and uh the possessive. So whenever you have an individual John this is John's. Okay. And they form sentences in the way you would expect. So whenever A and B are sentences then every AB sorry formulas then every AB is a formula where every binds the variable that the variable X that they share the same for at least lambda and the same for John's um all right so we fix a supervaluation structure and suppose that the domain D has a certain size you need a cardinal that is at least this big and then the conjunction will be an infinitiary conjunction of the size at most kappa. Um there is a deep philosophical reason for this choice which is that otherwise it doesn't work. So because you need to reproduce the kinavi construction that uses you know essentially infinitary um boo operator of that size. So it has to work like that. If the domain is finite then this is also a finite number. So no problem. Okay, formula are defined in the usual way. So these operators bind variables and form formula in that way. And I also assume a standard syntax for the infilitary language. Okay, now for the semantics. Variable assignments work exactly as in first order logic. It's a function that assigns an object in the domain for every variable of the language. And the x variant is exactly what it is in task semantics. Nothing uh special. Denotation also works exactly as it does in first order logic. You have a term T. If it is a variable, look at the variable assignment. Is it is a constant, look at the interpretation. Is it is a function then apply recursively. Okay. And now this is the semantics. Now suppose you have an atomic formula a predicate P applied to terms T1 to TN. Then it is um well if you have a ter you satisfy in a model under interpretation the formula a when a is atomic if the extension of uh that tpple belongs to the extension of the predicate. Instead if it is a negated atomic formula you satisfy that formula if the extension of that tpple belongs to the anti-extension of the atomic predicate. So here is trialent semantics just as usual. Um identity works in the same way. You remember we said at first that we need separate clauses for truth and falsity and we need that for the quantifier. So we need to do tr basically negated and non- negated clauses all the way. So double negation works in the expected way and double negation is satisfied if the formula without the double negation satisfied and conjunction and negated conjunction work in the usual way. So a big conjunction is satisfied if every conjunct is satisfied and a negated bit conjunction is satisfied if there is at least one conjunct such that this negation is satisfied. Okay. So everything is fine up until here. Now um this which is for the uh quantifiers I'm only going to comment on every because the other ones work essentially as expected. Okay. If your formula is every B is C, then you satisfy the formula in the model M under the interpretation J is every J prime that extends J is such that it preserves the subset relation just as I said before before I had the extension and this is a truth condition. If instead your formula is not every then it's satisfied in a module under interpretation. If you find okay um you find some counter example actually here I have for all J prime this can be simplified it doesn't have to be for all the semantics is done in such a way that if you find one it it keeps being a counter example so if you find a counter example somewhere so an individual who is B but not C where the negation is internal so it's not it's not the case that it is C but it is not C then if you find an individual here then your model and interpretation satisfy not every B is C. The other quantifiers work with Mandis in the usual way. Okay. And here is just the convention for the things that are a good uh as I said there is exactly one slide on truth. This is a super valuation semantics and if you restrict yourself to classical precification you get exactly kickfine style supervaluations. If P is of a predicate obtains exactly that treatment, you just extend it to all these quantifiers that can be defined. You know that supervaluation semantics can be um employed to deal with a self-applicable truth predicate and with paradoxes. It's the case that you can do it here too with I would like to say with some limitation but the honest truth is with many limitations you can do it here too. with many limitation. There is at least one super valuation structure uh and one interpretation for a truth language. So I have the truth predicate around such that true A is satisfied in that structure if and only if a is satisfied in that structure. Now I am purposefully being vague here because you see that this is a predicate and a here is a term that denotes a. So it means that there is a coding of the syntax but the language is infinitary. So you need to code an infinitary language. So certainly if your kapa is uncountable you cannot code it in arithmetic. There are works on the encodings of infinitary language. Um you can find uh delta def functions that are delta definable instead of c that do the encoding of essentially infinitiary languages of the shape l capa lambda. And what we need here is essentially l omega kapa plus the generalized point fire right and you find encoding of this much more general languages here that are definable in set theory. So you can encode the syntax that we are using here. I won't go into the details but it can be done. Okay, this is all I have to say about truth. Stop here. But now we want really to focus back on the quantifiers because you remember we started from some inferences and we were observing some properties and also perhaps some decided concerning the inferial behavior of quantifiers. Now in this particular semantics, this is the non-contraositive semantics. So all the things that I said uh might behave very non-classically at the beginning are recaptured in this semantic. So no controposition the homogeneity feature uh the empty restrictors and and so on. But those were very scattered observations. They were not very systematic. What we want is a um a more precise answer to the general question how good is this semantics at recovering classically valid inferences involved in the quantifiers. So to answer this question we need two things. But first we need a notion of logical consequence because we need to formalize inferences and then we need a well understood class of inferences on which there is a lot of work but formal and empirical and we we use it as a benchmark. Okay. And um for the logical consequence in supervaluation semantics you can define at least two notions of consequence. One the so-called local and one the so-called global. The local consequence is a classical one. So gamma locally entails a if for every structure and interpretation whenever such pair satisfies all the premises in gamma then such bear satisfies uh the conclusion. Okay, so that's classical consequence and for now we're going to use this and for B we will test our semantics on again a well understood class of inferences involving the quantifiers which is monotonicity properties. Now these are roughly the monotisticity properties and they have to do with preservation of the quantifier if you go up or down okay in either the scope or the restricture of the quantifier. Okay. So for example the quantifier has this monotonicity property here. If whenever you have the quantifier qab being true and you go up. So you take a superset of a the property remains instead the quantifier goes up in the second component if you do the very same but not for a but rather for b. Okay. And the same goes for the other monotonicity property and they combine in in the usual way. To give you a simple example, the quantifier every has this monotonicity property or as it is called it is upward and tailing in both restrictor and scope and this is a transitivity of the subset relation. So if you have um every a b then you can take that b and b is a subset of c then um every a is also c. Oh, sorry. Every has this property, right? So, every A is B, B is a subset of C, you can infer every A is C. But if you take that A Z is a subset of A, you also infer that every A z is B. So you have this property here. Sorry, I got it wrong. Um, so downward entailing in restrictor, upward entailing in SC. Okay. uh there is a lot of work in generalized quantifier theory on these monotony properties how they behave for example this is how they interact with negation I won't go into that I just want to take this as my benchmark and the problem for the current semantics is that we get very mixed results if you test monoticity properties in the current semantics you get some results right according to you know textbook benchmark you would want to have and some results not so right for example at least oh I'm going to formalize ize a subset using every because I need inferences. I need to test inferences of stuff in the object language at least is a um uh upward monotonic quantifier. And here the uh prediction um is actually correct is the theory correctly predicts that at least is upward uh entailing. But and as a corollery you have that plenty of quantifiers that are definable in terms of at least are also upward entailing. Sum is definable as at least one. So it's easy to define sum. Okay, you remember we have at least lambda for every lambda up to the cardality of our domain. So at least one is sum and at least omega is what I call I am for infinitely many. At least omega. Those are all special cases of at least and they're all monotonic in the right direction. So they're all upward entailing. Um however as soon as you have negation u you have failures of the intended monotony property. So the expected result was that this goes but it doesn't go. You can find counter examples and you have similar story for other stuff that you can define in terms of negated at least for example no not at least one you define the quantifier no should be monotonic in a certain way but it's not so you have a bit of a mixed bag here I have other examples here I'm just skip over them so a mixed picture at least and every satisfy their monotic properties but negating a basic quantifier disrupts the monotic property that it's predicted to have in GQT. So since you can define all these conservative type one quantifiers from the basic one by negation in conjunction monotonicity properties are not guaranteed to be preserved from the basic one going up and it's especially difficult to give a map a precise taxonomy of which monotony properties are actually preserved because for example if you take a quantifier as a conjunction of monotone quantifiers like every person and at least two cats that's a quantifier. Okay. Um, every person and at least two cats are in the room something like that. But few interesting quantifiers are defined like that. Most of the time you need negation to define interesting quantifiers. So interesting quantifiers have to be checked for monicity individually. For example, most is uh upward and tailing in the scope as it should be but it's not its negation should is not downward and taking in the scope but it should be. So this motiv yeah this is basically what we have understand behavior on the list this semantics here and this motivates a small variant of the semantics. I have been speaking for a long time so I'll be very short and I will just present you the result and the small variant is essentially imposing a modal structure that is more than just reflexivity and transitivity but is something similar to what you have in hyperintentional semantics. If you're familiar with Hannah's light system, hype for example, um hype can also be presented as a as via model structures. If you impose basically the hype model structures to the set of possible worlds that we have, which are interpretations, you get a different semantics and this is what goes on here. To define hype, you need to define the Rley star. But even glossing over all the details, what happens here is essentially that now your accessibility relation is much more complicated or at least much more stringent is reflexive transitive crucially it's convergent. So you start from here you go to different points and then you they they go back to at least one point and then it interacts exactly in the highway okay with the Rocky star. So this is very close to likelihood system high. Okay, it's for super valuations but it matters little. If you do here uh if you do that you can define the same semantics as before. You can actually simplify that because you do not need to split uh positive and negative clauses and you have um a nice result. If you define logical consequence in the global way this time not in the local way and the global way is when you expect satisfaction of all the structures for the premises then you have satisfaction by all the structures from the conclusion. If you do these two modifications, so superimposing the hype structure and defining consequence in the global rather than rather than the local way, you have very nice results for the monotony properties. In particular, you have all of them. So that's the nice result. So suppose that you take a monotonicity property P. So P there ranges over objects of that shape, right? For example, this monotonicity property and that monotonicity property and so on. So P is one of them. Let's say that Q your quantifier Q has the property P in a given semantics if the inferences encoding P are s are valid in that semantics. For example, every as I said before has this monoticity property P because these two inferences are valid in generalized point theory. Remember this is just the transitivity of the subset. So the result that you actually have is that for any conservative type on one quantifier and for any monotonicity property, the quantifier in question has that monotonicity property in the classical semantics if and only if the same quantifier has the same monoticity property in our semantics. So we have two competing systems. One that is way more non-classical, right? It doesn't have controposition. It has all the things that we said at the beginning. But because of the failure of controposition, the non-classity basically percolates to the monotisticity properties. And if you want monotonicity properties, you don't have them. Here instead, if you impose this stricter modal structure and you define consequence in a different way, you do not have all the non-class things that I said at the beginning. You're very classical there. So you give the classical response, but you have monotic properties. So it seems that Yeah. Okay. So this is the monotony behavior under oops old one under the KS3 semantics. Okay. Um so these are very preliminary results. I mean we haven't checked all the relevant classes of inferences from the literature but we have taken them off the shelf essentially and they do not yet show that this theory captures every possible inference involving quantifiers in the classical way. that he shows that a substantial portion of the classical theory of quantifiers can be recovered within a nonclass framework that accommodates weekly native predicate and all these supervaluation treatments of industry. So depending on how you adjust the model structure your supervaluation semantics can be further away from classical inference patterns or closer to classical inference patterns that all depend on adjusting the model structure. This is why I think it's important to view at least this brand of supervaluation as incorporating a model structure of of some sort. And um so I presented two systems as I said they are competing from the way that I presented you might think that I have wasted your time because it seems that the second one is better. But actually there is a genuine trade-off right and the genuine trade-off is this. If you take the first one, it's immediately possible to define the a conditional quant uh a conditional operator from every. So as soon as you have every a is b, you can translate that as you know uh if a and b for a conditional operator. And the conditional operator that you have is actually very nice. It obeys the deduction theorem. It's essentially an intuitionistic conditional even within this nonclassical framework. In the other fully much more classical framework as far as inferences go, you get a terrible conditional that doesn't behave very nicely. Certainly doesn't validate the deduction theorem. So it depends which way you want to go. You want to go very non-class, you buy a non-classical conditional that behaves very nicely. You buy the non-class solution to the examples that I presented before and you lose monotonicity. You want to go more classical, you stay classical with the response to the examples that I said at the beginning. you regain a very classical inferential behavior for the quantifiers. You cannot define a conditional from every. So if you think that there is a strict a very close connection between if then and every and you might want to define the conditional from the con the quantifier. I personally don't think that you should define a conditional from every. That is my take but people might have different views on that. So this is a bit more of a u picture in both you can have the truth predicate that I mentioned before. So both are non-classical. None of them coincides with classical logic. Otherwise you wouldn't have that truth predicate there. One is more classical when it comes to the monotonicity properties. The other one is more classical when it comes to the conditional that you define using the every uh quantifier. Um so maybe there's a trade-off. This is something of course to to be discussed. That's it. Thank you very much. >> Thanks. We take you Hey. Hey. Hey. Ah, ah. Hey, hey, hey. Hey. Hey. Hey. Hey, hey, hey. Hey. Hey. Hey. Hey. Hey. Hey. Heat. Heat. Hey. Hey. Hey. Hey, hey, hey. Okay. And also I can say out loud because there have been some folks online. So if you have questions coming in from online, don't hesitate. I am watching the chat. I can relay questions. Um good luck if you need to write formalism in the YouTube chat box. But other than that, >> that's it. Yeah. Actually, honestly, I I I read I read Latte, so you'll actually be fine. Um, I speak I speak Latte. >> So, is there any any question? >> I have some. >> You can. Um so yeah I think this is a very uh nice project and it seems very natural to to to how to find some determiny there and then try to deal with it. Uh um but I was like very fascinated by the two different semantics you get at the end. Um and it seems like it should not be a formal decision which one is the right one or something or it's for some property like it like in the first system you showed where when all these things fail >> and I think it's weird intuitively it's weird for me to see these as benchmark cases like oh logic has to do th those things you know you're in a kind of weird situation so we are exactly interesting interested in where the the the inferences are going to fail because of the indeterminacy uh like like counterposition. So I would rather think it natural to look for a a reason the reason why it fails and then see whether this is because there's there's no decisions making in the semantics into it that I see that are very unnatural or something. And so I I expect that I would expect that if you tell the story um like actually you could find convincing cases of of why why uh the benchmarks should fail. Uh that's a good thing. >> That's an excellent question. I don't have a satisfactory answer. I'm afraid I think that um so there is maybe I can split your question in two but maybe I can say first something about um if and why these inferences are expected to fail if at all and then how this links up with features of the semantics whether they're expected to fail it's an extremely tricky matter because it has to do with linguistic intuition and um even even there um first of all as far as I know I don't think that there is experimental evidence on these particular things just because as um as we were briefly discussing now um there isn't a lot of uh work on um generalized quantifiers in the presence of indeterminos there is very well established literature on general features um of generalized quantifiers and basic applications and so on but not so much on the non-class side let's say and how they can be defined uh going away from classical logic but try to retain at least some nice features inferential features of classical logic and so we have to rely on uh judgments here and here the judgments in a way are very um I don't know not very telling because uh there's a lot of phenomena that are stuck up like for instance here there's presupposition failure and if you insert like presuppositional sentences into an every construction and you have a certain kind of intuitions about presuppositional uh triggers then you you might end up rejecting uh sentence number two but it's you are testing a lot of things at the same time. You're testing the intuitions that you might have on presupposition in the context of the contraosed uh every statement right um here category mistakes category mistakes is a you know it's it's its own field and people have widely diverging opinions on what the right semantics is for category mistakes and here you might think that no in that case you know I I would be perfectly happy saying everything not on earth is um is not expensive because you know it's not for sale so for is neither expensive nor nor cheap. Obviously, you can retort, well, but I also want the inference that every non-expensive thing is cheap, right? Uh if you want that inference, then presumably you cannot answer it the way I just did. But it's extremely difficult I think to uh to find conclusive uh linguistic judgment in the absence of you know empirical work that has been extensively tested this this thing and also in the presence of a conflation of all these phenomena because quantifiers are higher level so they need a lot of input uh you know to be worked with and and here the kind of indeterminacy is in the input is not in the quantifier. the quantifier by itself doesn't uh doesn't do much. Um this one this case can actually be a little bit different because in this case the quantifier does something uh but this is really piggybacking on uh the conditional. So really what's going on there is that you have a lot of patterns that are well known from the literature on conditionals. Something I didn't mention for example is that you can find versions of the paradoxes of material implication for quantifiers right it's it's very easy and uh you would have similar counter examples so there you know the people who think that every and if then are closely connected will point to these kind of cases right um if you have something like this and I say well yes there is an indeterminacy in the quantifier but again uh first of all I think the majority of people who work on indicative conditions language think that indicative conditions do not contraose. But there is also a literature of people who think and argue that indicative conditions do contraose and there is an hidden concessive in apparent cases of failure of controposition. So when you have an apparent failure of controposition, you're talking about even if rather than if. There is a literature on this. So you might think that something like this is going on there. So this is just to say that uh we've tried to elicit some intuitions by considering examples but they are far from conclusive. So it's nearly impossible I think to take them to be a real benchmark that discriminates between the two semantics. That's why they they are like ways to get going in a way. But the second kind of cases the monotonic inferences who are which are very well established in the literature I think are more robust in from the point of view of where evidence comes from and what kind of evidence we can uh we can uh master. I don't know if that answers part of your question at least. >> Yeah, partly. But but it seems like it's not because our intuitions are like quite incoherent and so on. you're just bad at at at at lots of nesting and and and these things. I mean, when there's different linguistic phenomena or complicated, we have to combine and then we bit troubled and and intuitions go in all directions. But that doesn't mean that you as philosopher cannot give a like normative theory of why it should be uh uh uh controsive or whatever or monotonic in that sense and not and and I find it very interesting that for reasons of pure indeterminacy which you can take to be independently justified you lose this monotonicity features um >> and then you can like even if maybe intuitions are not supporting this at all, you can say no no but in fact it should um be it should fail this monity. Can I have a good argument for why it should fail? Why is it like actually incoherent uh um to to assume monicity in these cases? That's that's very interesting and it's connected to what I said because if you if you look at the proofs of the counter examples where you show that you can have um uh that you have structures that satisfy the premises of the monotony inferences but not the conclusion. If you look at the counter example obviously this is just one counter example so is as informative as you know it goes only a certain way but if you look at the counter example that we came up with is like oh if I only had contraosition I wouldn't have this counter example. that is closely connected uh to the failure of the proposition. Failure of the proposition makes you lose um monotonicity for the negative uh quantifiers for the negated quantifiers. So this is something that you can use to uh let's say bolster your case for uh let's say I should lose these things uh because they they stand up together and indeterminacy >> basically undermines them both. um I'm maybe a bit more hesitant um to give a strong normative tone to this work because it's linguistically motivated and so unless you have you know specific natural language independent goals in mind like you know I want to teach a machine to think along these lines or I want to set up an artificial language that for my scientific reasons behaves in this way. If you want to go back to natural language and say look you should take very seriously these cases of indeterminacy as seriously as they have been taken for vagueness, presupposition failure, anafhera and a lot of other language phenomena because they also arise in um in quantifiers. Uh but at the same time you should provide a theory that is a workable theory of natural language that accords with uh linguistic judgments more than it orients them. I mean this may be a very problematic discussion but I don't agree. I mean >> you would like a more normative but even for natural language. >> Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. So that we say um we have mixed intuitions about or intuitions might be wrong. But I have a good argument why they should be wrong and and and then it will guide you to better reasoning if you don't use contraosition in those cases or you don't use the monotonicity feature because there's a danger that you make an error >> um in natural language uh right because >> it's going to lead to errors right if you apply contraosition cases >> yeah you can apply along these lines and say well yeah if you reason fully classically in the cases where you would expect quantifier to be monotone um when it's not guaranteed. You end up with a classical conclusion, but the classical conclusion is wrong because maybe it's um it's something like that. The classical conclusion tells you that both these sentences are true and um it's it's clearly incoherent. So, you know, given the context, this is the context, these two uh sentences, I cannot envision a situation in which they're actually both a scenario in which they're actually both true. So, you would do a more normative slash metaphysical, I suppose, uh line of argument and say you wouldn't end up something like this. Uh and and that's actually a mistake. Yeah. Why what why can we do that? Why can I mean in in fairness to the classical theory these cases are discussed and there is uh you know a discussion of the it's recognized that uh you know the fact that they're both true is bad. Uh but um typically it's assumed that um quantifiers have a presupposition of a non- empty restrictor. So if I say every A is B is assumed um that um um um there is at least one A which might be strange in some occasions because you can say that um I don't know every unicorn lives here and every unicorn doesn't live here and you know they're both false uh or both true depending in the uh let's say traditional traditional uh situation uh would be both true in the arisatilian case in which you assume that there is at least one unicorn. Uh um in in the other case it would be both false. So yeah you you can argue more normatively like uh like that. I prefer to try and elicit um uh an intuition whereby in any given scenario exactly one of these is true and the other one is false. And it's not the case that they're both true and it's not the case that they're both false. um because I think it accords more with the uh intuition that come from the debate on interpreacy but that debate is also theory Latin so it's not natural one other way you can argue for this kind of conclusion is a sort of holism and you can say look we have uh a number of indeterminacy phenomenon we have a number of semantics to make them work in in in this or that way and this is a way of showing that the indeterminacy friendly semantics can go a long way towards capturing inferences that exceed um that exceed first order logic. >> Mhm. >> So yeah, >> great. >> This would be one way of arguing about this. >> Okay, cool. >> I have other questions, but you have >> Yeah, I have a little ones maybe just out of curio. You spoke about um type of quantifier one and one one. >> Yeah. >> Is there other in natural language? >> Ah uh it's a very difficult question and I'm very good prepared. >> So yes um >> yes but it's kind of difficult to come up with. Um so for example people have argued that reciprocals are uh what are they type one two quantifiers. I should check it up. I should look it up. But reciprocals like each other. If I say they hate each other or they like each other, um then you're basically talking about two people and a relation between the two people. So I think that's occasionally has been modeled as um a type one two quantifier. >> Um that was literally the best example I could come up with. I think that uh um definitely had it has been studied uh but I think the most common case is a type one just because it corresponds to the category of the determinant. So something you you use together with a noun with a noun to make a noun phrase like every dog and then you put a verb and you do a verb phrase. Um yeah. Yeah, I think so. Even even the purported counter examples to these claims um like to uh to the claims that um these constructions are conservative. They all involve type one um quantifiers because the counter examples have to be uh determiners or at least can be argued to work like determiners. So if you if you disagree with the conservativity universal, you have to come up with the counter example, but that also has to be something that functions like a determiner and that is modeled by a type one a type one quantifier. You you can cook up quantifiers mathematically of any kind. I mean it's super easy to quantify to to to cook up a um a type one quantifier that is non-conservative. Just the supererset relation is not conservative. The subset relation is but the superset relation is not. But the question is does it correspond to something in natural language? So in natural language in in in nature do we find these uh these quantifiers and the wisdom seems to be no for the conservative case and very little for the cases that exceed uh type one or type one or uh relatively low types anyway like for reciprocals. Yeah. um >> um yeah so I was wondering um whether um so if I understand well you can deal with both cases where your there's some phenomenon of determiny indeterminacy already there and then you add some uh uh um quantifiers on top of that you quantify >> in these kind of contexts over about these kind of contexts. >> Um >> that is one seemingly very useful application. Um and then there's something else where the quantification the the generalized quantification causes the indeterminance. Yeah. And I did not see in the semantics very well how this this difference like you attack I would expect like two levels of of of of um supervaluationism uh or something like that like you you have >> on the lower level just a propositional level already some supervaluationism. >> Yeah. Then when the quantifiers enter you also consider is that is that does that make any sense? >> Yes, it does. We didn't think about it this way but um because essentially what is supervaluated is mostly the quantifier in the sense that um if you look at the semantics I'm just going to keep take the first one for um for simplicity at the base level there's just the the trivalance in in a way right and then um when you quantif when you um supervaluate um you look at the extension for the quantifiers. >> U you don't have to do this. If you do this, what you end up with is basically uh strong cleaning logic for this framework for this uh for this language here and a supervaluated um a supervaluated quantifier semantics that gets you know definitely more classical than uh struct logic. For example, in a pure strong clean logic setting the sentence every A is a wouldn't be true >> whereas here it's true. uh because you know look at any extension that you want a is a subset itself. >> Um and um and if you take this to define a uh conditional you get essentially the intuitionistic conditional. So if you put them together you get to therson logic. So with the strongly base for the boolean and intuitionistic conditional at least for the um like the sequent calculus presentation you would get that. Um so you superval the weight only here. If you want you can also supervaluate here not >> and that's what feels more natural to me. >> You can do that you you can definitely do that and you can get essentially all the ranges of supervaluationism also at the basic level. So here uh you can supervaluate in the classical way and you get classical logic in the same sense of supervaluationists. So not classical semantics but the classical logical laws etc. Um if you don't instead you you have only here the supervaluation but you can do it >> both middle doesn't work in the logic >> here no >> but if you supervaluate here yes >> so for example >> yeah no no in that case it does the logic you presented in general excluded middle is not valid >> exactly because of the strong cleaning behavior but there is an easy modification if you do like whenever a is an atom or not an atom And then you you know you prove it um by induction rather than merely checking that this belongs to J+ you ask that it belongs to all the ones that extend J+ and you also ensure that every extension is total it's classical so they don't overlap but they exhaust the domain in that case you would have the excl middle and all the logical nodes but that's that's a very good observation because here it's really a hybrid between a highly non-class kiny line for the boolean connectives just because it's simpler so Because but the simplest would be that it's just classical >> on the level of non not qualifiers right so you can choose for >> to allow an indeterminacy there >> you can do that too it would lose you the possibility of having the truth why you could behave in this way >> of sure >> but you can you can and you can be fully classical at the ground level and super valuational at the quantifier level that actually is a very interesting observation because these two levels can be completely disconnected So if you want you can disconnect them entirely. >> So for example you can have um cases where um um yeah for every propositional A and B A or not A holds uh B or not B holds and every classical logical holds of A and B. But um uh you can have for example that it's not the case that every A is B or not every A is B. That's that's fine that you you can have that. >> Was there a reason why you did not uh go that way? >> Uh simplicity simplicity. But in the classical case, you mean? >> Yeah. Because people have always presented supervaluationism as a kind of a way to restore classicality and so on. >> And then it's kind of a bit schizophrenic or something to start already from a very weak logic. >> Yeah. Yeah. >> So that you'll never be able to recover classical logic anyway and you can recover some classicality. >> You can if you supervaluate the basic weak logic. >> Yeah. Super valuate. I do see that. >> Yeah. I I just wonder why you did not >> It's probably because we wanted to treat all the things simultaneously including the truth predications to behave >> in truth predication. >> Yeah. So this is a bit of a middle ground >> program. Yeah. Yeah. I see. >> So if you want that u that that's a natural way uh to do it. Um then of course uh you have to be a little bit careful because you can formulate carry- like paradoxes with the every u quantifier with the type one one quantifier that has been done there's a relatively recent paper by Bruno on on that and actually the the paper that I mentioned um in passing um before the paper with Michael Glansberg uh it's a paper where we do a formal theory of truth for generalized quantifiers but then there's no classicality recovering and not even supervaluation here it's super valuation but uh but I should emphasize that these two levels are at least in as a matter of principle independent and also the fact that if you do things like this you can isolate exactly the kind of um indeterminacy that arises exactly from the supervaluational quantifier because from the supervaluational quantifier you can have uh exactly the features that um I was mentioning at the beginning. So if you want exactly one of them to be true and the other one to be false, you can have it. Even if the basic logic is fully classical or the basic logic is fully strong cleanly, it doesn't matter. It's it's independent. So you can easily recover whichever intuition you prefer for that pair of sentences. >> Thanks. I should remember that >> the talk will be will be online so you'll have access to everything. >> Thanks. Okay. Um so sorry if you uh said so earlier but I was wondering and you said just before you developed another strategy at one point >> whyism we spoke about another strategy before. Uh >> oh. Why super valuation is here? >> I mean like could you have chosen the logic we spoke about this morning the >> oh our revision theory? >> Yeah I don't know like what was your process in it or is it that you were like okay look like the best tool? No no no no no my process was much simpler than that and it is that uh if you want to do a revision theory of truth for let's say non monotonic connectives it's already very difficult so if you want to do it for generalized quantifiers it's even more difficult so my my thinking was supervaluation isn't easy I do that the revision theory much less easy I won't do that no but this is a joke but it it's really something that is natural to mention Um actually when we submitted this paper here which is still under review round two um one referee said exactly what you said now one referee said oh this is nice but this is you know this is a strong cleanliny based so in the end you get all the issues for strong cleanliny based theories which are either very non-classical if you want uh certain behavior of the truth predicate or squeezed onto classical logic if you want a different behavior of truth predicate. one very you know new thing to do uh that would be quite interesting would be a revision semantics for all of that and um I I just don't know how uh that would go I mean presumably one can do that um really you have to um um yeah you have to combine um you have to combine the strong keys for the quantif ifier that we have there with a revision sequence. Um but developing this theory uh would take much much longer than it took to develop these and this is a very natural next step because essentially what we are doing is lifting to the generalized quantifier level certain approaches that have been explored um in detail at the propositional level and even the first order level. But the generalized quantifiers are very uh difficult to treat in full generality because a generalized quantifier by itself it's it's a relation. It's a set of ordered pairs. So it's an arbitrary object as it can get. So this is just a type one one quantifier. This is why we restricted oursel to the type one one quantifier because you can generate them in a relatively easy way using the Kenan Stavi theorem. the Kenan Stavi theorem. Um, yeah, I don't know how would that look like in a revision setting. I don't know. I have to think about it. On the top of my head, it looks really complicated. Um, yeah, but this is, as I said, a very natural question. So natural that the referee for this paper asked the very same question. And yeah, it's absolutely something to to be looked at. But we wanted to gather some evidence of how these quantified logics uh look like by first developing them in frameworks that are very wellnown like the weak cleaning the cleaning framework and the superpational framework. >> Yeah, thanks. This is an excellent question. >> I have a I have a really naive one. Um, so from a coming from a someone who well hasn't been thinking about this stuff in I don't know 15 years. Um, but one thing you said at one point, so you mentioned that that you uh, and forgive me because it went by fast, so I've already lost at which point in the talk it was, but you mentioned that there's a that that you keep tabs on the empirical work around the use of these kinds of, you know, around use of phrases like this in language. And so may come back a bit to this to this descriptive normative point that you were talking about with Peter earlier, but I'm just interested like >> how does that how does that inform what you do? How has that actually kind of what and and and sort of what is out there? Um I yeah I imagine this is so so this is stuff coming from the like like corpus linguists or like who who works on this stuff and like and in in in the sense of like figuring out how real humans use these kinds of these kinds of complex constructions >> okay that I am really quick to answer. So what I know is this is that there is a lot of work on the formal semantics and that's like a logic language interface. There is a lot of work on the um natural semantics and so capturing testing whether the theory captures uh ways of using the quantifiers that are actually out there like the monotonistic case would be a prime example of that >> and there is a lot of work on comparative linguistics testing for example whether the universal is actually tested across languages. So you go to languages whose um syntax don't even doesn't even have determiners working in the same way as they do in English for example and you check whether you can still explain um essentially something that takes the place of determinance in those languages by quantifiers that are uh conservative and there I I briefly mentioned that there seems to be a very strong evidence in support of the conservativity claim. I would say these are the three areas. So the logic language interface which would be where I would be more comfortable. Um the empirical semantics uh actually checking you know whether the predictions are right in a single language and then comparative linguistics checking whether they hold across languages. If you look for example the the textbook reference that I have been using is the Peters investors law um book is quantifiers language and logic. It was published in 2006 by Yup and there is a first part that is on syntax and also on a lot of different languages and quantifier construction. It was very fun. I remember reading about quantifier construction in uh native languages of the Canadian Arctic of which I know close to nothing. But what they were explicitly trying to do is gather data from languages that do not resemble each other or resemble each other as little as possible because they have different histories and and so on. and they they they do mention some applications uh there. That being said, it's just chapter one with some info. Uh and then the bulk of the work is the logic um language interface, but they do site a lot of relevant literature. Um another piece of literature that might be uh more empirical is the one that I briefly alluded to, the counter examples to conservativity. There is one counter example that is debated a lot. Even there was even though it's not super common to find um papers on generalized quantifiers in linguistic journals as it was maybe 30 years ago at least that's my impression. Uh there was a relatively recent paper discussing the case of only only um has been proposed as a possible counter example to the conservativity universal because arguably only if it is a quantifier corresponds to the superset relation not the subset relation if I say every so every is a subset every dog is friendly take a dog that dog is friendly if I say only dogs are friendly that's the superset individual right so take a friendly individual that individual will be a dog. So maybe some dogs are unfriendly, but definitely taking a friendly individual that's individual's friendly dog. Um, okay. And that's easy to show that it's non-conservative just because the in general conservativity fails there. Um, and people have wondered, okay, what do we do with this? Is this a genuine counter example to conservativity? And so we just throw it away because it has a counter example or we keep conservativity. And they have observed that every or most uh counter examples to conservativity tend to pattern like only. You have counter examples that work in a similar way and are really synonymous with only. And they have also observed that um these counter example in the case of only is especially clear. In other counter example it might be a bit more tricky but they tend to admit syntactic functionings and this would really be either intra the same language or infra different languages. They tend to admit syntactic constructions that um are um unavailable for genuine determiners. Take the sentence um really out of my depth here. Not a linguist, just a just a logician reading some language every now and then. Take the sentence um I saw him on Take the sentence, I saw him on the train. Right? If you take this sentence here, the word only can go essentially everywhere in that sentence. So you can say only I saw him on the train or I only saw him on the train. I saw only him on the train. I saw him only on the train. I saw him you can put it here but you know. >> Yeah. on only the train. I think it works. >> I saw him on the only train. Yes, I can put it here. Okay. So, you can put it almost everywhere in that sentence. Now, find a sentence where you can place every in as many places you can place only. >> Yeah. Yeah. >> That's not going to work. So, they have prop some linguists have proposed that only is not a determiner. It's what they call a polymorphic modifier because it modifies a sentence but in a polymorphic way. So it doesn't modify the sentence always in the same way. And so this is syntactic evidence that only shouldn't be treated as a determiner. So is not a quantifier. So is not a counter example to uh conservativity. So this is the kind of more empirical syntactic literature that that you can find. >> That's cool. Thanks. >> You're welcome. I have exhausted my knowledge of this. So this is perfectly fine. Thanks so much. >> Okay. >> Uh I have a question about the far the the far students could >> No, no, no. I assumed that you had none of that. That seems important for yourself. >> Yes. Uh so so as a relevantist uh this strikes me as not something you want to solve for valuation in aist way. >> Fair enough. >> Especially not given the fact how you justify it that it might fail, >> right? >> Or it might be indeterminate because you said like Yeah. But uh um um if he if there had been for students then Mateo would have interviewed them. Um and so what you're looking for is uh an intentional relation >> between uh the far the being an applicant and specifically Far applicant but but that doesn't matter then and the fact that it's all handled or interviewed by by the good right. >> Yes. And so um so like that's my intuition that you should not have a simple supervaluationist supervisationist solution for it. And my first question would be do you agree or or or like how you settle on this? But um how is is there any literature about more intentional um generalized quantifiers? Um, okay. On intentional and generalized quantifiers, I I I I said yes because I thought you were going to say intentional quantifiers and I said yes, there is generalized quantifiers. No. >> Yeah. Yeah. But that's what I'm very interested in. It seems uh important for to understand language that we get these >> definitely >> more interesting. >> I I think we might be closer in our respective positions than u than it looks because uh in a way I wanted this to be intentional. Okay, let me say this. uh there is a literature on intentional quantifier. I mean not and don't just have in mind linear logic or or these other uh technical literature but for example recent work by Linbo and others on non-instantial generality. So their idea is that there are forms of generality that are not based on the instances. And so here um it's not the case that um this is true because we have our applicants and they're interviewed by Mateo or we have her applicants and they're not interviewed by Mateo but it's to do with the relation between being a foreign applicant and being interviewed by Mateo. And so this is not based on the incidents >> and this is perfectly fine. We wanted to recover that phenomenon by being less radical than Linbo is and possibly less radical than some relevantist approaches are because we wanted to say yeah you can do that and you can view it intentionally where intentions are cached out as you know possible worlds which are each of them uh is an extension. So here you can do that you can imagine that this is an intentional treatment because it's different extensions for the functions corresponding to the quantifiers and the meaning of the quantifier however is given by the intentions quantifying in the relevant model way over all the accessible functions. So is not that there is uh let's say a relation that is irreducible to extensions. It is reducible to extensions but in a quantified manner and the meaning of the quantifier far applicant for example is given by that intention. >> So it's more classical carnapian intentionality than it is relevantist intentionality or uh the potentialist intentionality that Linbo has in mind or or one of these things. So it's still intentional but in a way it's reducible to sets. So is then right that that the thing I interrupted you about during the talk uh whether there actually is a difference between this age and the smaller than relation becomes very important here >> you wouldn't be able to deal >> with just intentional cases if it wasn't merely super evaluationist project it's a modal thing >> definitely absolutely right it has to be a modal supervaluation of semantics Yeah. Yeah. Yeah. That is very interesting. >> I think this is and this is something that my co author uh has developed independently in a paper of his like this way of doing supervaluationism uh in this sort of modal fashion and you can apply it uh to a variety of contexts including uh conditionals modals now generalized quantifiers. Um, and I think that maybe this is the to my view at least is the more natural way of doing super valuation is because you consider the space of resistation as if it were a modern space. So you're much more free to be classical, non-class, classical at certain points, not classical at other points. What matters is that there is a structure on um the space of precifications, a modal structure. The simplest one is reflexive and transitive and it gives you a widely known classical behavior. That answers possibly part of your previous question. Where does the indeterminacy of quantifiers come from? It comes from the structure of the supervaluational space. If you instead impose something more classicalizing like the negation via the routley star convergence on the frame and so on like light does in a system hype uh then you become closer you get closer uh to classical semantics. But really the indeterminacy in the quantifiers comes from the modal space and so it's still an intentional semantics >> right >> but it's a reducible intentional semantic in the end everything boils down to sets which is what we want because we are sort of classically minded if you elinable for example wouldn't want that would say no there is a non-setbased generality the generality has to do with something that has nothing to do with set membership our position is more classical in that respect but you do construct sets of sets of sets And you give the quantifiers an intention in addition to you know what happens in the classical case. >> Good. Thanks >> I should remember that too was very good observation have to has find its way into the motivation. I think >> you have questions anymore. >> No. >> Do I have other questions? Well, um, let me note they are very pressing. Uh, not as pressing as this irrelevancy that shocks me. Uh, well, no, maybe continuing on on on the previous conversation like so there's some intentionality you can handle. >> Yeah. >> You have any idea how much intentionality? I mean is there is there a kind of cases that that clearly call for intentional uh generalized quantifiers >> that you can handle that obviously existing projects can't >> yeah I mean the cases would be exactly these exactly where you have empty restrictures empty restrictures I I think there there is like an entire chapter of the theory of quantification that should be rewritten That's my opinion on the anti-restricture issue. The empty restrictor is treated in in bulk. Either they're all true or they're all false. And there are pragmatic criteria as to why it's uh inappropriate to to utter um you know qualified sentences with empty restrictures. But there I think there are a lot of cases in which you can form all the US inspired counterfactual examples uh involving empty restrictures. So um this would be a case of I think an intentional not the quantifier is intentional but a case in which the interpretation of the quantifier reveals the intentionality. Um suppose that I say every person who walks more than um I don't know 10,000 km in a day walks more than 5,000 km in a day. Now, no one walks more than 10,000 km in a day. But that seems analytically true just because 10k is more than 5k. But if I tell you every person who walks more than 10k walks more than 20k, >> is that true? That's weird, right? I mean, depends. It seems that there is a counterfactual consideration in there. So, if there were to be a person who works more than 10k, right? you go to the single closest world where this is true and it's certainly there you have walked more than 5k but maybe you have not walked more than 20k because you go to the closest one so you have maybe walked uh you know 10k plus uh 1 meter or something then your eastern consideration can you know it's 10k plus 1 meter plus half meter blah blah blah so you maybe do a set of possible but regardless of the specifics this is really a case where you want like an empty restrictor um case like this to behave in two different ways like 10 more than five yes 10 more than 20 probably not and this I think shows that there is an intentionality in there you can capture that here um but you're reducing everything to sets so in a very classical fashion you're constructing these intentionality out of sets uh you might think that this is not intentional enough and absolutely fair enough theory >> it depends if you can get the results that >> you can definitely get this out that one but like it's the question like how I expect that you would get some paradoxes at some point or or very counterintuitive cases uh if you make it complex enough but I don't I cannot justify and that's why I asked the question like how how much can you do with this classical solution >> that I don't know because we haven't explored that far but what I can tell you is that definitely this semantics here which is basically a variant of at least for the model structure not for the specific clauses but the modal structure is a variant of hype. This is well equipped to deal with hyperintentional operators. So if you want to add a cing operator assuming that C is a hyperintentional verb uh then you can add it here. I have no idea how far that goes from quantifier. So I can think of examples of intentionality in quantifiers. I don't know if there are examples of hyperintentionality quantifiers. uh if if there are then you you're living somewhere in the vicinity of this. >> Yeah, that's good. >> But if you have an example of a clear hyperintentionality case, maybe you can cook up one like that. >> Maybe one can cook up one using hyperintentional verbs like see. >> Yeah. Yeah. Yeah. Yeah. Sure. >> Yeah. Hyper intentionality there however wouldn't arise from the qualifier as it does in the case of the empty restrictor for the intentionality. It would be more like the first family of examples where you take ready and indeterminacy trigger presupposition failure category mistake what have you and you put it into the context of a quantified sentence. >> Yeah. >> Yeah. But I also have to probably distinguish that that's a very helpful thing the cases that you said before like where the indeterminacy comes from what you feed to the quantifier or where the indeterminacy comes from the way the quantifier behaves like a >> different project. >> You're right. Yeah, I agree. I agree with that. Yeah, >> questions. Thanks again for coming. >> Thanks to all of you for having me. It was such a pleasure. Thank you.