 Okay, we might get out of the way. Welcome everybody to this week's linguistics department seminar. Today we have Dr Rick Nowen from Utrecht University and Rick's area of specialisation in linguistics is formal semantics and pragmatics and he's mostly worked in the area of interactions, intersections between philosophy of language, cognitive psychology and formal semantics and pragmatics. He has an impressive website, no one.org which I was very impressed to have a look at, a very long list of papers and articles and books that he's published and in addition to that he's also editor of the Journal of Semantics, one of the top class journals in the field of linguistics. So we're very fortunate to have him here today to talk to us about modified numerals. So thank you very much for the invitation and a nice introduction and I'm very happy to be here and I'm very curious what you think of my work. I've just heard you don't see formal semantics that often so I'm curious to see what you think of my work. I've heard that acoustics are slightly weird here because of the, so whenever you can't hear me anymore just raise your hand or, well just raising your hand is probably not very helpful just say louder or something like that. I prefer not to use the microphone but if I have to I'll do it. So this is a talk about modified numerals and what I decided to do is I'm going to give you an overview of sort of, I'm going to give you my view of the past ten years of research into the field of modified numerals and my main goal is to show you what is interesting about them. So because it's of course a very specific topic but I think especially if you're interested in the relation between meaning and form and use of meanings then modified numerals is maybe a nice place to look and I'll show you why. I've also kept the formal semantics to an absolute minimum. If you want more details just let me know and I'll tell you what's under the hood. Okay so here's what this is about. So this is about expressions of quantity. So we've got words for numbers, numerals and we also have expressions for relations between numbers. These are the kind of things that I want to look at. We call them modified numerals. So these are things like more than ten, fewer than ten, at least ten, at most ten. And our intuition is that this somehow relates to these kinds of arithmetic relations. So more than ten means that the number you're talking about is strictly greater than ten. So at first glance it's completely not obvious why you would want to look at the semantics of these things because the semantics seem so incredibly simple. So this just seems to be something about you have ten and you've got everything that's more than ten. Now what we'll see later is that actually if you look at the deeper semantics of these things these things are very mysterious. But before I go on one thing, so I call them modified numerals and I call these guys things like more than and things like at least modifiers but bear in mind that I don't use the term modifier in any technical way. So they might not be modifiers. It's just a handy catch-all phrase for these kind of things. So these guys are probably modifiers. There are some arguments that say that these guys aren't strictly speaking modifiers but I'll call all of these modifiers. So why do we want to study modified numerals? Well, one reason is that the lexical forms that we use to modify numerals come in a great variety. So you see here's the picture from English. You can say more than ten. You also have things like over ten, from ten, at least ten, ten or more, minimally ten. And if you want to talk about upper bounds you have the similar stuff over here. But what is strange is that this variety of ways of expressing things really still expresses these four relations, right? So either an inclusive or an exclusive comparison between numbers. So these simple relations really go with quite a strange variety of expressions. And if you look a bit closer at what these expressions look like you see that there really is no dedicated morphology for these things, right? So what we really do when we modify numerals is we borrow from other domains of the grammar. So we use comparatives or prepositions or superlatives or disjunctions, right? Or we use these kinds of adverbs that you use in other areas too. Now the idea is that modified numerals, you can use modified numerals now to really study what happens when you use lexical material from other domains in this very narrow domain of numerals, right? So what happens, so we know a lot about comparatives. What happens if you take comparative morphology and the meaning of comparative morphology and suddenly start using it with numbers, yeah? And the same with the superlative, the same with prepositions, etc. So this is the plan. So first of all I'm going to zoom in on the lexical form of these things. So why do we see this borrowed vocabulary for modified numeral forms? And in particular I'm going to zoom in on spatial numerals. So things like over 10, up to 10, under 10, from 10, yeah? So the question is why do we use these prepositions here to talk about numerals, right? Prepositions talk about space, right? And what happens if you use a form that normally talks about space to now talk about quantity? And then in the second part I'll say a little bit about the meaning of these things and there will be an interesting connection between these two parts. So what is actually the meaning of something like over 10, or under 100, yeah? And we'll see that the lexical form really matters here, that you can actually find... So for instance the difference between comparative and superlative morphology really matters to the semantics of a modified numeral. That's going to be one of the claims that I'll make. So here's part one, so the form of modified numerals. This took me 10 seconds to Google. I wanted to find a sentence that had psoas in it and that had a modified numeral with over in it and really this was the second hit. Psoas has over 115 postgraduate programs taught on campus. Everybody knows what this means, right? Why is there a spatial preposition here? That's the question now. And we can think about this a little bit more abstractly. What would it mean to compare thinking about numbers to thinking about space? And here's where we should start then. Does anybody know what this is? Nobody knows about this, it's really nice, I like this. I'm always afraid that everybody's going, yeah, that's a number form. This is a number form. The concept of number forms appeared in two publications by Francis Galton in the 19th century and he described them as follows. A number form is that this peculiarity consists in the sudden and automatic appearance of a vivid and invariable form in the mental field of view whenever a numeral is thought of and in which each numeral has its own definite place. Do people get a feeling what this is already? So the idea is that when you think of a number, you see something in your mental view, right? So you see the numbers in a certain shape. So when you think about the number 20, you see them in relation to the nearby number say 10 and 30 but 10 is somewhere here and 30 somewhere there or 10 is somewhere here and 30 somewhere there. You give these numbers a location in space when you think about them. So what Galton did is he interviewed his friends and he asked them, do you have this? Have you got some spatial ideas about the numbers that you think about? And so in these interviews, things like this came up, so this is his friend TM. The representation I carry in my mind of the numerical series is quite distinct to me so much so that I cannot think of any number but I at once see it as it were in its peculiar place in the diagram. So these diagrams come in various forms, right? So that everybody has his own number form, right? Everybody who has a number form has his own or her own number form and some are quite exotic like this one and some are even more exotic like this one for instance where it's not just a two-dimensional shape, it's actually a three-dimensional shape so the person is sort of looking up towards the numbers and then they sort of go a hundred is quite far away and to the right and you see 12 is to the left. What is very common is that there is like very obvious spatial features that the direction changes around 10 and it changes around 100, right? And it might even change around 12 again or something like that, right? So there are some things that are quite common to all of these things but as it becomes clear from this picture, they're also strikingly different. So number forms really is a type of synesthesia. It's quite common though, it's 12%, right? So this is data that is quite recent. Does anybody recognize this? Does anybody have this? So keep on thinking about it. Most people don't realize they have it and next week you might suddenly find your number form. So this is really something that you have. You have it, it's conscious but you might not be aware of it. So it's not subconscious but you might simply not realize that you're doing it. And sometimes you might be doing it only when you think about numbers in a certain context for instance when you think of the days of the month or the days of the week, right? So for most people when they think of the days of the week they think of a circle, right? Or when they think of the year they think of a circle, right? So there's an obvious way in which if you have this you might give different shape to these kinds of number forms in different contexts. Okay. So people who have this, I've said this, can become aware of it. Now what is this? What do we do with this? Well the answer is not very much, right? So what does this suggest? It suggests that there can be a cognitive relation between number and space, yeah? But this relation seems to be incredibly unsystematic, yeah? We can't really say anything about it because for some people a hundred is over there and for other people it's over there. I mean what are we going to study then, right? Okay. So number forms are interesting to start a talk with but that's about it, yeah? So let's look at something that might tell us a little bit more about the relation between space and quantity. This is the snark with a C. A snark stands for Spatial Numerical Association of Response Codes and it was studied extensively in the 90s by Stanislaus Dahana, a French cognitive psychologist. And what he did was this. So he had a fairly simple experiment. So the experiment was like this. You have a computer screen in front of you and in the middle of the computer screen you're going to see a number between 1 and 9. Or 0 and 9, it doesn't really matter. Yeah, and you know that the number will be between 0 and 9, okay? You've got two buttons and all you have to do is decide whether the number is odd or even, yeah? If it's odd you press left, if it's even you press right, yeah? There's another group which has the same task but with the button switched, right? So odd is then right and even is left, yeah? Now what happens is that if you see a 3 in the middle of your screen, 3 is odd, yeah? Then you're quicker if this is your response rule, if you have to press with the left button, yeah? If you see a 7 which is also odd, then you're quicker with the second response rule when you have to press the right button. Now the interpretation of this is as follows. You know you're going to see a number between 0 and 9 and you picture this as a line with the 0 on your left-hand side and with a 9 on your right-hand side, yeah? And so when you see that 7 you have a bias towards the right and when you see that 3 you have a bias towards the left, yeah? Okay. This has been replicated many times but there's something very interesting about it. There's clear evidence that the snark is actually influenced by reading systems, yeah? So this works if you read from left to right. If you read from right to left, you also have a snark. Also you have a similar effect but it's the other way around. If you read from top to bottom you'll have a vertical snark, yeah? So this relation between quantity and space is one that really has to do with how we're used to ordering things in space, right? So when you teach your child how to count from 0 to 10 you buy them one of those toys, maybe one of those puzzles, right? Where you have puzzle pieces and what the child will see is really that the 1 is on the left and the 9 is on the right, right? So that's at least if you read from left to right. So that basically is an effect not really of quantity but it's an effect of ordering, yeah? And that ordering is culturally influenced. Now, so what we've seen is that there is some cognitive interaction between number and space but it doesn't really, it's not very hopeful that we're going to find anything linguistically interesting here. Because we haven't found any linguistic influence yet, right? All we found is that how you use to ordering things influences things and you might have this vague number form, right? That's the way you picture numbers. But so far no link to language. So let's now have a look at language. And then we have to talk about metaphors. And in particular, the only really serious work I know about this is the, I think, famous Lake of Van Johnson book about metaphor. And they propose the metaphor of the heap, not to be confused with the paradox of the heap, which is something completely different. And the metaphor of the heap is a way of thinking about how we talk about quantity. And the idea is as follows. When we talk about quantity, we conceptualize quantity as a pile of stuff. And if there's more stuff, then the pile will become bigger. In particular, it will become higher. So what we're interested in when we talk about stuff, the quantity of stuff, is how high is the pile? That's the metaphor, right? So when we talk about quantity, we talk about a vertical axis. So this is already, you see, there's something different going on here, because at least for somebody like me who's used to reading from left to right, I have this snark effect that is horizontal. So what Lake of Van Johnson say, when you talk about quantity using natural language, you will do this vertically. And here's some evidence of what this looks like. So this is the kind of thing they're looking at. When you're talking about how high your income is, did you hear what I said? How high your income is, right? You say your income rises, right? My income rose, right? So the idea is that it goes up. And so I've already used three metaphors, right? I said how high is it? It goes up. It's rising, right? So when we talk about quantity, we immediately talk about a vertical axis, right? We can say the number of arrows he made is low. He's underage, right? So there's a preposition here that's also vertical, right? His blood pressure is high. He turned the heat down, yeah? So all these expressions basically have to do with... If you simply abstract away of what the sentence says, and you look at the relevant prepositions here and the relevant adjectives here, they really have to do with the vertical axis. And we're so used to using these things that we don't see them anymore. Okay, now modified numerals. Modified numerals, you see exactly the same thing, right? So you can say under 100 pages, you can say up to 100 pages, but you can't say next to 100 pages, or in front of 100 pages, or behind 100 pages. All the expressions that you use to combine with a numeral, if they have a spatial character, that spatial character will have to do something with a vertical axis. So that's the idea. That's the idea that when we look at anything in language that has to do with quantity and space, then we'll see verticality. Now, there's a paper in Linghua that looks at this, especially in Dutch and quite a few other languages, too, by Koffer and Schwartz. And they say the number scale is metaphorically related to verticality. We find vertical prepositions like above, below in prepositional numerals, but not horizontal prepositions like in front of or next. Dynamic prepositions like over and against lose their motion sense and keep only the ordering sense and proximity sense, respectively. The important thing is that the lexical semantics of numerical prepositions is very close to their spatial sources. And this is going to be the intuition that we're going to need for the rest of the talk. So this final sentence says the lexical semantics of these prepositions is very close to their spatial source. So what these authors actually suggest is that when you use a preposition and you combine it with a numeral, so you use something that you really use to talk about space and suddenly you start using it to talk about quantity, really you're not really changing that much to the meaning. You're really still using that spatial meaning even in the quantity domain. So we're not changing the preposition. We're changing our conceptualization of quantity into something that resembles space. So it's really that we're reforming our way of thinking about quantity instead of our way of thinking about prepositions. Just a few notes before I show what this looks like. There are some seemingly counter examples of this verticality hypothesis. But I don't think they're very severe. So in American English you can say things like he's earning north of $50,000. I've been told. In a way this is still vertical, right? Because the north, we think of the north-south axis as something that's vertical. So here's my hypothesis. You won't find a language that will say he's earning east of $50,000. Correct me if I would be very interested to hear if you know of a language that says this. Also you could think about things like he earned between $10,000 and $20,000. Is this vertical? Well, at least it's compatible with a vertical axis. So I'm standing between the floor and the ceiling. Between the floor and the ceiling. The sausage is between the two sides of the bone, the two bits of the bone. You can turn these things into vertical things. Same with a round. Of course a round has a horizontal use, but it also has a vertical use. So that's the main thing. There are these kinds of examples that you see in not many languages, but in some way where you sometimes see that you have things like left and right to talk about these kinds of proximal things. So he earns left right of $50,000. There are quite a few languages that do that. I don't know where to place these because there's no way that that is vertical. So that's usually these proximal things that are quite difficult to put into this vertical hypothesis. So I'm honest. This is not a completely watertight generalization, but it seems to be quite general. So the conclusion here is number is metaphorically conceptualized vertically and this is why spatial prepositions can modify numerals because we can think of quantity in a spatial way. But they can only do this if they are compatible with vertical orientation. So here's the idea then. This is the only slide with formal semantics in it. So the cat is on the table. It's true if and only if. The location of the cat is on the table and there's a formula behind this, you can decide whether this is true or false simply by checking whether the cat is indeed under the table. Now if I have a sentence like this, Chatham has under 300 residents. All I have to do is the table becomes 300 and the cat becomes the number of residents of Chatham, which is 273. Then what you need to do is you need to check whether the cat is under the table and then the sentence. In principle, we don't have to change the semantics of the preposition. The only thing we have to do is we have to find the right representation of quantity. So now for the catch. So what I think this shows is some sort of plasticity of lexical semantics so that you can just take the semantics in one domain and apply it in a different domain as long as that different domain has the same kind of features or at least can be made to have the same kind of features. So the idea is that the domain of quantity and the domain of space have enough in common for you to use locative prepositions in both domains. So at the heart of this some kind of metaphorical mechanism where a formal semantics like me doesn't know what to do with metaphorical mechanisms. So we simply have to assume that these things are somehow in place. We don't really know how to model them correctly. Lekov and Johnson are not formal semantics that needless to say, right? Okay, so here's what this picture looks like, the third bullet point. So the linguistic link between space and number appears to be limited to the verticality of the spatial representation. But then you start to wonder what about other aspects of spatial representation about prepositions. So in particular, one salient feature of prepositions of spatial prepositions is that there's a distinction between locative and directional prepositions or sometimes called dynamic. I call them directional because that's how I call them. So here's that distinction. A locative preposition expresses the location of the subject. This is a Mickey Mouse overview of the difference between locative and prepositions. As many of you probably know, a lot more complicated than this, but I just want to make a main point. So a locative preposition talks about a location. So the cat is on the mat, the cat is behind the house, the cat is on the table. A directional preposition expresses the path that describes the motion of a subject. So a directional preposition is really meant to express motion, not location. And so these kinds of prepositions are incompatible with locative spatial relations. So that's why you can't say something like the cat is to the mat or the cat is up to the house or the car was parked from the house to the church. The only way you can read this is if it's a very long car. So immediately you start thinking of something that is a path, right? So what you need to do, you need to bring in an element of motion like the cat walks to the mat, the cat was carried up to the house, the cat drove from the house to the church. So this is an interesting, quite obvious feature of prepositions. And if we go back to the Corvair and Swartz paper, they actually suggest that this distinction disappears whenever you look at modified numerals. And one of the reasons that they did this was because they looked at something like over in English, like over a hundred. So over has two senses in English. You can fly over a bridge, which is directional and a cloud can hang over something or somebody. And so the second sense really is locative and that's also the real vertical sense of over. The flying over the bridge is actually sort of horizontal, right? So there has to be something under it, but the main direction is horizontal. So they said over is maybe just simply evidence that directionality is nothing to do with modified numerals, right? So if you have this number line, you know with Chatham with 300 and 273, that's clearly just a very static two-dimensional representation. There's no place there you would think for motion. Okay, so here's something we could try. If, this is very strange, this is a garden path I think, if what matters for modified numerals, if all that matters for modified numerals is verticality, if you only need verticality to talk about modified numerals, then up to and under should sort of be similar to each other, right? So up to is sort of the directional version of under in the numerical domain. But it actually appears that this isn't a case. So let's have a look at this. So this is a little bit of reasoning that you can do. So formal semantics do that, right? So Chatham has 273 inhabitants and so Chatham has under 300 inhabitants. That's a valid bit of reasoning. I hope you all agree. But what about this one? Chatham has 273 inhabitants and so Chatham has up to 300 inhabitants. Anybody think that this is good? Okay, so it's all good. You might not think it's totally bad, right? But I hope that most of you will see that there's something going on here. This one is clearly less happy than this one, right? So let's not give it a number or a name. There's something going on here. Here's another contrast that I want to try out on you. When I looked up the exact number of followers I have on Twitter, it turned out I still have under a thousand followers versus when I looked up the exact number of followers I have on Twitter, it turned out I still have up to a thousand followers. Now, this one might be, this one is a bit strange. If you think that this one isn't strange, I hope you also, then I hope that you agree that this one means something different from this one. And we'll have some crisper things later. These examples are teases for you to see that what you can't do once you start thinking about spatial prepositions combining with numerals, you can't just think, I take the spatial preposition, I bash it until it's locative, I take all the motion out of it, I just take the verticality, I also ignore anything else that's going, so locative vertical, that's all I'm interested in. And then I have my semantics that I will apply in a numerical domain. That's probably not going to happen. Okay. So this means that we need to have a, this is my bridge to the semantic part of the talk. This really means that we now need to have a look at the proper semantics of these things. So what do these things really mean? So what we've seen is that up to a certain extent there's plasticity of lexical meaning. So apart from orientation, it might be that also the dynamics of a preposition can interact with numerical scales, but it's not immediately clear what that would be and what that would mean. So the next part is really the semantics. What does it mean? So what could it mean that a spatial preposition has a directional meaning in a numerical domain? Okay. So what is the semantics of more than NAB? So more than 10 students pass the exam. That's it. Well, here is what you would find say 20 years ago. More than NAB is true if and only if the number of A that B exceeds N. That's a very simple semantics. So you look at those entities that have both these properties, you count them, and it has to be higher than N. With at least you do exactly the same, but you have an inclusive relation. So at least NAB is true if and only if the number of A that B is N or higher. And so that means that this is sort of your semantics then for all these expressions. They simply mean these kinds of relations. So they simply talk about either inclusive or exclusive relations to a number and either upwards or downwards. Now here's a prediction. The prediction is something like this, that the meaning of more than 3 is equivalent to the meaning of at least 4. And the meaning of fewer than 10 is equivalent to the meaning of at most 9. This is if you talk about count nouns. If you talk about kilograms or if you talk about average children then this doesn't work. But if we just talk about the number of students that passed the exam then this should work. So we doubted this quite a few years ago but Kurt and me in a paper from 2007 and we did a very simple experiment. This was again one of those reasoning experiments. So we gave people a premise like Beryl had three sherrys and then we asked them whether not all four at the same time we asked subjects whether these sentences followed. So does it follow from Beryl had three sherrys that Beryl had more than two sherrys? And yeah, 100% said yes. Does it follow that she had fewer than five sherrys? 92% yes. Up to this day we really have no clue what happened to those 8% but I propose to ignore it for now. The interesting stuff happens here did Beryl have at most four sherrys and people are very reluctant to accept this. Did Beryl have at least three sherrys? Then you get sort of chance level response. It should be clear that these superlative forms behave completely different from these comparative forms. And you might have the same intuition if you were subject in this simple experiment. So this clearly... it makes you doubt this hypothesis that these superlative and these comparative modified numerals are somehow equivalent semantically because when you use them in an experiment they give completely different results. And you can get intuitions that might inform this too. I have more than one child. That's true if I say it. What happens if I say I have at least two children? So it suggests that... do you get it? I'm a rock star, right? Or a sperm donor. One of the two, right? But I'm not quite capable of deciding how many children I have then, right? And that's quite strange because most of us I would presume know how many children we have. So I have two children, but it would be very strange to say this, right? It would be okay for me to say I have more than one child but it would be strange for me to say that I have at least two children. So here's one way to describe this and this is to use the term ignorance. So comparative modified numerals, so more than, fewer than, less than are compatible with speaker ignorance but superlative modified numerals really require it, yeah? Of course you could say I have more than one child if you don't know how many children you have but if you say I have at least two children you really need to not know how many children you have. You need to be ignorant about these things. Here's another way of showing this. I know exactly how much memory my laptop has and it's more than two gigabytes. That seems to be fine but something like I know exactly how much memory my laptop has and it's more than two gigabytes. Wait a minute, that should be at least, right? Okay, so this should be at least, right? I know exactly how much memory my laptop has and it's at least two gigabytes. It almost sounds like I make you guess how much it is, right? So that I want you to remain ignorant. There's some level of ignorance here. Sorry for the typo. But it's not just ignorance. There are other effects as well. So a triangle has fewer than ten sides is fine but a triangle has at most nine sides. Who finds this acceptable? I should maybe just get rid of all the... Nobody finds this acceptable. So I did this talk in front of an audience of logicians. Everybody found it fine, right? So it really matters in which... what kind of frame of mind you are but I think the intuition here with the second sentence is either you don't know how many sides a triangle has which is strange, right? Or you think, and this is even stranger, that there are different kinds of triangles and the kind with the most number of sides has nine sides, right? Which is even stranger, yeah? Okay. I have an example later where this actually works. I should have had it here but I'll show it to you later. So not everything might be ignorance and here you see it even clearer. So my laptop has at least two gigabytes of memory. I suggest that I don't know how much memory my laptop has but if I'm a salesperson you come into the shop and I tell you the laptops we sell all have at least two gigabytes of memory. Now either you're suggesting that this is a bad salesperson, right? Who doesn't know how much memory there's in the laptops but the more likely reading is this. This shop doesn't sell laptops with fewer than two gigabytes of memory, right? So here there's not necessarily an ignorance stuff going on, right? So here's another one. John has accumulated at least 110 ECTs versus John is required to accumulate at least 120 ECTs. So here you're saying I don't know how many he has but it's more than 110 and here you're not saying I don't know how many he has to have. Now here there's no ignorance. You're basically saying the rule is once you have 120 credits you're fine. So there's no ignorance there. And so what you might observe is there are these operators in this, these quantifiers in these sentences that actually get rid of the ignorance. So this is a very simple sentence, right, with not much going on in it. Here suddenly we have this quantifier and we have a plural. Here nothing special going on. Here we have a modal. A modal is also a quantifier. So the descriptive generalization that comes out of this is the following. Modified numerals of the modifiers, like at least, need to be associated to a range of values. And this will account for the observations that we've made. And this works as follows. So if I say I have at least two children, then there's a range here and that's an epistemic range. So I'm undecided about how many children I have, but I have a range in my head. So within the possibilities that I'm still entertaining, two is the lowest. So in this sentence that's the kind of range that we have. If you have a quantifier, like all laptops have at least two gigabytes of memory, it's actually the quantifier that gives you the range, right? So what happens is you simply line up all the laptops, you look at all the different kinds of, different amounts of memory that these laptops have, and then you conclude that the lowest is two gigabytes, but there's other stuff as well. So there the range is really due to a range of laptops. You must get at least 120 ECTS. Here you have a deontic range. So here you've got a deontic modal, must, and that deontic modal will give you all the options. So the lowest possible option is 120, but there are also other options. So if you get 122 credits, that's also fine, and you also pass. But if you get 119, that's not good enough. So here the range is deontic in HA. So here's the triangle example again. Triangle has at most nine sides. It's strange. Ignore the star. I don't know what it says is star, but it should have, I don't know what it should have, a question mark or a hashtag or whatever. It's somehow unacceptable. So why is this? Well, because we can't really think of a range that would make sense. If we try the epistemic range, that's a bit weird, because you would expect the speaker to know how many sides a triangle has. And also, we can't really think of a range of different kind of triangles with different numbers of sides, because we know that these kinds of figures have a fixed number of sides. So here's the example that does work. So the base of a pyramid has at least three sides, right? So because the base of a pyramid can be a triangle, it can be a square, it can be whatever, right? As long as it's not a line, because then you have a triangle and not a pyramid. Right? So here we have a range. We can put all the kinds of pyramids next to each other, and then the pyramid with the simplest base will be with a base of three, right? And next to it will be a base of four and a base of five. So that will be our range now. Okay, so here's the conclusion about this. This is descriptive generalization. At least it needs to be associated to a range of values, and more than lacks this requirement. So this is the semantics now. So I'm not under the hood, but this is what captures the data. A range for things like at least, and no range for things like more than. Now we have something to grasp hold of, right? So now we can go back at all these different kinds of modified numerals and test whether they are of the range kind or of the non-range kind. And so let's just give them labels. So modified numerals that do not have the requirement are called class A. So those are things like more than. And modified numerals with the requirement. So with these range effects we call class B. And then English looks like this. So we have the comparative here, the superlative here. We've got the locative prepositions here. We've got the directional prepositions here. We've got this junction here. We've got these adverbs here. This one here between is a difficult one. I'm not actually not sure whether this belongs here. So that's some... Here's the Dutch table, and for those of you who don't speak Dutch, this is an exact copy of the English table, right? Which maybe isn't that big of a surprise. But what might be a surprise is that it's so neat, right? So it's, again, the forms that you encounter are comparative, locative, superlative, directional disjunction. And in several other languages that we've tried, we see exactly the same thing in Italian. I'll give you one other example in Japanese. Did you do this? I don't know who to blame for this. But what is interesting here is that this superlative here looks different from the superlative in English, right? I'm right, right? There's no real superlative morphology here. There's a sort of dedicated superlative... I don't know what to call it, even quantifier, I suppose. So what you see really is that whatever a language uses to talk about this superlative possibility or to talk about the comparison, that's what's going to end up in this contrast. And that is going to give you the contrast that you... that I talked about in English. So the hypothesis is really that this is something that is quite stable. We haven't tested this. I mean, we've tested this maybe on 10, 15 languages, but nothing deep. So the hypothesis is that class A will always look like comparatives and locative prepositions, and class B will look like superlatives, directional prepositions, disjunctions, and adverbs of minimality and maximality. What we did test was the directional prepositions. So I had a student look at this, and she took a random sample of 13 languages, and it was very easy. You just ask people, how do you, say, drive up to the church or something like that? Okay, now take that preposition. Can you stick that in front of a number word? And then, if yes, okay, then try the centers with a triangle. You find it acceptable and compare it to something else. It was really seamless. Every interview went exactly the same way. People just found it very easy to find these expressions, to show that they're directional, and to show that they have these effects, these range effects. So what could account for this? So why do we see superlatives, directionals have these range requirements and disjunctions have these range requirements, and comparatives and locatives don't have this? So here's my stab at this. These class B expressions are expressions that for whatever reason, so that reason might be different for every item that you look at, they have a range requirement. So the range requirement is really lexical, but it might be for a different, it might have a different, it might look different for each of these items. So another way of saying that, these are expressions that resist singular values or specific values. So they're anti-specifics, and to use some terminology that you see, all the way. So what it means then, is if it has this requirement in its original domain, as soon as you transpose the meaning onto the numerical domain, that requirement will still be in place, and that will give you the effects that we've seen. And I'm going to very quickly show you how that would work with superlatives, directional prepositions, and disjunction. So this is to show you that when we use a superlative, not in a numerical domain, but in its original adjectival domain, it also has things like a range requirement. It just looks a little bit different. So have a look at the first sentence. The tallest screen of the Netherlands is called Maxima. This is a strange sentence because we only have one queen, and she's not just the tallest queen. She's also the shortest queen. She's the only queen. So this is very strange. Why would you call her the tallest queen? So what it means is that we can't use an adjective in superlative form to talk about things that are only one. So you need a range of things, and then you need to pick out the tallest. So this also goes for these superlative adverbs like maximum or adjectives like maximum. The maximum number of wheels on my car is four. That's just ridiculous, right? Why did you not just say that the number of wheels on your car is four? Why stick in that maximum, right? There's just one number here, and that's four. So you can't really compare a single number to a maximum or a minimum. A minimum or a maximum is something that you apply to a set, a range of numbers, not to a number by itself. That's what goes wrong here. So you can save these kinds of sentences. So the tallest queen the Netherlands ever had is called Beatrix, right? So where you now start talking about a range of queens, right, and then you pick the tallest one, that's fine. Or the maximum number of wheels I can fit on the back seat is four, yeah? So where again you talk about a range of modal possibilities. So here you see immediately that a modal can actually introduce this, right? So we saw that before, too. These modals can introduce ranges. That's exactly what happened. I gave a talk similar to this in Belgium. And people didn't understand this first sentence because everybody thought it was fine. And that's because in Belgium, when a king dies or abdicates and the queen remains alive, she remains queen. So Belgium right now has three queens, which is something I didn't know. So in Belgium, the shortest queen of Belgium is called Fabiola. It's probably a true sentence, and it's a felicitous sentence because Belgium has three queens. And my guess is Fabiola is the shortest one, but I'm not sure. Let's get this in the interest of time. So what I think this shows is that in its original domain, superlatives already have this range requirement, right? And so the only thing that we then need to assume is that this range requirement also holds in the numerical domain, and that gives you the effects that we saw. Now let's see path expressions do something similar. So there are two kinds of path expressions. We have path expressions of space and path expressions of time, and most languages use the same preposition to talk about both. English is actually strange because it distinguishes up to and until. So these things are not compatible with something that is not directional. So Jasper is standing up to the goal line as strange, or Jasper arrived until 10 p.m. is also strange. So you've got this something punctual where you use something path-like, interval-like to talk about that. That's not fine. So what you get is something like this. So what saves it is if you have something like sleep, Jasper slept until 9 a.m. That's fine, and that's because sleeping has a number of consecutive points in time that you're talking about. And this seems to be the... So this is also what you see in the literature. So the requirement really for something like until is that the Jasper verb bit holds for an interval i and it also holds at each sub-interval of i. So this doesn't work for arrive, because arrive is punctual and you can't carve it up into multiple arrive bits. But with sleeping you can do that, and that's why it's felicitous. So we sometimes call this homogeneity, but it should be clear that in principle this is again a kind of range requirement. Or at least a range requirement is at the basis of this. So in order to use something like until, you really have to have a range of things. And there might be some additional things that you also need, like this homogeneity, but to start with you need to have some kind of structure. In an interesting paper by Chris Pignon, he shows that this thinking in temporal semantics about until you can actually show that in the spatial semantics you see similar things. So you might think of something like spatial homogeneity. So you can't say things like he relocated up to Amsterdam. So the only way you can interpret this is we're up to our two prepositions and we're up is basically the direction. So he was so say the Hague is south of Amsterdam and so Amsterdam is upwards from the Hague and so he moved, he relocated upwards to Amsterdam. So in Dutch this is easier because you don't have this simple preposition tot which means up to. The sign points up to the auditorium. So think of a sign that points like this. So if a sign points like this it might be fine but if the sign simply points to the auditorium that seems to be really bad. Or I crossed up to the north side of the past that also seems to be bad. And the requirement really seems to be again that you have homogeneity but what goes wrong here is that these things like crossing and relocating and pointing they aren't homogeneous. Because if you relocate from the Hague to Amsterdam then you're not also relocating to all the towns in between. You're basically taking one point and you're going to the next point and if I point to the auditorium I'm not pointing to the tree that's in the way. I'm really pointing you towards that single thing and the same with crossing, right? So you don't cross to the halfway point you only cross to the final point. Finally here's another range requirement that we see and that's the range requirement of disjunction. The typos I was getting tired here I think and maybe you too. Oh no this actually makes sense. Disjunction has to be part of a range. That does make sense. So there's a lot of work about disjunction and especially about the effects that are described here but they're never really described in any sense as a range requirement but the goal of this slide is to show you that basically this is the same kind of thing. It might be a different mechanism that is at the heart of it but still the effect is that there is a range. So again so let's take this first example John 8 and Apollo Repair the normal way to interpret this is that the speaker doesn't know what John 8. So here again there's an epistemic range and this epistemic range reflects the disjuncts. So the disjuncts form a range and that range respects something epistemic. Similarly and this is the free choice effect of disjunction John may eat an apple or a pear what that means is that John has the permission of eating a pear and he has the permission of eating an apple. And here again what we have is a deontic range so the disjuncts now fall in what is deontically possible. Similarly with the universal quantifier everybody ate an apple or a pear that could mean that you line everybody up and you simply write down what they ate and then you get a list of things and that range is going to be this and this. So again you have some sort of range here and that range then reflects the quantification here. By the way this one and actually this one too you could also read with an epistemic range so if I ask you what did everybody eat so everybody ate the same thing but I forgot what it is and you say everybody ate an apple or a pear but I forgot which one. Similarly here the speaker may be giving permission to eat something but he forgot what it is that people actually have permission to have. What this shows you is as long as there is some kind of range then the sentence is felicitous. But for instance if you know that John ate an apple and he didn't eat a pear then this sentence is in felicitous. Similarly if you know that John is allowed to eat an apple but he's not allowed to eat a pear then this sentence is also not acceptable. So here's the conclusion. Put simply something like up to 10 doesn't correspond to something like this and at least 10 doesn't correspond to something like that. So the semantics and pragmatics of these things are completely non-trivial and that's because the forms that we use to form these modified numerals carry a lot of baggage. They carry certain requirements that they have in the original domain and those requirements still hold in the numerical domain. So this is at this point. So the semantic profile of these things really applies across domains. And so the reason why you see this especially in the domain of quantity is because the domain of quantity is nice and easy to mold. We have this metaphor that we can use but also it's scalar so whenever we have something like space which is also scalar which also works with orderings and stuff like that we can easily transpose it to quantity. There's one other thing that I haven't talked about and that is that this is sort of something I'm quite enthusiastic about always a decomposition or analysis pays off. So when you do semantics you shouldn't look at at least 10 and then just look at them to the same strength. And what we found is that this is a superlative modified numeral. Implicatures are very good. People do a lot. The comparative modified numerals are significantly less often and people find them less good. But they're not bad. So I can't explain you the whole experiment but you see that this is a significant difference but what you would actually expect maybe is that this is all the way to multiply. So what we found is that actually comparative modified numerals behave almost the same as comparative modified numerals with respect to the scalar of images. So this is actually in the sense it's not a very surprising finding but the next step would really be now to look at these free choice styles in princesses and for instance the ignorance in princesses to see whether they're the effects are much greater. Now one reason why this is interesting is because some free choice, I mean there are theories of free choice of images that basically almost reduce that couch them into mechanisms that are used for scalar of images. So you can use this domain as a vehicle to study different kinds of images. Actually that was my answer. You showed us that the superlative requires a range with the Dutch coin example but the comparative counter part of this sentence is as we understand but you want to say the comparative doesn't require a range. Is that correct? So what you are saying is that the comparative does not require a range? That's a very good question. You can't compare them here because the comparative compares things. So you need something to compare it to. Does that mean that the comparatives require a certain kind of range? They need two values but the second value doesn't need to be part of a range. So here you also compare this is really sort of entrenched in the semantics of these things. So here you're also comparing the height of the queen to the height of others. But the question is now what are these others? And so here you really need a range of things. So if you look at these kind of we have a very good example. So here we've got two numbers. We've got the number of children and we've got the number one and that's what you're comparing. And that's enough for the comparative to have two values. For the superlative that's not enough. You really need to assume that the thing you're comparing this to is itself a range. So what you might actually end up saying is that this example that I have with the queen so you might be right that this can't tell me anything because it can't give you a similar example with the comparative that would be fine here. But I kind of feel tricked here because the height of the queen is fixed in this case. And in the child example if this carries over to the child example the number of the kids I have is fixed. And the range should be in the number rather than in the number of children I have. The height of the queen is like the number in the multi-numeral. Oh that's normal. So that's not something that's... Do you see what I mean? And you're comparing to that to a state of affairs and that would be say a range of queens and their heights. That's the comparison class in your semantics of the superlative. But I really have to think hard about so your challenges now give me a comparative sentence that I could stick next to this that would show that this behavior is different. But you might be right that this is quite hard Yeah, so I have to think about this the only thing I can say is that of course there is no simple way of comparing this immediately to the numeral domain because in the numeral domain one of the things is really both the number and the thing you're comparing it to are really given in the sentence. Here the comparison class is sort of implicit but it's not something... There's also a uniqueness difference. Queens are unique whereas children... Yeah. Right, but that's... Exactly, yeah. So I think what you just just pointed out is what if I have the taller queen of the families that it's called the maximum? Yeah, but the comparison doesn't have a range of requirements so why is that? Yeah, but then I don't have my minimal pair. Can I ask you a quick question? From a topological perspective it would be interesting to know what you're talking about is extensive. The study that you mentioned all of the languages there what are they in the European spoken in West Africa or South East Asia and see if you can replicate them. Yeah. Yeah, absolutely. It would be very nice. And you can also think about so what happens if you don't really have a kind of comparative apology that's in English, preposition systems might be slightly different in some ways. That's definitely something that we want to do. I'm on a European grant and it's very hard for us to go through so I need sort of the requirements I know, but so I really need to find my... so I need help finding good info. So if you have suggestions where to look? We'll take any bus in one minute. I actually I had a colleague who did this and he just spoke to a group of people who said really cool that's your mother tongue. Oh, okay. I'm also curious there's lots of Isomophism between temporal and spatial domains. You're saying that they're actually very different in the sense that you don't get the temporal... You don't get it in English, almost exactly. No, so they're not different. Oh, it's just English, but weird. In English I say until which is very strange. I saw it though, yeah. But in all the languages that we looked at are like we can say the temporal domain. Actually the temporal domain is interesting because there metaphors might be slightly different so you can really take forward the vertical domain. So there's the spatial metaphors for time for the spatial so that brings an extra I have to really look at those. So most of the when people tell me about counter examples of verticality, they usually shift to temporal uses of the top and we said that it's problematic or maybe a challenge still for formal semantics to talk about this metaphoric transfer from location space to temporal space to this. So in your analysis, how do you present the positions? Is there a way that the parallelism comes out or are you just assuming it's a constant and that's what it is? So the honest answer is that this metaphorical story accounts for why you can use one meaning in a different domain but what most formal semantics will then do is they still give you two different meanings, one that can apply to numbers and one that can apply to spaces and then it just points out that if you look at them from a metaphorical point of view, they're actually the same meaning. So what I tried pointing out is that it's just not something we do in formal semantics and maybe we should do it but we don't so we simply think that so the semantics of up to in the numerical domain if I were to write it down for most students I would simply still use a comparison that's involved to compare numbers but the understanding is that at the heart of this is something that is spatial but why should we use vector space semantics to talk about the relation between two numbers if we can very simply write it down as B is greater than 2 so it also has to do with a level of complexity so this is how it goes if we wanted to we could use the semantics our spatial semantics to talk about these things but in reality it would be easier just to stick with our numerical relations that we know in the understanding that this has come about by this metaphorical bridge it would be interesting to see if anything happens if we really apply the full spatial semantics of these things I mean because spatial semantics isn't just orientation and direction there's often a lot more going on than in a high perspective it's also like just that the medical operator is greater than us so they don't know what we need to compare them what do they know I mean the operator is like 2 plus 2 equals 4 but that's nothing in this needs that I suppose we need to write things down but write a semantics down but I don't think you need any math language as long as you have something that looks like a comparative it doesn't have to be comparative that does the job of a comparative I would think could you go back to the side of the barrel head for you to share this the question you're asking can you explain if I could no no this one's surprising that should be a wonder so these two are so low the last one for you to share this I can't understand how you don't understand why you would so you would say yes to this I don't understand this do you have to be strategic so this one this room will really split in half some people will find it completely off so this suggests in the other half they say no so why why this splits because if you look at the proposal this should be 0% so actually so from my proposal of semantics I would be able to account for why you have this intuition because your intuition according to my semantics should be this says that she might have had four shares but the premise says she didn't right we changed this later to exactly three shares and the results are exactly the same yeah so we thought so we thought that this was because better than three shares maybe read in a brief way so three or more right so we then changed it and we chinsed into better than exactly three shares and I think the percentage just went down a little bit but no one nears you know I think it might be that people were unclear as to whether they are asking a question is it true as opposed to I would say it so for me actually the 92% I think I would be with the eight in the sense that well you know she had fewer than four but not really fewer than five wow you can inflate this is supposed not to exist but that's fine I completely buy that and you might be completely right the only thing we use this for back then and I would like to use it now too is that there is a clear contrast between these responses and these responses so the conclusion is we have to do something and this could be very much a pragmatic thing so that's why this could be chance level not because people think that this is false but because you have sure she had at least three shares but I would never say that and then of course she would get half of it half of it going either way but this has been replicated many times by the way with different variations here there's a paper by William Cuttsos and Chris Cummins where they try to do this a little bit more professionally and they show that I can't remember but they really show that this is a pragmatic effect and so they use acceptance scales and they train, I think they train I might be telling a lie here I think they train people to use the middle value for things that are not and they use the bottom value things that are false and they'll ask me how can you train people to do that but what they see is that people with these kind of things they go for I suggest we actually carry on an experiment with Rick had three shares so join me in thanking Rick for a very interesting talk