 Rwy'n fawr i ddweud y bydd yng ngoswm yn adrodd yn y ddweud. Fel ydych chi'n cerdd. Felly dywed i'r rhaid i'r hwn yn yma i'r lleffordd yma i'r ddweud yma i'r cyllid yma i'r rhaid i'r cyllid yn yr yddwynt. Rwy'n ymgyrch yn yma i'r cyllid yn yma i'r ysgolod ac yn y dyma i'r cyllid yn y ddweud, i'r ddweud i'r cyllid yn y ddweud i'r cyllid, i'r cyllid yn yr ydwym, Ac here he is presenting some joint work which is joined in a telephone board with details and the parallels and differences between fact and least and more. So yeah, thanks a lot. OK. Thank you. I hope you can hear me. OK. So this is joint work with Marie Christine Meyer. We are not related. I'm not quite sure how much semantics background there is here, so I was hoping to make things a bit simpler for this presentation, but due to the fact that I was ill for two weeks, I was not really able to do that. So that means I will try to go slowly and probably not go through the whole talk actually, so that at least I can what I present can get across. But it also means that if you have any questions of understanding in between that you feel are necessary, that you need an answer to necessarily please interrupt me, right? Okay, so I will talk about at least and more than, and specifically at least and more than as modifiers of numerals, so at least three, more than three and so on. Because as some of you might know, there has been a quite extensive debate about these modified numerals in over the last, well, almost ten years and pragmatics. If you look at a sentence like John had at least, has at least four kids, then Manfred Kriffger, and I think this is sort of basically the earliest reference here, has noticed that if we think of scalar implications to be derived by negating relevant alternatives to a given assertion, and then we might expect that John has at least four kids would come out as meaning that John has exactly four kids. Because a relevant alternative to John has at least four kids would be John has at least five kids. And John has at least five kids is a stronger statement that you could make. It entails that John has at least four kids. Whenever John has at least five kids, he must also have at least four, we would think. However, we do not find this, right? This inference is absent from that sentence. John has at least four kids does not have the scalar implication that John has exactly four kids, right? However, there are other types of inferences to come with at least. So if I say I have at least two kids, it's a bit odd. It suggests that I don't really know how many kids I have. Similarly, when I say a hexagon has at most eight sides, it has six. It's also strange. And why is it strange? Well, intuitively speaking. The first part seems to convey that I don't know how many sides a hexagon has, and then I'm stating, well, but I actually do know because it has exactly six. And that's quite contradictory in a way to say something like this. And similarly, if I say, trust me, I know how big this apartment is. It has at least 40 square meters. There might be context where this is fine, but out of the blue this doesn't seem very coherent statement. And again, if I ask you to trust me, because I know exactly how big this apartment is, and then I say, well, it's at least 40, then there's again something contradictory going on here. And this fact has been taken to suggest that at least comes with an obligatory uncertainty inference that the speaker is not certain what exactly is the case. Now, let's come to the other. So there are two things, right? At least does not have a scalar implicatures, but it has uncertainty inferences or sometimes called uncertainty implicatures. Let's look to the other side of the debate. More than, similarly to at least seems to not carry a scalar implicatures. Namely, when I say John has more than four kids, under the same reasoning as before, John has more than five kids would be a stronger possible statement that I could make whenever John has more than five kids. He also has more than four. Thereby, by Griesian reasoning, you might expect that given that I've said John has more than four kids. It's not true that John has more than five kids there, which means that he has exactly four kids because he has more than four but not more than five. He must have exactly four. Again, this is not what the sentence means, right? So we would think that both types of modifiers of numerals, the at least modifiers, the more than modifiers, do not give rise to scalar implicatures and they should be... So the next question to ask is, does more than similarly to at least come with obligatory uncertainty inferences? Because if that's the case, then we would think, well, they're pretty much the same, right? I mean, they mean something else but the general analysis should be the same. The problem is, this doesn't seem to be the case. If I say, I have more than two kids that's a perfectly coherent statement in certain context, right? Where a hexagon has fewer than eight sides, it has six. There is no oddness about this. Compared to a hexagon has at least... has at most eight sides, it has six, which was strange, but this here is fine. Similarly, if I say, trust me, I know how big this apartment is, it has more than 40 square metres. Perfectly fine. So more than does not seem to come with obligatory uncertainty inferences, whereas at least seems to do. So the general conclusion that is drawn in the literature as far as we are aware is that superliff modifiers, like at least, I call them superliff because they seem to have a superliff morphology built in, come with an obligatory uncertainty component, whereas comparative modifiers do not. They don't have this uncertainty component. Now, that mean, and moreover, that is taken to mean that more than, even though it does not have scalar implicatures, it should be analysed differently from at least in the sense that more than has a simple semantics probably along the lines that all of us here would expect. However, superliff modifiers are a real conundrum because this uncertainty inference must be derived from something. Since it's not part of more than, it shouldn't be derived by, let's say, general pragmatic mechanisms. It should be part of the semantics. That's the usual conclusion. Part of the semantics means, some people say it's part of a modal component in there, or some people say we should go to completely different compositional semantics for superliff modifiers and things like that. Our claim is that this is not a complete picture. Namely, when we consider utterances of more than in specific contexts, uncertainty does seem to arise. Namely, when I say how many kids do you have, or you ask me how many kids do you have, and I answer I have more than three kids, somehow strange again. It seems like either I'm trying to mislead you somehow, or be evasive, or it seems to suggest that I don't know how many kids I have. So there is the uncertainty. That's context of type one. A context of type two where uncertainty does not arise, we claim would be one like this. Imagine you're at some office and the clerk there asks you, I need to double check if you qualify for these benefits. Do you have three kids? And then I say yes, I have more than three kids. That's perfectly fine. And here there is no uncertainty. I can say yes, I have more than three kids, because the only thing that's relevant is what I have three or less than three actually. In other words, we claim that in those types of contexts, like context two, that's the types of context where uncertainty does not arise with more than. In contexts like type one, uncertainty will arise, contrary to what has been claimed in literature. Now let's see this more clearly. Imagine, let's focus in on context one. Someone asks, what is the distance between Ramallah and Jerusalem? And then A answers, it's more than 10 kilometers. That seems as a chest that the speaker is not quite sure how many kilometers it is, but it's definitely more than 10, 11, 12, 15, whatever. Similarly, in that type of context when I say it's at least 10 kilometers, again the speaker is not sure what it is, but the lower bound that the speaker considers possible is 10. So again, in this type of context, both at least and more than have uncertainty, we think therefore that uncertainty should be derived in a general way as a pragmatic inference. And once this has been done, we will come back to contexts of type two, where more than does not deliver uncertainty. So pearlive modifiers never give rise to scalar implicatures. They always give rise to uncertainty, and comparative modifiers never give rise to scalar implicatures, but they sometimes give rise to uncertainty. What we will try to do is argue for an implicature-based account of these inferences, basically along the lines that I've independently suggested, and also Bernhard Schwarz has suggested, and earlier, though very different in a very different way, Daniel Buehring has suggested. However, they always remain certain problems, which we will attempt to solve here. And the idea is that the absence of the scalar implicature inference, both with at least and more than, is tightly connected to the presence or absence of uncertainty inferences. That is, the uncertainty implicature arises when a scalar implicature does not arise. So they're kind of in complementary distribution. It's not quite right at the moment. And that means, and to derive this, we will argue for a very specific type of semantic pragmatic interface, one of these with an interesting division of labour. And in order to be able to do so, to make this work, we will show that, we will argue that the notion of alternative, that is, alternative utterances that a speaker might make, has to be revisited and has to be made very precise. The actual reason for this is, as you will later see, that this part of the, this is only one part of the at least more than problem. There is a whole other problem which has to do with the co-occurrence of those modified numerals with modals, which is a very confusing empirical field. So we will first focus on modified numerals without any modals inside. And then we will bring them in these models. Okay. So let's consider, we will first, what the theoretical framework is that we will be assuming. I try to be very, well, not too technical, let's see. So we will say that this will later on be modified, but we will say that at least and more than, have a very simple semantics. This very simple semantics can get us a long way, we will find out, but not all the way. In particular, at least, a statement like John has at least three kids means there is a potentially plural individual, right? Which has the cardinality of three or higher that John has. And it's kids, right? And similar, more than means, John has more than three kids, is that there is a potentially plural individual that is larger than three, whose cardinality is larger than three, sorry. And those are kids, and John has those kids, and they are kids of John's in other words. And John has three kids, simply means that there is a, that there is a maximal unique, there's a unique individual with the cardinality of exactly three, such that they are kids of John. So in other words, John has exactly three kids. So we will start with this, and in the background, I haven't said anything about at most and fewer than, right? Which are the negative versions of at least and more than. For the moment, we'll assume they are simply negations of that. We'll turn out that that's not quite right. But for the moment, that's fine. Okay, now, there has been a big debate about how the semantics-pragmatics division should be framed in recent years. We will make an argument here that scalar implicatures are derived in a semantics, whereas the pragmatics derives uncertainty implicatures, uncertainty inferences, which are weaker than scalar implicatures we will see. In particular, when you say, let's make this precise by considering disjunction. Disjunction is like one of these cases that's very often discussed. When I say something like A or B, John drinks wine or John drinks beer. The basic meaning of such a disjunction is that of an inclusive disjunction. It doesn't say that John doesn't drink both. It could be that he drinks both. It just says that one of these is true, at least. Now, we basically assume that there are two possible syntactic structures available for such a sentence. Namely, the plain one over here, which will give you, in the semantics, this inclusive meaning that I just talked about, plus one where you have an operator on top here. This is an exhaustive operator, which is similar in semantics to a word only, in the sense that it excludes certain or negates certain alternatives. In particular, the semantics of the exhaustification of A or B relative to a set of alternatives is the assertion of A or B, that must be true because I'm saying it, plus the negation of all innocently excludable alternatives to A or B. Now, what are the innocently excludable alternatives? It's fairly easy to see. I will show you this in a second. An alternative can be negated if it is in the intersection of all those maximal sets, alpha one, alpha two, bla bla bla, such that when I negate all of them, alpha one, alpha two, alpha three, and conjoin them with the assertion phi, I do not get a contradiction. What does it mean? Consider our sentence John or Mary called a disjunction, right? And we have an exhaustive operator on top, which is similar to only. It's supposed to say something like, it's only the case that John or Mary called. The alternatives to the sentence from now are John called, Mary called, or John and Mary called, right? The simple disjuncts plus the conjunction. All of them are logically stronger than the disjunction. If John called, it must be the case that John or Mary called. If John and Mary called it, of course it must be the case that one of them called. Now, since we're trying to see what the exhaustification of that sentence is, we have to ask ourselves what are the maximally excludable alternatives? Well, there's two ways you could go. Since we're saying that John or Mary called, we could negate John called. We could negate John and Mary called. What would be the outcome? Well, if I'm saying that John or Mary called, but it's not the case that John called, then it must be the case that Mary called, right? Of the negation of this. Adding to this, the negation of Mary called would be a contradiction. I can't say John or Mary called, but John didn't call and Mary didn't call, right? So this is the sense of a maximally excludable set of alternatives. It's that set when, as soon as you add one more alternative and negate it, you get a contradiction, right? So this is maximal. This is also maximal, right? So John or Mary called, but Mary didn't call, and it's not the case that John or Mary called. That would end up being saying that John called. Again, you can't add the negation of Mary called to this. And now, the innocently excludable alternative turns out to be just John and Mary called. Because it's in both sets. It's the only alternative that is in both sets. And the intuition behind this is that it is the only alternative such that when you negate it and assert John or Mary called, it does not force the truth of another alternative. So, right. Let's think once more. If I say John or Mary called and I negate John called, then I force the truth of Mary called. Vice versa. If I say John or Mary called and negate Mary called, then I force the truth of John called, right? So the negation of one forces inclusion of another one. With John or Mary called, however, that's not the case. So therefore it's innocently excludable. That's why it's in both sets essentially, right? This is just a way to make this intuition precise formally. And the resulting meaning is just the exclusive disjunction. John or Mary called, but not both. So this is what the semantics does. Let's assume this for a moment, right? It negates those alternatives that it can negate. That means that after we've exhaustified, we get the exclusive meaning for or. If we don't exhaustify, we stick with the inclusive meaning in the semantics. And then both of them could be sent to the pragmatics. I mean, just one is going to be sent to the pragmatics because usually we don't other two things twice. But one of them, right? And then the pragmatics does whatever pragmatics is assumed to do. Following basically Gries, we assume a co-operative speaker, right? Let's quickly remind ourselves what that essentially means. It says that in a given context and to sentences Phi and Psi, the speaker should choose Psi over Phi as an assertion if the following holds. Phi and Psi are both relevant in the context. Psi asymmetrically entails Phi, so it's a stronger statement, more informative. Psi is an alternative. It's not just any type of sentence, right? And the speaker is more over certain that both Phi and Psi are true. In that case, you should choose Psi. If you nevertheless choose Phi, then we as heroes draw the conclusion that the speaker is not certain that Psi holds. Which is this. And that's essentially the ignorance, the uncertainty inference, right? If you're not certain what Psi holds, then you were... Well, you answered. Okay, so uncertainty inferences are derived in the pragmatics. Scale implicatures are derived in the semantics. And now we will show what this does for at least more than. Let's come back to our actual puzzle. We must make one more thing precise. I've already... You remember when I said John or Mary... Was it smokes or something, right? We said that the alternatives to that sentence are not just that John and Mary smokes, but also the simple disjuncts, John smokes and Mary smokes. And how do these disjuncts come in there, right? A simple grize or basically what Larry Horne suggested was that or and and form scale alternatives. But it has become clear through the work of Saul and other people and then Katzir and Fox and Katzir that we also want the disjuncts in there for more complex disjunctions. And so we need to restate the alternative... How should I say it? The algorithm that derives alternatives. And the way we do this is by saying that the alternatives of a sentence phi are those sentences psi which can be derived from phi by these two ways. Namely, by substitution of a node in phi with an element from the lexicon or by substitution of a node in phi with a subconstituent of phi itself. What does it mean? I've shown this here for at least, right? Consider at least two. This is the node V here. The first case where I substitute with an element from the lexicon, I could for instance go and substitute two with three, thereby deriving alternatives at least three. But I could also go and substitute the whole node V with something from the lexicon, V4, right? But there's also something else allowed. I could take this node, the sub node and substitute it for the whole thing which would be equivalent in that case to taking something from the lexicon out I have the liberty to take something from the structure, so to speak. And this is what we saw with this junction, right? When I had John or Mary's smokes, I took one disjunct and replaced it for the whole thing. There by deriving John's smokes. And John's smokes is not a lexical item. So I have to take this from them. This is the way why we assume that syntax gives us alternatives, right? Okay, and now we're basically there. This means that for at least three we get alternatives of the form at least four, at least five, but also exactly three, exactly four, exactly five. And similarly for more than, we actually get more alternatives, but these are the ones that are relevant, right? More than four, more than five, exactly four, exactly five and so on. And now, remember, we want to do two things, well actually three, but first we want to do two things. We want to first explain why scalar implicators are absent with at least and more than. Then we want to derive why uncertainty inferences are there with both. And then later on we want to show that uncertainty inferences are not always there with more than, and why that is the case. So in order to show why scalar implicators are not there, we have to consider the exhaustification of a sentence like John had at least five beers. Why? Because we said that the scalar implicators are derived in semantics, right? So what does it mean? So he gets complicated. Is everyone with me so far or not? I don't know, I'm not seeing people. You can ask questions, right? Remember. So this is one of the most crucial points. Once you've seen the logic, I think it's fairly intuitive. The alternatives we've just shown, the alternatives for John under these assumptions, right? For John, had at least five beers, are things like John had at least six beers, John had at least seven beers, John had exactly five beers, he had exactly six beers and things like that, right? Which is also intuitive in a way, but. Now we have to see which of these alternatives can be negated by our only like operator, the exhaustive operator. And the way we do this is by seeing what are the maximally consistent alternatives that we can negate and list all these sets. It turns out, this is an infinite set of, there's an infinite number of such sets. When I'm saying John had at least five beers, I could for instance negate John had at least six beers. I could negate exactly six, I could negate at least seven and so on. What would be the outcome? It would mean because of the assertion that John had at least five beers and I'm negating that he has at least six that he had exactly five, right? This is the implication that we started out with and said this sentence does not have this implication. Now notice, from this set, we've left out one crucial alternative, namely the alternative that John had exactly five beers. And in fact, we couldn't add it here because I've just told you, right, with this set, we get the meaning that John had exactly five beers. If we added it, we would be negating it and we would derive a contradiction, so we can't add it. But what we can do is we can start another set and take out John had at least six beers and John had exactly six beers. What we are now saying, what we now arrive at is that John had at least five, but he hadn't exactly five. He hadn't at least seven, he didn't have at least seven and so on. In other words, he had exactly six, right? Let's go through one more step. Now, I could take out the at least sevens alternative and the exactly seven alternative and put back in the six alternatives, right? What I would be saying then is that John had at least five beers, he didn't have exactly five, he didn't have at least six, he didn't have exactly six, but he didn't have at least eight either. That means he had exactly seven and then I go on like this, right? Now, which alternative is in all of these sets? Well, none. There is none. This is the solution to the puzzle, right? If you don't do it this way, you can still derive the facts, but for this simple case, but we will not derive it in other cases. That's a problem, that's why we go through this complicated solution here, which has been argued to be accurate in other ways. That means, okay, so I should state what the outcome is more clearly. There is no, these sets do not share a member, so that means the exhaustive operator or only, the covert only, let's call it, cannot negate anything. That means exhaustification is vacuous, there is no scalar implicature, and John had at least five beers just means what the lexical semantics gives us. That means that John had five beers or more, right? So, first step is derived. The same outcome is derived for John had more than five beers, right? We would just be replacing this non-strict comparison with a strict comparison that lifts everything one step up, but it's exactly the same. Now, we want to derive, on the other hand, that when scalar implicatures are not there, that we get uncertainty inferences. And remember, we said these are derived in the pragmatics. So, we've seen that John, the exhaustification of John had at least five beers means he had five or more. Now, since we're operating in the assumption that the speaker is cooperative, there is now a, you as a hero of that sentence, will ask yourself, why didn't the speaker say that John had exactly five beers? Why didn't he say that John had at least six beers or exactly six? These are all more informative, as we've determined, right? Since he didn't say that, the speaker must be uncertain about them. That means he doesn't believe that this is true. He doesn't believe that this is true. And we have the uncertainty inference. So, the outcome is, after the pragmatics, that John had at least five beers means John had five or more. That's what the speaker is certain about. But he's not certain what's the number above five beers from five on such that John has had them beer. It could be six, could be seven, five, whatever. And similarly, for more than, right? Exactly the same conclusion. So, this is good. Given the way we define the alternatives, the exhaustification of an unembedded superliff modified at means without any modals or anything is vacuous. Given the constant set of alternatives as we've defined it, unembedded superliff modifiers will, however, give rise to uncertainty implications. And obligatorily so, if we don't say anything further. And even once we've said anything further. The question is, remember, now we have to come back to our old question. Why do comparative modifiers not give rise to uncertainty in all cases? Remember. So, let's recapitulate. Yes, no, you're right. But I was actually looking for another word. Let's remind ourselves of the facts that we started out with. I wanted to say, recapitulate. And then I thought that's actually not the right word. So, in a context where someone asks, how much is it? How much does it cost, right? The address he answers, it's more than 10. It costs more than 10. It costs at least 10. We've seen it both more than and at least have uncertainty. This is what we've derived. That's good. This is what no one else has derived in their account. Because everyone else tries to have more than at least strictly separate. What we have not derived is why in a context where someone says, how much is it? Is it 10? And someone answers, it's more than 10. Why there's no uncertainty? Because at the moment we predict uncertainty all over the place. And this is good for at least because there's still uncertainty, but not with more than. So, what is, I will give you the intuition. I think there's, I mean this is a very, well, it's not an easy problem. Again, it has to do with the alternatives that are relevant. In a context where someone says, how much is it? Is it 10? We basically want to say that the only alternatives that are actually relevant are the ones that you see there. 10 more than 10. More than 10 because it's in the assertion of it's more than 10. 10 also because it's in the context, right? Someone asks, is it 10? That is, while we so far in our computations assume this huge set of structural alternatives that you generate, in particular context you act, of course, restrict these alternatives, right? We don't, we don't, we don't, when I ask you, is it 10, you don't bother thinking about a million, right? Or something like that, usually. And now the claim is, or what we want to say is that this restriction of the set of relevant alternatives is actually licensed with the, when you're altering more than 10, but not in the case of at least 10. I'm the reason, now we have to ask why, right? There has been, again, this is formalized again here in a maybe not so clear way at the beginning, but the intuition behind it, I think they're fairly clear and they have been around. So the first part is found in literature all over the place, the second part we added, but we will give more, we will defend the second condition for independent reasons. So there is a conditional alternative pruning. That's basically a condition that says when you can restrict the set of all alternatives to a set of, a potential set of restricted alternatives, right? And it says that when you have a structure or a sentence by, the alternatives of phi that you derive automatically in the way that we talked about, they can be pruned to a subset of that set, only if one and two hold. First, there is no distinct alternative psi that is in the original set, such that the exhaustification of phi with respect to the new set is identical to psi. That means the intuition is complicated stated, but the intuition is don't do via scalar implacature what you could do via an overt assertion of some alternative, right? Essentially, that's what it says. And that intuition is around all over the place. It's also found in Pialet literature. The second intuition that we want to add here is that all the alternatives in the original set that are part of the sentence that we utter, so to speak, they must be in the smallest set. So when I say it's more than 10, an alternative life, like it's exactly 10, must still remain, because I have other more than 10, and so I've thereby made relevant, basically, it's exactly 10. It's an alternative. But you can get rid of other alternatives, right? So what does this do? Oh, why the second? So, okay, this first part has been argued for the second part. I think we need four independent reasons anyway. Let's make a step back once more to this junction. If I say John O' Mary Cole, and then I say namely John, that's very strange, right? Why utter this junction and then utter something like that? Well, it should be good if we could restrict our set of alternatives like John Cole, Mary Cole, John O' Mary Cole to simply John O' Mary Cole. That's basically the old horn alternative. Namely, when I say John O' Mary Cole and then I exhaustify and get not John O' Mary Cole, why not add namely John? There's nothing that prohibits that, right? That means, in other words, that the uncertainty inference that we found with or seems to be as obligatory as the uncertainty inference with at least. And that's for a particular reason, right? Namely, for the fact that you can't get rid of alternatives that are sort of given to you by the very fact that you are asserting at this junction and that would be an alternative like John Cole and also an alternative like Mary Cole. So let's see what this does for more than 10 and at least 10. Remember, the relevant context or context of type 2 where someone asks, how much is it? Is it 10? And then you answer, it's more than 10. We assume this exhaustification here. We know that the alternatives are more than 10, more than 11, more than 12, exactly 10, exactly 11 and so on. And we ask ourselves, can we restrict this set of alternatives to that smaller one where there's just 10 and more than 10 in it? Well, why not? The only alternative that I actually, from this set, that I could negate is 10, right? Because more than 10 is asserted, so I can't negate it. So let's do that. That means it's more than 10, but it's not exactly 10. Well, that's already entailed by the assertion that there's more than 10. So you get back, it's more than 10. Exhaustification, in other words, is vacuous with respect to that smaller set. Now, that means that nothing here is violated in our conditional pruning. First, since we get back to the meaning that we had without exhaustification, the resulting meaning is not equivalent to some other distinct alternative in the set. It's more than 10 is not equivalent to it's more than 11, or it's exactly 10 or exactly 11. So that's good, the first part. The second part, or the second sub-condition that is not violated is that in this smaller set we have all the alternatives in there that we can derive from it's more than 10 by simply replacing more than 10 with sub-constitutions. We can go and replace more than 10 with 10. So that part is also not violated. Okay, in other words, cut a long story short, we are allowed to restrict a set of alternatives thereby for more than 10, thereby not derive a scalar implicature, which is as we want, but in addition also not derive an uncertainty inference. Essentially you do not derive the uncertainty inference here because there are no relevant alternatives left, basically. That's the idea. Now, consider at least. Again in the context where someone says, how much is it? Is it 10? And then you answer, it's at least 10. The intuition is first of all, it's maybe not so good this answer. Second, the uncertainty inference doesn't go away, right? Very clearly. So we have to ask ourselves again, can we restrict this set of alternatives to the smaller set as with more than 10? Well again, with it, we could, but now we get a problem, namely, the only alternative that we can negate by accessification is 10, because at least 10 is other, so I can't negate it would be a contradiction. If I say, if however I say at least 10, but not exactly 10, what does that mean? It means it's at least 11, right? So the accessification of it's at least 10 with respect to the smallest set gives you a meaning that you could have expressed by taking an alternative without accessification. It's at least 11. And this goes via the first part of the conditional pruning, right? Don't do via accessification what you could do by just making something, making a stronger statement with an overt meaning, so to speak. Okay, so we have derived this, right? Or so it seems that we cannot prune the set of alternatives with at least 10 to the smallest set, and thereby we don't get uncertainty influence. But hey, wait a minute, you might ask now, right? This was a trick, right? Namely, this only worked because of the context that I gave you, right? How much is it 10, and then I said it's at least 10. But what if I said, is it 9, and you answered it's at least 10? That's, first of all, a good answer. Second, it still has the uncertainty, huh? And now you might ask yourself now, right? If we restricted the set of alternatives to at least 9, 9, at least 10, I don't know about at least 9, but definitely 9 is in there, right? Because it made relevant here in the question. And then we ask ourselves, well, when we now exhaustify, it's at least 10 with respect to the smaller set. What do we get? We can't negate it's at least 9, because at least 10 entails at least 9. But we can negate exactly 9, right? You with me? We can negate exactly 9. Saying it's, and that would mean that when we negate exactly 9, that we get back it's at least 10. Because if it's at least 10, it might, but a very meaning of it's at least 10, it can't be 9, right? So it would seem that exhaustification, in that case, is again, vacuous as with more than. And therefore we should be able to restrict our set to this smaller set and not get an uncertainty inference with at least. However, this violates another, the second condition, this is why we have the second condition, right? Because it's crucial that for this to work, that in this set we left out the alternative exactly 10. And we say, well, you don't just restrict your set of alternatives to the ones that are mentioned in the preceding discourse. You also must carry those along that are sort of made relevant by making an assumption, by making an assertion, like it's at least 10. On that it makes relevant exactly 10. So you can't actually, you will never have this set. And then we're done. Because then we have all, as far as I can see, all the context where at least 10 might not have an uncertainty inference and we've taken care of them by the conditional improvement. So what does it mean? What is the result? In context where someone asks, how many children do you have? Both the unimbedded comparative and the unimbedded superlative modifiers do not give rise to scale implications according to this account. But they give rise to uncertainty inferences. And we've derived this in a principled way. However, in the second type of context, namely where someone asks, do you have 10 children? Only the comparative modifiers allows restriction of the set of alternatives that get exhaustified, so to speak. And therefore, only the comparative modifier can ever show up without an uncertainty inference. So this way of thinking, so let's step back for a second or so. Why do this that way? Well, if we hadn't shown, how should I say it? Consider, I have not given you alternative views on this, really. I mean, I haven't discussed them in detail. But I said that so far, people take at least three and more than three to be completely different animals. This could be. But that casts doubt, if that were the case, it would cast significant doubt on an approach to the uncertainty inference coming with at least that is based on such a general mechanism like pragmatics. So what people had to do was, well, okay, more than doesn't have certain alternatives, by lexical stipulations, but that gets you into trouble in certain cases, as I will show in a minute. So, by having shown that empirically you do get actually uncertain inferences with more than, when you look carefully, we've given support to the view that both at least and more than should allow for a derivation of uncertainty by a principle pragmatic mechanism, so to speak. And then we only tried to show that in certain cases these go away. No, I don't know. We've essentially now only done the ground work that allows us to go into the the puzzle that everyone working in this area is really interested in and that has to do with the combinations of modified numerals and other logical elements like modals. And in particular, and the things are there, the empirical picture there is so complicated that it has not, it's basically not clear whether you can do this, derive the right meanings, the attested meanings with what would be set out to do here. But I mean the time is a bit, I don't know, how much time do I have? I don't know. The question is I can maybe give you a 5 minute version of, I won't go through all this, I give you a 5 minute version of the because we started a little late anyway. I give you a 5 minute version of a hint of what the puzzle is and I will not talk about fewer than than at most. I mean we have something interesting to say and I can ask questions about them but there are things that get complicated. So let's start with an observation. I mean an observation about a fact that follows from what I've said. When we have a modal co-occurring with a modified numeral like John must drink at least three beers let's say. Then there is in principle two structures available at least you can either scope the modal modified numeral above the modal or below it. However given the fact that when the modal the modified numeral scopes above the modal it is so to speak unembedded once again we will derive for this case exactly the facts that we've derived so far is even if there was no modal there. That means that in all of these cases where the modal has white scope the modified numeral where the modified numeral has white scope the exhaustification will be back here we don't get a scalar implicature and there will always be an uncertainty inference except for these little tweaking with more than that we've seen. However when the modified numeral takes scope below the modal well we don't know things might be very different and we will see that the exhaustification there is not always back here that means you do get a scalar implicature but by our by our initial hypothesis that also means that the uncertainty goes away because there are complementary distribution right? So let's see that. I told you we would modify a bit the semantics for the modified numeral. In particular for at least three or at least right? We assume a degree semantics. That means at least three says there is a degree such that the degree is there has been again this is formalized again here in a maybe not so clear way at the beginning the intuition behind it I think they are fairly clear and they have been around. So the first part is found in literature all over the place the second part we added but we will defend the second condition for independent reasons. So there is a conditional alternative pruning that is basically a condition that says when you can restrict the set of all alternatives to a set of potential set of restricted alternatives and it says that when you have a structure or a sentence by the alternatives of by that you derive automatically in the way that we talked about they can be pruned to a subset of that set only if one and two hold first there is no distinct alternative Psi that is in the original set such that the exhaustification of Phi with respect to the new set is identical to Psi that means the intuition is complicated stated but the intuition is don't do via scalar and placature what you could do via an overt assertion of some alternative essentially that's what it says and that intuition is around all over the place it's also found in the NPI literature the second intuition that we want to add here is that all the alternatives in the original set that are that are part of the sentence that we utter so to speak they must be in the in the smaller set so when I say it's more than 10 it's exactly 10 must still remain because I've uttered more than 10 so I've thereby made relevant basically it's exactly 10 but you can get rid of other alternatives so what does this do this first part has been argued for the second part I think we need for independent reasons anyway let's make a step back once more to disjunction if I say John or Mary called and then I say namely John that's very strange why utter the disjunction and then utter something like that well it should be good if we could restrict our set of alternatives like John called, Mary called John and Mary called to simply John and Mary called that's basically the old horn alternative namely when I say John or Mary called and then I exhaustify and get not John and Mary called why not add namely John there's nothing that prohibits that that means in other words that the uncertainty inference that we found with or seems to be as obligatory as the uncertainty inference with at least and that's for a particular reason namely for the fact that you can't get rid of alternatives that are given to you by the very fact that you are asserting a disjunction and that would be an alternative like John called and also an alternative like Mary called so let's see what this does for more than 10 and at least 10 remember the relevant context or context of type 2 where someone asks how much is it, is it 10? and then you answer it's more than 10 we assume this is a big classification here we know that the alternatives are more than 10, more than 11, more than 12 exactly 10, exactly 11 and so on and we ask ourselves can we restrict this set of alternatives to that smaller one where there's just 10 and more than 10 in it well why not the only alternative that I actually from this set that I could negate is 10 because more than 10 is asserted so I can't negate it so let's do that that means it's more than 10 but it's not exactly 10 well that's already entailed by the assertion that there's more than 10 so you get back it's more than 10 exhaustification in other words is vacuous with respect to that smaller set now that means that nothing here is violated in our conditional pruning first since we get back the meaning that we had without exhaustification it's not the resulting meaning is not equivalent to some other distinct alternative in the set it's more than 10 is not equivalent to more than 11 or exactly 10 or exactly 11 so that's good the first part the second sub-condition that is not violated is that in this smaller set we have all the alternatives in there that we can derive from it's more than 10 by simply replacing more than 10 with sub-constitutions we can go and replace more than 10 with 10 so that part is also not violated in other words cut a long story short we are allowed to restrict a set of alternatives thereby for more than 10 thereby not derive a scalar implicature which is as we want but in addition also not derive an uncertainty inference essentially you do not derive the uncertainty inference here because there are no relevant alternative alternatives left basically that's the idea now consider at least again in the context where someone says how much is it is it 10 and then you answer it's at least 10 the intuition is first of all it's maybe not so good this answer second the uncertainty inference doesn't go away very clearly so we have to ask ourselves again can we restrict this set of alternatives to a smaller set as with more than 10 well again with it we could but now we get a problem namely the only alternative that we can negate by accessification is 10 because at least 10 is other so I can't negate it would be a contradiction if I say if however I say at least 10 but not exactly 10 what does that mean it means it's at least 11 right so the accessification of it's at least 10 with respect to the smallest set gives you a meaning that you could have expressed by taking an alternative without accessification at least 11 and this goes via the first part of the conditional pruning right don't do via accessification what you could do by just making something by making a stronger statement with an overt meaning so to speak okay so we have derived this right or so it seems that we do not get we cannot prune the set of alternatives with at least 10 to the smallest set thereby we don't get on a certain influence but hey wait a minute you might ask now right this was a trick right namely this only worked because of the context that I gave you right how much is it 10 and then I said it's at least 10 but what if I said is it 9 and you answered it's at least 10 that's first of all a good answer second it still has the uncertainty and now though you might ask yourself now right if we restricted the set of alternatives to at least 9 9 at least 10 I don't know about at least 9 but definitely 9 is in there right because it made relevant here in the end in the question and then we ask ourselves well when we now exhaustify it's at least 10 with respect to the smallest set what do we get we can't negate it's at least 9 because at least 10 entails at least 9 but we can negate exactly 9 right you with me saying it's and that would mean that when we negate exactly 9 that we get back it's at least 10 because if it's at least 10 it by by but a very meaning of it's at least 10 it can't be 9 right so it would seem that exhaustification in that case is again vacuous as with more than 10 and therefore we should be able to restrict our set to this smaller set and not get an uncertainty inference with at least however this violates another second condition on this is why we have the second condition because it's crucial that for this to work that in this set we left out the alternative exactly 10 and we say well you don't just restrict your set role of alternatives to the ones that are mentioned in the preceding discourse you also must carry those along that are sort of made relevant by making an assumption like it's at least 10 on that makes relevant exactly 10 so you can't actually you will never have this set and then we're done because this is then we have all as far as I can see all the context where at least 10 might not have uncertainty inference and we've taken care of them by the conditional improving so so what does it mean, what is the result insert in context where someone asks how many children do you have both the unimbedded comparative and the unimbedded superlative modifiers do not give rise to scale implications according to this account but to give rise to uncertainty inferences and we've derived this in a principle way however in the second type of context namely where someone asks do you have 10 children only the comparative modifiers allows restriction of the set of alternatives that get exhaustified so to speak and therefore only the comparative modifier can ever show up without an uncertainty inference so this way of thinking let's step back for a second or so why do this that way well if we hadn't shown how should I say it consider I have not given you alternative views on this really I mean I haven't discussed them in detail but I said that so far people take at least three and more than three to be completely different animals this could be but that casts doubt if that were the case it would cast significant doubt on an an approach to the uncertainty inference coming with at least that is based on such a general mechanism like pragmatics so what people had to do was ok more than doesn't have certain alternatives by lexical stipulations but that gets you into trouble in the certain cases as I will show in a minute so by having shown that empirically you do get actually uncertain the inferences with more than when you look carefully we've given support to the view that both at least and more than should allow for a derivation of uncertainty by a principle pragmatic mechanism so to speak and then we only try to show that in certain cases these go away now I don't know we've we've essentially now only than the ground work that allows us to go into the the puzzle the puzzles that everyone working in this area is really interested in and that has to do with the combinations of other of modified numerals and other logical elements like modals and in particular and the things are there the empirical picture there is so complicated that it has not it's basically not clear whether you can do this derive the right meanings the attested meanings with what would be set out to do here but I mean the time is a bit I don't know how much time do I have well I don't know I can also the question is I can maybe give you a like a 5 minute version of I won't go through all this I give you a 5 minute version of the because we started a little late anyway so I give you like a 5 minute version of a hint of what the puzzle is and I will not talk about fewer than than at most I mean we have something interesting to say but there are things that get complicated so let's start with an observation I mean an observation about a fact that follows from what I've said when we have a modal co-occurring with a modified numeral like John must drink at least three beers let's say then there is in principle two structures available at least you can either scope the modal modified numeral above the modal or below it however given the fact that when the modal the modified numeral scopes above the modal it is so to speak unembedded once again we will derive for this case exactly the the facts that we've derived so far even if there was no modal there that means that in all of these cases when a modal has white scope a modified numeral the accessification will be vacuous we don't get a scale of amplification and there will always be an uncertainty inference except for these little tweaking with more than we've seen however when the modified numeral takes scope below the modal well we don't know things might be very different and we will see that the accessification there is not always vacuous that means you do get a scale of amplicature but by our by our initial hypothesis that also means that the uncertainty amplicature goes away because they are in a bind complementary distribution right so let's see that I told you we would modify a bit the semantics for for the modified numeral in particular for at least 3 or at least right we assume a degree semantics that means at least 3 says there is a degree such that the degree is okay so this is a very interesting question this is the maybe that's a slide a relevant slide here I'm not quite sure but John had John read more than John read more than 12 books you might think in easily constructible context that doesn't mean that he did not read more than 20 while it still doesn't mean he did not read more than 13 so it seems that there is something to be said about the granularity of the alternatives that play a role right and in most cases sometimes we talked about 10 maybe we talked about children how many kids do you have the granularity is always the smallest one so to speak but in certain cases like this it seems that you are only considering the smaller granularity but the scalar seems to re-emerge for coarser granularity and at the moment we can only say that this is actually a huge problem for everyone who tries to drive this on pragmatic terms because it's not so clear how you I'm not explaining this well what you would like to say is maybe I can now write here so if you have like say 8 something before 9 6, 7, 8 and you say John John read more than 6 books intuitively you would like to say well things that are relevant are in this type of context that we are discussing here very close numerals 6, 7, 8, 9 but as soon as 9 is relevant why isn't 10 not relevant anymore formally how do you get rid of it but once you have 10 relevant how is 11 not relevant it's not clear at all how you can so we would like to say that when we make when we assert in a given context a a sentence with a without this course granularity that alternatives that would correspond to the course of granularity are simply not available but it's not quite clear to me how to do that there must be a way but I really don't know so it's a very good question and that these inferences do show up has actually also been shown experimentally by the people who we are citing here and they also don't actually have an explanation for this I think there was also ok if we are right if anything like if anything of the sort that we are suggesting here you don't even have to buy the complete picture if anything of the sort of an implication based account for say at least is correct then you would like to know when I say John had at least 5 beers you draw the conclusion that he didn't half a million well not even 20 right we don't even have to go to these outrageous numbers but if you know John and you think we are going to believe this of course but probably he didn't have 50 I mean what I'm saying is no matter how you slice it that problem doesn't go easily away there is a certain cut of range which will not be the same for everyone so say we had this Ramallah Jerusalem case and say someone doesn't know the geography of Israel very well for instance me I have no idea really then I mean I know it a bit then someone who says who gets the answer what's the distance and says it's more than 10 kilometers then easily a person who knows Israel quite well let's say will draw the conclusion that it's probably not more than 25 or 30 but someone who has no idea really about how how large Israel is or something like that isn't that technically in Australia to ask oh how far is it at a distance and somebody says it's more than 10 kilometers then to me that means okay probably I'm asking because of the golden and to hire a car to buy petrol and get some water so if you say more than 10 then I buy a bottle of water because I know it will take me maybe half an hour to draw it because then I can assume it's going to be 10 or 15 but if it's like 2,000 kilometers then your answer is irrelevant of course of course because I then make the I was not meaning to suggest that first of all you perfectly right that it's very context things are very context dependent but what I'm trying to say is one person might be when we stick to this Israel case might be considering alternatives of the form 10 15 20, 25 the next type of person might consider alternatives of the form 10, 20 30, 40 that means you predict a varying cut of range of what of what the higher limit is with people what I was trying to say before was you don't expect that everyone will assume that John didn't have more than 20 beers people who know John very well might think he might have actually had between 20 and 25 because he is a big drink but other people might not think so that's all I was trying to say but you're right things are very context dependent certain things are never considered for certain reasons because they would not be practical well you would not be a cooperative speaker you're right in that case you discuss one question do you know how many beat brazilians by at least two girls five or seven impossible yes if there aren't any more questions that suggest that we should go to the test and continue a more formal discussion at the institute far for at least one and thank them once again for this talk thank you