 Welcome back. In my e-lecture, Predicate Logic 1, I dealt with the basic machinery of predicate logic, that is, with predicates and their arguments. We saw how simple propositions can be broken up into predications, we saw that predications consist of predicates and arguments, and we discussed the various types of arguments and the principles of argument structuring. This e-lecture expands the basic machinery of predicate logic by principles of quantification, and it discusses the central attributes of quantification, namely restriction and scope. Well, and furthermore, we will use several examples to illustrate our main points. But before we dive into all this, let's repeat the outcome of predicate logic 1 using some examples. Here is the first. The proposition John is tall can be converted into a simple predication with a one-place predicate tall, the predicate, and John, the argument. Or take this proposition, Jane loves Paul, where the predication contains a two-place predicate love and two arguments, namely Jane and Paul. Note that a different sequence of the arguments would result in a different proposition. This sequence here would stand for the proposition Paul loves Jane. Well, and our third example is a little bit more complex. The teacher saw that the children were reading a book. Now here we have a sentence that consists of two propositions. The teacher saw something, and the children were reading a book. And this can be converted into a predication with a two-place predicate C, which has two arguments. The first is teacher, and the second is the proposition S. And this proposition can be broken down into the second predication with a two-place predicate read and two arguments, children, and book. But beyond these simple propositions, there are problems. Consider the following propositions. Here is the first. All girls love Paul. Now the predication would simply represent this in terms of a two-place predicate love with two arguments, girls and Paul. And we have the following relationship. There is a set that contains all the girls in the world, and all these girls love Paul. Let's compare that with the next one. Some girls love Paul. Now the predication would be the same because we have no option of integrating some. But in terms of a set representation, we have only some of these girls that are in love with Paul, but not all of them. And in our third example, girls don't love Paul. We have in fact no girl that is included in this set, but the predication would still be the same. In all three cases, the simple predications represent identical relations between entities. But this does not reflect the different meanings. What we need is the addition of a mechanism that allows the formalization of relationships between sets and not between simple entities. Such a mechanism is referred to as quantification. And in fact, there are three quantifiers that are used in predicate logic. The first one is represented by the upside down a symbol and it is referred to as the universal quantifier. It denotes that all elements of a set are included within another set. So here is a set, the set of all x's and this is the representation. All x's are included within another set. Well, and the universal quantifier can be read like this. For all x, it holds that and then the predication follows. The next quantifier is this special reverse e-symbol, the so-called existential quantifier. Now, the existential quantifier stands for sum and it denotes that there is at least one element that two sets have in common. So some elements x are part of another set. The existential quantifier can be read like this. There is at least one element x such that. Well, here is the last quantifier, the so-called negative quantifier. No or not would be the ordinary language equivalent and it denotes that there is no intersection between two sets. So this time we have two different sets and no x is involved or included in that second set. Well, and here is how we can read the negative quantifier for no element x, it holds that. Let us now integrate these quantifiers into some predications and let's start with the universal quantifier all. Now the example I've chosen is a well-known one in semantics and predicate logic. It's this one, the proposition, all linguists are bald. Here, the set of bald people includes the set of linguists. So this is the sort of relationship. All bald people are included in the set of linguists. In other words, there is no linguist who is not bald. Well, and this is how we represent this in terms of predicate logic. And this has to be read like as follows. For all elements x, it holds that. We've already know the interpretation of the universal quantifier. For all element x, it holds that. If x is a linguist, then x is also bald. Note that the arrow is the logical connective if then, which you should know from propositional logic. Predicate logic uses the same inventory of logical connectives as propositional logic to combine simple predications into more complex formulae. Here is another example. All girls love Paul. Now here Paul is in the focus of all girls. Hence Paul is included in the set of girls. We can drag Paul over here. And in other words, there is no girl who not also loves Paul. Well, and the representation in predicate logic can be read as follows. For all elements x, it holds that. If x is a girl, then x also loves Paul. So to summarize the universal quantifier is an operator in predicate logic that stands for any individual or object in the universe of discourse. And the universe of discourse includes everything that we speak about in a certain context. In ordinary language, it corresponds roughly to all and every. The next quantifier, the existential quantifier is an operator in predicate logic that states that at least one entity in the universe of discourse satisfies the expression that follows the quantifier. It does implies the existence of such an entity. Let us illustrate using our example again. Now, this time we have the proposition some linguists are bald. And here the set of bald people intersects the set of linguists. So we have this sort of relationship. In other words, there is at least one linguist who is also bald. Well, and this can be read as follows. There is at least one element x such that if x is a linguist, x is also bald. Well, and here is our second example. Some girls love Paul. Now here Paul is in the focus of a subset of all girls. So we can draw Paul into this subset. In other words, there is at least one girl who also loves Paul. Well, and this predicate logic representation, now we should know that, can be read as follows. There is at least one element x such that x is a girl and x loves Paul. Let us finally discuss the negative quantifier. Now for the negative quantifier, we have two alternative symbols that can be used. We have decided to use this one, but the tilde would also be possible. Now in this example, we have the proposition no linguist is bald. And now these two sets do not intersect. We cannot even move them. So the negative quantifier states that there is not a single entity in the universe of discourse that satisfies the expression that follows the quantifier. It corresponds to no or not in ordinary language. So in no linguist is bald, the set of linguists has no common elements with a set of bald people. In other words, there are no linguists who are also bald and this is the representation in predicate logic and it can be read as follows. There is no element x such that if x is a linguist, then x is also bald. Well, and here is our second example. No girl loves Paul. Here Paul is not in the focus of any girl. In other words, there is no girl. So no girl here in this set that also loves Paul. So Paul is not in the focus of any girl. Well, and again, we should now understand how we spell out this predicate logic representation. There is no element x such that if x is a girl, then x also loves Paul. Okay, that's basically it. However, some general remarks about quantifiers have to be added. The first remark concerns their restriction. Quantifiers cannot refer to anything by themselves. It is always necessary to specify what is being quantified. The restriction establishes the set of entities that are affected or restricted by the quantifier. In natural language, restriction sets are denoted by common nouns accompanying the quantifiers. Here, the common noun linguist. Further restrictions may be added by using certain kinds of, let's say, adjectives. If you want to express that all German linguists are bald, you have to add a conjunction, linguist x and German x. So for all element x, it holds. If x is a linguist and x is German, then x is also bald. Or take this example, all linguists who live in Marburg. And again, you use a conjunction here, linguist x and in Marburg x. So for all x, it holds. If x is a linguist and x lives in Marburg, then x is bald. Or you can take a relative clause, all linguists who smoke. And again, you have a conjunction here. For all x, it holds. If x is a linguist and x smokes, then x is also bald. The second remark I want to make concerns the scope of quantifiers. The scope of a quantifier is the set of entities denoted by the main predication of the proposition in which the quantifier appears. In other words, it expresses what is true of the entities referred to by the quantified noun phrase. In our example, all linguists are bald. Bald x is the scope and linguist x the restriction. We've just talked about the restriction. In some girls love Paul, love x Paul is the scope and girls x the restriction. It's quite simple, I think. Let us finally discuss some examples. Here they are. Here's the first one. Everybody is happy. And now we can easily spell this out. For all elements x, it holds. If x is a person, then x is also happy. I think you should have got it by now. Here is the next one. John likes some animals. Well, some is of course represented by the existential quantifier. There is at least one element x such that x is an animal and x is liked by John. So John likes x, John likes some animals. Well, and finally an example that uses the negative quantifier. Nobody likes Mary. Well, for no x, it holds that if x is a person, then x likes Mary. And here we have an alternative. The alternative could be, could use the existential quantifier and negate it. In which case we would read this as it is not the case that there is at least one element x for which it holds that x is a person and x likes Mary. Note that the quantifier plus its restriction corresponds to the quantified noun phrase. In our cases, everybody, some animals and nobody. And that the items in the quantifier scope are propositional functions that can be turned into propositions. Well, I hope that now the central ideas underlying predicate logic should be clear. Using its machinery that is combining predicates and their arguments to more or less complex predications, which in turn can incorporate principles of quantification, we can break up propositions and formalize sentence meaning. Even though there are limitations, for example, the exclusion of non-declarative aspects such as the function of sentences in particular contexts, propositional and predicate logic are suitable tools for the formalization of sentence meaning. I hope that I've shown this at least to some extent. So, thank you and see you again.