 Hi, here we are again with our series of sentence analysis videos where we analyse sentence meaning from propositions to predications. In this exercise, we will deal with propositions that involve the existential quantifier, and as a prerequisite, I recommend to have done my e-lectures Predicate Logic 1 and Predicate Logic 2. Well, and this is our task. We have to convert the three sentences, some swans are white and small, Peter is buying some new books and Roberto loves himself, first into propositions and then into their predications. If you want to try on your own first, pause the video here and compare your solutions with mine later on. Okay, here is our first sentence, some swans are white and small, and our first sentence constitutes a complex proposition which can be broken up into two simple ones linked by and. P, some swans are white and Q, some swans are small. Both propositions can be converted into simple predications which both involve a one place predicate which assigns the attributes white and small to swan. But how do we integrate the set sum into our predication? Well, in order to make a statement about sets, we need a variable, X. And X is combined with the existential quantifier and assigns properties to its arguments to X by means of three simple predications. By the way, the symbol for the existential quantifier is the mirror image of capital E. So, here are our three predications. X is a swan, X is white and X is small. Well, and this is now the term where as usual the existential quantifier or the quantifier stands outside the bracket, is combined with its argument X and now we put the various predications inside the bracket. Well, and this time it seems suitable to link them by and so that we get as a final result the complex predication which can be read as follows. There is at least one element X such that X is a swan and X is white and X is small. And this is the solution to our first sentence. In our second example we have a similar case where a simple proposition, Peter is buying some new books, can be broken up into three predications. X is a book, X is new and Peter buys X. And again, all these predications involve a variable for a quantifier to operate on. Well, and as a formula we get for there is at least one element X such that and now we link them all by and again. So we get the final solution. There is at least one element X such that X is a book and X is new and Peter buys X. So this is a result. Well, and our last example which can be converted into a single proposition Roberto loves himself is tricky. A simple solution would be this. Love Roberto Roberto. Roberto loves himself. However, this very simple solution implies that there is only one Roberto in the universe who loves himself. However, if you want to create the possibility that any Roberto can love himself, then we have to break up our simple proposition into two predications with the same two place predicate and the same arguments. However, the arguments are in reverse order. Love Roberto X and love X Roberto. And the formula can now be set up like this. Love Roberto X and love X Roberto. The formula itself can now be set up as an equivalent relationship, equivalence relationship. And it can be read as follows. There is at least one element X for which it holds. If and only if X loves Roberto, then Roberto also loves X. And this is an explicit realization of reflexivity. Okay, that's it again. And here are all solutions again. And as you know, in the VLC e-lecture library, you can download them in the PDF format. And once you're logged into the Virtual Linguistics Campus, you will see this link, the VLC e-lecture library. And that leads you to the place on the VLC where you can see all the e-lectures and all the e-lecture support material. So thank you and see you again.