 Welcome back to our series of sentence analysis videos. As you know, our goal is the formal analysis of sentence meaning from propositions to predications. And in this exercise here, we will deal with simple propositions that involve the universal quantifier. As a prerequisite, I recommend to have gone through the e-lectures predicate logic 1 and predicate logic 2. And here is our task. We have to convert these three sentences, all swans are white, all exciting matches are sold out, and Roberto loves all Italian women, first into propositions and then into predications. If you want to try on your own first, pause the video here and compare your solutions with mine later on. Okay, let's start. All swans are white. Now this sentence constitutes a very simple proposition, namely all swans are white. And as a predication, it involves a simple one-place predicate which assigns the attribute white to its argument swan. The central question, however, is how do we integrate the amount all into our predication? Well, in order to make statements about amounts or sets, we need a variable, a variable such as x. And this variable can be assigned properties by means of two simple predications. The first one is x is a swan, predicate swan, variable x. And the second one is x is white, predicate white and variable x. So x is a swan is the first one, x is white is the second one. And now we can apply the universal quantifier, you know, the upside down A symbol to both predications. And the convention to do this is quite simple. Put the quantifier and its variable outside a bracket from where it can bind all elements with the same variable within the bracket. Well, and these are our elements, swan x and white x. And how do we combine the two terms in the bracket? Well, we could try the logical connective AND. But this would mean that x is a swan and x is white. In other words, it would mean that all elements x in the universe are swans and all elements x are white. Well, is this what we really want? Not at all. So we cannot take the conjunction, but we need the implication. And now this term can be read as follows. For all elements x it holds, if x is a swan, then x is also white. So as a result, this is the complex predication which realizes the proposition all swans are white in terms of predicate logic. In our second example we have a similar case where a simple proposition involves a quantifier. All exciting matches are sold out. This time, however, the proposition can be broken up into three predications. And again they all involve a variable for the quantifier to operate on. As a formula, now here again you see the formula with the universal quantifier and its variable outside it. Well, now we can move these elements into the formula. We can now combine the first two terms by means of logical AND. Oops, there we are. And to make this term absolutely precise we put another bracket around it. This is a bit big, isn't it? Well, and this first term, x is exciting and x is a match, can now be combined with the second one in terms of an implication. Which then can be read, for all x it holds, if x is exciting and x is a match, then x is sold out. Well, and this is a reasonable result. Our last example is similar. The proposition Roberto loves all Italian women can be translated into three predications, where the last one involves a two-place predicate with one variable and one constant as its argument. So love Roberto x, Roberto loves something. And the formula again involves a conjunction and an implication. So we have this relationship again, just like in the previous cases we can put brackets around it and have a conjunction as the first term, as the antecedent, and a simple predication as the consequent. Well, and this is the result. For all elements x it holds. If x is Italian and x is a woman, then Roberto loves x. Simple, isn't it? Okay, that's it. Here are the solutions. And in the VLC e-lecture library you can download all these solutions in the PDF format. So thanks and see you again. Well, this is what the e-lecture library looks like on your personal VLC start site. So a click on the VLC e-lecture library URL will lead you to all the VLC support material. Thank you very much.