[Man] Now let’s talk about the syntax of first order logic, and like in propositional logic, we have sentences which describe facts that are true or false. But unlike propositional logic, we also have terms which describe objects. Now, the atomic sentences are predicates corresponding to relations, so we can say vowel (A) is an atomic sentence or above (A, B). And we also have a distinguished relation–the equality relation. We can say 2=2 and the equality relation is always in every model, and sentences can be combined with all the operators from propositional logic so that’s and, or, not, implies, equivalent, and parentheses. Now, terms, which refer to objects, can be constants, like A, B, and 2. They can be variables. We normally use lowercase, like x and y. And they can be functions, like number of A, which is just another name or another expression that refers to the same object as 1, at least in the model that we showed previously. And then, there’s 1 more type of complex sentence besides the sentences we get by combining operators, that makes first order logic unique, and these are the quantifiers. And there are two quantifiers for all, which we write with an upside-down A followed by a variable that it introduces and there exists, which we write with an upside-down E followed by the variable that it introduces. So for example, we could say for all x, if x is a vowel, then the number of (x) is equal to 1, and that’s the valid sentence in first order logic. Or we could say there exists in x such that the number of (x) is equal to 2, and this is saying that there’s some object in the domain to which the number of function applies and has a value of 2, but we’re not saying what that object is. Now, another note is that sometimes as an abbreviation, we’ll omit the quantifier, and when we do that, you can just assume that it means for all; that’s left out just as a shortcut. And I should say that these forms, or these sentences are typical, and you’ll see these form over and over again, so typically, whenever we have a “for all” quantifier introduced, it tends to go with a conditional like vowel of (x) implies number of (x)=1, and the reason is because we usually don’t want to say something about every object in the domain, since the objects can be so different, but rather, we want to say something about a particular type of object, say, in this case, vowels. And also, typically, when we have an exists an x, or an exists any variable, that typically goes with just a form like this, and not with a conditional, because we’re talking about just 1 object that we want to describe.