Because some arguments are impossible to model exhaustively — there is an infinite number of ways the world in question could be —it is harder to show that these arguments are true. Some such arguments are categorical ones. For instance, we could argue that “All apples are fruits. Therefore, if everything is an apple, everything is a fruit.” Even though this argument is clearly valid, there is no way to model every world in which our premise is true. But if we make the argument: “All apples are blue. Therefore, everything that is blue is an apple.” We could imagine a world in which there is a pear and an apple, and they are both blue. In this world, the argument is invalid, since every apple is blue, there is something blue that is not an apple. (in fact, we don’t even need an apple in our world to prove this argument is false.) As with our previous, simpler arguments, a valid argument is one whose conclusion is true in every world in which its premises are true. Thus, it is suffic