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
Determining whether an argument is valid is easy to mechanically test for in some cases. For example, there is the argument: “If the apple I am holding is red then I am happy. The apple I am holding is red. Therefore, I am happy.” This is an easy one to test for since there are only two possibilities for whether or not the sentence “the apple I am holding is red” is true and two possibilities for whether or not the sentence “I am happy” is true. Thus, there are only 4 ways the “world” could be. Either my apple is red and I am happy, my apple is not red and I am happy, my apple is red and I am not happy, or my apple is not red and I am not happy. Now, on to the argument. In the argument, we are assuming the conditional “if the apple I am holding is red then I am happy’ and that “the apple I am holding is red”. Our second assumption limits us to two possibilities: either the apple I am holding is red and I am happy, or the apple I am holding is red and I am not happy. F