Logic

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 sufficient to show that the argument does not hold in this one world to prove it invalid.

                We could analyze arguments (and sentences in general) more deeply by considering predicates as sets. Rather than considering the predicate “x is an apple”, for example, we could simply have the set of all things that are apples. Using the same method, we could state that “the set of apples is a subset of the set of blue things” rather than “all apples are blue”. Then, to show that our previous argument is invalid, we could imagine that there are other elements in the set of blue things that aren’t apples. In other words, these sets don’t have to be identical. This idea of subsets is a very good one to use when you have more complicated arguments and works for almost everything. It does, however, run into some problems. The most basic problem comes in the form of a contradiction and can be derived from defining a set through the predicate: “all sets that do not contain themselves”. If the set does contain itself, then the predicate tells us that it must not contain itself, and if the set does not contain itself, the predicate tells us that it must. Another potential problem is infinitely telescoping—sets that are infinitely large and contain themselves. For example, there is the set of all sets that “contain more than one element”. This set has an infinite number of elements, meaning that it must be a member of itself. Thus, not only does this set have infinitely many subsets, but it contains a copy of itself, meaning that it must have more elements than itself.