In this post, I seek to establish that there is at least one necessary existent. I do not seek to establish any properties other than necessity and existence.

Consider the set S of every grammatically correct sentence of the form "X exists." Each of these sentences must be true or false. Let E be the set of all true sentences in S. Since we can observe that something exists, we know that E is not the empty set.

Every sentence in E either does or does not depend on the truth of other sentences in E. What I mean by "depend on" is that the existence of the subject of the prior sentence enables the existence of the subject of the second sentence. For example, the sentence "Chelsea Clinton exists" depends in this way on "Hillary Clinton exists," since Hillary's existence enables Chelsea's existence.

Let C be the set of all sentences in E that depend for their truth on others, and N be those that do not. Further, let C' and N' be the sets of the subjects of the sentences in C and N respectively. So C' is the set of everything that exists contingently; N' is the set of everything that exists of necessity.

Now, let us consider the sentence S1, "Every item in C' exists." S1 is syntactically correct and of the form "X exists," so S1 is in S. It is true and therefore in E, because we have defined C' to include only items with true existence claims.

S1 must either be in C or N. If S1 is in N, then we can answer the question immediately: the subject of S1 is a necessary existent.

If S1 is in C, then there must be some other sentence S2 which S1 depends upon for its truth. Let's suppose that S2 is also in C. That means the subject of S2 is in C' and already part of the truth-claim of S1, which in turn means that S1 stands or falls on its own - it does not depend on another sentence for its truth. This puts S1 in N, which is a contradiction. So S2 cannot be in C and must be in N, which means the subject of S2 is a necessary existent.

In either case, there is a necessary existent.