Identity and the Excluded Middle
But to go back to where we were, there are a couple of variations on the Principle of Contradiction, which have their own names.
The Principle of Identity states that what is is what it is. You could also say that what is true is true, or that you know what you know. Obviously, if something were not what it was, then the Principle of Contradiction would not be true--not that this "proves" it; it simply reveals that this is another way of looking at the basic law of our minds.
A third way of looking at the Principle is called the Principle of the Excluded Middle, which states that there is no middle ground between being and not being; or in other words, statements cannot be neither true nor false, or you can't be neither thinking nor not thinking what you are thinking, or something can't neither be a fact nor not be a fact. If it isn't a fact, it ain't, and if it is, it is.
Notice that sentences can sometimes be neither true nor false, if they are not statements. For example, "Go shut the door" is not true (because it doesn't express the fact that you're going to shut the door) nor false (because it doesn't express the fact that you're not going to); it expresses what I want done, not a fact.
But if a statement were neither true nor false, then either we are doing something strange with language, or the statement would also be both true and false.
That is, suppose "There is something" is neither true nor false. Since it isn't true, this would mean that what it expresses is not a fact (unless you are supposing that "not true" means something different from "isn't a fact," which is the strange use of language). But since it isn't false, this would mean that what it expresses is a fact (unless "not false" gets you into a linguistic never-never land that doesn't mean "is true"--in which case, what could it mean?). Hence, if it's neither true nor false, it's both; or words mean very funny things indeed(1) .
Well, but what about the statement, "This statement is false"? Since the statement says it's false, it's got to be either both true and false (since it's false if it's true, because it says it is, and true if its false, because it says it's false) or neither true nor false. Since by the Principle of Contradiction, it can't be both, then it has to be neither.
No, the answer to this (and it's complicated) is that this, though it has the form of a statement, is not a statement. How could it be stating a fact, which is what statements do? It would be stating, if it meant anything, what would be known to be a non-factual fact, and you can't think a factual non-fact. It can't express either a thought or a fact (because, in my view, of its content, not its form); and hence is not a statement.
Books could be written about this; but let this suffice.Next
Notes1. Of course, this is not a proof of the Principle of the Excluded Middle, because the rejection of the strange use of language is on the basis of the Principle; it just is a way of showing how we have it presupposed in everything we think.