Two semantical paradoxes

Extract

Consider the statement:

(1) What I am saying cannot be proved.

Let us suppose that this statement can be proved. Then it must be true, i.e., in its own words, it cannot be proved, which contradicts our supposition.

Therefore the statement cannot be proved, since our supposition has led to a contradiction. In other words, the statement is true. In this way we have proved the statement.

Therefore the statement both can and cannot be proved.

In the second part of the argument, if we speak of proving (1) in a given formal system S, then we cannot draw the conclusion that we have proved the statement in S. For it is possible that the argument cannot be formalized in S. Indeed, as we know, Gödel shows in his famous article of 1931 that for suitable systems S, it is possible to construct in S a statement saying in effect that the statement itself is not provable in S. The conclusion Gödel draws, we may recall, is that the statement thus constructed is true but not provable in S. Thus no contradictions arise in this way if we are confined to provability in a given system.

A dual of (1) is the following:

(2) What I am saying can be refuted.

Let us suppose that this statement is true, or, in its own words, that it can be refuted. Then it must be false, which contradicts our supposition.

Therefore the statement is false, since our supposition has led to a contradiction. In this way, we have refuted the statement. We see that the statement can be refuted; in other words, it is true.