An Explanation of the Infamous Liar Paradox

The Liar Paradox is a statement that is expressed, variously, as "This statement is false", "This sentence is false", or "I am lying". We are told that the statement is paradoxical because:

* if it is false, as it claims, then it is true,
* if it is true, then it is false, as it claims.

Either way, we are told, the statement is both true and false at the same time. Logicians and philosophers profess their mystification.

Every student of logic has been introduced to logical propositions, which are statements that are either true or false, but never both true and false. If students of logic understood why propositions are never both true and false, they would understand why the statement of the Liar Paradox is not a proposition.

A true statement is an accurate statement about reality, whereas a false statement is an inaccurate statement about reality, e.g.,

* The Moon is a sphere, vs.
* The Moon is a cube.

These are statements of fact that can be verified by observing reality. But the Liar Paradox cannot be verified because it makes no statement about reality.

The Paradox is a non-propositional statement whose form imitates the form of an actual proposition. Consider three propositions:

1. Lead is denser than zinc.
2. Statement 1 is true.
3. Statement 1 is false.

The truth of statements 2 and 3 depends on the truth of statement 1. To say that statement 1 is true, is to restate statement 1, i.e.,

Statement 1 is true, i.e.,
It is true that lead is denser than zinc, i.e.,
Lead is denser than zinc.

Similarly, To say that statement 1 is false, is to restate statement 1 negatively, i.e.,

Statement 1 is false, i.e.,
It is false that lead is denser than zinc, i.e.,
Lead is not denser than zinc.

Thus, statements 2 and 3 are only valid propositions because they restate statement 1, which is a valid proposition. In contrast, the Liar Paradox is not a valid proposition because it does not restate any valid proposition. Consider two statements that resemble the Paradox:

1. Statement 0 is false.
2. Statement 2 is false.

We see that statement 1 is nonsense because it pretends to restate a statement that does not exist. Since it does not restate a statement of fact (a proposition), it is not a proposition.

Similarly, statement 2, which is the Liar Paradox and which pretends to refer to itself, is also nonsense because it pretends to restate a statement of fact that does not exist.

The Liar Paradox, then, is nothing more than a clever misuse of language.

Eugene Paul