﻿ The Liar Paradox - curiouser.co.uk
curiouser.co.uk

Epimenides, the celebrated Greek prophet, lived in Crete during the sixth century BC. A statement made by him is believed to be the origin of the oldest of logical paradoxes. He is reputed to have said:

"All Cretans are liars."

Epimenides was himself a Cretan. His statement therefore implies that he was a liar himself. If he is a liar, does that make his statement false? And if his statement is false, does that then mean that he is telling the truth?

In order to analyse his statement further one must first define exactly what is meant by the term "liar".

If one defines a liar as someone who sometimes tells the truth and sometimes doesn't, one can clearly see that no paradox exists, as the statement "all Cretans are liars" can be true without contradicting the status of the speaker as someone who sometimes lies and sometimes tells the truth.

However, if one defines a liar as someone who always lies, the analysis is less straightforward. Under this definition, the statement cannot be true, because if all Cretans always lied, then Epimendides would have had to be a liar who told the truth, which is not possible. However, no such contradiction arises if one assumes the statement to be a false. If "all Cretans are liars" is a lie, then one can logically infer that "Not all Cretans are liars", which means that at least one Cretan tells the truth at least some of the time. This is not inconsistent with Epimendides lying on this occasion.

In this instance the statement "all Cretans are liars" can be seen to be necessarily false, but not paradoxcial, as it may intially have appeared.

## False Analysis

Some texts (including this site originally) have incorrectly stated that Epimenides's statement was paradoxical.

The (invalid) argument then follows:

In order to elliminate questions about the nature of liars, such as whether a liar ALWAYS lies, the statement can be re-written as:

"All statements by Cretans are false."

1) All statements by Cretans are false.
2) Statement 1) was made by a Cretan.
3) Therefore statement 1) is false.
4) Therefore all statements made by Cretans are not false.

Clearly statement 1) and 4) cannot both be true; they contradict one another. But, since statement 4) follows logically from statement 1), statement 1) can be seen to be self-contradictory, and thus paradoxical.

The error of this argument is that "all statements made by Cretans are not false" cannot be logically deduced from the fact that the statement "all statements by Cretans are false" is not true. The negation of "all statements by Cretans are false" is "not all statements by Cretans are false" or "at least one statement by at least one Cretan is not false".

There is however a genuine paradox which arises from a similar statment. It is known as the Liar Paradox, and is attributed to Eubulides of Miletus, a Greek philosopher who lived in the fourth Century BC. His question was, "A man says that he is lying; is what he says true or false?"

This paradox can be written more simpy as:

"This statement is not true."

Clearly, no truth value can be consistently assigned to the statement. If one assumes it to be true it tells us that it is false, but if one assumes it to be false it tells us that it is true. Any assumption about the statement seems leads contradiction.

The philosopher George Edward Moore was once asked whether he always told the truth. After moment's thought, he replied, "No".

Bertrand Russell suggested that George Edward Moore only ever told one lie.