Epimenides Paradox Revealed - Paradox inconsistence

The solution of the paradox according to the interpretation of Aristodemos from Elea

Paradoxes can be of two types. They may be illusions, magic tricks performed on the thread of logic. In this case, the "logical leaps" are not naive mistakes, but well camouflaged "special effects".

The other type of paradox, more importantly, is made up of "special cases" or "singularity" that demonstrate the weakness of rules or logical assertions that we believe to have a universal value.

The Paradox of Epimenides has the characteristics of both types: it uses a well concealed logical leap and at the same time highlights a singularity.

The paradox of Epimenides (some authors prefer to call it antinomy but I find this an improper choice of language) is a declaration made by a Cretan, named Epimenides, who says:

"All Cretans are liars."

The question is: This statment is true or false? If we look for an answer we come to a paradoxical conclusion.

We can explain the paradox in two ways:

First way : Deciding if Epimenides is sincere (truth-teller) or liar.

We can ask whether Epimenides is sincere or if he is a liar. The conclusions are as follows:

If Epimenides is sincere then his statement (in which he says that the Cretans are liars) is true. If the statement is true then we must conclude that Epimenides, being a Cretan, is a liar;

If Epimenides is a liar then his statement is false and therefore the Cretans are not liars but sincere. Epimenides is Cretan and so we must conclude that Epimenides is sincere.

Second way: Deciding whether the statement of Epimenides is true or false.

If the statement is true then the people of Crete are liars, but if the Cretans are liars and Cretan Epimenides is a liar, how can we believe that what he says is true?

If the statement is false, then the people of Crete are not liars but sincere. If the Cretans are sincere then Epimenides must be sincere and we must ask ourselves how he could make a false statement!

************ If you're in a hurry looking for the solution of the Epimenides Paradox you should skip to the "THREE KINDS OF LIARS" section.

********** If you are also interested in other variations of the Paradox, then you can directly skip to the section "SOME VARIANTS OF THE PARADOX".

Otherwise you can continue (expect a boring read!) on the page where we will try to demonstrate the inconsistency of the paradox. The paradox is inconsistent if it is believed that a Liar is always obliged to declare false sentences.

The inconsistency will be revealed soon.

THREE EASY PIECES

1) - STATUS -

First of all we must agree about the meaning of the phrase "All Cretans are liars." In particular, it must be said that the predicate "are liars" does not constitute an assignment operation of value. In computer jargon, value assignment operations are called "set" operations.

If the statement was an assignment of value, then the propensity of the Cretans to tell lies or the truth would depend on Epimenides' sentence, something impossible to belive. Epimenides was said to be a kind of magician, but it certainly was not a god that could affect the character of the Cretans. Epimenides could only declare a quality that already existed. Basically, Epimenides could only declare a pre-existant condition or "status" in order to clarify the propensity of the Cretans, and even himself, to lie. The significance of the statement is as if to say: "The Cretans have a particular gene CRET in chromosome 13 that predisposes them to lie." So the first important conclusion to make is that the statement of Epimenides is nothing more than the recognition of a Cretan "status". In computer terms,"flag status", specific of Cretans and of Epimenides.

The word "flag" just indicates a particular status.

To try to illustrate what a "flag" is, we can refer to the warriors of the films of Kurosawa. Each warrior wore a little flag, on their backs, with the insignia of the army to which they belonged. So, imagine that each Cretan wore a flag showing membership to a group. The words written on the flag were LIAR or SINCERE (truth-teller) depending on if the Cretan belonged to the group of liars or to the sincere group.

The statement of Epimenides says that the Cretans carry, on their shoulders, a flag stating LIAR.

2) - MODUS OPERANDI -

We can ask if the significance of Epimenides' statement is only an attribution of a label to the people of Crete, or if there is more to be discovered? Formulated this way, the question presumes that there's more.

But what is it specifically?

It has never been told in clear terms but Aristotle had already sensed that paradoxes keep a hidden or double meaning. Aristotle spoke of "mention" and "use". The modern authors have interpreted "mention" as literal: what you put in quotation marks. In computer terms is defined as "string" (a sequence of characters).

They interpreted "use" as a semantic value, which is connected to the meaning of the sentence.

It seems, to us, that in this context, the distinction between use and mention constitutes a significant step forward but is not then used to propose a solution to the paradox.

William of Ockham came closer to the truth when he talked about METAVALUES, ie values overlapping those commonly considered.

In any case, also this thinker does not specify the metavalues hidden in the paradox of Epimenides.

Our analysis focused on the term " liar " and we found that this attribute is not just a label but also has a hidden meaning. The word "liar" has a "pregnant or meaningful" meaning that identifies a way of working, that is a "modus operandi". The term "liar" in the context of a sentence that has to do with the values of truth, is not a neutral attribute such as blond or bald. The liar is a falsifier of values of truth, and since the values of truth, in classical logic, are two and exclusive, true or false, the liar is one that "inverts" the truth values. The liar then becomes an "inverter" or a particular type of logical operator that corresponds to the Operator-NOT.

The Operator-NOT is one of the classic operators of Human Logic.

A value of truth, subjected to the action of an Operator-NOT, then "denied", takes on a logic value reversed.

Example: 0 ---> Operator-NOT ---> 1 or False ---> Operator-NOT ---> True

1 ---> Operator-NOT ---> 0 or True ---> Operator-NOT ---> False

The Operator-NOT is represented graphically as a triangle that carries a small circle at the exit point:

the circle indicates the negation operation, that is to say that each input logical value is output with a value of truth reversed.

Conversely, we can say that a sincere person, declaring the truth values without changing them, acts as an Identity Operator . An identity operator is usually called a "Buffer". In this context and for the sake of clarity we will call it by the term "Operator-ID".

Example: 0 ---> Operator-ID ---> 0 or False ---> Operator-ID ---> False

1 ---> Operator-ID ---> 1 or True ---> Operator-ID ---> True

The Operator-ID is graphically represented as a simple triangle.

You may notice that the output is missing the small circle that represents the negation.

We have discovered that the "status" associated with the word "liar" is composed not only by a label but also by a specific way of working or mode of operation. It 's as if a Commander of the armies of Kurosawa had decided to be able to identify the modus operandi of his warriors in ruling that the soldiers were marked with a flag with a red eagle drawn on a white field and knights were marked with a blue eagle in white field. To obtain this additional information on the modus operandi, the Commander not only needs the information provided by the picture (the eagle is the coat-of arms), but also other information provided by the colour of the picture.

Similarly, a liar has a status encoded by two elements, one of which represents the name or label and the other the way of operating. To summarise, the status of the Cretans and Epimenides is "structured", by having a name and an associated function.

With two binary parameters that constitute the status, in theory, we can have four combinations and therefore four possible "status".

Well, we have formalized the problem but the analytical work is not yet finished. It requires a third step that we are now going to describe on the next page.