Revealing the paradox 2

3) - Interaction between LABEL and MODUS OPERANDI : the problem of CONSISTENCE.

The third step is to analyse the possible relationship between the label and the modus operandi of a Liar.

This analysis will unveil the leap of logic leading to the paradox.

Let's say that the human mind usually establishes a link between name and modus operandi.

The connection is logical as well as semantic.

Analysing the possible couplings between the internal label of Epimenides and the related way of working, we have four configurations, as mentioned in the previous page. We see that some couplings are illogical or contradictory, others are acceptable or not contradictory.

We'll call the first type inconsistent coupling and the latter consistent coupling.

The pair (A) LIAR-LABEL <-------> OPERATOR-ID

Is of course an inconsistent pair because there is a total contradiction between modus operandi and labelling: an Operator-ID is an operator that does not falsify the values of truth ​​and therefore should not have the label of liar.

Also inconsistent is this other possibility of coupling:

The coupling (B) SINCERE-LABEL <-------> OPERATOR-NOT

This coupling is inconsistent because an Operator-Not, that falsifies the values of truth​​, can not be labelled as SINCERE, a characteristic of an Operator-ID.

The other two consistent pairs are the following:

Coupling (C) SINCERE-LABEL <------> OPERATOR-ID;

Coupling (D) LIAR-LABEL <-----> OPERATOR-NOT.

If the statement about the status of the Cretans was guided by the criterion of consistence, Epimenides could only say, based on the coupling (C), "Cretans are Sincere" and according to the coupling (D) " Cretans are Sincere". The coupling (D) should be "canonical" because the label of liar corresponds to the "natural" way of operating liars, that is precisely to falsify the truth values. In this case (D), during the declaration of the value of internal status (LIAR), the Cretan would be forced (by it's modus operandi) to reverse the value of truth of the internal status and therefore is obliged to say that he is SINCERE.

Basically, if the statement of Epimenides had been guided by the criterion of consistence, ie a criterion of logical consistence, the only statement that he would have been able to make would have been "Cretans are Sincere."

As Epimenides said "Cretans are Liars", it is clear that the only way of doing this is the use of a inconsistent coupling configuration, as shown in coupling (A) and coupling (B).

The paradox is powered by a contradictory coupling, namely the statment of Epimenides hide an inconsistent coupling between label and modus operandi.

We try to represent graphically how Epimenides can communicate, to the external world, his internal status. In the example below the internal status is constituted by the LIAR-NOT combination in the first image; while in the second image it is constituted by a SINCERE-ID status. In both cases, the internal configuration is consistent. Note that in the case of the coupling LIAR-NOT (CASE D) the status, at the time of the statement, is inverted to SINCERE.

The communication, on the outside, of the internal status does not change when the transfer is implemented by an Operator-ID, as in the case represented by the coupling SINCERE-ID (CASE C).

Different is the situation when the information transfer is implemented by an Operator-NOT. In this case the values ​​of truth in the internal status are reversed out during the statement. Thus we have the following changes:

The internal status of SINCERE-NOT (case B, not consistent) will be modified during the statement to LIAR-ID;

The internal status of LIAR-NOT (case D, consistent) will be changed to SINCERE-ID.

The latter case (case D) is very important because it shows that, if the subject who made the statement had a consistent status of LIAR_NOT, then he would not be able to declare that Cretans are Liars.

The Cretan said by Epimenides is an hybrid LIAR-ID and does not have a "consistent" personality. It is a composite object, a kind of MINOTAUR. To make this statement, Epimenides should display a LIAR flag and behave as if he were SINCERE (Truth-teller).

It would be like seeing a foot-soldier of Kurosawa that exposes his flag with a picture of a red eagle in a white field (standard infantry) while riding a horse!

If Epimenides had a label of liar and a "modus operandi" typical of a Liar (coupling D), then its "internal configuration" would be consistent, but in this case he could not declare "The Cretans are Liars." In fact, the modus operandi of an Operator-NOT typically swaps the values ​​of truth and thus reverses, at the time of the declaration, the value of his status.

This results in a first, immediate conclusion: an Operator-NOT (LIAR) can not declare his status in a direct way. This fact, of great importance, was not understood with sufficient clarity. Many Scholars distinguish between Epimenides and the rest of the Cretan, stating that if Epimenides says False, then the other Cretans can say True. According to our understanding there is no difference between the status of Epimenides and the status of the other Cretans. The statement of Epimenides is "illegal" because it does not observe the rules of logic (this is the Original SIN) and yet has universal value and includes both the status of the Epimenides and the status of the Cretans. All, Epimenides and other Cretans, have the same status declared. It must be an illegally declared status, namely a status of LIAR_Operator-ID, while their inconsistent, internal status, could be either LIAR_ID or SINCERE_NOT.

If we accept the inconsistent status (LIAR_ID) declared by Epimenides, we must conclude that the internal status of Epimenides is undecidable but exclusive (LIAR_ID or SINCERE_NOT). A Cretan can not have, at the same time, both configurations of status.

The status of Epimenides and Cretans would be impossible to decide even if he had made a "consistent" statement saying, "Cretans are Sincere (truth-tellers)" . In this case we would have two possible configurations or internal couplings equally probable. This type of statement could either be made by a Cretan with internal status SINCERE_ID or by a Cretan with internal status LIAR_NOT.

This means that an ID-Operator can declare his status in a direct way but we must be aware that the declared label SINCERE may also be the result of a statement provided by a NOT-Operator (LIAR_NOT).

Returning to the classification of the paradoxes described at the beginning of the chapter, we would like to conclude by saying that the paradox is based on a very intelligent, logical leap. Nevertheless the Paradox highlights a singularity.

The logical leap is contained in the statement, "Cretans are Liars", a statement which is equivalent to declaring "I am a liar." We have shown that, according to the strict path of logic, such a statement can not be made. So, any attempt to verify, with logical procedures, if the statement is true or false, in the end, cannot lead at a logical conclusion. We have shown that the status of Epimenides is impossible to decide, having the internal status of LIAR_ID the same probability of the other status SINCERE_NOT. The Paradox was created by an illogical coupling and stands on an illegal statement . This hidden "original sin" undermines any subsequent logical conclusions.

What is really paradoxical is the fact that a quasi-perfect logic machine, the human brain, is unaware of building an inconsistent statement when saying "I am a liar".

The Paradox tell us that also a legal statement such as "I am SINCERE" may have not an unambiguous interpretation: it may come from a men having an inner status of SINCERE_ID or alternatively a status of LIAR_NOT.

Nevertheless the paradox identifies a singularity that we hope we have made clear: an Operator-NOT can not declare his status in a direct way.

This is a fact of considerable importance, not only in the field of logic but it extends, in our opinion, to the field of epistemology and of philosophy.

If you are very interested in this paradox you can jump to another my paper (more complex):

https://sites.google.com/site/epimenidetoggleflipflop/home