Epimenides Paradox Revealed - THREE KINDS OF LIARS

THREE TYPES OF LIAR

It is easy to say Liar but the human mind does not pay attention to the various meanings it attributes to this word. Let's see what we can mean by the term "LIAR".

The simplest definition is to say that a registrant is a Liar because he is declaring false sentences.

It would be better to be more specific and to distinguish by saying that a declarer is Liar because:

1) will declare false sentences (this is what we expect and for this reason he is appointed Liar);

2) declared false sentences (we evaluated his statement and it turned out to be FALSE

The mind not only evaluates what happened (post-evaluation) but often has expectations about the event (pre-judgment). And it is precisely in these expectations that the core that leads to the Paradox is hidden.

A Liar's statements should undergo an ex-post evaluation while they are often subjected to a preconditioned, expectation-based evaluation. It is a widespread belief that a Liar must always say False sentences. A Liar's statement is always false by default. We will see that this does not always happen.

Based on the elements described above, we can identify three types of Liars.

Type 1: OCCASIONAL LIAR ----- (NON-CONDITIONED DECLARER)

Type 2: ABSOLUTE LIAR ---- (PRE-CONDITIONED DECLARER)

Type 3: INVERTER LIAR ----- (PRE-CONDITIONED DECLARER)

--- The Type 1 Liar, the Occasional Liar, is a "normal" declarer, who follows a "classic" logic and is not subject to behavioral constraints. The name he bears is a label that does not oblige him to say false sentences. Sometimes he declares false sentences but he can also declare true sentences. The logical value of its declarations is evaluated through an XNOR logic gate also known as Exclusive-NOR. This type of Logic Operator has the same truth table as the "Double Implication" connective.

This "connective" is also called biconditional (↔) and corresponds to the premise "if and only if". To decide whether a sentence is TRUE or FALSE, among the available true-functional connectives, the use of the XNOR logic gate is the one that seems most suitable when trying to resolve the Epimenides paradox. Prof. Tarski also indicated this operator as the most suitable for verifying the truth value of a statement. Prof. Tarski had written that the statement "snow is white" is to be considered true if and only if, snow is white.

The logical operator XNOR produces a logical TRUE output when the truth values brought to its input are concordant i.e. both TRUE or both FALSE. The XNOR operator is in short a input CONCORDANCE logic detector.

When using the XNOR Operator or the biconditional connective, if the declared status does not correspond to the status (label) actually possessed, the statement is False. Reciprocally, If the declared status corresponds to the status possessed, then the declaration is considered true. Example:

If a person with blond hair declares "I am blond" , declares a true sentence because the declared quality-characteristics correspond to the quality possessed.

Similarly, if an Occasional Liar declarer declares “ I am a Liar ” he declares a sentence true because the declared quality corresponds to the quality (identified by the Liar label) actually possessed.

In this context, the word Liar is comparable to the word Blond. The Occasional Liar has no one operational conditioning that force him to falsify a truth value.

The behavior and logic followed by the Occasional Liar is represented graphically in Figure 35, statement-1.

FIGURE 35: graphic presentation of the three types of liars

--- The Type 2 Liar, the Absolute Liar , is a "particular" declarer, who follows a "particular" logic and is subjected to severe behavioral constraints. From the Absolute Liar we always expect false sentences (prejudice). The sentence declared by an Absolute Liar is ever false. Even when according to traditional logic (Classical Logic) the sentence should be considered true. For the simple fact that it is declared by an Absolute Liar, the declaration will be evaluated as false. The statement of this LIAR is considered false when it would turn out to be so according to common logic and is considered false even when, according to common logic, it would be considered true.

The evaluation of a statement made by an Absolute Liar is therefore subjected to a prejudice which must meet the expectations of the human mind. And to obtain this result the Absolute Liar must operate a logical pre-conditioning on the declaration. It is a preconditioning that places it outside the Classical Logic.

A graphical representation of the behavior of the Absolute Liar and the logic followed by this declarer is represented Figure 35, statement-2 . In this figure it is evident that the Absolute Liar, to satisfy the expectations of the human mind, must follow a particular logic, different from the Standard Logic. The evaluation of the truth value of the sentence is not done using a single logic gate XNOR as it happens in the Classical logic observed in statement 1.

In the graphic representation of the logic followed by the Absolute Liar, a new logic operator appears, constituted by the XOR (exlusive OR) gate. Through this variation the declarer with the status of Absolute Liar is able to declare a false sentence even when the declared status (Liar) corresponds to the status possessed (Liar).

The logical XOR operator works in the opposite way to the XNOR operator: it outputs a logical true value only when the values placed at its input have an opposite truth value. In practice, the XOR Logical Operator is a input DISCORDANCE logic detector.

The behavior and logic followed by the Absolute Liar is represented graphically in Figure 35, statement-2.

--- The Type 3 Liar, the Inverter Liar, is also a conditioned declarer, who has his own particular way of manipulating truth values.

The INVERTER declarant does not always declare false sentences, unlike the Absolute Liar.The logic of the Inverter Declarant is in opposition to the Classical Logic.

In fact, the Liar-Inverter declares false sentences when, according to Classical Logic, they should be considered true.

Reciprocally, the NON-Liar-Inverter declares true sentences when, according to classical logic (XNOR), they should be considered false.

The Liar-Inverter mode of operation is implemented through the XOR Logic Operator. In fact, if we think of a logical post-conditioning, we can imagine that the XNOR Operator, used by Classical Logic, is followed by a NOT Operator which inverts its truth value. From this concatenation the resulting Logical Operator is the XOR Logical Operator.

In statement 3 of Figure 35 we also wanted to represent the possibility of logical pre-conditioning using a NAND logic gate followed by the XNOR logic gate.

Evaluations similar to those set out above are also valid for the NON-LIAR status, of which 3 types can be identified, comparable to those of the Liar. The graphical representation of the NON-LIAR declarer appears in the "LET'S GO ON TRACK" section.

THE MIND BUILDS THE LOGICAL TRAP

Let's try to follow the deductive path made by the human mind to reach the paradoxical conclusion. In reality the paths are two and in opposite direction:

1) the mind starts by assuming a truth value of the statement and comes to the conclusion that the truth value appropriate for said statement must be of opposite value;

2) the mind starts by assuming that the declarer has a predefined status (Liar / Non-Liar) and comes to the conclusion that the declarer must have a status opposite to that hypothesized.

In practice, the reasoning follows these four options:

1) if the declaration is TRUE then the declarer is LIAR; ........................ but if it is LIAR then the declaration is FALSE

2) If the declaration is TRUE then the declarer is NON-LIAR...... but if it is NON-LIAR then the declaration is False

3) if the declaration is FALSE then the declarer is NON-LIAR; ......... but if it is NON-LIAR then the declaration is True

4) if the declaration is FALSE then the declarer is LIAR; ........ but if it is LIAR then the declaration is True

The mind does not realize that, in its reasoning, it collects in the same term "LIAR" ( NON-LIAR ) different declarants who follow different logics.

When the mind says "if the sentence is TRUE, then the declarer has a LIAR status", the mind refers to the type of Occasional Liar represented in declaration 1 of Figure 35. This LIAR operates in the logical world of the XNOR Operator.

When it then moves on to the second deduction, saying "if the declarer is LIAR then the sentence is FALSE", the mind takes the license to make a logical leap. In fact, the LIAR of this second deduction, the one from which we expect false sentences, is not the same as that of the first deduction. It has the same label but operates in a different logical world. Whether the mind thinks of the Absolute Liar or the Inverter Liar, in any case the mind makes a leap to registrants who follow different logic from the Occasional Liar.

In summary we can say that the truth value of the declaration is not unique because it is evaluated from different perspectives.

*** In the case of the statement considered true, the evaluation of the truth value is concentrated on the logical structure of the statement itself.

*** In the case of the declaration considered false, the truth value is based on a "belief", ( "habit" according to the thesis of Charles Sanders Peirce ) conditioned by a prejudice: the mind bases its evaluation exclusively on the authoritativeness of the declarant. The mind is so sure that the statement made by a "Liar" is false that it does not bother to evaluate the logical structure of the statement.

In reality, this confusion between Liars of different types hides a further subtlety.

If we pay attention and go deeper, we see that there is no leap from one type of liar to another type. Let's deepen the analysis.

If we try to classify Epimenides into one of the three types of Liar described above, we see that he does not belong to any of the three.

In fact, Epimenides is a SPECIAL LIAR. He is capable of making the same statement both true and false at the same time. This does not happen to any of the three types of liars listed. What logical scheme can we attribute to Epimenides?

The logical scheme attributed to the Absolute Liar comes to our aid. The logic diagram of this Liar shows us that, before the final meeting in a single exit (in the last door XNOR), the logical path splits into two exits, one of which is True and the other is False. That's what we need.

CORE OF THE PARADOX

Now we realize that the core of the paradox was constituted by the claim to convey in a single output and at the same time the true logical value and the false logical value.

The two types of liars, the one acting through the XNOR Operator and the one acting through the XOR Operator, coexist within Epimenides, merged into a single Logical Operator. This Operator is represented in CONFG. 3 of Figure 36, shown below.

The representation of Epimenides as a Logical Operator (Liar - NON-Liar) with two outputs solves the paradox (see Config. 3 of Figure below). This interpretation should not cause a scandal: for a long time we have known Logic Operators with two outputs, such as the FLIP-FLOP. These Operators have two distinct outputs with opposite logic values. Epimenides' statement is True and False at the same time. But this does not happen on the same logic output. It can happen if the mind follows two different logical paths based on different Logical Operators (connectors).

In Figure 36, below, the logical configuration of Epimenides is represented.

Epimenides is a two-output Logical Operator. This structure allows him to make a true statement and a false statement at the same time.

LET'S GO ON THE TRACK

Figure 37 below shows the logical structure of the paths used by the mind to move from a conclusion with a defined truth value to another conclusion with an opposite logical value.

The explanation of the Epimenides Paradox is not yet complete.

In fact, it remains to represent which path the mind takes when transforming the status of the declarer from Liar to NON-Liar and vice versa.

With two releases, attributed both to the LIAR and to the NON-Liar, it is possible to let your mind wander at will, according to its whims.

With two exits we can build a TRACK (see Figure above) that the mind can travel in both directions and where it can pass from true sentence to false sentence and from Liar to NON-Liar.

Let's take a ride on this track.

The start is located at the Liar point. From this point the mind chooses to head to the "false declaration curve" (red dotted line) through the XOR Logical Operator. Shortly after, he takes the second part of the "false declaration" and arrives in the territory of the NON-Liar. Through the XNOR Operator it goes back to the NON-LIAR status. From here the mind go through the XOR door to the "true declaration curve" (blue dotted line), go through the first part and then, having passed the territory of the NON-Liar, pass the last part of the curve and go up again, through the XNOR Logical Operator, at the starting point where the LIAR declarer is located. The circuit is closed, the mind has made a counterclockwise turn. At will, the mind can make the laps of the track it wants.

If he had initially chosen to head towards the " true declaration curve ” he could have completed the tour by going clockwise.

In this case, clockwise, the sequence would be as follows: Liar → first part of "true statement" → second part of "true statement" → NON-Liar → first part of "false statement" → second part of "false statement" → Liar.

What is striking, looking at the track, is that there are always two directional options. If, for example, the mind had entered the track at the level of the true bid curve, it would have had the option of heading towards either the LIAR or the NON-LIAR territory. Similarly, she would still have had two directional options even if she entered the track at the level of the false declaration curve.

We note another feature: in the simulation of the logical path along the track, we have seen that the status of the declarer alternates, passing from the status of Liar to that of NON-Liar. This alternation of status generates a "toggle" type alternation.

In the past, a great logic specialist such as Thomas Fowler (1869) had already noted that "Epimenides the Cretan says, 'that all the Cretans are liars,' but Epimenides is himself a Cretan; therefore he is himself a liar. But if he be a liar, what he says is untrue, and consequently the Cretans are veracious; but Epimenides is a Cretan, and therefore what he says is true; hence the Cretans are liars, Epimenides is himself a liar, and what he says is untrue. Thus we may go on alternately proving that Epimenides and the Cretans are truthful and untruthful."

CONSIDERATIONS

The graphic scheme of the track helps to understand how the Mind is pushed to make logical deductions that are disposed in an endless vicious circle. It also clearly shows where the errors of deductions supported by the XOR operator are located. These deductions are not comparable to those supported by the XNOR Operator.

How come the Mind has fallen into error by basing certain deductions on "BELIEFS"?

The fact is that these "beliefs" are deeply imprinted in logical thinking. These beliefs are of two types:

1) a TRUE sentence is necessarily and exclusively declared by a NON-Lie declarant, and vice versa;

2) a FALSE statement is necessarily and exclusively declared by a Liar declarant, and vice versa.

In essence, the mind places the relationship between the TRUE sentence and the NON-Liar declarant as a "DOUBLE implication", which can be represented as follows:

TRUE sentence ⇔ NON-Liar.

Likewise, it places the relationship between the FALSE statement and the declarant Liar as a "DOUBLE implication", which can be represented as follows:

FALSE statement ⇔ Liar.

These "beliefs" appear well founded and have quasi-general value but do not hold in the context of the statement "I am a Liar" made by a Liar. This exceptional case, this singularity, is shown in the Figure 39, below.

In the Figure 39 it can be seen that the relationships actually constituted by a valid bidirectional implication are those that are positioned in Quadrant 1 and in Quadrant 3.

The relation of Quadrant 2 highlights that, in the context of the Paradox, the double implication sentence TRUE ⇔ NOT-Liar does not hold. Likewise,

the relation of Quadrant 4 shows that, in the context of the Paradox, the double implication sentence FALSE ⇔ Liar does not hold.

Therefore, only the relationships represented in Quadrant 1 and Quadrant 3 remain valid.

LET'S TAKE AN ELEVATOR

To complete the reasoning we show a three-dimensional graphic image of the two levels of representation used by the Mind in evaluating the Epimenides Paradox. Now we can penetrate the epistemological process that hides in the paradox

Figure 49, below, shows that the mind uses two levels of evaluation in solving the paradox:

1) Logical level;

2) Semantic level.

At the Logical Level the Mind operates strictly on "labels" and is not interested in the meaning of these labels. The label Liar is a string of characters, a symbol comparable to an icon, used to identify an object. At this level the Logical Operator used is XNOR. The XNOR Operator compares the correspondence of these labels without claiming to grasp their meaning (the mechanism is similar to that which occurs when verifying a password).

We can be even more rigorous. If we want the term Liar of the logical level to be "pure", deprived of any semantic meaning, then we must conclude that this Liar should not be defined as Occasional Liar (semantic value-1). An Occasional Liar is already interpreted by the mind as an object with semantic meaning. So the Liar of the logical level is simply an object identified by the character string "L-i-a-r ".

Let's see what happens when we take the elevator down to the ground floor.

At the Semantic Level the mind is interested in giving a "trascendent" meaning to these labels. On a semantic level the label becomes a word. At this level Liar indicates a declarer who declares false sentences. The Liar can be an Absolute Lier who always declares false sentences (semantic value-2) or he can be an Inverter Liar who inverts the truth value of the declaration (semantic value-3).

The semantic level is the XOR Operator level. In order for the Liar of the semantic level to correspond to a false declaration (as expected) it is necessary to use the XOR Operator. The conferral of a semantic meaning to the label "Liar" satisfies the mind but involves, in the context of the statement in question, the descent to a different evaluation level where the logic of the XOR Operator applies. The logic followed by the XOR Operator does not follow classical logic when trying to define a truth value. The XOR Method is different and could be defined as PARALOGIC.

LEVELS OF ASSESSMENTS

Figure 49 showed that the Mind uses two different planes when analyzing the Paradox. The logical evaluation level remains single (according to Classical Logic) but the semantic evaluation level can be broken down into multiple levels, given that multiple meanings can be attributed to a single word.

We can therefore list multiple levels of evaluation in this way:

Case-1 ----- Logic level: l-i-a-r (string-literal)

Case-2 ----- Semantic level-1: liar (occasional)

Case-3 ----- Semantic level-2: liar (absolute)

Case-4 ----- Semantic level-3: liar (inverter)

Case-5 ----- Semantic level-4: liar (Pinocchio: red nose when he lies)

Case-6 ----- Semantic level-5: liar (ethical: lies only for good purposes)

Case-7 ----- Semantic level-7: liar (??????? ..)

At the beginning of this session we had classified the Liar into only three types. These three types were useful for understanding the fundamental differences that distinguish a conditioned Liar from a non-conditioned Liar (to generate false sentences).

In reality, on a semantic level we can also list other types of Liar, for example the Pinocchio-Liar who reddens his nose when he lies or the ethical-Liar who lies only for good reasons. And so on ...

The identification of different levels of evaluation allows us to understand more clearly the core of the Epimenides Paradox and probably also of other paradoxes. Now we know how this Paradox was created:

1) it is necessary to create a statement where the status of the declarer has the ability to give that statement, on a semantic level, a truth value of FALSE (..if the declarer is a Liar then the statement is FALSE.....);

2) this FALSE truth value is compared with the truth value that emerges when, on a logical level, the declarant's status is considered as a string of characters, devoid of any semantic meaning (........ .if the declarer is L-i-a-r then the declaration is TRUE ). In this case the truth value of the statement is TRUE.

In the end the veil that hid the mechanism of the Paradox has been removed. Now it is very clear that the endless deduction, already analyzed when we got on track, is generated by subsequent evaluations that take place on different levels:

**** a LOGICAL evaluation (literal-free-XNOR) is alternated (and contrasted) with a PARALOGICAL evaluation (semantic-conditional-XOR)

CONCLUSIONS

The purpose of our investigation into the Epimenides Paradox was to find a satisfactory solution on the Logical level, that is on the level of Classical Logic. We believe that this type of Logic uses the XNOR Logical Operator when it needs to identify the truth value of a statement. We therefore do not allow "contaminations" that bring other Logical Operators into play such as the XOR Operator.

The Epimenides paradox leads to two conclusions with opposite truth values but this happens because evaluations, that are positioned on different levels, are compared.

One of these levels is compatible with Classical Logic and is the "literal" level. In this plan it is possible that a "L-i-a-r" can declare its status, as long as L-i-a-r is evaluated as a string of characters, deprived of any semantic implication usually attributed to the term Liar. In this case, a L-i-a-r who declares to be L-i-a-r, is a simple declarer, not forced to declare false sentences. In fact it declares a true sentence, well placed in Classical Logic.

The statement, evaluated on this level, is logically acceptable, is operationally expressible and is not "ineffable".

The other level of evaluation is the semantic one. In this case the term Liar implies a declarer who declares false statements. At this level the statement "I am a Liar", made by a Liar, can only be declared false if one operates outside of Classical Logic. In addition we can say that, at this semantic level, the statement "I am a Liar", made by a Liar, can never be evaluated as true. The statement "I am a Liar" can only be evaluated as true if made by a NON-Liar declarant. So no Liar can actually state a sentence that is true and false at the same time. He can declare a sentence that is considered false on paralogical grounds but he can never declare a true sentence. If this is the case, how can we maintain that the statement "I am a Liar" made by a Liar has the characteristics of a logical paradox?

Therefore our conclusions, based on the tracks of Classical Logic, are simple. We can conclude like this:

1) if the statement is TRUE then the declarer is a L-i-a-r, and vice versa;

2) if the statement is FALSE then the declarer is a NON-L-i-a-r, and vice versa.

The paradox has disappeared. The conclusions are merely counterintuitive. The evaluation is limited, as it must be, to what happens on the logical level, that of Classical Logic (XNOR).