ON THE LIAR'S PARADOX

THE LIAR'S PARADOX IN ITS FORMAL ESSENCE

“ This sentence is False “

REFERENCE MODEL

The essential characteristics recognized in the Liar's Paradox, expressed in the form "this sentence is false", are mainly three:

1) self-declaring;

2) self-denial;

3) self-reference.

I will try to analyze each of these characteristics in detail to see the effects of these features.

SELF-DECLARING

In search of what is the characterizing element of a self-declaration, what is striking is that the declared object is the declaration itself. To make the self-declaration even more understandable, the sentence "this sentence is false" could be transformed into logically equivalent sentences such as "I sentence am false" or "I am a false sentence".

The evaluation of this sentence is made difficult by the fact that the declared object and the declaration are expressed in a single verbal configuration, namely: "this sentence is false".

The declared object and the declaration are tangled in a single skein. Therefore, the first task to be done will be to "unravel" the skein in order to have a structured sentence that can be analyzed logically.

We must give credit to Alfred Tarski not only for identifying two hidden components in the declaration but also for suggesting the use of two types of language for a thorough analysis of the statement. Tarski spoke of an object language useful for analyzing the declared object and of the need for a metalanguage to formalize the declaration. According to Tarsky, the declaration can evaluate itself as an object only by resorting to a metalanguage.

We believe, however, that the evaluative statement can be expressed in the normal language of Logic. The structure of the evaluative statement, in our opinion, is "((this sentence is false)) is false." The expression in parentheses represents the declared object, the expression outside the parentheses represents the predicate.

By replacing the word "this sentence", present in the declaration, with whatever is declared (i.e. whatever constitutes the declared object), we arrive at this structure of the declaration: "((this sentence is false)) is false".

This conclusion can be reached if we realize that a semantic compression is hidden within the statement. In the Paradox what happens is that more data is inserted into the same amount of space.

The decompression led to the isolation, from the same writing, of two elements that differ by a few letters. We have the object declared: ((this sentence is false)) and we have the declaration "((this sentence is false)) is false". The bold part is common between object and declaration. The Paradox is a compressed form of these two elements.

In computer science, data COMPRESSION refers to the process of reducing the amount of data needed to transmit information.

In Computer Science, data compression works by identifying and eliminating statistical REDUNDANCY. In the Paradox, REDUNDANCY is given by the fact that the phrase "this sentence is false" is repeated, unchanged, twice: it appears both in the declared object and in the declaration.

Without a DECOMPRESSION operation it appears that the truth value of the statement refers to a declaration identical to the declared object. In reality this reference is "apparent". It actually refers to the unzipped form "((this sentence is false)) is false". The expression "is false" works as a predicate in the declaration and as a character string within the object.

Now we can restructure the statement into a form that can be analyzed using the common language used in logic. If we understand the META-meaning hidden in the declaration, we can interpret the declaration in a way that does not need a META-language.

Restructured, the statement looks like this:

1)"((this sentence is false)) " [ declared object ]

2)"((this sentence is false)) is false" [ valuation statement ]

The expression “is false”, inside the object, is to be considered as a character string and the whole object can be equated to a variable name. Outside of the object, "is false" is to be considered as a verbal predicate of the statement.

Now we have a structured statement. Let's see how we can evaluate it.

EVALUATION TOOLS

To evaluate the statement “((this sentence is false)) is false ", we use the XNOR logical operator. This logical operator is not found in logic textbooks but is widely used in electronics textbooks. The XNOR operator is equivalent to the Double Implication operator, which is widely used in logic textbooks. XNOR and Double Implication have the same truth table. The XNOR operator allows a graphical representation of the evaluation process. This operator has two inputs and one output. A graphical representation of this Operator is shown in Fig.1.

Fig-1 shows the graphical representation of the XNOR Operator.

The truth table of a logical operator describes its operation. The XNOR operator is a concordance detector, meaning it outputs a truth value = True whenever both input values are concordant, that is, both true or both false. In any other case (conflicting inputs), the output value is = False.

This XNOR Operator is well suited to the evaluation of a statement. In fact, it evaluates the correspondence or concordance between the truth value declared for the object under examination ( Input1 ) and the truth value actually possessed by the object under examination ( Input2 ). The Input values can be briefly defined as "status" values as they define the characteristic that concerns the object under examination. If there is correspondence between declared status and possessed status, the declaration is True. It is False otherwise.

Thus in the sentence “snow is white” we have that the object in question is made up of snow. The declared status for this object is “white”. If the status possessed by the snow is that of being actually white, or if there is correspondence between the declared status and the status possessed, then we can say that the statement is True. The two input arms of the XNOR Operator receive the status values: one receives the declared value, the other the value actually possessed. The Operator output generates the truth value of the statement. The evaluation is performed according to the rule established by the truth table. If the status values match, then the statement is True. If the Status values diverge, then the statement is False.

Fig 2 shows how the XNOR operator behaves in the evaluation of the statement "Snow is white".

To comment on Fig. 2, let's say that the declared object is Snow. The expression " is white" constitutes the declared status. This status is compared with the status actually possessed by Snow. Since the declared status and the possessed status agree, the truth value of the statement "snow is white" is = True.

Using the XNOR Operator, it is possible not only to determine the truth value of the statement but it is possible to see and follow the logical path of the statement.

SELF-NEGATION

The use of the XNOR operator, chosen to evaluate the Truth value of the declaration, not only is it able to evaluate the truth value of the sentence but it also helps us understand an important thing, namely why the paradox, in order to work, must be formulated by selecting, for the declared status, the value of “False”. In fact, if we assign an input arm, the one representing the declared status, with a truth value set to False, then we see that the XNOR Operator behaves exactly like a NOT Operator. The NOT operator is just what you need to negate a truth value. Excellent!!

The XNOR Operator has offered us two useful functions, one of which was programmed (that of revealing the truth value of the statement) while the other (that of functioning as a NOT Operator) was not initially foreseen: this is a pleasant surprise.

Fig. 3 shows that the XNOR Operator, in the configuration where the declared value has a status = False, behaves the same way as a NOT Operator.

Fig. 3: XNOR Operator behaves the same way as a NOT Operator

One thing to note is that, in this configuration, the truth value (status) declared for the object, when compared with the status actually possessed by the object, determines a sentence truth value (output) that is always the opposite of the possessed status by the object.

The consequence of this configuration, applied to the statement “((this sentence is false)) is false " is the following:

1) If the status owned by the Object " ((this sentence is false))" is = False, then the truth value of the statement is = True;

2) If the status owned by the Object " ((this sentence is false))" is = True, then the truth value of the statement is = False.

As you can see, the truth value of the statement is opposite to the status value actually possessed by the declared object. This disagreement in values is considered paradoxical, but this is not a correct interpretation. Divergent truth values concern different entities that, moreover, operate at different logical levels. It is wrong to confuse the truth value of the declaration with the status value of the object .

To conclude this subsection, let's say that, for the statement to appear paradoxical, it must be self-negating. This is not a coincidence, but rather a necessary condition for the XNOR operator to behave like the NOT operator and thus force the statement to have a value opposite to the declared object's status value. In fact, the XNOR operator, in the configuration where the declared status is set to False, behaves like a NOT operator and is therefore capable of implementing self-negation. Self-negation implies that the truth value of the statement always has a sign opposite to the status really possessed by the object. This divergence in values is interpreted as a paradox. In reality, it is not a paradox because the divergent values do not concern the same logical entity but rather two different logical entities. In conclusion, self-denial is not a sufficient condition to generate a Paradox.

SELF-REFERENCE

Self-reference concerns the property of a statement to refer to itself, either directly or indirectly. In the Liar's paradox, a self-reference can be generated by feeding back into the input the truth value obtained at the output. We said that the XNOR operator behaves the same way as a NOT operator. Figure 4 shows what happens if the output value is fed back into the input of a NOT operator.

Fig.4

When the output value from the XNOR Operator is directly regressed to the input, then an Oscillator is generated. The truth value of the statement no longer has a stable state but continuously varies between True (T) and False (F).

This condition seems paradoxical, but in reality it involves truth values that change over time. In the Epimenides Paradox, the supposed paradoxical element could be due to the fact that, in the same logical entity, there could simultaneously be opposite truth values. In this case of the Liar's Paradox, we have oscillations of the truth value that manifest themselves over TIME. But it is not a real paradox. Self-reference is not a sufficient condition to generate a Paradox. Self-reference generates oscillation.

This type of self-reference is also present in the version of the Paradox attributed to Buridan. In this case, the self-reference mechanism is indirect.

THE LIAR'S PARADOX IN BURIDAN'S VERSION

In Buridan's version, the Paradox "breaks" into two statements and presents itself as follows:

**** Socrates' Statement (SD): "Plato's statement is False";

**** Plato's Statement (PD): "Socrates' statement is True."

These two statements, placed in relation to each other, determine an instability of truth values. This instability has been defined as a Paradox. In reality, we will see that it is due to an oscillation of truth values between True and False. In this version, there are two distinct retrogressions and neither refers directly to itself. Each statement retrogresses to itself through the interposition of the other statement. The scheme of the Paradox according to Buridan's version is shown in Fig. 71. In electronics, this logical scheme is called "sequential," meaning it is a logic that develops "over time." Buridan was already aware of this fact when he stated that the same statement can have different truth values at different times.

Fig. 71 shows a graphical representation of the Buridan's version.

Explanation of Fig. 71

The truth value of the statements is evaluated through XNOR Operators. The output of one XNOR Operator is fed into the input of the other XNOR Operator, in order to create the mutual reference. The XNOR Operator shown above collects Socrates' statement; the bottom one is Plato's statement.

In Socrates' statement , the "status" declared for the truth value of Plato's statement is set = FALSE ( XNOR Operator's red arm ).

Conversely, in Plato's statement, the "status" declared for the truth value of Socrates' statement is set = TRUE ( XNOR Operator's blue arm )

If we analyze the logical function of Socrates' XNOR Operator we see that it behaves like a NOT Operator: at the output, it inverts the logical value it receives from Plato's Logical Operator. Conversely, Plato's Logical Operator repeats as output the logical value it receives from Socrates. Plato's Logical Operator behaves like an ID Operator, an operator that does not invert the logical value it receives as input. This type of operator in electronic jargon is called a "buffer", i.e. a repeater which does not invert the logical signal and which simultaneously enhances the electrical signal. To facilitate the reader's understanding I have arbitrarily called this type of operator with the label "ID" (identity). The colors of the arrows coming out of the Logical Operators indicate the logical status: the red arrow indicates False; the blue arrow indicates TRUE.

Let's see how the logical scheme works. Let's initially assume that the input signal to the Socrates XNOR Operator is FALSE (snippet of red arrow (1)). In these conditions the signal emitted by said Socrates Operator will be = TRUE. Plato receives the TRUE signal and does not modify this logical value at the output. This TRUE logical value is sent once again to Socrates (blue arrow 2). We said that Socrates' XNOR logical operator behaves like a NOT operator. Therefore the output value from said logical operator changes and becomes FALSE. When Plato receives the False signal, he retransmits it unchanged to Socrates but at this point Socrates changes the output signal and outputs a logical value = TRUE. In conclusion, a continuous oscillation of the truth values of the statements is generated.

On the left side of the graph, the equivalent circuit of the logical representation of Socrates and Plato is represented in miniaturized form. The equivalent circuit is very simple: we have a circular structure made up of a NOT Operator concatenated with an ID Operator. This structure is called " ring oscillator ".

In the case of Buridan, an ID operator is inserted in the feedback loop. This does not change the operation of the circuit which continues to swing the same way. This structuring, however, modifies the form in which the Pradox is presented and perceived. Buridan introduces an "innovation" which is well described by Wikipedia:

"during Scholasticism, it had always been thought that the logical problems deriving from the paradox of the liar derived from the character of self-reference. Buridan demonstrated that the problem was not self-reference, but mutual reference, called circular, developing a paradox in which self-reference was, so to speak, broken in two".

Buridan's introduction of the ID Operator has the cunning to change the perception of the Paradox with a formulario that apparently introduces innovations but, in reality, does not change the essence of the problem.

WHY IS BURIDAN'S FORMULATION CONSIDERED A PARADOX?

It should be clear that an Astable Oscillator is not supported by paradoxical logic. Oscillators created using the NOT Operator respect logic and, moreover, work very well. So where did the confusion arise?

The paradoxical interpretation arose from the fact that, in the logical process, the analysts stopped at the "first round" of reasoning. The reasoning went something like this:

" If Plato's statement is false, then Socrates' statement is true. But if Socrates' statement is true, then Plato's statement is also true." STOP. The conclusion, stopped at this point, is clearly paradoxical. But if the analysts had continued their investigation with other logical "rounds," they would have discovered that the essence of Buridan's version is not the Paradox of Truth Values but the continuous oscillation of the truth values of statement.

CONCLUSIONS

In analyzing the Liar's Paradox, we saw that no formal feature, whether referring to self-denial or self-reference, can demonstrate that it is a paradox.

The most difficult part to analyze is the self-declaration.

The self-declaring conceals within itself an information compression mechanism that is not immediately apparent. In its decompressed form, the "self-declared" statement can be analyzed using the normal language of logic, so the use of a metalanguage does not seem necessary. Once we have regenerated an analyzable statement, the characteristics of self-negation and self-reference are easily understood.

Self-denial does not generate a paradox. Self-negation has the effect of causing the truth value of the statement and the status value possessed by the object to have opposite logical values. This divergence of logical values does not constitute a paradox because it refers to different logical entities, which, moreover, operate at a different logical level.

Self-reference also does not generate paradox. Self-reference, whether direct or indirect (see Buridan), generates an oscillation of truth values. This result is determined because the feedback is applied to an XNOR operator, which, in a state of self-negation, behaves like a NOT operator.

The final conclusion is that the statement “this sentence is false,” unpacked and regenerated in the way that has been proposed, should not be considered a Paradox. The essential features of the Liar's Paradox, namely self-denial and self-reference, are not decisive in generating a contradictory statement.

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