Some Variants of Paradox

1) This sentence is false

2) Paradox of the Liar according to Buridan's formulation

PREMISE

Three Values of Truth

In every judgement on the truth value of a sentence, consisting of an object and a predicate, not two but three truth values are involved. The first and the second values concern the object, the third value concerns the sentence.

As an example we have provided a schematic analysis of the statement "2>3 is False".

First value of truth Second value of truth Third value of truth (CALCULATED_VALUE)

inherent truth value declared truth value sentence truth value

"2>3" is inherently False "is False" "2>3 is False" = True

The truth value of the sentence (third value) is a CALCULATED_VALUE.

Such CALCULATED_VALUE derives from the processing of the two truth values (first and second) that must be taken for granted.

The processing of truth values acquired is configured as an operation of this kind:

If there is a match (no difference in the value of truth) between the two INPUT values (declared and intrinsic), then the truth value processed, or the value of the OUTPUT, corresponding to the truth value of the sentence, is True. The truth value of the sentence is not only influenced by the two INPUT values but, to be identified, needs a logic operation taken out on them.

It is also clear that the three truth values are so closely connected that if we know the truth value of the sentence and one of the two INPUT values, the value of the other INPUT can be identified by logic deduction.

Is there a Logic Operator capable of performing this function?

Well, this Logic Operator does in fact exist. For long time it has appeared amongst the instruments used by Logicians and is called XNOR_Operator,

otherwise said NOR-EXCLUSIVE_Operator.

An XNOR_Operator results from the combination of two operators: an XOR_Operator (OR-EXCLUSIVE) followed by a NOT_Operator.

The XOR_Operator is a detector of diversity so it makes a comparison between two truth values (input values) and brings out the value of zero (False) if the truth values being compared are equal, and 1 (True) if the input values being compared are different. "Equal" means that both inputs must have the same logical value, then they will both be with the value of either True or the value False.

The operator that we need, to check the truth value of a sentence, is a detector of non-diversity. This is precisely what the XNOR_Operator does, a logic operator which corresponds to an XOR_Operator chained to a NOT_Operator.

Logic Professors do not use the XNOR Logic Operator. Instead they use the "connector" called DOUBLE IMPLICATION.

The two operators (XNOR and DOUBLE IMPLICATION), although with different names, are equivalent: the truth table of one corresponds exactly to the truth table of the other.

In the drawing here below we show the representation of an XNOR Operator. This representation is used in electronics.

At the end we will have that the Output is the result of an XNOR operation between Input1 and Input2 so we have:

Truth value of the sentence (OUTPUT) = Intrinsic_Truth_Value of the object (INPUT-1) XNOR Declared_Truth_Value of the object (INPUT-2)

So whether the two truth values (intrinsic and declared) are both true or both false, the output from the XNOR logic gate generates in each case a truth value that is True. The sentence is True whenever the XNOR operation produces 1 (True) on output.

The XNOR_Operator is our TRUTH MACHINE. We can arrange (consistently) the three truth values according to the grid or truth-table of the XNOR_Operator, with two separate input and one output:

Intrinsic value Declared value Sentence value

INPUT-1 INPUT-2 OUTPUT

case 1) True True True

case 2) True False False

case 3) False True False

case 4) False False True

The statment "this sentence is false" is usually considered a logic variant of the Epimenide's paradox. This variant is now universally known as the Liar Paradox

In the expression "This sentence is false" there is only a single truth value given to us. This truth value corresponds to the logical value declared with the predicate "is false." In the table this declared value corresponds to INPUT-2. Knowing the logical value of an input-2, the other logical value needed to apply the truth table does not necessarily have to be the logical value of input-1. The other logical value can be the logical value of the sentence. The other logical value may be the answer given by our partner when asked about the truth value of the sentence. By answering "the sentence is True" or "the sentence is False", our partner will provide the other truth value. That value of the answer results from a logical operation and is actually a CALCULATED_VALUE. In the Table, the truth value of the answer provided by our partner corresponds to the OUTPUT. We must point out that if our partner does not know the intrinsic truth value of the object (INPUT-1), he does not have the data to conclude whether the sentence is true or false: he can only guess.

Knowing the logical value of INPUT-2 and the logical value of the statement (OUTPUT), the third value, to be deducted, can not be anything other than the intrinsic truth value of the object, corresponding to INPUT-1. With the availability of a truth table, the deduction will be easy.

So, since in the statement "This sentence is false" the declared value is equal to False, we have only two possibilities of immediate interest.

- If the partner says that the phrase is False then we must look at case 2 of our truth table and the intrinsic truth value of INPUT-1 is True;

- If the partner says that the phrase is True, then we must look at case 4 and the intrinsic truth value of INPUT-1 is False.

As you can see, there is no paradox! Everything goes according to logic because everything meets the requirements of the logical XNOR_Operator.

So, if the sentence is True, then The inherent Truth value (a STATUS value) of the object declared ( 'this sentence ' ) must be False.

On the other hand, if the inherent status of the declared object ( this sentence ) is False, then the Truth value of the sentence( 'this sentence is False ' ) must be True.

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We now have all the tools to analyze the expression "this sentence is false", even when expressed in self-reporting mode.

THIS SENTENCE IS FALSE:

( SENTENCE VIVISECTION )

The statement "this sentence is false" is judged to be different if the sentence is considered self-declaring, of this type: "I am a false sentence". In this case the statement is usually considered paradoxical. The conclusions drawn are of this type:

"if the sentence is false then it is true and.... if it is true then it is false."

THIS SENTENCE IS FALSE will now be interpreted as a self-declaring phrase like

I AM A FALSE SENTENCE

In the self-declared statement, the entire sentence (this sentence is false ) is usually regarded as the declared object, so the expression ' is false' is part of the object. In this case the entire sentence corresponds to the declared object. This way of interpreting the paradox is called a "strengthened" interpretation of the statement. Consequentially the paradox is regarded as the "strengthened Liar paradox".

In this context we can isolate three elements:

THIS SENTENCE IS FALSE ------> STATEMENT (self-declaring)

IS FALSE ---------------------------> PREDICATE

THIS SENTENCE IS FALSE ------> OBJECT DECLARED

This new situation is strange. In fact the expression ' is false' is not logically separated from the object (as usually happens) but is embedded in the object itself. This doesn't happen in a common statement. Embedded in the object, the expression "is false" must be considered a simple string of characters.

In order to consider 'is false' as the predicate of this sentence, the declaration should usually look like this:

THIS SENTENCE IS FALSE (object declared) is false .

This is the actual (real) format of the statement, the one that must be subjected to logical analysis.

The expression "is false" appears twice, once incorporated into the declared object and a second time as a predicate of the declaration.

The only way to get out of this confusion is to consider the underlined expression as an identifying element of the declarant or as a possible name of the declaring phrase.

Only at this point can we glimpse a certain parallelism between Epimenides' statement and that of the sentence.

Epimenides declaring says "I Epimenides declaring I am a liar" while the declaring sentence says "I declaring sentence am false (the name of the declared object) , I am false".

The idea that this statement is paradoxical hides a subtle error that I will try to show with the graphic example shown in the Figures below, Figure 45 A and Figure 45B.

In Figure 45 A, in the image above, there is a self-declaring sentence that declares "this sentence is written in FreeMono font". The phrase declares its specific quality or its specific STATUS.

In Figure 45 A the statement “this sentence is written in FreeMono font” is expanded (almost vivisected) and exposed into its constituent components. The statement "this sentence is written in FreeMono font", once developed, is broken down into two parts, namely the declared object and the predicate (it is written in FreeMono font). But there is a peculiarity. The amazing feature of self-declaration is that the declared Object is the entire statement, so the real declared object is the entire sentence i.e. "this sentence is written in FreeMono font". At this point all that remains is to compare, in the XNOR Operaor, the declared status (written in FreeMono font) with the status actually possessed by the object. Since declared status and possessed status (actually written in FreeMono font) correspond, the statement "this sentence is written in FreeMono font" is TRUE, as shown by the dotted arrow at the top of the figure. It is of fundamental importance to understand that the inferred truth value (TRUE) refers to the statement "this sentence is written in FreeMono font" and not to the declared object "this sentence is written in FreeMono font".

Another characteristic of this paradox consists in the fact that the status of the declarant (written in FreeMono font) does not pre-exist the declaration but is configured at the moment of the declaration or more precisely in the declaration.

In Figure 45 B the self-declaring sentence "this sentence is false" is developed. The analysis procedure is developed along the lines of Figure 45 A. In this case we have that the statement consists of "this sentence is false" and similarly the declared object consists of "this sentence is false". Now we do not have objective confirmation to know whether the status actually possessed by the declared object corresponds to the declared status or not (False). In Figure 45 B we hypothesized that the status actually possessed is = False. In this case the XNOR Operator tells us that the truth value of the statement is = TRUE. Also in this case there is no need to get confused: the truth value found refers to the statement and not to the object. The fact that both have the same verbal structure leads to confusion and suggests a paradox.

In reality we cannot say "if the sentence is true then the sentence is false". It must be said:

if the status value of the object "this sentence is false" is really = False, then the truth value of the sentence "this sentence is false" is = True.

The truth values diverge but do not refer to the same logical entity.

We have two possibilities:

1) If we assume that the declared object (I am a false sentence) actually has the status of FALSE, then we must conclude that the declaration (I am a false sentence) is TRUE.

2) If we assume that the declared object (I am a false sentence) actually has the status of TRUE, then we must conclude that the statement (I am a false sentence) is FALSE.

It should be kept in mind that the value of False, declared in the sentence, is only a declaration of status and cannot claim to be extended as the truth value of the declaration. Considering the predicate "is false" as referring to the truth value of the statement is a fatal error of evaluation for the interpretation of the paradox.

Very Important note:

1) No self-declaring sentence can declare the truth value of the statement it is making!!!!

2) A self-declaring sentence can only self-declare a status value ( INPUT ).

3) The status value of the declared object and the truth value of the statement are placed at different logical levels.

The status value of the declared object is an input, the truth value of the declaration is an output.

CAN THE TRUTH VALUE OF THE SENTENCE OSCILLATE BETWEEN TRUE AND FALSE?

Returning to the conventional interpretation which considers the statement "this sentence is false" as a paradox, it is necessary to understand how the belief has been consolidated which states "if the sentence is true then it is false... and if it is false then it is true."

The genesis of this circular deduction can be identified in the graphic representation of Figure 62, and Figure 71 shown below.

Figure 71 is a graphic representation of the Liar's Paradox according to the statement made by Burudan:

Socrates says: "Plato's statement is False";

Plato says: "Socrates' statement is true".

In Figure 62 we see that this oscillation can actually be triggered if the status value of the declared objectis is not defined autonomously but is derived and assimilated to the truth value of the sentence. If the truth value of the sentence, expressed as OUTPUT, is fed back as INPUT to define the STATUS value really possessed by the object, then the oscillation is triggered. Under these conditions the XNOR logical operator becomes an oscillator that sequentially generates opposite truth values. The XNOR Operator behaves like a NOT Operator where the logical output value is fed back into the input.

While in the classic version of the Epimenides Paradox the essence of the paradox was constituted by the fact that the statement appeared, at the same time, True and False, in the version represented in Figure 62 and Figure 71 an alternation of opposing truth values is generated.

Fig 62

This is a graphic representation of the Logic contained in the statement "this sentence is false" and which generates an oscillation of the truth value between True and False

BURIDAN PARADOX EXPLANATION

Paradox of the Liar according to Buridan's formulation:

Socrates states: "Plato's statement is False";

Plato states: "Socrates' statement is True"

The combination of these two statements generates a cross-reference that causes a continuous alternation of truth values.

In Fig. 71, drawn below, we have a graphic representation of the Buridan Paradox and the logic that controls it

Explanation of Figure 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. Conversely, in Plato's statement the "status" declared for the truth value of Socrates' statement is set = TRUE.

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". 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 (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.

In its essential form an Oscillator can be constructed with a single NOT Operator in which the output signal is fed back directly into the input (see Fig. 62).

In the case of Buridan ( see Fig. 71 ) an ID operator is inserted in the feedback loop. This does not change the operation of the circuit which continues to oscillate in 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 statements.

CONCLUSIONS

If we look for the distinctive characteristic associated with a self-declaring phrase, what can we say?

This thought comes to mind: the declared object is the declaration itself!

It therefore turns out that the verbal composition of the declared object (I am a false sentence) is identical to that of the declaration (I am a false sentence). This is a verbal coincidence generated by self-declaration and this fact creates a confusion that is the basis of the Paradox. If the status value of the declared object (I am a false sentence) is actually false, the mind is led to think that the truth value of the declaration (I am a false sentence) must be false. The mind thinks they are the same thing, since they have the same verbal structure (I am a false sentence). And yet they are not the same thing. They are different logical entities and, on a logical level, no confusion is permitted.

Now we have the elements to say that the conclusion "if the sentence is true then it is false and..... if the sentence is false then it is true" is based on incorrect logical evaluations.

A correct logical evaluation, based on the XNOR Operator, should say like this:

1) "if the status value of the declared object (this sentence is false) is actually False, then the truth value of the statement (this sentence is false) is True";

2) "if the status value of the declared object (this sentence is false) is actually True, then the truth value of the statement (this sentence is false) is False".

The truth value of the sentence emerges on a plane that overrides the plane of the truth value of the declared object.

The logical contrast does not exist because it is not the same logical entity that has divergent truth values.

The two truth values (status value of the declared object and truth value of the statement) cannot be confused because they indicate different entities and, above all, because they operate at a different logical level.

Additionally, if we then interpret the Liar's Paradox as a logical structure where the truth value of the sentence (Output) is fed back into the Input as "Status" of the declared object, then we have a continuous oscillation of the truth value, as shown in Fig. 62 and Fig. 71.

NOTE: In no treatise on Electronics will you find that the Logic that supports an Oscillator is defined as "paradoxical".

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