Aristotle and the Paradoxes of Logic

By Gilbert Voeten

#1 Aristotle's Laws of Logic
   #1.1. The Principle of Alternity
   #1.2. The Principle of Antinity
   #1.3. The truth-functional connective NIL
   #1.4. The truth-functional connective VAL
   #1.5. The Conjugation
   #1.6. The Law of the Excluded Middle
   #1.7. Summary

#2. The Antinomies
   #2.1. Russell's Paradox
   #2.2. The Barber Paradox 
   #2.4. Grelling's Paradox
   #2.5. Zeno's Paradoxes


Introduction

At about the same time that set theory began to influence other branches of mathematics, various contradictions, called paradoxes were discovered. A paradox however is only due to a striking violation of at least one of Aristotle's laws of logic. The purpose of the present paper is to provide tools in order to eliminate those contradictions. The material is divided into two sections. In section 1 a more adequate form is given to Aristotle's laws. For that purpose several operators that hitherto were neglected in Boolean logic, are introduced. In section 2 it is shown how easy logical paradoxes can be eliminated if one only seriously takes Aristotle's laws of logic into account.

 

1.   Aristotle's Laws of Logic

Years ago my friend Hubert explained me how formal Logic is a contradictory science because it is not able to deal with the panta rei of life. "Look", he said, "I'm now 37 years old. Once I was a kid of seven. But now I'm not that boy anymore. How must I express this in formal logic? Hubert37 = NOT-Hubert7. But what does that eventually mean? Hubert is not Hubert. And that violates the law of identity."

Immediately after Hubert's departure I started to search for a refutation. And so I contemplated Aristotle's laws of logic.

(1)  (p p)   law of identity.

(2)  [p ( p)] law of noncontradiction.

(3)  p ( p) law of the excluded middle.

The variable p stands here for individual. The law of identity can be written as follows:

[p p] [(p p) (p p)]  

Hence

          ( p p ) (p p )

Thus

(4)  (p p) (p p ) ( p p)

But the conjunction

(5)  p p

violates the law of noncontradiction. Thus the law of identity and the law of noncontradiction seem to contradict each other. So, Hubert was right after all? Took me a long while to find a solution.

Eventually I asked myself: Has the expression ' p' in (2) and (4) the same logical meaning? Suppose that meaning differs. In this case there should be no contradiction at all. To emphasize that semantic difference the only way I saw was to introduce two supplementary connectives for the negation. So I adapted (2) and (5) .

            (2') [p (ANTI- p)]

(5') p (ALTER- p)

And now I'm of course obliged to explain this more fully.

 

1.1.                 The Principle of Alternity

The same individual Pi may have several names: {N1, N2 ... Ni}. The name is NOT the individual. And two different names are NOT the same name. For the above negation 'NOT' I use the connective 'ALTER'.

N1 = ALTER-Pi;  N2 = ALTER- Pi   The name is NOT the individual.

N1 = ALTER- N2;  N2 = ALTER- N1 Different names are NOT the same name.

A concise definition:

(1) DEF (ALTER: Pi, N1, N2...Ni)

By definition (1) we define a set which members are an individual Pi and all its different names Ni. In fact we define a set of distinct but identical elements. In such a set following double negation is permitted.

(2) (ALTER-p) = p

Although the name Ni is not the individual Pi following statement is valid:

(3) Pi Ni

According to definition (1) :

 Ni = ALTER- Pi

Thus from (3)

(4) Pi ALTER- Pi

But ALTER- Pi is a name for Pi. We use double negation to emphasize that individual and name are not the same thing.

  (ALTER-Pi) = Pi

Finally we obtain the congruent tautology:

(5) Pi Pi

We are able to identify an individual with a name while we deny that name and individual are identical.

May be one will object that Pi is also a name. So, we are always referring to a name and not to an individual? To this objection I can only reply with Juliet's words:

What's in a name? that which we call a rose
By any other name would smell as sweet.

The law of identity can now be written as follows:

(6) [p p]  [(p p) (p ALTER- p ) (ALTER- p ALTER- p)]

Hubert's problem deals with the consecutive stages of life. Let Hi be the timeless individual 'Hubert' to whom one always refers; Hp the present Hubert and H7 the kid of seven years old. Consider following definition:

DEF (ALTER: Hi, H7, Hp)

We substitute the above variables into (6)

[Hi Hi] [(Hi Hi) (Hi ALTER- Hi) (ALTER-Hi ALTER- Hi)]

Hence

[Hi Hi] [(Hi Hi) (Hi Hp) (Hp H7)]

Hubert may deny that he is a boy of seven. But it is now clear that the principle of identity is safeguarded. We are able to deny our past and present identity without violating the laws of identity and noncontradiction. Most likely that's what Heraclitus meant with: 'We are and are not'. He saw the principle of alternity in life. Parmenides and Aristotle saw only the principle of antinity. That last principle we will now examine.

 

1.2.                 The Principle of Antinity

Consider two distinct entities P1 and P2. In section 1.1 we have seen that, whenever the definition

(1)  DEF (ALTER: P1, P2)

holds, we must consider both distinct entities as identical. Of course distinct entities are not always identical. For that purpose I use the definition

(2)  DEF (ANTI: P1, P2)

Such that: P1 = ANTI- P2 and P2 = ANTI- P1. What's now the difference in use between ALTER and ANTI? The conjunction

(3)  P1 ALTER- P1

does not violate the law of noncontradiction and this unlike with

(4)  P1 ANTI- P1

which violates the law of noncontradiction. In standard Logic there was hitherto only one way to formalize (3) and (4)

(5)  P1 ( P1)

The conjunction (5) however gives lesser information and is therefore ambiguous. By the definition

(6)  DEF (ANTI: P1, P2, ...Pi)

we define a set of distinct and non-identical entities. Double negation is permitted. 

 

1.3.                 The truth-functional connective NIL

Consider the law of noncontradiction

(1)  [p (ANTI- p)]

What is the function of the operator '' in (1) ? We have to consider that operator as a truth-functional connective. For that purpose I introduce the connective 'NIL'. So, I rewrite (1)

(2)  NIL- [p (ANTI- p)]

Suppose that p and ANTI- p are two different variables. Then we read: 'It is not true that two different variables are one and the same variable'. If p and ANTI- p are contrary statements then we read: 'Two contrary statements are not one and the same statement'. The real meaning of (2) is

(3)  NIL- [p (ANTI- p)] = 0

Here zero indicates that the statement is not valid and therefore must be nullified. By NIL we multiply variables, names and statements by zero. We may compare the connective 'NIL'   to a rubber used to erase false or invalid statements.

 

1.4.                 The truth-functional connective VAL

It is now natural to introduce an operator in order to emphasize the validity of variables and statements. For that purpose I introduce the connective 'VAL'. For instance the statement

(s) Aristotle was a Greek philosopher

is a true statement.

For a false statement p we would of course use

Since multiplication by zero always results in zero we attribute priority to the operator 'NIL'.

Consider the argument

(1)  Sentence (2) is true

(2)  Sentence (1) is false

If we use the operators VAL and NIL

(1)  VAL- (2)

(2)  NIL- (1) 

Hence

(1) VAL- [NIL- (1)] = NIL- (1) = 0

(2) NIL- [VAL -(2)] = NIL- (2) = 0

It will now be clear how void the above argument is. The most familiar Liar Sentence is the following 'self-referential' sentence:

(3)  This sentence is false.

Sentence (3) however coincides with the truth-functional connective 'NIL'! So we can translate:

(3) NIL- (3)

How absurd it was, trying to assign a truth-value to a mere connective! Consider the false arithmetical expression '3 + 4 = 17'. Would you assign a truth-value to the '+' operator?

 

1.5.                 The Conjugation

If we examine the laws of identity and noncontradiction

then we are aware that the deeper meaning is:

The two laws respectively permit and forbid conjugation. But that conjugation doesn't coincide with the conjunction used in Boolean logic. It is a different connective. For that connective I propose the name conjugation and the symbol ''. That symbol we can call conjugator. And so I present our two laws in the following form:        

(1)  VAL- (p ALTER- p)

(2)  NIL- (p ANTI- p)

The only thing that annoys me yet is the fact that (1) is a prescription while (2) is a prohibition. For the sake of conformity I propose to transform (1) into a prohibition.

(3)  NIL- [p ( ) ALTER- p]

What is now the meaning of this? From a pure formal point of view: It is forbidden to apply the negation '' to the conjugator ''. From a semantical point of view: It is forbidden to deny the identity of an entity.

 

1.6.                 The Law of the Excluded Middle

Till now we have not dealt with the law of the excluded middle.

Seen our previous experiences there are three ways to write the contradictory of the law of the excluded middle.

(I)              p   ALTER- p

(II)            p   ANTI- p

(III)          p   NIL- p

Possibility (I) coincides of course with the law of identity and (II) is already forbidden by the law of noncontradiction. Remains (III) . If p is an entity then we read that p can be something and nothing at the same time. Which is absurd of course. If p is a logical proposition then we read that p can be true and false at the same time. Which is also absurd. To avoid those contradictions we ought to state

And that's the form that I will give to the law of the excluded middle.

 

1.7.                 Summary

 

I present you the three daughters of Aristotle in their new dress:  

How striking is the affinity with three sons of Common Sense.  

It's about time to treat the paradoxes that have defied common sense for a long time now.

 

2.   The Antinomies

2.1.                 Russell's Paradox

2.1.1.  Set and Element

Let S  be any non-empty set. By m we denote an element and not a set. Following definition is valid:  

(1)  DEF (ANTI: S, m)

According to the law of noncontradiction (A2)  

(2)  NIL- (S m) =  

2.1.2.  The Paradox 

Let A be the set of all sets that do contain themselves as members. And let Z be the set of all sets which do not contain themselves as members, that is,

(3)  Z = {S | S S}

Does Z belong to itself or not? If Z does not belong to Z then by definition Z does belong to itself. Furthermore, if Z does belong to Z then by definition Z does not belong to itself. In either case we are led to a contradiction.

Although S is a set it is by definition (3) also considered as an element m. By statement (2) however S m = hence Z = . And that means that A is the Set of all sets. We discover the sneaky trick of the paradox: Since the set belongs to any set it does belong to itself but it also belongs to the Set of all sets! Obviously Russell's argument leads to a contradiction because it violates the law of noncontradiction.

 

2.2.                 The Barber Paradox 

The crew of a ship consists only of men. No man let grow a beard. There is also a barber on board who claims that he shaves only and all those men who don't shave themselves. Who shaves the barber? If he doesn't shave himself then he does. And if he shaves himself then he does not.  

By s we denote a man who shaves himself and by b we denote the barber. For the barber holds (when he shaves himself)  

Hence  

The barber is not aware that his claim violates both the laws of noncontradiction and identity. Suppose that he shaves himself. But remember that the barber is not supposed to shave a man who shaves himself. So, the barber cannot support his claim without violating the law of identity.  

Suppose the captain shaves the barber. However it is supposed that the barber is the only man who shaves men who don't shave themselves. So, the barber cannot support his claim without violating the law of noncontradiction.

According to the laws A1 and A2

The barber ought to say: "If you don't count me then I shave all those men who don't shave themselves."

 

2.3.                 The Liar Paradox

Once upon a time a certain Epimenides said:

(M) All Cretans are absolute liars;
(m) I, Epimenides, am a Cretan;
(C) thus I am an absolute liar?

If the above syllogism is valid then holds the identity:

       VAL- [(Epimenides) (Cretan) (Absolute Liar)]

Suppose VAL-M. Could Epimenides utter VAL-M? Could a Cretan    who is an absolute liar    pronounce a true sentence? No! Thus Epimenides was an ANTI-Cretan and he was lying when he pronounced m.
Suppose NIL-M. It doesn't matter now if Epimenides was a Cretan or not: he was lying when he pronounced M.
In both cases Epimenides was a liar. But he was not an absolute liar. It's now easy to see that the two premises cannot be true at the same time without violating the three fundamental laws of logic.

 

2.4.                 Grelling's Paradox

If an adjective truly describes itself, call it 'autological'; otherwise call it 'heterological'. For example, 'polysyllabic' is autological, while 'monosyllabic' is heterological. Is 'heterological' heterological? If it is, then it isn't; if it isn't, then it is.

The definition for adjective: Any of a class of words used to modify a noun.

The word heterological however modifies adjectives but not nouns. So, 'heterological' is not an adjective. Obviously

The paradox is based on following contradiction:

One has to consider such paradoxes as a warning that an element m is neither the member of a set S nor of the complementary set S'.

 

2.5.                 Zeno's Paradoxes

2.5.1.  Achilles and the Tortoise

In a race in which the tortoise has a head start, the swifter-running Achilles can never overtake the tortoise. Before he comes up to the point at which the tortoise started, the tortoise will have got a little way, and so on ad infinitum.

We have to take two relative motions into account.

Mts: The motion of the tortoise relative to the point at which he started. The distance Start-Tortoise is continuous made longer.

Mta: The motion of the tortoise relative to Achilles. The distance Achilles-Tortoise is continuous reduced.

Obviously we have:

The paradox is based on following contradiction:

And according to law A2

 

2.5.2.  The flying Arrow

The flying arrow is at rest. At any given moment it is in a space equal to its own size, and therefore is at rest at that moment. So, it's at rest at all moments.

Here also we have to take two relative motions into account.

       A flying arrow moves relative to an observer. We denote this by VAL-m.

       The space occupied by the arrow is the arrow self. A flying arrow is at rest relative to itself! There is no motion in this case and we denote this by NIL-m.

The paradox is based on following assumption:

And that is a violation of the law of the excluded middle.

 

Conclusion

I hope to have convinced the reader how easy paradoxes can be eliminated on the bench of Aristotle's laws of logic.

 


Gilbert Voeten 2002

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