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

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.

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) Ø
[*p* Ù
(Ø
*p*)] — *law of noncontradiction.*

(3) *p *Ú (Ø *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*) Ù (*p *¬ *p*)]

Hence

(Ø *p* Ú *p *) Ù (*p* Ú Ø *p *)

Thus

(4) (*p *Ù* p*) Ú (*p* Ù Ø* p *) Ú (Ø *p *Ù* *Ø *p*)

But the
conjunction

(5) *p* Ù
Ø* 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 'Ø *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')
Ø [*p* Ù (ANTI- *p*)]

(5') *p* Ù (ALTER-* p*)

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

The same individual *Pi* may have
several names: {*N _{1}, N_{2} ... N_{i}*}. 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'.

*N _{1} *= ALTER-

*N _{1}*
= ALTER-

A concise definition:

(1) DEF (ALTER: *Pi*, *N _{1},
N_{2}...N_{i}*)

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

(2)
Ø (ALTER-*p*) = *p*

Although the name *N _{i}* is
not the individual

(3) *Pi* Ù *N _{i}*

According to definition (1) :

*N _{i}* = ALTER-

Thus from (3)

(4) *Pi* Ù 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.

Ø (ALTER-*Pi*) = *Pi*

Finally we obtain the congruent tautology:

(5) *Pi
*Ù *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 roseBy any other name would smell as sweet.
*

The law of identity can now be written as follows:

(6) [*p* « *p*] « [(*p *Ù* p*) Ú (*p* Ù ALTER-* p *) Ú (ALTER- *p *Ù* *ALTER- *p*)]

Hubert's problem deals with the consecutive
stages of life. Let *Hi* be the *timeless *individual 'Hubert' to whom
one always refers; *H _{p}* the present Hubert and

DEF (ALTER: *Hi, H _{7},
H_{p}*)

We substitute the above variables into (6)

[*Hi* « *Hi*] « [(*Hi *Ù* Hi*) Ú (*Hi* Ù ALTER-* Hi*) Ú (ALTER-*Hi *Ù* *ALTER- *Hi*)]

Hence

[*Hi* « *Hi*] « [(*Hi *Ù* Hi*) Ú (*Hi* Ù* H _{p}*) Ú (

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.

Consider
two distinct entities *P _{1} *and

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: *P _{1},
P_{2}*)

Such that:
*P _{1} *= ANTI-

(3) *P _{1 }*Ù
ALTER-

does not
violate the law of noncontradiction and this unlike with

(4) *P _{1 }*Ù ANTI-

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

(5) *P _{1 }*Ù
(Ø

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

(6) DEF (ANTI: *P _{1}, P_{2},
...P_{i}*)

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

- Ø (ANTI-
*p*) =*p*

Consider
the law of noncontradiction

(1) Ø
[*p* Ù
(ANTI- *p*)]

What is the function of the operator
'Ø' 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*
Ù
(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* Ù (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.

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.

- VAL-
*s*

For a
false statement *p *we would of course use

- NIL-
*p*

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

- VAL- (NIL-
*p*) = NIL-*p* - NIL- (VAL-
*p*) = NIL-*p*

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:

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?

If we
examine the laws of identity and noncontradiction

- (
*p*Ù*p*) Ú (*p*Ù ALTER-*p*) Ú (ALTER-*p*Ù*p*) - NIL- (
*p*Ù ANTI-*p*)

then we
are aware that the deeper meaning is:

- (
*p*=*p*) Ú (*p*= ALTER-*p*) Ú (ALTER-*p*=*p*) *p*¹*p*

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*
(Ø ¤) ALTER- *p*]

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

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

*p*Ú (Ø*p*)

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

- NIL- (VAL-
*p*¤ NIL-*p*)

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

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

*A1:*NIL- [*p*(Ø ¤) ALTER-*p*]*A2:*NIL- [*p*¤ ANTI-*p*]*A3:*NIL- [VAL-*p*¤ NIL-*p*]

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

- NIL- (I am not I)
- NIL- (I am somebody and somebody
else)
- NIL- (I am somebody and nobody)

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

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*) = Æ

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 *Ï
*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 **=
*Æ hence
*Z* = Æ. And
that means that *A* is the Set of all sets. We discover the sneaky trick of
the paradox: Since the set Æ 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.

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)

- DEF (ALTER:
*b, s*)

Hence

- VAL-
(
*b*¤*s*)

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.

*b*(Ø¤)*s*

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.

*b*¤ ANTI-*b*

According
to the laws *A1 *and* A2*

- NIL- [
*b*(Ø¤)*s*] - NIL- [
*b*¤ ANTI-*b*].

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

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 ¾ who is an absolute liar ¾ 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.

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

- DEF (ANTI:
*adjective, heterological*)

The paradox is based on following
contradiction:

*Adjective*¤ ANTI-*Adjective*

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'*.

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:

- DEF (ANTI:
*Mts, Mta*)

The paradox is based on following
contradiction:

*Mts*¤*Mta*

And
according to law *A2 *

- NIL- (
*Mts*¤*Mta*)

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:

- VAL-
*m*¤*m*

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

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