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: {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) Ø (ALTER-p) = p
Although the name Ni is not the individual Pi following statement is valid:
(3) Pi Ù Ni
According to definition (1) :
Ni = ALTER- Pi
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 rose
By 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; 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 Ù Hi) Ú (Hi Ù ALTER- Hi) Ú (ALTER-Hi Ù ALTER- Hi)]
Hence
[Hi « Hi] « [(Hi Ù Hi) Ú (Hi Ù Hp) Ú (Hp Ù 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.
Consider
two distinct entities P1 and P2. In section 1.1
we have seen that, whenever the definition
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 Ù
ALTER- P1
does
not violate the law of noncontradiction and this unlike with
(4)
P1 Ù
ANTI- P1
which
violates the law of noncontradiction. In standard Logic there was hitherto only
one way to formalize (3)
and (4)
(5)
P1 Ù
(Ø
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.
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.
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:
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
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 (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.
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.
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.
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)
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."
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
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'.
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
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.
I hope
to have convinced the reader how easy paradoxes can be eliminated on the bench
of Aristotle's laws of logic.
#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
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