Heinrich C. Kuhn wrote:
> Alexander Zenkin wrote many interesting lines, most
> of which I read with great interest. However: it may
> be a missunderstanding, but I have some difficulties
> with the following statement:
>
> > POSTULATE 2. If a premise A in the formal inference
> > A =3D=3D > B is FALSE,
> > then any its consequence B is necessarily FALSE too, or shortly:
> > FALSE =3D=3D> FALSE (only!). IFF the Aristotle's Logic deductive rule=
s are
> > applied correctly.
> >
> > Why did Great Aristotle's not formulate explicitly such obvious
> > Postulate 2? I think the reason is purely psychological:
>
> I have some doubts. I'd suggest that Arsitotle did not
> formulate it, because he believed it to be false. Aristotle's
> logic is a syllogistic one. Each syllogism has *two* premises.
> And a syllogism with a wrong premise can nevertheless render a
> true result:
>
> <Example 1>
> All Renaissance philosophers where great mathematicians (F)
> Cardano was a Renaissance philosopher (T)
> Cardano was a great mathematician (T)
> </Example 1>
>
> And even syllogisms with two false premises can have a true
> result:
>
> <Example 2>
> Palatinate philosophers are and were always great mathematicians (F)
> Cardano was a Palatinate philosopher (F)
> Cardano was a great mathematician (T)
> </Example 2>
In my new logical Super-Induction method {see: A.A.Zenkin,
Superinduction: A New Method For Proving General Mathematical Statements
With A Computer. - Doklady Mathematics, Vol.55, No.3, pp. 410-413 (1997).
Translated from Doklady Akademii Nauk, Vol 354, No. 5, 1997, pp. 587 -
589.}, I introduced the following generalization of the traditional logic=
al
notion "ALL":
"ALL, EXCEPT FOR an explicitly defined FINITE set of counter-examples=
".
That generalization of the habitual logical notion "ALL" allows highl=
y
successfully to solve common Number Theory problem of the kind "Whether P=
(n)
holds for ALL n ?" (where P(n) is a number-theoretical predicate defined =
for
all natural numbers n >=3D 1) even if there is a FINITE set of exceptions
(counter-examples) for P(n). In Mathematics, such the generalization work=
s
fine. I think the following variant of your counter-example 1 will be qui=
te
interesting from that point of view:
<Example 1'>
ALL Renaissance philosophers (EXCEPT FOR Cardano only!) were great
mathematicians (L)
Cardano was a Renaissance philosopher (T)
Cardano was a great mathematician (T)
</Example 1'>
That is, by not great wish, it is possible to introduce in classical
syllogisms not only false premises, but even contradictory ones. Is not s=
o?
: -)
Now, about your objection.
At the end of my message (Subject: RE: As to Aristotel's "Infinitum
Actu Non Datur" Thesis; Date: Wed, 24 Feb 1999 02:54:11), I write:
The huge scientific experience during about 2300 years showed that th=
e
"inference" FALSE =3D=3D > TRUE, can be realized in the following two ca=
ses
only:
a) when A is a non-essential (i.e., removable) "premise" for B, and i=
n
such the case the "premise" A must be, simply and smoothly, removed from
such the "proof". So, we have here the trivial gross logical error "It do=
es
not follow" (in Russian "Ne sleduet");
b) in the proof process itself, the other (usually, much more fine)
logical error takes place - the error "substitution of notions (terms) " =
(in
Russian "Podmena ponjatii").
It allows to state that the "inference" ("proof") of the kind FALSE =
=3D=3D
> TRUE is possible in Classical Logic or in Classical Mathematics, based
upon that Logic, only if such the "inference" ("proof") breaks grossly th=
e
main deductive inference rules of Aristotel's Logic.
Then I formulated the following quite strong empirical statement
(axiom).
POSTULATE 2. If a premis A in the formal inference A =3D=3D > B is FA=
LSE,
then any its consequence B is necessarily FALSE too, or shortly: FALSE =3D=
=3D >
FALSE (only!). IFF the Aristotel Logic deductive rules are applyed
correctly.
Two your counter-examples (and a lot of alike ones) from the Aristotl=
e's
syllogistics area, seemingly, disprove my POSTULATE 2 in that area (remar=
k
that the syllogistics is not the all Aristotle's Logics). There are two
obvious ways: either my POSTULATE 2 is wrong for the Aristotle's
syllogistics, or your (and alike) counter-examples are some incorrect. At
any case, as a professional scientist with a length of service, I would n=
ot
like that it were possible to obtain the scientific TRUTH from a FALSE ev=
er
and anywhere in the science (since any false is non-scientific). I am sur=
e
all scientists support me in that. Therefore I am forced to disprove your
(and alike ) counter-examples as to the Aristotle's syllogistics area.
Let us consider some other version of your (counter-)example 1.
<Example 1a>
{1} 10% of all Renaissance philosophers were great mathematicians (almost=
T
?)
{2} Cardano was a Renaissance philosopher (T)
{3} Cardano was a great mathematician (T)
</Example 1a>
Though the conclusion {3} may be true, every Aristotle's syllogistics
expert will claim that here {3} simply does not result from this syllogis=
m,
i.e., that "inference" contains the logical error "it does not follow" (i=
n
Russian "ne sleduet"), because the Example 1a roughly violates the main
logical rule of the given syllogistic figure usage, viz the first premise
{1} must be here a common assertion. And he will be certrainly right: it =
is
inadmissible in science to violate the laws of the Classical Logic and th=
e
RULES and the CONDITIONS of their correct USAGE.
REMARK. Moreover, a QUANTITY (90%, 50%, or 1%) of "great mathematicia=
ns"
among "Renaissance philosophers" is not a direct real reason why "Cardano
was <or became> a great mathematician", so that, in Exs 1, 1a, the
conclusions {3} does not follow from the premise {1} even from the common
informal point of view.
Now, what is the main logical rule for the correct usage of Aristotle=
's
syllogistics as a whole? Aristotle states: the syllogistics gives the
RELIABLE TRUTH, IF AND ONLY IF (IFF) all premises are RELIABLY TRUE. Just
that "IFF" allows us to state that the application itself of the Aristotl=
e's
syllogistics to FALS premises is a rough logical error of the type "it do=
es
not follow".
So, I state that the Aristotle's syllogistics is defined on the true
premises set only. For the false or contradictory premises, the Aristotle=
's
syllogistics simply is not defined, i.e., any conclusions obtained by mea=
ns
of the application of the Aristotle's syllogistics to false or contradict=
ory
premises have no logical sense. Whether such the statement is too strong?=
I
don't think so, because alike "algorithmical" limitations meet in
Mathematics and Logic quite frequently. Here are two examples.
1) MATHEMATICS. Consider the REAL function, say, sqrt(x) of the real
variable x. It is obvious that the function sqrt(x) is defined for x >=3D=
0
only. For x<0 the REAL function sqrt(x) is not defined, i.e. it has no
mathematical sense for x<0.
2) MATHEMATICAL LOGIC. In the most modern formal system, there is the
modus ponens rule: IF A is a deducible formula (i.e., a true one by an
interpretation), and IF A =3D=3D > B is a deducible formula, THEN B is a
deducible formula too. It means that the modus ponens rule is defined on =
the
deducible formulas set only. To apply the modus ponens rule to non-deduci=
ble
(in a framework of a given formal system) formulas (i.e., to false or
contradictory ones by an interpretation), in my opinion, yet took into
nobody's scientific head.
Summarizing the said above, I formulate the
POSTULATE 3. The Aristotle's syllogistics as a whole is defined on the
authentic true premises set only. For the false or contradictory premises=
,
the Aristotle's syllogistics simply is not defined, i.e., ANY conclusions
obtained by means of the application of the Aristotle's syllogistics to
false or contradictory premises have no logical sense.
But often they say: we not always sure that all premises are true.
Well, in such the case, the question arises: for what aim is such the pow=
er
logical tool as the Aristotle's syllogistics, which, by Aristotle's "IFF"=
,
works correctly only on the authentic true premises set, applyed to
unreliable (in particular, false or contradictory) premises? - In order t=
o
obtain (by virtue of that Aristotle's "IFF") a new unreliable (in
particular, false or contradictory) knowledge =85 ?
- I think the Aristotle's syllogistics leads never to errors in real Scie=
nce
just because every scientist, following to the Aristotle's "IFF", in the
beginning PROVES the TRUTH of ALL his premises, and only after that he us=
es
corresponding figures of the Aristotle's syllogistics in order to obtain =
a
new authentic scientific TRUTH.
In one word, I believe that a lot of "counter-examples" (like your
Examples 1 and 2), well known in syllogistics, roughly violate the main
logical rule of the correct usage of the syllogistics itself. All such
"inferences" of the kind FALSE =3D=3D > TRUTH are simply not a correct lo=
gical
deductions, i.e., all they contain the error "it (i.e., any conclusion) d=
oes
not logically follow (from such false premises)".
Therefore Aristotle, I think, completely could state that it is the
FALSE only what can DEDUCTIVELY follows a FALSE, or shortly, FALSE =3D=3D=
>
FALSE. That is Aristotle himself could formulate my POSTULATES 2 and 3 :
-).
You write further:
> Besides: I have the impression that when writing about the
> impossibility of infinity Aristotle's main focus was on
> *physics* and *not* on mathematics. AZ wrote that in Aristotle's
> context there was a "very abstract problem as whether the
> Infinity is actual or potential": I'd assume that for
> Aristotle it was not a "very abstract" problem. It concerns
> e.g. the question whether movement is possible at all (remember
> the running contest with the turtle ...) ... .
You are absolutely right as to "Aristotle's main focus was on *physic=
s*
and *not* on mathematics". Why? Because in Aristotle's time, just the
(speculative, and therefore "naive") *physics* was an only understandable=
to
all scientific language. But I think when Aristotle, taking up the Infini=
ty,
speaks even about the physical "air", he means some abstraction filling s=
ome
abstract space, but not a concrete physical mixture of O2, N2, CO2, and s=
o
on under a given temperature, humidity and pressure, i.e., he uses the
"air" and "space" just as abstract concepts.
As to such the problem as "whether the Infinity is actual or potentia=
l",
I believe that these notions even today are the most abstract notions of
modern Mathematics.
As to "Achilles and the Turtle ". I don't remember who first said that
Achilles and the Turtle make their "running contest " well, but it is a
human-beeing only who takes the real initial distance between them,
transforms that real distance, in his abstract imagination, into an abstr=
act
segment, say, [0,1], and begins to divide it abstractly in his abstract
imagination by means of an abstract bisection process up to the abstract =
(of
course, potential) Infinitum. Since, according to Aristotle (and Samuel S.
Kutler : -) ), "A Dia Ex Hodos =3D No way out by going through", Zeno (an=
d we)
obtains the abstract result: the abstract movement in our abstract
imagination is, abstractly, impossible at all.
Thank you very much for your non-trivial, deep and very interesting
objections.
Other objections and comments are welcome.
LAST REMARK.
Why do I believe that POSTULATE 2 has the important value for the modern
science?
Today, there is a lot of grandiose projects-XXI and programs-XXI concerni=
ng
a global formalization of all mathematical knowledge-XX. The last was
historically obtained and is based today on the Classical Aristotle's Log=
ic.
But all these grandiose projects-XXI and programs-XXI use the formal syst=
ems
technique of the modern meta-mathematics. It is naturally by obvious
reasons. But I state that the modern meta-mathematics which, in contrast =
to
the POSTULATE 2, considers the inferences *FALSE =3D=3D > TRUTH* and *A,=
not-A
|- B* as the basic true inferences, is a not quite adequate description
(formalization) of the Classical Aristotle's Logic. I am sure that in the
near future, an essential improvement of the modern meta-mathematics as a
whole lies ahead. Else it will be able to prevent to achieve that high ai=
ms
which were formulated in these grandiose projects and programs of the XXI
Centure.
Regards
A.Z.
############################################
Prof. Alexander A. Zenkin,
Doctor of Physical and Mathematical Sciences,
Leading Research Scientist of the Computer Center
of the Russian Academy of Sciences.
e-mail: alexzen@com2com.ru
WEB-Site http://www.com2com.ru/alexzen
############################################
"Infinitum Actu Non Datur" - Aristotel.
--------------6FA533A27C8787B3DF456283
Content-Type: text/html; charset=koi8-r
Content-Transfer-Encoding: quoted-printable
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CAA29205
Heinrich C. Kuhn wrote:
Alexander Zenkin wrote many interesting lines, mo= st
of which I read with great interest. However: it may
be a missunderstanding, but I have some difficulties
with the following statement:> POSTULATE 2. If a premise A in the formal inference
> A =3D=3D > B is FALSE,
> then any its consequence B is necessarily FALSE too, or shortly:
> FALSE =3D=3D> FALSE (only!). IFF the Aristotle's Logic deductive ru= les are
> applied correctly.
>
> Why did Great Aristotle's not formulate exp= licitly such obvious
> Postulate 2? I think the reason is purely psychological:I have some doubts. I'd suggest that Arsitotle did not
formulate it, because he believed it to be false. Aristotle's
logic is a syllogistic one. Each syllogism has *two* premises.
And a syllogism with a wrong premise can nevertheless render a
true result:<Example 1>
All Renaissance philosophers where great mathematicians (F)
Cardano was a Renaissance philosopher (T)
Cardano was a great mathematician (T)
</Example 1>And even syllogisms with two false premises can have a true
result:<Example 2>
Palatinate philosophers are and were always great mathematicians (F)
Cardano was a Palatinate philosopher (F)
Cardano was a great mathematician (T)
</Example 2>
In my new logical Super-Induction method {see: A.A.=
Zenkin,
Superinduction: A New Method For Proving General Mathematical Statements
With A Computer. - Doklady Mathematics, Vol.55, No.3, pp. 410-413 (1997).
Translated from Doklady Akademii Nauk, Vol 354, No. 5, 1997, pp. 587 -
589.}, I introduced the following generalization of the traditional logic=
al
notion "ALL":
"ALL, EXCEPT FOR an explicitly defined FINITE set
of counter-examples".
That generalization of the habitual logical notion
"ALL" allows highly successfully to solve common Number Theory problem
of the kind "Whether P(n) holds for ALL n ?" (where P(n) is a number-theo=
retical
predicate defined for all natural numbers n >=3D 1) even if there is a FI=
NITE
set of exceptions (counter-examples) for P(n). In Mathematics, such the
generalization works fine. I think the following variant of your counter-=
example
1 will be quite interesting from that point of view:
<Example 1'>
ALL Renaissance philosophers (EXCEPT FOR Cardano only!) were great
mathematicians (L)
Cardano was a Renaissance philosopher (T)
Cardano was a great mathematician (T)
</Example 1'>
That is, by not great wish, it is possible to introduce in classical syllogisms not only false premises, but even contradictory ones. Is not so? : -)
Now, about your objection.
At the end of my message (Subject: RE: As to Aristotel's "Infinitum Actu Non Datur" Thesis; Date: Wed, 24 Feb 1999 02:= 54:11), I write:
The huge scientific experience during about 2300
years showed that the "inference" FALSE =3D=3D > TRUE, can be reali=
zed
in the following two cases only:
a) when A is a non-essential (i.e., removable) "pr=
emise"
for B, and in such the case the "premise" A must be, simply and smoothly,
removed from such the "proof". So, we have here the trivial gross logical
error "It does not follow" (in Russian "Ne sleduet");
b) in the proof process itself, the other (usually=
,
much more fine) logical error takes place - the error "substitution of
notions (terms) " (in Russian "Podmena ponjatii").
It allows to state that the "inference" ("proof")
of the kind FALSE =3D=3D > TRUE is possible in Classical Logic or i=
n
Classical Mathematics, based upon that Logic, only if such the "inference=
"
("proof") breaks grossly the main deductive inference rules of Aristotel'=
s
Logic.
Then I formulated the following quite strong empir=
ical
statement (axiom).
POSTULATE 2. If a premis A in the formal inference A =3D=3D > B is FALSE, then any its= consequence B is necessarily FALSE too, or shortly: FALSE =3D=3D > FALSE (only!). IFF= the Aristotel Logic deductive rules are applyed correctly.
Two your counter-examples (and a lot of alike ones) from the Aristotle's syllogistics area, seemingly, disprove my POSTULATE 2 in that area (remark that the syllogistics is not the all Aristotle's Logics). There are two obvious ways: either my POSTULATE 2 is wrong for the Aristotle's syllogistics, or your (and alike) counter-examples are some incorrect. At any case, as a professional scientist with a length of service, I would not like that it were possible to obtain the scientif= ic TRUTH from a FALSE ever and anywhere in the science (since any false is non-scientific). I am sure all scientists support me in that. Therefore I am forced to disprove your (and alike ) counter-examples as to the Aris= totle's syllogistics area.
Let us consider some other version of your (counter-)example 1.
<Example 1a>
{1} 10% of all Renaissance philosophers were great mathematicians (al=
most
T ?)
{2} Cardano was a Renaissance philosopher (T)
{3} Cardano was a great mathematician (T)
</Example 1a>
Though the conclusion {3} may be true, every Aristo= tle's syllogistics expert will claim that here {3} simply does not result from this syllogism, i.e., that "inference" contains the logical error "it doe= s not follow" (in Russian "ne sleduet"), because the Example 1a roughly vio= lates the main logical rule of the given syllogistic figure usage, viz the firs= t premise {1} must be here a common assertion. And he will be certrainly right: it is inadmissible in science to violate the laws of the Classical Logic and the RULES and the CONDITIONS of their correct USAGE.
REMARK. Moreover, a QUANTITY (90%, 50%, or 1%) of "great mathematicians" among "Renaissance philosophers" is not a direct real reason why "Cardano was <or became> a great mathematician", so that, in Exs 1, 1a, the conclusions {3} does not follow from the premise {1} even from the common informal point of view.
Now, what is the main logical rule for the correct
usage of Aristotle's syllogistics as a whole? Aristotle states: the syllo=
gistics
gives the RELIABLE TRUTH, IF AND ONLY IF (IFF) all premises are RELIABLY
TRUE. Just that "IFF" allows us to state that the application itself of
the Aristotle's syllogistics to FALS premises is a rough logical error
of the type "it does not follow".
So, I state that the Aristotle's syllogistics is
defined on the true premises set only. For the false or contradictory pre=
mises,
the Aristotle's syllogistics simply is not defined, i.e., any conclusions
obtained by means of the application of the Aristotle's syllogistics to
false or contradictory premises have no logical sense. Whether such the
statement is too strong? I don't think so, because alike "algorithmical"
limitations meet in Mathematics and Logic quite frequently. Here are two
examples.
1) MATHEMATICS. Consider the REAL function, say, sqrt(x)
of the real variable x. It is obvious that the function sqrt(x) is define=
d
for x >=3D0 only. For x<0 the REAL function sqrt(x) is not defined, i.=
e.
it has no mathematical sense for x<0.
2) MATHEMATICAL LOGIC. In the most modern formal system,
there is the modus ponens rule: IF A is a deducible formula (i.e., a true
one by an interpretation), and IF A =3D=3D > B is a deducible formula, TH=
EN
B is a deducible formula too. It means that the modus ponens rule is defi=
ned
on the deducible formulas set only. To apply the modus ponens rule to non=
-deducible
(in a framework of a given formal system) formulas (i.e., to false or con=
tradictory
ones by an interpretation), in my opinion, yet took into nobody's scienti=
fic
head.
Summarizing the said above, I formulate the
POSTULATE 3. The Aristotle's syllogistics as a whole is defined on the authentic true prem= ises set only. For the false or contradictory premises, the Aristotle's syllog= istics simply is not defined, i.e., ANY conclusions obtained by means of the app= lication of the Aristotle's syllogistics to false or contradictory premises have no logical sense.
But often they say: we not always sure that
all premises are true. Well, in such the case, the question arises: for
what aim is such the power logical tool as the Aristotle's syllogistics,
which, by Aristotle's "IFF", works correctly only on the authentic true
premises set, applyed to unreliable (in particular, false or contradictor=
y)
premises? - In order to obtain (by virtue of that Aristotle's "IFF") a
new unreliable (in particular, false or contradictory) knowledge =85 ?
- I think the Aristotle's syllogistics leads never to errors in real
Science just because every scientist, following to the Aristotle's "IFF",
in the beginning PROVES the TRUTH of ALL his premises, and only after tha=
t
he uses corresponding figures of the Aristotle's syllogistics in order
to obtain a new authentic scientific TRUTH.
In one word, I believe that a lot of "counter-exam=
ples"
(like your Examples 1 and 2), well known in syllogistics, roughly violate
the main logical rule of the correct usage of the syllogistics itself.
All such "inferences" of the kind FALSE =3D=3D > TRUTH are simply not a c=
orrect
logical deductions, i.e., all they contain the error "it (i.e., any concl=
usion)
does not logically follow (from such false premises)".
Therefore Aristotle, I think, completely could stat= e that it is the FALSE only what can DEDUCTIVELY follows a FALSE, or shortl= y, FALSE =3D=3D > FALSE. That is Aristotle himself could formulate my POSTUL= ATES 2 and 3 : -).
You write further:
Besides: I have the impression that when writing about the
impossibility of infinity Aristotle's main focus was on
*physics* and *not* on mathematics. AZ wrote that in Aristotle's
context there was a "very abstract problem as whether the
Infinity is actual or potential": I'd assume that for
Aristotle it was not a "very abstract" problem. It concerns
e.g. the question whether movement is possible at all (remember
the running contest with the turtle ...) ... .
You are absolutely right as to "Aristotle's main
focus was on *physics* and *not* on mathematics". Why? Because in Aristot=
le's
time, just the (speculative, and therefore "naive") *physics* was an only
understandable to all scientific language. But I think when Aristotle,
taking up the Infinity, speaks even about the physical "air", he means
some abstraction filling some abstract space, but not a concrete physical
mixture of O2, N2, CO2, and so on under a given temperature, humidity and=
pressure, i.e., he uses the "air" and "space" just as abstract concepts.
As to such the problem as "whether the Infinity
is actual or potential", I believe that these notions even today are the
most abstract notions of modern Mathematics.
As to "Achilles and the Turtle ". I don't remember who
first said that Achilles and the Turtle make their "running contest
" well, but it is a human-beeing only who takes the real initial distance
between them, transforms that real distance, in his abstract imagination,
into an abstract segment, say, [0,1], and begins to divide it abstractly
in his abstract imagination by means of an abstract bisection process up
to the abstract (of course, potential) Infinitum. Since, according to Ari=
stotle
(and Samuel S. Kutler : -) ), "A Dia Ex Hodos =3D No way out by going thr=
ough",
Zeno (and we) obtains the abstract result: the abstract movement in our
abstract imagination is, abstractly, impossible at all.
Thank you very much for your non-trivial, deep and very interesting objections.
Other objections and comments are welcome.
LAST REMARK.
Why do I believe that POSTULATE 2 has the important value for the mode=
rn
science?
Today, there is a lot of grandiose projects-XXI and programs-XXI conc=
erning
a global formalization of all mathematical knowledge-XX. The last was his=
torically
obtained and is based today on the Classical Aristotle's Logic. But all
these grandiose projects-XXI and programs-XXI use the formal systems tech=
nique
of the modern meta-mathematics. It is naturally by obvious reasons. But
I state that the modern meta-mathematics which, in contrast to the POSTUL=
ATE
2, considers the inferences *FALSE =3D=3D > TRUTH* and *A, not-A |-=
B*
as the basic true inferences, is a not quite adequate description (formal=
ization)
of the Classical Aristotle's Logic. I am sure that in the near future,
an essential improvement of the modern meta-mathematics as a whole lies
ahead. Else it will be able to prevent to achieve that high aims which
were formulated in these grandiose projects and programs of the XXI Centu=
re.
Regards
A.Z.
############################################
Prof. Alexander A. Zenkin,
Doctor of Physical and Mathematical Sciences,
Leading Research Scientist of the Computer Center
of the Russian Academy of Sciences.
e-mail: alexzen@com2com.ru
WEB-Site =
http://www.com2com.ru/alexzen
############################################
"Infinitum Actu Non Datur=
"
- Aristotel.
--------------6FA533A27C8787B3DF456283--