> Alexander Zenkin wrote many interesting lines, most of which I read
> with great interest. However: it may be a misunderstanding, but I
> have some difficulties with the following statement:
>
>> POSTULATE 2. If a premise A in the formal inference A ==> B is FALSE,
>> then any its consequence B is necessarily FALSE too, or shortly:
>> FALSE ==> FALSE (only!). IFF the Aristotle's Logic deductive rules
>> are applied correctly.
>>
>> Why did Great Aristotle not formulate explicitly such obvious
>> Postulate 2? I think the reason is purely psychological:
>
> I have some doubts. I'd suggest that Aristotle 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 were 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, Super-
induction: 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, pp.
587-589 (1997)}, I introduced the following generalization of the
traditional logical 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 problems of the kind
"Whether P(n) holds for ALL n ?" (where P(n) is a number-theoretical
predicate defined for all natural numbers n >= 1) even if there is a
FINITE set of exceptions (counter-examples) for P(n). In Mathematics,
such a 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 it not so? :-)
Now, about your objection.
At the end of my message (Subject: Re: [HM] Aristotle's "Infinitum
Actu Non Datur" Thesis; Date: Wed, 24 Feb 1999), I wrote:
The huge scientific experience during about 2300 years showed that
the "inference" FALSE ==> TRUE, can be realized in the following two
cases only:
a) when A is a non-essential (i.e., removable) "premise" for B, and
in such a case the "premise" A must be, simply and smoothly, removed
from such a "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
==> TRUE is possible in Classical Logic or in Classical Mathematics,
based upon that Logic, only if such a "inference" ("proof") breaks
grossly the main deductive inference rules of Aristotle's Logic.
Then I formulated the following quite strong empirical statement
(axiom).
POSTULATE 2. If a premise A in the formal inference A ==> B is FALSE,
then any its consequence B is necessarily FALSE too, or shortly:
FALSE ==> FALSE (only!). IFF the Aristotle Logic deductive rules are
applied 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 some of your (and alike) counter-
examples are incorrect. At any case, as a professional scientist with
a length of service, I would not like that it were possible to obtain
the scientific 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 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 syllogism, i.e., that "inference" contains the logical
error "it does not follow" (in 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 certainly 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 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 Aristotle's syllogistics to FALSE 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 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 be false or contradictory premises have no logical sense. If such
a 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 >=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 ==> 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-deducible (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 a case, the question arises: for what aim is such a
power logical tool as the Aristotle's syllogistics, which, by Aristotle's
"IFF", works correctly only on the authentic true premises set, applied
to unreliable (in particular, false or contradictory) premises? - In
order to obtain (by virtue of that Aristotle's "IFF") a new unreliable
(in particular, false or contradictory) knowledge...?
- I think the Aristotle's syllogistics leads never to errors in real
Science just because every scientist, following Aristotle's "IFF",
in the beginning, PROVES the TRUTH of ALL his premises, and only after
that he uses corresponding figures of 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 ==> TRUTH are simply not correct logical
deductions, i.e., all they contain the error "it (i.e., any conclusion)
does 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 ==> 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
*physics* and *not* on mathematics". Why? Because in Aristotle's time,
just the (speculative, and therefore "naive") *physics* was the only
understandable of all scientific languages. 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 problems such 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-being 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 Aristotle (and Samuel S. Kutler :-) ), "A Dia Ex Hodos = No way out
by going through", 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 modern
science?
Today, there is a lot of grandiose projects-XXI and programs-XXI
concerning a global formalization of all mathematical knowledge-XX.
The last was historically obtained and is based today on the
Classical Aristotle's Logic. But all these grandiose projects-XXI
and programs-XXI use the formal systems technique of the modern
meta-mathematics. It is naturally by obvious reasons. But I state
that modern meta-mathematics which, in contrast to POSTULATE 2,
considers the inferences *FALSE ==> 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 aims which were formulated in these
grandiose projects and programs of the XXIth Century.
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" - Aristotle.