Modern interpretations of Aristotle's logic
pyrrho
The discussions here have pettered out a bit, maybe I can get it going
again with some questions.
(1) In history of science, Aristotelain logic and modern logic are
seen to stemm from two different traditions.
``There is a continuous tradition in formal European logic from
Aristotle to Kant. However, in England, this tradition ran dry long
before Kant and started afresh and from different grounds by George
Boole, Augustus de Morgan and William Hamilton. Frege and Russell
belong to this English tradition of logic.''
Quoted from: Volker Peckhaus, "19th Century Logic between Philosophy
and Mathematics", Bulletin of Symbolic Logic, No. 4, Vol.5,,1999.
When talking to the history of science folks, I find this opinion is
quite widespread. But what are the formal differences between the `two
traditions'? Why do they exist?
(2) Aristotle did not formalize his logic. If it is reconstructed as a
formal system today, it can be done in various ways: Lukasiewicz,
Quine, Corcoran,... However different, these reconstructions seem to
share a common assumption. Namely, that some rules of deduction of
Aristotelian logic are not generally valid,
but can be turned valid by additional assumptions. What these
assumptions are depends on how Aristotelian logic is formally
reconstructed. In the reconstruction by Quine (in Methods of logic p.
107) what is needed is ``a small reinforcing premise'' In the formal
semantics of Aristotelian logic by Corcoran, it is the the
stipulation, that the sets, which the terms semantically denote, must
be non-empty. It is only after correction, that Aristotelian logic
turns into a trivial part of modern logic.
What do you think about that?
Thank you
claus
pyrrho
>Perhaps, there are logic notes, as asides, in Aristotle's non logic
>works. He mentions that the opinions of very experienced persons,
>due to vast experience at a relatively early age, or a long life to
>gather experience, should be accepted without the usual requirements
>for proof. NE, book 6, chapter 11. Folks that want to only know
>Aristotle's Logic, but not his other works equally well, are following
>the folks who brought about the dark ages, which were caused by the
>burning of classical antiquity's works, in the West, but preserved
>Aristotle's logical works only. Aristotle's formal standards for
>proof, as evidenced in Euclid's' Logic, are found in his
>Metaphysics. And perhaps scattered as asides or other comments
>throughout his works. The greatest logician of modern times, is
>Charles Sanders Pierce, much of whose works are unpublished. Pierce
>states "My Philosophy is that of Aristotle plus science, and the
>evidence is that he studied Aristotle, very carefully." Many try to
>trace the word pragmatism to Emmanuel Kant, perhaps a mistake, as
>Aristotle uses practical in a way similar to the later term moral, and
>practical pretty well is a more common term for pragmatic. When
>pragmatic became too much used in literature, Pierce suggested getting
>away from the literary version of pragmatism by using the new term
>pragmaticism, so ugly no man of letters would use it. Aristotle did
>so much, and died not many years after the age 50, that he had not
>the time to fine tune his system.
Zeno777 makes two arguments:
(1) to understand Aristotle's logic, we should not only look at the
Organon, but also his Metaphysics and other texts.
(2) The relation between modern formal logic and Aristotelian logic is
one of fine-tuning.
To support (2) with (1) James offers NE, book 6, chapter 11.
I deeply agree with (1). And because I do, I cannot view the
relationship of modern to ancient logic as one of "fine-tuning". NE,
book 6, chapter 11 refers -- to all I can see -- to practical
syllogism and a practical mode of reasoning, which is a coherent "sub-
theory" of the Aristotelian theory of knowledge and reasoning. (1)
deeply relates to what the mathematicians did with Aristotle's texts,
that is the process of formalization. Aristotle has not provided us
with a formal logic but with a new perspective and a rich field to
ponder about. Mathematicians like Corcoran zoom in some aspects which
are easy to formalize and "blow them up" into a formal system. To
assume that they have "formalized" THE meaning of Aristotelian logic
would be a misunderstanding. And (2) is based on exactly this
assumption.
That Corcoran formalized some aspects of Aristotelian logic only is
readily acknowledged by himself. When he introduced his seminal
semantics he was criticized by Mary Mulhern ("Corcoran on Aristotle's
Logical Theory", 1974) on the point that his formalization ignored the
finer points of Aristotle's logic. What was his response? ``Even if
subsequent research shows that these opinions are incorrect, our model
need not be changed. However, its significance will change. Inclusion
of proper nouns, adjectives, relatives and/or indefinite propositions
would imply only additions to our model; no other changes would be
required. Our language seems to be a sub-language, at least, of any
faithful analogue of the abstract language of Aristotle's
System'' (in: "Aristotle's Natural Deduction System") This is exactly
the point. Up to now, formal "reconstructions" of Aristotelian logic
formalized only a sub-language. The question which interests me is:
the sub-language of what? How does a formal language look like of
which Corcoran's language is only a sub-language? To find out, (I
think) we must not only take into account the non-logical texts by
Aristotle himself, as you suggest. Aristotle handed over his "logical
torch" to others who continued his work. Taping this tradition should
give us a much better chance to answer this question.
-----
waveletter
Thanks for your interesting post. Yes, I've been a little distracted
from Aristotle lately. My job has been fairly hectic, and I've been
doing some plain translation work for the Suda On Line project. I'll
try to get back to the "modern interpretations of Aristotle's logic"
thread soon.
To everyone else, I think Prof. Peckhaus's paper is available on the
web here:
http://www.math.ucla.edu/~asl/bsl/0504/0504-001.ps
This is a PostScript file, of course, but if you go through Google on
the link, it should give the option of viewing it as text. That's what
I did. There are also some .pdf versions on the web, and I think from
Prof. Peckhaus's own website:
I couldn't find the quote below ("There is a continuous tradition...")
in the paper however. But it does seem to be consistent with what
Peckhaus says in the article posted.
Anyway, without having yet carefully read Peckhaus's paper, I'm not
sure how truly separate the Continental and British traditions in
logic happen to be. For example, medieval logicians spent a lot of
time "debugging" Aristotle's logic, and they seem to have come up with
the same answer that we do nowadays, namely that the terms can't be
mapped by the semantics to empty sets. I don't think that we should
bash Aristotle because of this assumption, whether he was explicit
about it or not. After all, in their arithmetic, the Greeks didn't
even consider 1 and 0 to be numbers (although Aristotle seems to say
that 1 is a number, and he mulls over calling 0 a number in the
Metaphysics, but then rejects the notion). So there seems to be a
concern with logical rigor, systematization, and proof--not just
analysis of individual predicates, in the "continental" tradition.
(I'm writing this without access to my library right now. This is all
ad lib. Apologies if it's all bogus!)
Also, I think I would say that Aristotle *did* formalize his logic.
Sure, he doesn't come close to what we would consider to be a formal
system nowadays, but this was the first systematization of the forms
of argument. It includes variables, proof structures, syntax, and
semantics. There is even a reduction of one set of syllogisms to
another, and it seems like Aristotle thought that he had shown
completeness, even if there was no formal argument fo it. (Hard to
fault him, as there was no clear statement and proof of this by anyone
until Corcoran and Smiley in the mid-late 20th century!) Yes, the
syllogistic is not rigorously defined, but nothing was until the late
19th century.
And, you can make the case that the Megarians also attempted to
formalize the logic of propositions. To me, it seems like Prof.
Peckhaus makes a good point, but goes too far in splitting the
traditions. If Aristotle stopped with the "Categories" and perhaps
"Posterior Analytics", then Peckhaus's thesis would make more sense.
Then Aristotle would be the stereotypical continental logician,
analyzing terms and natural kinds along with Kant, and so forth.
But, A. did specify the forms of his narrow slice of correct thought,
and this is what the British tradition (Boole, et al.) were to do
centuries later. Similarly, the Stoics with propositional logic. And
then, there is the whole medieval tradition that follows Aristotle,
restates the syllogistic schemata, and fixes the empty terms problems.
So those are three "mathematizing" logical traditions on the
continent, to me at least.
Well, perhaps I have been a little bit charmed by Corcoran's
rehabilitation of Aristotle. Let me have a little more time to look at
Peckhaus's article. But this is my first reaction to the concept of
"dual" traditions in logic. Thanks again!
--Ron
pyrrho
You made so many points in your email that you left me breathless. I
can only but pick on some.
Prof. Peckhaus' texts can be downloaded from:
>I couldn't find the quote below ("There is a continuous tradition...")
>in the paper however. But it does seem to be consistent with what
>Peckhaus says in the article posted.
I checked and rechecked other papers, but I could not find the quote
any longer. Then I contacted Prof. Peckhaus and he said that this
quote is not from him, things are much more complicated. Therefore I
apologize to you all for giving you a wrong quote! I promise to be
more careful in the future. Thank you Ron for checking!
>Also, I think I would say that Aristotle *did* formalize his logic.
>Sure, he doesn't come close to what we would consider to be a formal
>system nowadays, but this was the first systematization of the forms
>of argument.
OK -- I can agree with that. Today it is required that `a formal
logic' must consist of a deductive theory and a formal semantics. Of
course with Aristotle a formal semantics is totally absent. That would
be asking too much. But he provided us with a reference system for a
equivalence class of deductive theories. Why an equivalence class?
>From the modern, deductive point of view, a formal system A has the
same deductive power as a system B, if all deducible sequences of
system $A$ are deducible in system $B$ and the other way round. And
Aristotle has characterized a certain type of deduction. "Deduction"
here must not be identified with the specific rules given by
Aristotle. There can be other systems (some more simple, some more
complex) with the same deductive properties. However the criteria
`same deductive power' is rather crude. Two systems can belong to the
same deductive equivalence class, yet their `interesting' properties
can vary widely. That is why there are so many different formal
reconstructions of Aristotelian logic around. Some exceptions
excluded, all the modern systems found in literature can rightly claim
to have formalized Aristotelian logic -- even if they *semantically*
violate everything Aristotle said. That is why a formal semantics is
so important. It is needed to distinguish a deductive theory A from
another one (B), even if both have the same semantic power. But didn't
Corcoran and Smiley provide a formal semantics? In the face of it they
did. They gave a system which satisfies all the formal requirements
for a semantics. But basically the situation is as follows: Modern
logic and its employed notion of deduction, is strongly connected to
the notion of a "set". If you think Aristotle talks about sets, then
you have to ramify the Aristotelian rules. Alternatively you can keep
the generality of the Aristotelian rules if you restrict the semantic
domain appropriately. And that is what Corcoran did. The
"rehabilitation" of Aristotelian logic requires unrestricted
generality on both, the syntactic and the semantic level and
preferably a strong relation of the semantic objects to fundamental
mathematical structures.
>medieval logicians spent a lot of time "debugging" Aristotle's logic, and they seem to have come up with
>the same answer that we do nowadays, namely that the terms can't be mapped by the semantics to empty sets. I don't think that >we should bash Aristotle because of this assumption, whether he was explicit about it or not.
Who is making what assumptions? Did Aristotle assume sets? Or do
modern logicians make the same assumptions as their medieval
breathers, namely that Aristotle spoke about sets? And why this
fixation on sets? In mathematics sets are just some kind of abstract
objects with specific structural properties. There are infinitely many
others. The mathematical theory which studies how certain types of
abstract objects are related to others is called -- in honor of
Aristotle -- Category-theory. Now assume Aristotle *did* think of
sets as denotations of terms. The consequences is a well known petitio
principii: ``It must be granted that in every syllogism, considered as
an argument to prove the conclusion, there is a *petitio principii*.
When we say, All men are mortal, Socrates in a man, therefore Socrates
is mortal; it is unanswerably urged by the adversaries of the
syllogistic theory, that the proposition, Socrates is mortal, is
presupposed in the more general assumption, All men are mortal: that
we cannot be assured of the mortality of all men, unless we are
already certain of the mortality of every individual man: that if it
be still doubtful whether Socrates, or any other individual we choose
to name, be mortal or not, the same degree of uncertainty must hang
over the assertion, All men are mortal: that the general principle,
instead of being given as evidence of the particular case cannot
itself be taken for true without exception, until every shadow of
doubt which could affect any case comprised with it, is dispelled by
evidence \emph{aliunde}; and then what remains for the syllogism to
prove? That, in short, no reasoning from generals to particulars can,
as such, prove anything, since from a general principle we cannot
infer any particulars, but those which the principle itself assumes as
known.'' (J. St. Mill, System of Logic Ratiocinative and Inductive, p.
120) If you think Aristotle thought about sets (empty or not) then
this circle is inevitable. Sets are "wholes" precisely determined by
their extent (If two sets consist of the same elements, they are
identical (as sets)). If follows that either Aristotle was a dunce
because he didn't see this circle, or terms must designate other kind
of abstract objects. Take your pick!.
Thanks
Claus
On 22 Aug., 23:50, waveletter <wavel...@pacbell.net> wrote:
> Hi Claus:
>
> Thanks for your interesting post. Yes, I've been a little distracted
> from Aristotle lately. My job has been fairly hectic, and I've been
> doing some plain translation work for the Suda On Line project. I'll
> try to get back to the "modern interpretations of Aristotle's logic"
> thread soon.
>
> To everyone else, I think Prof. Peckhaus's paper is available on the
> web here:
>
> http://www.math.ucla.edu/~asl/bsl/0504/0504-001.ps
>
> This is a PostScript file, of course, but if you go through Google on
> the link, it should give the option of viewing it as text. That's what
> I did. There are also some .pdf versions on the web, and I think from
> Prof. Peckhaus's own website:
>
> http://www.uni-paderborn.de/fileadmin/kw/Institute/Philosophie/Person...
Ron Allen
Thanks for your reply & for checking out the quotation
with Prof. Peckhaus. I hope it's OK if I insert my
replies into your quoted text below.
--- pyrrho <c.bril...@googlemail.com> wrote:
>
> Hi Ron:
>
> You made so many points in your email that you left
> me breathless. I
> can only but pick on some.
Ha! Well, I was just rambling along, reacting to your
earlier post. I hope that at least some of the points
I made were good ones, and your being breathless was
not because of your shock and dismay at my banality!
>
> Prof. Peckhaus' texts can be downloaded from:
>
>
http://www.uni-paderborn.de/en/fakultaeten/kw/institute-einrichtungen/institut-fuer-humanwissenschaften/philosophie/personal/peckhaus/texte-zum-download/
>
> >I couldn't find the quote below ("There is a
> continuous tradition...")
> >in the paper however. But it does seem to be
> consistent with what
> >Peckhaus says in the article posted.
>
> I checked and rechecked other papers, but I could
> not find the quote
> any longer. Then I contacted Prof. Peckhaus and he
> said that this
> quote is not from him, things are much more
> complicated. Therefore I
> apologize to you all for giving you a wrong quote! I
> promise to be
> more careful in the future. Thank you Ron for
> checking!
No problem! It's an interesting quotation. The source
will turn up, eventually. (Ha! Between you and me,
Claus, I have to laugh: a typical response from a
university professor. After we put down a concise
statement that summarizes some position he takes in a
paper: no, it's much more complicated than that!
Actually, I do think that it's more complicated, in
agreement with Prof. Peckhaus. But the mysterious
quote does seem to point to a central moment of
continental vs. English logical development; it
gestures at an abstract tendency, however nuanced, and
it carries some explanatory power.)
>
> >Also, I think I would say that Aristotle *did*
> formalize his logic.
> >Sure, he doesn't come close to what we would
> consider to be a formal
> >system nowadays, but this was the first
> systematization of the forms
> >of argument.
>
> OK -- I can agree with that. Today it is required
> that `a formal
> logic' must consist of a deductive theory and a
> formal semantics. Of
> course with Aristotle a formal semantics is totally
> absent. That would
> be asking too much. But he provided us with a
> reference system for a
> equivalence class of deductive theories. Why an
> equivalence class?
(Well, I'd like to keep the replies short, so that
folks can read them in a brief sitting. So, let me
reply to this point right here. I'll try to catch up
to your post, Claus, in some later replies. I hope
this format works for you and the rest of the group.
Somewhere, Socrates (through Plato) says, "More
thinking, less talking;" so that's what I'm trying to
do for myself! (In the Protagoras?)
I think that we can find in Aristotle a deductive
theory. I think that I'd agree with you that Aristotle
does not provide a formal semantics. But, I would
hasten to add: how could he? The semantics involves
the definition of a mapping from terms of the language
to objects external to the language. For example, the
semantics of a first-order language is defined in
Shoenfield, "Mathematical Logic," Reading, MA:
Addison-Wesley, 1967, section 2.5. There is a universe
of objects, and there is mapping that associates each
predicate symbol of the language with a predicate on
the universe of objects.
I think it's fairly clear that Aristotle does not
achieve this kind of formal precision. For instance,
A. does not even have the notion of a function--a rule
that associates each thing from one set to a
particular thing within another set. I don't think
that this mathematical abstraction was realized until
fairly late in mathematical practice, perhaps with the
development of the calculus. So, it's hard to fault
Aristotle for not having what we would call a formal
semantics.
It's pretty clear, to me at least, that Aristotle
understood that there were natural things that could
be grouped together under a term, such as 'man'. He
does not make this explicit in his "Prior Analytics,"
I suppose, but it is fairly clear in the "Categories."
So, intuitively, Aristotle has the notion of a set of
things defined by a common term. He calls this set of
natural objects a species. So the collection of all
human beings is associated (informally) with the word
'man' or 'anthropos', and the collection of all horses
with the the word 'horse' or 'hippos', and then
Aristotle has collections of collections, which he
calls genera. For example, the collection of all
animals, including the human beings and the horses and
the amoebas (of which A. was ignorant) was the
collection of animals, 'zoa', from the Greek word
'zoon'.
So, I would suggest that Aristotle does, in his
"Categories," lay out what ought to be the semantics
for his logic in the "Prior Analytics."
Yes, it's not formal, like we might find in
Shoenfield's book. But, it's just up to us,
Aristotle's interpreters, beginning with his own
student, Theophrastus, to sort these things out. That
is: construct a formal system consonant with
Aristotle's own development, in its original,
historical context.
Now, it could be that Aristotle's system in its
original context has fundamental problems.
One problem that was identified early on was empty
predicates. If A. allows these, then it seems that
some of his proof theory does not work. So the
medieval logicians concentrated on this.
Aristotle does seem to know that some predicates have
no referent. In a number of places, he considers the
"goat-stag," an imaginary half-goat, half-deer
(unicorn?), which does not exist. (Actually, this is
in "Posterior Analytics," II.7, Bekker no. 92b4 ff;
also in "Prior Analytics" 49a24; cf. Plato's
"Republic" 488a, Aristophanes "Frogs" 937.)
So, does Aristotle realize that the empty predicates
have to be disallowed or not? This is a good question,
and it's one that would be great to explore on this
list.
Well, we know that they do have to be disallowed in
order for his deductions to work correctly! OK, so
this is what his interpreters worked on for hundreds
of years. Fine. Now, after the medieval logicians have
worked over Aristotle, we modern logicians can employ
our own tools to analyze The Philosopher:
(1) like Lukasiewicz and Patzig did, by casting
Aristotle's system into modern first-order predicate
logic; or
(2) reconstructing his proof theory and semantics into
a formal system that is consonant with Aristotle's own
writings, as did Corcoran and Smiley.
At first, I thought that on this list I'd be
developing (1), Lukasiewicz's interpretation of
Aristotle's syllogistic. But then I learned of (2)
Corcoran's interpretation, and I liked it. So this is
what I've been blogging out, albeit in piecemeal
fashion.
Well, this is too long a post. Let me break off here.
You have made a number of intriguing points in the
attached text that follows, Claus. I will attempt to
respond to them tomorrow.
You may be way ahead of me on this.
Thanks! This is great.
--Ron
=== message truncated ===
waveletter
Well, I can agree with some of these points.
To begin with, sure, we can read Aristotle's non-logical works for
insight into his logical works. But, in general, I think that this is
backwards. Aristotle was nothing if not methodical. And he developed
his philosophical logic first as a foundation for his formal science
and his higher level thinking on politics and ethics. So, we don't
expect that something in an ethical work would supersede some point in
one of the logical works, unless there was a good reason to argue that
Aristotle had revised his position.
It's hard to pin down what Zeno777 says, as it is topical and highly
schematic, but I can state a few points of difference with his remarks
(which Claus has kindly transmitted to us below):
1. In my opinion, Aristotle's formal standards for proof are found in
the "Prior Analytics", a logical work, one of the key texts of the
Organon, not in his "Metaphysics". There are several instances of
Aristotle's proofs in the "Prior Analytics"; but where are there
instances of syllogisms in the "Metaphysics"?
2. C.S. Peirce was a great logician, but not the greatest logician of
modern times. If he was the greatest, as Zeno777 apparently contends,
then Zeno777 would probably take the care to spell his name correctly.
The 'e' comes before the 'i'.
3. Well, OK, so what. Who is the "greatest logician of modern times"?
You know, it's hard to say, except that it really can't be Peirce.
4. Frege certainly comes to mind. He developed a logic that superceded
Aristotle's logic, and it is the grandfather of our modern first-order
predicate logic.
5. Next, you have to admit that Bertrand Russell, who found flaws in
Frege's set theory, and constructed all of mathematics on a rigorous
basis, would have to be a candidate for greatest logician.
6. Finally, a generation later, who could argue against Kurt Godel as
the greatest logician? He proved the completeness of 1st-order logic,
the incompleteness of arithmetic, the incompleteness of extensions of
arithmetic, constructed the Constructible Universe V = L of set
theory, and thereby showed that the Continuum Hypothesis and the Axiom
of Choice were consistent with set theory...and it was only a
generation later that these were shown to be independent of the axioms
of set theory.
7. But nobody supplanted the rigor of Aristotle's syllogistic (well,
maybe propositional logic with the Megarians and the Stoics and then
Boole; I am not so sure) until Frege came along. So, if I had to say
who was the greatest logician of modern times, I would say it was the
guy who established the new logic that Russell and Godel were able to
investigate and exploit, the logic that extended both Aristotle's and
the Stoics': that would be Gottlob Frege. Yeah, yeah, nasty notation
and a glitch in the set theory. But so what.
8. Peirce was a pragmatist. Kant was not a pragmatist. But I believe
the term 'pragmatism' was really promulgated by William James, not by
Peirce, as Zeno777 seems to have suggested in the attached text below.
I don't know of anyone who traces the word 'pragmatism' to Kant.
9. Hmm...I tend to agree with Claus that modern logic is *not* a fine-
tuning of ancient (Aristotelian, categorical) logic. It's really a
fundamental revision and extension of ancient logic.
10. Claus also says, "Mathematicians like Corcoran zoom in some
aspects which are easy to formalize and 'blow them up' into a formal
system." This I disagree with. Corcoran is not a mathematician, for
one thing; he's a philosopher. This statement really applies to
Lukasiewicz more than Corcoran, and L. *is* a mathematician. L's
theory tends to cast Aristotle's logic into modern, Fregean, first-
order logic without taking Aristotle's own evident proof theory into
account. So L has "zoomed in" upon some parts of Aristotelian logic
and reworked them in modern form (and it's really valid,
mathematically, but not exegetically) for us to admire. But Corcoran
has endeavored to explain Aristotle's proof theory in Aristotle's own
terms, and this makes all the difference. It allows us to see
Aristotle's logic at work, in its original form, but with a modern
vantage point. Lukasiewicz takes away the original form and
substitutes a modern proof schema, and in so doing, he has to discard
some of Aristotle's own results. Conclusion: either Aristotle is
grievously mistaken or Lukasiewicz's interpretation is anachronistic.
Sure, Aristotle's logic is weaker than first-order predicate logic.
So, it's not a surprise that someone could recast categorical logic
into a subset of first-order logic and exhibit all of the proof
schemas as deductions in 1st-order proof theory. With Corcoran, you
get an interpretation that consonant with Aristotle's own logical
writings and is logically rigorous to boot. Lukasiewicz is logically
rigorous, but it's just not consonant with Aristotle's legacy. As
Gadamer might say, Corcoran's interpretation has a transparency to the
original author (Aristotle) that Lukasiewicz's interpretation does
not.
Just rambling along....
Thanks!
--Ron
> > claus- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -
pyrrho
nice you are back! But now strait into the discussion ..
First of all one must distinguish what is first relative to perception
(kata aisqhsin) and what is first relative to reasoning (kata logon).
[Meta 1018b 29-34].
According to Aristotle, knowledge is obtained by learning and learning
is something which in time begins with perceptions of concrete
individuals. But knowledge is more than the knowledge of singular
facts. It consists also in the knowledge of principles (archai) under
which the particular cases of cognition fall. And it is these
principles which are prior to knowledge. An indicator for that is,
that principles are so convincing. Proofs, which start from these
principles become less convincing the longer they are [Ana.Post I 2
72a 25]. But proofs and the principles of proofs belong to the final
stage of a long process of learning which starts off much more humble.
A child can learn how to count and to add quite early in its life, if
addition is explained with the help of perceivable concrete
individuals, maybe apples. But what are two apples plus tree pears, if
anything? Only later will the child progress to the concept of number.
Later again, it may even be able to make the important distinction
between numeral and number, and later still, it may comprehend the
idea of a proof. And with the proof comes the comprehension, that it
is totally impossible that 2+3=5 today and 6 tomorrow. Thus comes the
insight into impossibility and necessity. And because necessity
implies that something cannot be otherwise, it must be generally
(kaqolou) so. Thus learning starts out de facto with perception and
individuals and proceeds to the general. But from the point of
reasoning, the general and general principles come first.
What is obtained by learning is knowledge. One has knowledge if one
knows the reason why something is and possesses the insight, that it
cannot be otherwise (Ana.Po first book, ch. 2). One way to obtain such
a knowledge is by proof. And proofs must rely on first necessary
principles (axioms) which are true, first, direct,very convincing and
reasons (aition). Sciences are always particular sciences which
investigate "that what is" from a particular point of view and use a
particular set of axioms suitable adapted to their field. Mathematics
investigate "that what is" as a multitude of discrete or continuous
quantities. Physics investigates "that what is" as moving, biology
investigates that what is as far as it is alive. And your "higher
sciences" politics and ethics are even more particular. Scientist can
proceed without considering logic, because logic is already "built
into" the particular axioms and in the scientific methods. A scientist
who does not adhere to logic simply produces bad science.
There is however one first science which investigates "that what is"
not relative to any respect but "as it is" "as such": ``There is a
science which investigates that which is as far as it is and also by
which principles it is ruled as such'' (Meta 1003a) This "first
science" was later called "metaphysics". And what are its objects and
principle? That what appears when one looks across the single
sciences. Only then do common [koina; AnaPo I 10;76a 38] principles
like the "law of identity" appear and they hold "analog" [Meta IV
3;1005a-25]. (A good comentary on that is Kurt von Fritz "Die ARCHAI
in der griechischen Mathematik" p.335ff and 348-349. Or I.Schüßler,
"Aristoteles: Philosophie und Wissenschaft" p.65-67)
>To begin with, sure, we can read Aristotle's non-logical works for
>insight into his logical works. But, in general, I think that this is
>backwards.
My comment: ???
>Aristotle was nothing if not methodical.
Yep!.
>And he developed his philosophical logic first as a foundation for his formal science
>and his higher level thinking on politics and ethics.
It is exactly the other way round. Science does not need philosophy.
(Just look at what is happening today.) Philosophy comes later. And
later still comes logic. Logic in the Aristotelian mould cannot be
mathematical only. It applies to everything, not just to mathematical
entities. That is, it is ontological. And Aristotle's logical lecture
notes (notes is only what was transmitted in history) are, according
to some interpreters, his latest writings. In particular "de
interpretatione".
>9.I tend to agree with Claus that modern logic is *not* a fine-
>tuning of ancient (Aristotelian, categorical) logic. It's really a
>fundamental revision and extension of ancient logic.
My opinion is: it can be read that way but it should not. It s a
completely different beast. Certainly it is just a nucleolus of
something. But of what?
>10.. Corcoran is not a mathematician, for one thing; he's a philosopher.
Oh dear.
A man went to a tailor at Seville Row to get a new suit. They where
all of very high quality and therefore expensive. "Don't you have
anything cheaper?" the man asked the tailor. "Well there is suit done
by my apprentice. It just might fit you." The man tried it on, it
fitted, and the man was well pleased with the low price quoted. "But",
he said, "one leg is longer than the other". "This is no problem" said
the tailor. "You just bend your knee a bit and nobody will notice."
"But the right arm is longer than the other one". "Also no problem.
You just retract your arm slightly." Satisfied the man left the store.
Outside were two men. "Look at the poor cripple", one said. "Yes, a
poor cripple. But he has an excellent tailor."
Ron, you only have eyes for Corcoran the excellent tailor:
>With Corcoran, you get an interpretation that consonant with Aristotle's own logical writings and is logically rigorous to >boot.
But I happen to see the poor crippled Aristotle. Lets straighten him
up a bit, shall we? The "categories", chapter 3 starts as follows:
"Whenever one thing is predicated of another as of a subject, all
things said of what is predicated will be said of the subject also.
For example man is predicated of the individual man, and animal of
man; so animal will be predicated of the individual man also - for the
individual man is both man and animal." (Translator: J.L. Ackrill, The
Complete Works of Aristotle, Oxford 1985)
Corcoran maps this relation to the subset relation between sets. E.g.
{1} subset {1,2} subset {1,2,3}. Let us instantiate: {1,2} is a
predicate said of the subject {1}, because the set {1} is both subset
of {1,2} as well as of {1}. So far so good. But Aristotle said: "all
things said of what is predicated will be said of the subject also."
And the subset predicate is just one thing (one property of sets). Now
let us say another one. "{1,2} has two elements." But {1} of course
does not have two elements. Therefore Corcoran's interpretation is
simply not consonant with Aristotle's own logical writings. This looks
mad, but I insist because this is the touchstone of Aristotelian logic
for the next 2000 years. Later this relation became the intension/
extension relation. A thing like "man" is under "animal" (in the
extent of) because animal" is in "man" (in the intension of). And
there are formal objects where this relation holds too. But sets
aren't.
Understanding, according to Plato, requires analyses. That, means,
that problems and things must be broken up at their ``natural
joints'', taking care not to break the natural parts themselves,
``after the manner of a bad carver''[Phaedrus, 265e.] Corcoran carved
up the problem at rather unnatural places.
Just an answer to Ron's Ramblings ...
Thanks!
Claus
Ron Allen
You make some good points. I'm not sure where we agree
or disagree or why. In other places, I'm not sure if
you're saying that this or that is Aristotle's
position, or it's your position. So, I'll just comment
along.
--- pyrrho <c.bril...@googlemail.com> wrote:
>
> Hallo Ron, hallo Zeno777,
>
> nice you are back! But now strait into the
> discussion ..
>
> First of all one must distinguish what is first
> relative to perception
> (kata aisqhsin) and what is first relative to
> reasoning (kata logon).
> [Meta 1018b 29-34].
Here, in my English translations, A. is talking about
his notion of "priority" ("protera" in Attic).
>
> According to Aristotle, knowledge is obtained by
> learning and learning
> is something which in time begins with perceptions
> of concrete
> individuals. But knowledge is more than the
> knowledge of singular
> facts. It consists also in the knowledge of
> principles (archai) under
> which the particular cases of cognition fall. And it
> is these
> principles which are prior to knowledge.
I'm not sure whether I understand what you write
above. You seem to be saying there are principles that
are prior to knowledge, so it has a Kantian ring to
me. Aristotle says in the passage you refer to that
"Universals are prior in formula, but particulars in
perception." (1018b33-34) So, it's my reading that A.
is giving a reciprocal relation, not a one-way
principles-are-prior-to-any-knowledge.
(A little further down, I note that you come back to
something very close to what Aristotle says; it could
be that you're just looking at one side of the
reciprocal relationship of priority with universals
and particulars.)
An
> indicator for that is,
> that principles are so convincing. Proofs, which
> start from these
> principles become less convincing the longer they
> are [Ana.Post I 2
> 72a 25]. But proofs and the principles of proofs
> belong to the final
> stage of a long process of learning which starts off
> much more humble.
Well, mathematical proof is convincing, but so is
sensation. If someone shows me the cat on the mat,
well, then I wouldn't be more convinced of that fact
if they had some kind of axiom system and a derivation
that the cat was on the mat.
Is yeast an animal or a plant? Well, we don't need to
know the precise definition of the abstract terms
'animal' and 'plant' in order to know what yeast is,
that you can make bread rise and beer ferment by
adding it at the right temperature and so on.
> A child can learn how to count and to add quite
> early in its life, if
> addition is explained with the help of perceivable
> concrete
> individuals, maybe apples. But what are two apples
> plus tree pears, if
> anything?
Here ^ it seems like you're saying that the concrete
individuals are prior to the abstractions of
arithmetic. So, this is one of the possible directions
of priority that A. talks about, and it seems to go
against your earlier statement that the principles
precede the perceptions. Or, maybe I misunderstand
you.
Only later will the child progress to the
> concept of number.
Yes, and so with any number of other examples.
Plato, of course, argued against this, saying that the
abstract notion of say Justice or Goodness or
Triangularity was prior to the concrete individuals
and even more prior to the perception of the
individuals.
> Later again, it may even be able to make the
> important distinction
> between numeral and number, and later still, it may
> comprehend the
> idea of a proof. And with the proof comes the
> comprehension, that it
> is totally impossible that 2+3=5 today and 6
> tomorrow. Thus comes the
> insight into impossibility and necessity. And
> because necessity
> implies that something cannot be otherwise, it must
> be generally
> (kaqolou) so. Thus learning starts out de facto with
> perception and
> individuals and proceeds to the general. But from
> the point of
> reasoning, the general and general principles come
> first.
OK, this ^ is where it looks like you are very
close--if not spot on--to what A. says in the passage
we're looking at (1018b29ff)
There is a nice statement of A's position at the
beginning of his "Physics" (184a10ff): "The path of
investigation must lie from what is more immediately
cognizable and clear to us, to what is clearer and
more intimately cognizable in its own nature.... Now
the things most obvious and immediately cognizable by
us are concrete and particular, rather than abstract
and general; whereas elements and principles are only
accessible to us afterwards.... So we must advance
from the concrete whole to the several constituents
which it embraces...."
And so on.
>
> There is however one first science which
> investigates "that what is"
> not relative to any respect but "as it is" "as
> such": ``There is a
> science which investigates that which is as far as
> it is and also by
> which principles it is ruled as such'' (Meta 1003a)
> This "first
> science" was later called "metaphysics".
Actually, the subject was called "Metaphysics" because
that particular text came *after the Physics* in the
standard order of Aristotle's works. It really didn't
mean "better than physics" or "above physics" in any
sense in Aristotle's time.
The passage you are quoting is 1003a21, I believe.
And what
> are its objects and
> principle? That what appears when one looks across
> the single
> sciences. Only then do common [koina; AnaPo I 10;76a
> 38] principles
> like the "law of identity" appear and they hold
> "analog" [Meta IV
> 3;1005a-25]. (A good comentary on that is Kurt von
> Fritz "Die ARCHAI
> in der griechischen Mathematik" p.335ff and 348-349.
> Or I.Schüßler,
> "Aristoteles: Philosophie und Wissenschaft" p.65-67)
Hmm...my German is probably not good enough to get me
through these. I'll try to scope these out, though.
Thanks for the references!
>
>
> >To begin with, sure, we can read Aristotle's
> non-logical works for
> >insight into his logical works. But, in general, I
> think that this is
> >backwards.
>
> My comment: ???
Well, what the heck was I trying to say there?? I
think I meant that, we really don't know the order of
composition of A's works. They are all probably
lecture notes, and they may not have been written
directly by A himself. It's probably true that the the
Organon was developed later, but that when A.
discovered his syllogistic and worked out its
development, he would see it as the foundation of a
rigorous science. So inside the "Parts of Animals" or
"Meteorlogica" will be instances of argumentation that
follow his logic, and thus the logic explains the
elaboraton of the science, but *not* that the
elaboration of the particluar science would be a guide
to what's going on in the "Categories" or "On
Interpretation" or "Prior Analytics". Similarly, we
don't read a book on mechanical engineering to find
out about linear algebra. Similarly, A. sees the
science of "Being qua Being" to be worked out
independently of the other sciences.
>
>
> >Aristotle was nothing if not methodical.
>
> Yep!.
>
> >And he developed his philosophical logic first as a
> foundation for his formal science
> >and his higher level thinking on politics and
> ethics.
Yes, thank you. What I wrote ^ there is just wrong, I
think. I'm pretty sure that A. developed his syllogism
fairly late, and he may have even embarked on some
rudiments of propositional logic and then abandoned
its deeper study in favor of his term logic, when he
discovered its details.
What I would rather have said is that he developed his
logic later, but saw it as a foundation for his
physical and social sciences.
>
> It is exactly the other way round. Science does not
> need philosophy.
> (Just look at what is happening today.) Philosophy
> comes later. And
> later still comes logic. Logic in the Aristotelian
> mould cannot be
> mathematical only. It applies to everything, not
> just to mathematical
> entities. That is, it is ontological. And
> Aristotle's logical lecture
> notes (notes is only what was transmitted in
> history) are, according
> to some interpreters, his latest writings. In
> particular "de
> interpretatione".
Here, I think you are basically right, and I was
wrong. But, it seems to me that "Categories" and "On
Interpretation" are fairly old and independent of
other things. I would guess that "Topics" and
"Sophistical Refutations" are the older part of the
Organon. Then, the first two works, and finally the
"Analytics". But, I'm not an expert.
>
> >9.I tend to agree with Claus that modern logic is
> *not* a fine-
> >tuning of ancient (Aristotelian, categorical)
> logic. It's really a
> >fundamental revision and extension of ancient
> logic.
>
> My opinion is: it can be read that way but it should
> not. It s a
> completely different beast. Certainly it is just a
> nucleolus of
> something. But of what?
OK.
>
> >10.. Corcoran is not a mathematician, for one
> thing; he's a philosopher.
>
> Oh dear.
I agree. What kind of point was I trying to make?
Corcoran is clearly both, as, say, was Russell.
>
> A man went to a tailor at Seville Row to get a new
> suit. They where
> all of very high quality and therefore expensive.
> "Don't you have
> anything cheaper?" the man asked the tailor. "Well
> there is suit done
> by my apprentice. It just might fit you." The man
> tried it on, it
> fitted, and the man was well pleased with the low
> price quoted. "But",
> he said, "one leg is longer than the other". "This
> is no problem" said
> the tailor. "You just bend your knee a bit and
> nobody will notice."
> "But the right arm is longer than the other one".
> "Also no problem.
> You just retract your arm slightly." Satisfied the
> man left the store.
> Outside were two men. "Look at the poor cripple",
> one said. "Yes, a
> poor cripple. But he has an excellent tailor."
>
> Ron, you only have eyes for Corcoran the excellent
> tailor:
Perhaps, but in what way do you find Corcoran to be
the clumsy apprentice tailor?
>
> >With Corcoran, you get an interpretation that
> consonant with Aristotle's own logical writings and
> is logically rigorous to >boot.
>
> But I happen to see the poor crippled Aristotle.
> Lets straighten him
> up a bit, shall we? The "categories", chapter 3
> starts as follows:
Oh, crud. My stupid email service has cut off the
attached reply (below). I'll have to continue this
later, right here when we start in with Corcoran.
Thanks & bye for now!
--Ron
>
> "Whenever one thing is predicated of another as of a
> subject, all
>
=== message truncated ===
Ron Allen
particular:
Last post from me was broken off just as it began to
get good! What happens is that my email
provider--AT&T, the huge US telecom
conglomerate--somehow thinks that quoted reply texts
only need to be so long...and a post of moderate
length, such as Claus's recent one, gets cut off. So
what I have to do is open up the post I'm replying to,
copy out the entire text to a text editor outside the
browser, and then reply by pasting in the text and
putting in my own > marks. A pain.
Oh well, I'm going to cut out the earlier part of
Claus's post and insert the part that was chopped by
AT&T/Yahoo. Just so it's clear, I'll flag my new
remarks with [ron].
--- pyrrho <c.bril...@googlemail.com> wrote:
>
> Hallo Ron, hallo Zeno777,
>
[snip]
> Ron, you only have eyes for Corcoran the excellent
tailor:
>>With Corcoran, you get an interpretation that
consonant with
Aristotle's own logical writings and is logically
rigorous to >boot.
>But I happen to see the poor crippled Aristotle. Lets
straighten him
up a bit, shall we? The "categories", chapter 3 starts
as follows:
>"Whenever one thing is predicated of another as of a
subject, all
things said of what is predicated will be said of the
subject also.
For example man is predicated of the individual man,
and animal of
man; so animal will be predicated of the individual
man also - for the
individual man is both man and animal." (Translator:
J.L. Ackrill, The
Complete Works of Aristotle, Oxford 1985)
[ron] Two comments: (1) you have to understand
Aristotle's "one thing predicated of another as of a
subject" correctly, and (2) altough the transfer of
predication works for individuals in the "Categories",
Aristotle's syllogistic does not allow this, because
the variables do not stand for individuals, but only
for sets (species or genera) of individuals.
[ron] I think Claus is headed towards a mistake...and
here it comes:
>Corcoran maps this relation to the subset relation
between sets. E.g.
{1} subset {1,2} subset {1,2,3}. Let us instantiate:
{1,2} is a
predicate said of the subject {1}, because the set {1}
is both subset
of {1,2} as well as of {1}. So far so good. But
Aristotle said: "all
things said of what is predicated will be said of the
subject also."
And the subset predicate is just one thing (one
property of sets).
[ron] OK so far.
>Now
let us say another one. "{1,2} has two elements."
[ron] This ^ is the mistake, a serious one, but it's
an easy one to make. The predication "has two
elements" is applies to the *set*, not to the things
making up the set. It is important to see Aristotle's
glib statement in the categories as applying to the
elements of the classes, not to the classes
themselves.
[ron] That is "has two elements" is something that
applies to sets, not to individuals. So "has two
elements" is not something that would be predicated of
{1, 2} as subject, because what Aristotle means is to
predicate it of the elements of the class.
[ron] Actually, if Claus's interpretation is correct,
Aristotle is as much a fool as Corcoran. For Aristotle
could clearly have seen that "contains a pig" is
predicated of Animal, and so contains a pig is
predicated of Man as well as of Socrates himself.
>But {1} of course
does not have two elements. Therefore Corcoran's
interpretation is
simply not consonant with Aristotle's own logical
writings.
[ron] Well, no, for the reason I gave above. Corcoran
maps the logical variables to sets of objects. A valid
predicate for {1,2,3} would be something that
describes the individuals in the genus...say "less
than 4". Then, the predicate "less than 4" applies to
the genus {1,2,3}, and hence it applies to the species
{1,2}. This is the limit to how far you can do this in
Aristotle's syllogistic.
[ron] If you invoke the remark from the "Categories",
then you can go further, but you are *outside of the
syllogistic* in so doing! You can say that "less than
4" applies to the individual 1.
[ron] And I think this mistake is cause to reiterate
what I said in the previous two posts about taking
stuff from outside the logic (I mean the "Analytics"
here, specifically) and then using it to interpret the
logic. You can find stuff elsewhere that is correct,
but does not apply to the formal syllogistic...such as
the predication on individuals. Aristotle does not
develop his formal logic with individual
symbols...that would be an extension of syllogistic,
that, as far as I know, has not been done. But it
seems kind of straightforward. Anyway, the reason that
there are not individuals in the syllogistic is that
then the conversion rules would not work, I believe.
You would have to say things like "Socrates does not
apply to Animal", which is nonsense. Only abstractions
are denoted by variables in Aristotle's logic, not
individuals (as in modern predicate logic).
[ron] I'm not saying, don't read the other stuff and
relate it to the syllogistic. I'm saying that the
logical works are read largely independently (they are
"prior", "protera" in Aristotle's sense) to the other
scientific works, because the logical works can stand
alone, without the others, e.g. "Nicomachean Ethics",
"Parts of Animals", etc. And if we consider the others
as instances of rigorous science, then we should see
them as depending on, and therefore posterior to, the
logical works.
>This looks
mad, but I insist because this is the touchstone of
Aristotelian logic
for the next 2000 years. Later this relation became
the intension/
extension relation. A thing like "man" is under
"animal" (in the
extent of) because animal" is in "man" (in the
intension of). And
there are formal objects where this relation holds
too. But sets
aren't.
>Understanding, according to Plato, requires analyses.
That, means,
that problems and things must be broken up at their
``natural
joints'', taking care not to break the natural parts
themselves,
``after the manner of a bad carver''[Phaedrus, 265e.]
Corcoran carved
up the problem at rather unnatural places.
[ron] I'm still sticking with Corcoran. But, we can
continue this!
>Just an answer to Ron's Ramblings ...
>Thanks!
>Claus
[ron] Thank you, and sorry to have to split up the
reply like this. Well, it almost happened at a natural
place in the text. Maybe I should thank AT&T after
all....
--Ron
pyrrho
Thanks for your answer! So, you found a mistake in my considerations?
>This ^ is the mistake, a serious one, but it's
> an easy one to make. ...
>From your framework of sets and classes, variables etc. you are
completly right! But what I can see now is that I try to explain a
complicated matter with too few words.
>ron] I'm still sticking with Corcoran.
I would so too, if I would not have a better semantics in the drawer.I
have to sort that out first.
Some final remarks and more canditates for the ongoing beauty-contest:
who is the greatest logician?
Logic did not stop with Gödel's incompeleteness. Gregory Chaitin
(still alive) extended and quantified Gödel's approach.
He introduced a complexity index for calculi and theorems called
"algorithmic information content". If that index of a theorem you wish
to proof in a certain calculus is lower than the index of the calculus
then the theorem is too complex and you can't prove it. You may add
this theorem as a new axiom however. The complexity index of the new
calculus is now higher that the previous one, but it won't do you much
good.
While Chaitin introduced a quantitative measure of information into
"logic", the logician Dana Scott (also still alive) introduced what he
called, "qualitative information" into logic.For this he received the
Turing award in 1976. I can't explain it in a few words howerver. A
good introduction to all that can be found in "Handbook on the
Philosophy of Information", Ed. J.v. Benthem, P. Advioans, Elsevier,
especially the article by Abramsky. I do not know if this book has
appeard yet, I think it is just in the the process of appearing.
Ron, it was geat fun talking to you!
Thanks!
Claus
On 2 Okt., 09:08, Ron Allen <wavel...@pacbell.net> wrote:
> Hello again aristotle-logic group and Claus in
> particular:
>
> Last post from me was broken off just as it began to
> get good! What happens is that my email
> provider--AT&T, the huge US telecom
> conglomerate--somehow thinks that quoted reply texts
> only need to be so long...and a post of moderate
> length, such as Claus's recent one, gets cut off. So
> what I have to do is open up the post I'm replying to,
> copy out the entire text to a text editor outside the
> browser, and then reply by pasting in the text and
> putting in my own > marks. A pain.
>
> Oh well, I'm going to cut out the earlier part of
> Claus's post and insert the part that was chopped by
> AT&T/Yahoo. Just so it's clear, I'll flag my new
> remarks with [ron].
>