"Prior Analytics" Book A, Chap. 4 -- First Figure
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syllogistic.
In Ch. 4 of the first book of the "Pr. An.", The Philosopher lays out
four syllogisms, all of which happen to be perfect. The Attic Greek
word that we translate as 'perfect' is 'teleios', which primarily means
'complete', 'finished', 'fulfilled', or 'accomplished'. You might
recognize the Greek root in such English words as 'teleology'. It also
means 'perfect' or 'entire' in a secondary sense. The Tredennick
translation in the Loeb edition uses the conventional 'perfect'.
A syllogism is made perfect by using the conversions (25a14ff) and
inserting missing premises that are deduced from prior ones in the
account by way of perfect syllogisms of the First Figure.
The conversions are as follows:
(1) if A belongs to no B, then B will belong to no A (25a15);
(2) if A belongs to all B, then B belongs to some A (25a19);
(3) if A belongs to some B, then B belongs to some A (25a21).
The (perfect) First Figure syllogisms are:
Barbara: if A belongs to all B, and B to all C, then A belongs to all C
(25b39);
Celarent: if A belongs to no B, and B to all C, then A belongs to no C
(25b40-26a1);
Darii: if A belongs to all B, and B to some C, then A belongs to some
C (26a23-25);
Ferio: if A belongs to no B, and B belongs to some C, then A does not
belong to some C (26a26-27).
So, tomorrow I hope to give some examples of valid syllogisms that are
not perfect. We'll make them perfect by using the conversion rules and
inserting missing premisses into the syllogistic logos (account).
Thanks!
--Ron
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or "teleios" according to Aristotle. In the previous post on the First
Figure syllogisms (attached below), I explained the three rules of
conversion. These are basically "single-premise" syllogisms, if you
want to call them that. They are always valid, as long as the terms are
not vacuous, and I think that Aristotle generally considers them to be
non-vacuous. The conversions, together with the procedure of adding in
premises that are left out of a syllogism, are the two steps that we
can apply to perfect a syllogism.
(1) For example, suppose I had a "Barbara chain": If A belongs to all
B, and B to all C, and C to all D, then A belongs to all D.
This is valid, but it's not perfect. There is a missing predication,
namely, that A belongs to all C. If I add this statement to the
account, then, together with C belongs to all D, we conclude perfectly
that A belongs to all D. The original syllogism gets perfected by
supplying the missing conclusion in the first Barbara premise of the
chain, and this becomes the missing premise of the second Barbara in
the chain.
(2) If A belongs to all B, and B to all C, and A belongs to no D, then
D belongs to no C.
It's really hard to see if this second one is valid or not without
working out the perfection steps!
First, we have the Barbara premises in the first part, so that there is
a missing statement that A belongs to all C. We can insert the omitted
premise wherever we want. We insert it just before the conclusion.
Thus, we have
(2a) If A belongs to all B, and B to all C, and A belongs to no D, and
A to all C [by Barbara], then D belongs to no C.
Now we can convert the third premise in (2a) to get
(2b) If A belongs to all B, and B to all C, and D belongs to no A [by
conversion], and A to all C [by Barbara], then D belongs to no C.
The syllogism is perfect because of Celarent on the end.
Thanks!
--Ron
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do this, we take some sequence of categorical statements and
(1) convert sentences to their equivalent forms;
(2) insert conclusions that follow from previous statements by one of
the First Figure syllogisms (Barbara, Celarent, Darii, or Ferio).
The conversions and perfect (complete) First Figure syllogisms are
given below in the quoted text.
Aristotle carries out the perfection of imperfect syllogisms in the
"Prior Analytics" (at Bekker numbers 27a1-18). These happen to be
syllogisms of the Second Figure, Cesare and Camestres.
A good explanation of Aristotle's procedure is given by L. Rose,
"Aristotle's Syllogistic," Springfield, IL: Charles Thomas, 1968,
Chapter V, pp. 34ff.
What do the four syllogisms of the First Figure have in common? Well,
in all of the syllogisms one of the terms is special in that it is both
contained in (or excluded from) one of the others and contains
(excludes) one of the others. This is the middle term.
Here is Barbara again:
If A belongs to all B, and B to all C, then A belongs to all C (25b39).
The middle term is B. It is contained in A and it contains C. Aristotle
has the awkward (to us, perhaps) way of expressing "all Bs are As" by
"A belongs to all B". An example is "Animal belongs to all Bison". We
might say this more clearly as "Every Bison is an Animal," but The
Philosopher has chosen another way to express the same idea. Thus, in
Barbara, the term B is both a subject in "A belongs to all B" and a
predicate in "B [belongs] to all C." So the First Figure syllogisms are
those in which the middle (common) term is a subject of one premise and
a predicate of the other.
One question is why Aristotle thinks that the First Figure syllogisms
are perfect or complete (teleios, in Greek), but the deductions of the
other Figures are imperfect. They're all categorically valid, so
there's nothing wrong about the Second and Third Figures. To me it
seems that the alternating nature of the middle term in the First
Figure is the key. When the middle term is first a subject and then a
predicate, it constitutes a kind of trace path for the inclusion or
exclusion of terms. This makes the validity of the syllogisms of the
First Figure particularly clear.
We haven't covered the Second and Third Figure syllogisms yet. However,
the middle term of a Second Figure syllogism is a predicate both times,
and the middle term of a Third Figure syllogism is a subject in both
premises.
In the next few posts, I'll cover the Second & Third Figures, and some
alternatives to the direct perfection technique for showing a
categorical account to be valid.
Thanks!
--Ron
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I've been on vacation for the last two weeks, but I reread the
"Analytics" as well as Bochenski's book, "Ancient Formal Logic,"
Amsterdam: North Holland, 1951, while on the river cruise; now I'm
ready to cover the other two Figures of Aristotle's syllogistic.
Just to review:
(1st Figure) Syllogisms where the middle term is a subject of one
premise and a predicate of the other.
(2nd Figure) Syllogims where the middle term is a predicate in both
premises.
(3rd Figure) Syllogisms where the middle term is a subject in both
premises.
I'll start new threads for the 2nd and 3rd Figures.
Also, you can see--perhaps--why Aristotle only had only three Figures.
The above schemata exhaust all the possibilities. If there is no middle
term, there can be no syllogism. If there is a middle term, then it can
be either twice a predicate (2nd Figure), twice a subject (3rd Figure),
or once a subject and once a predicate (1st Figure). In any of the
Figures, Aristotle does not consider the order of the premises to be
significant; indeed, the conclusion necessarily follows whatever the
order of the premisses. Hence, there are only three Figures. Aristotle
does not make this clear, but it seems to follow from his presentation
in the "Prior Analytics."
Thanks!
--Ron
The Philosopher considers the second Figure beginning in
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Before covering the 2nd and 3rd Figures, I wanted to comment on some
First Figure issues. But I will also be brief (and hopefully not
incorrect!).
(1) What does Aristotle do with the rest of Chapter 4 of Book A of the
"Prior Analytics?"
(2) What is the evidence that Aristotle views syllogisms to fall into
schemata, or Figures, that are defined by a triple A-B-C?
(3) Is Aristotle's notation for a syllogistic triple A-B-C consistent
with the notation he uses for a proposition, such as might form one of
the premises of his syllogistic accounts?
(a) On the first question, Aristotle stabs us with Barbara and Celarent
at (25b39-26a1), but then The Philosopher dwells on some arrangements
of terms that admit no syllogism. Thus, he considers syllogistic
arguments of the 1st Figure format, where the middle term is a subject
and then a predicate, but with the quantifications all-none (A-E) and
none-none (E-E). You can find this at (26a2-17). These aren't First
Figure accounts; they're not valid, if fact. But it's interesting that
Aristotle spends more time on the invalid arguments than on the valid
ones.
>From a modern perspective, it seems to me that Aristotle offers
model-theoretic arguments as opposed to logico-deductive arguments that
certain A-B-C accounts are valid or invalid. It seems further to me
that Aristotle is implicitly invoking a kind of completeness argument,
to the effect that if his categorical theory has a model, then it is
consistent; and further, if the categorical account is refuted by some
model, then there is no deduction from an A-B-C account, with the
middle term B first a subject then a predicate, and, for example,
quantifiers being A-E or E-E. Thus, so far, Aristotle is taking his
possible axiom systems of categorical theory to be already modeled by
some configuration of categories in the real world.
This is clearly a modern perspective on Aristotle's logic, now
twenty-five centuries old. The clear distinction between semantic and
syntactic developments has only been made in the 20th century, and we
can't expect any ancient logicians to anticipate these results. So
there is a blurring of the two in the "Prior Analytics." But, it's a
correct, if blurred, elaboration, nevertheless.
For the bulk of Chapter 4, Aristotle is making the metalogical argument
that his perfect First Figure syllogisms comprise precisely those that
are valid among premise-conclusion categorical accounts. He comes back
again to Darii and Ferio--and again it's a sharp, quick remark--at
26a24-28.
(b) For some reason, Aristotle's preference for using triple schemata
of the form A-B-C are shown not in the perfect syllogisms of the First
Figure, but rather in the model-theoretic counter-examples where he
rejects this or that account as invalid.
(c) Actually, yes, at 26a29 Aristotle identifies a proposition by BC
(beta-gamma Greek), and this, together with his habit of giving
examples in terms of instantiated triples, such as "animal-man-horse"
at 26a9, seem to show that he's thinking of propositons as symbolic
pairs A-B and syllogisms as symbolic triples A-B-C.
I'll continue this discussion tomorrow and provide some references to
the scholarly literature. Hope this isn't too abstract!
Thanks!
--Ron
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I'm continuing to address the second question below: Does Aristotle
view syllogisms as defined by a triple of categories, A-B-C?
The first place in the "Prior Analytics" that Aristotle uses the triple
to refer to a syllogism is at 26a9, where he says "The positive
relation of the extremes may be illustrated by the terms
animal-man-horse; the negative relation by animal-man-stone."
(Tredennick translation, in the Loeb edition of Aristotle's works)
Aristotle has just finished enunciating Barbara and Celarent. Barbara:
"For if A is predicated of all B, and B of all C, A must necessarily be
predicated of all C" (25b39).
In the Greek: "ei gar to A kata pantos tou B kai to B kata pantos tou
G, anagke to A kata pantos tou G kategoreisthai...."
Or, translating literally (my rash mistake): "For if the A according to
all of the B and the B according to all of the G, necessarily the A
according to all of the G is to be predicated"
You can see that Tredennick makes the terse Greek a lot more readable
in the English. In the Greek, the verb "to be" is implicit, and there
is only the infinitive "kategoreisthai", and it applies to all of the
preceding clauses which have the "kata pantos" or "according to all"
phrases in them. Ah, the Greek alphabet, kind of like the Russian, goes
alpha, beta, gamma, or A-B-G, not A-B-C, like our Latin-derived
alphabet in English.
Anyway, in the paragraph that follows, Aristotle says that "If,
however, the first term applies to all of the middle, and the middle to
none of the last, the extremes cannot admit of syllogism..." (26a2-3).
This would be the putative syllogism with premises: "A belongs to all
B" and "B belongs to no C". That is, the premises are AaB and BeC. It
is clear, if you draw a Venn diagram picture of the statement of the
premises, that Aristotle is right: no necessary conclusion follows. For
the C category items, which are disjoint from the B category
substances, might lie outside of A or they might lie inside of A, but
outside of B. Thus, the relation between A and C is not determined by
the premises. Aristotle says this too: "...it is possible for the first
term to apply to either to all or to none of the last..." (26a5) (in
the Greek: "...kai gar panti kai medeni endechetai to proton to eschato
huparchein...")
Aristotle goes on to wrap up this paragraph with some triple-examples:
animal-man-horse and animal-man-stone. These triples represent the
positive and negative relations, respectively, that are implied by the
triplicate form of the syllogism.
Thus, we have animal-man-horse as an example of the positive relation
with premises AaB and BeC, but where C is inside of A (horses are
animals), but C is disjoint from B (horses are not human beings). We
also have animal-man-stone as an example of the negative relation with
the very same premises AaB and BeC, but where C is outside of A (stones
are not animals). This is what Aristotle is trying to say by his
triple-examples just before 26a10.
So this is the first evidence I would point to that shows that
Aristotle views the syllogism's structure, or scheme, as defined by a
triple of categorical terms. It's strange that he uses the triples to
provide examples of non-syllogisms, rather than valid syllogisms, but
that's just the way the text reads. Also, the quantifiers, "belongs to
all", "belongs to no", are left out of the triples. You just have to
read the prior text to figure out what Aristotle is talking about.
Thus, if we see the syllogistic Figures as governed by the triples
A-B-C, then there are just three cases: the middle category B is first
a subject, then a predicate (First); a predicate in both premises
(Second); and, finally, a subject twice (Third Figure).
Then, Aristotle accepts or rejects the possible syllogistic forms by
inserting quantifiers between the categories of the triple.
L. Rose, "Aristotle's Syllogistic," Springfield, IL: Thomas, 1968,
argues for a triplicate form of the definition of the syllogism. If
this argument is correct, then it explains why there are just three
syllogistic Figures. See Chapter 3 of Rose's book (p. 16ff) for a
discussion of why there are just three Figures. On the cover of my copy
of Rose's book, there is a blurb: "This is the first book to examine
Aristotle's syllogistic from the point of view that Aristotle regarded
the syllogism as a rectilinear array of three terms."
So, I've been trying to support Rose's view in working through the
"Prior Analytics" in the last few posts to the list.
Thanks!
--rla
Zeno777
translations. The word "Perfect" implies a use of some technological
measuring device to measure things, and adherence to some model,
engineering wise to the latest fraction of a metric or English unit.
The Japanese company that makes all Japanese auto transmissions uses a
metric measure so fine that American engineers can find no deviations
from one unit to another, prompting them to say "Such a transmission
will last forever--meaning in engineer-speak, very much longer than
transmissions whose parts are machined to a lower standard of
accuracy--but even these are not perfect, they just ramp up the
standards of measurment, and resquire finer measuring devices such as
finer micrometers and perhaps electronic micrometers, to find any
deviations or differences amongst the individual machined parts. Thus
the search for the "Perfect" is a never-ending, never-complete,
never-finished process, engineering-wise. Such is the meaning of
"Perfect" in Western European languages, especially English. You can't
make a thing both 'Perfect" and complete and finished, and then go on
forever. You always have to kep fiddling with it, and be laughed at,
or rediculed for claiming "Perfection", and engineering wise, the
detractors are correct.
You really want "Complete and Finished," and a going on to more
Different, more Interesting and more Inspiring Tasks.
Aristotle started his philosophy with Rhetoric, as Athenians had to be
their own lawyers, and the asssembly, so far as I know, served as the
court as well as a legislature. One has to defend one's property
there, and to educate wealthy young men to protect their property, was
the major reason by which the Academy justified its existence and
gathered fees from wealthy young students. Aristotle was also the
Academy's Mastser Publicist. He was also know as the greatest popular
writer of Ancient Times, serving with his popular writings as the rople
model for the Roman writer, Cicero. Unfortunately Aristotle's greatly
praised popular writings are mostly lost. The Aristotle we know today,
is through his secret writings, open only to rich young Greeks who
payed the Lyceum a Hefty Fee.