"Prior Analytics" Book A
waveletter
We definitely need a thread on the "Analytics."
Aristotle viewed the "Prior Analytics" and the "Posterior Analytics" as
parts of one overall treatise. We know that the "Prior Analytics" is
about deductions, in particular syllogisms, whereas the "Posterior
Analytics" covers demonstrative sciences. The basic difference between
the two is that a syllogism is correct, but can be empty of content,
whereas a demonstration has to have true, referential premisses.
Now, the first sentence in the "Prior Analytics" is as follows:
"Our first duty is to state the scope of our inquiry, and to what
science it pertains; that it is concerned with demonstration, and
pertains to a demonstrative science." (24a10-11) [I'm quoting from the
Loeb edition of Aristotle's works, "Aristotle: Categories, On
Interpretation, Prior Analytics," trans. H. Tredennick, Cambridge, MA:
Harvard, 1996.]
The word that Tredennick translates as 'demonstration' is the Greek
word 'apodeixis' or 'a showing forth' or 'a proof'. It's not the
technical term that Aristotle will elaborate upon throughout the rest
of the "Pr. An." The scope of the text is "demonstrative science"
("epistêmês apodeiktikês," accusative case in Greek), and thus it
looks like Aristotle is introducing the "Posterior Analytics." But
right after that he proposes the tasks of defining a premise, a term, a
syllogism (sullogismos), perfect and imperfect syllogisms, how one term
is or is not contained in another term, and what is meant by predicated
of all or none. The Greek word he uses for 'predicated' is
'katêgoreîsthai', the passive infinitive, and it is cognate to the
word 'katêgoriai' or 'categories'; I like to think of the word as
saying "to be categorized as", although the standard translation is
"predicated". After the first sentence states that the enquiry is about
demonstrative science, Aristotle goes right into the task of
explicating his purely logical machinery.
I would guess that the first sentence is one that was written with the
anticipation that the two "Analytics" would be grouped together, and
was left in place in the "Pr. An." when later redactors split the
original treatise. In any case, it is clear that The Philosopher will
found his demonstrative science on a method of deductive reasoning.
Among the pieces comprising this method of deductive reasoning are:
(1) terms;
(2) premisses;
(3) syllogisms, which can be perfect or imperfect;
(4) a notion of term containment;
(5) a notion of predication, which can apply to all or none of
something or other.
Well, now, we are only into a few sentences of the "Pr. An.", but it
seems that Aristotle has named, if not yet specified, a logical
machinery that is not unlike what we get in modern textbooks. We have
to read further to see exactly what Aristotle will present, but in the
mathematical logic that was worked out in the 20th century, predicate
logic had terms; deductive rules, which have premisses; and even
concepts of interpretation of terms and predicates, for which
Aristotle's ideas of containment and predication seem to gesture. It's
an intriguing parallel. Does it carry through in the text?
Next time: syllogisms, perfect and imperfect.
--Ron
waveletter
An.", as I'll abbreviate it, covers deduction--or syllogism,
sullogismos in Greek--and it provides a foundation for the
demonstrative science that will be explicated in the "Posterior
Analytics." We could cover some of the components of a formal system of
deduction, but I propose to rush right into Aristotle's notion of a
syllogism.
Despite its first sentence, which does appear to spotlight the purpose
of the analytics as a whole, Aristotle's "Prior Analytics" is about the
syllogism. We all know the standard example of a syllogism:
(1) All men are mortal; Socrates is a man; and therefore Socrates is
mortal.
An influential book, based on research that began decades earlier, but
appearing in English only in the 1950s, reawakened interest in
Aristotle's "Organon"--his logical works. But this book assiduously
attacked our standard syllogistic exemplar (1), arguing instead that
the form proposed by Aristotle was in fact an if-then statement:
(2) If all men are mortal and if Socrates is a man, then Socrates is
mortal.
This book, J. Lukasiewicz, "Aristotle's Syllogistic From the
Standpoint of Modern Formal Logic," Oxford: Clarendon, 1951, sparked an
interest and debate in Aristotle's logic that continues to this day.
Actually, Lukasiewicz argues that singular terms, such as 'Socrates',
appearing in the above syllogisms--whichever one happens to be a bona
fide syllogism!--do not belong there. Faithful to The Philosopher's
logic would be something like,
(3) If all men are mortal, and if all Greeks are men, then all Greeks
are mortal.
This syllogism has no singular terms. The Greeks are a species, as it
were, and they belong a genus, the genus of men. We note in passing
that this syllogism isn't very good demonstrative science, since some
Greeks are not men--they are women--and if they were only men, then
after a while there wouldn't be any Greeks at all, the ultimate in
singularity. Well, enough jejune deconstruction of Lukasiewicz's
example. What does Aristotle say of syllogism?
Chapter 1 of the Book A of the "Pr. An." Aristotle says that
"A syllogism is a form of words in which, when certain assumptions are
made, something other than what has been assumed necessarily follows
from the fact that the assumtions are such. By 'from the fact that they
are such' I mean that it is because of them that the conclusion
follows; and by this I mean that there is no need of any further term
to render the conclusion necessary." (24b19-22) [Aristotle, "Prior
Analytics," in vol. I of Loeb Aristotle series, trans. H. Tredennick,
Cambridge: Harvard, 1996]
Some things to note from this definition:
1. The "form of words" is actually the overloaded Greek word 'logos',
which in this context might be better rendered as 'account'.
2. The syllogism has assumptions and a conclusion, and the conclusion
differs from the assumptions. Thus, in modern propositional logic,
something implies itself, another way to say that (not A) or A, but
this doesn't seem to be allowed by Aristotle as a syllogism.
3. The assumptions alone are sufficient for the conclusion. Actually,
Aristotle says that "...there is no need for any further term to render
the conclusion necessary" (24b20-22). An "imperfect" syllogism
"...requires one or more propositions which, although they necessarily
follow from the terms which have been laid down, are not comprised in
the premisses" (24b25-27).
4. The conclusion follows of necessity.
5. There does not seem to be any restriction of the assumptions to only
two parts, unlike our stereotypical syllogism (with or without Prof.
Lukasiewicz's corrections to the form). Thus, at this point, it seems
that a syllogism might have a tripartite, or even more complex,
premise.
--RLA
waveletter
In the previous post, I listed the logical pieces that comprise a
syllogism. The smallest piece is the term. Quoting Tredennick's
translation of the "Prior Analytics" from the Loeb edition of
Aristotle's works,
"By a term I mean that into which the premiss can be analysed, viz.,
the predicate and the subject, with the addition or removal of the verb
to be or not to be." (24b17-19)
It's a little obscure, this passage--or perhaps the translation. But
the premiss (protasis, in Greek) of a syllogism is broken down
(dialuetai, in Greek, passive voice) into two terms, a subject and a
predicate. What these are might not be clear from the English, but the
Greek is "to te katêgoroumenon kai to kath' hoû katêgoreîtai...."
Roughly, I would translate this literally as "both that which has been
predicated and that according to which it is predicated."
But the useful clue is the word 'category' or 'predicate' in the Greek,
which points us back to the "Categories." For example, in Book III of
Aristotle's "Categories" we have the example of "a human being is an
animal". The terms are instances of categorizations. That which has
been predicated is the predicate and that according to which it is
predicated is the subject. So the Tredennick translation is in line
with this, even if it is not that literal. Another example from Book
VIII of "On Interpretation" is "a human being is pale." These could be
premisses of a syllogism.
There are three types of premisses:
(1) "...the premiss will be demonstrative if it is true and based upon
fundamental postulates...." (24a31) Here Aristotle uses the overloaded
term archê, the premiss is true ex archês, which I would rather
translate as "from fundamental principles". We encountered this word in
Book I of the "Topics," and we could stipulate that the "Posterior
Analytics" is about reasoning with demonstrative premisses.
(2) "...the dialectical premise will be, for the interrogator, an
answer to the question which of two contradictory statements is to be
acceptd, and fo rthe logical reasoner, an assumption of what is
aparently true and generally accepted,--as has been stated in the
'Topics.' (24b10-13) Thus, the dialectical reasoning explained in the
first book of the "Topics" is put into its proper place here at the
beginning of the "Prior Analytics." We could stipulate that the
"Topics" is about reasoning from dialectical premisses.
(3) "...a syllogistic premiss will be simply the affirmation or
negation of some predicate of the subject...." (24a28-30) And finally,
we could stipulate that the "Prior Analytics" is about reasoning from
deductive premisses.
--RLA
waveletter
There are two components of Aristotle's formal logic that I haven't
covered yet--at least as far as we've read in the first book of the
"Prior Analytics:" these are the notion of term containment, which was
the fourth item I pointed out above, and the notion of predication,
which can (fifth item) apply to all or none of something or other.
Once again working from Trendennick's translation, in the Loeb edition
of Aristotle's works, we read that "For one term to be wholly contained
in another is the same as for the latter to be predicated of all of the
former. We say that one term is predicated of all of another when no
examples of the subject can be found of which the other term cannot be
asserted. In the same way we say that one term is predicated on none of
another." (24b28-31)
To me, at least, the thing that comes out here is the shift from the
logic language to the referents. There is in fact, I would suggest, a
kind of model-theoretic criterion at work here: "...no examples of the
subject can be found...."
For that matter, how would you define one predicate in predicate logic
to be contained in another? This could in fact be done syntactically:
(PQ) (for all)(x)(P(x) -> Q(x)),
which is to say that Q is contained in P.
But it also means that in any model of the theory with this axiom (PQ),
the things having being Q are contained in the things having being P.
You can't really fault someone 25 centuries ago for not clearly
separating the language-predicates from the model-predicates, but it
does seem like Aristotle has the idea in mind, if not explicitly
exposing in his text formulation.
I'm reminded of Quine's example of "All bachelors are unmarried" from
'Two dogmas of empricism,' in "From a Logical Point of View," Harvard,
1953, where he asks whether it's true of the terms 'bachelor' and
'unmarried' that one implies the other. How does one resolve this? Just
make a convention? But who did that? Nobody. It seems that one is
naturally inclined to point to the extensions of each term and ask if
one includes the other. Indeed, the language of containment seems to
gesture at sets of things rather than at logical implication. This idea
goes back to Aristotle, and it stops and begins right here in the
"Prior Analytics."
The next element of the syllogism is quantification, at least in
Aristotle's terms, and I'll touch upon that next time.
--Ron
waveletter
The second paragraph of Book A, Chapter 1 of Aristotle's "Prior
Analytics" explains what we might like to call the "quantification"
aspects of The Philosopher's logic. Quoting from the Loeb edition of
Aristotle's works, trans. H. Tredennick, we find
"A premiss is an affirmative or negative statement of something about
some subject." (24a16-17)
Negation has been covered in an earlier (I suppose so, at least!) work,
"On Interpretation," also included in the same volume of the Loeb
series as the "Pr. An." See Book VII of "On Interpretation", Bekker
numbers 17a39 ff. for another formulation of these ideas. I'll
eventually pick up "De Intepretatione", as "On Interp." is sometimes
known by Latinophiles and medieval scholars, in a subsequent thread.
Suffice it to say for now that the terms Aristotle considers are
elaborated in the "Categories," the element of negation in "On
Interp.," and the rest of his foundational logic in the "Pr. An."
"This statement may be universal or particular or indefinite. By
universal I mean a statement which applies to all, or to none, of the
subject; by particular, a statement which applies to some of the
subject, or does not apply to some, or does not apply to all...."
(24a17-20)
Although Aristotle does introduce letter variables for the terms in his
logical apparatus, he does not have a notation for these quantification
elements. In medieval times--precisely when is unclear--notations were
adopted for universal and particular, affirmative and negative. If S
and P are terms, subject and predicate, then we have
SaP : "all S are P" : universal affirmative
SeP : "no S is P" : universal negative
SiP : "some S is P : particular affirmative
SoP : "some S is not P" : particular negative.
Some historical notes on how these mnemonics arose are given by I.M.
Bochenski, "A History of Formal Logic," trans. I. Thomas, Notre Dame,
IN: Notre Dame University Press, 1961, pp. 210 ff. I hope to comment on
this next time.
Now that we have the mnemonics for syllogistic quantification, as it
were, we can turn to Barbara, Celarent, Darii, and the rest of those
Latin codewords for the moods of the syllogisms. We have to read some
more Aristotle first, though: Chapter 4 of Book A of the "Prior
Analytics" (25b26 ff.)
Thanks!
--Ron
waveletter
There is a small (large?) point of translation upon which Aristotle's
concept of syllogism turns in Book A, Chapter 4 of the "Prior
Analytics."
At the beginning of the chapter The Philosopher says that "Having drawn
these distinctions we can now state by what means, and when, and how
every syllogism is effected." (Bekker numbers 25b26-27)
It seems like Aristotle is going to tell us the form of every
syllogism. Earlier I noted that there was some reason to think that
Aristotle does not necessarily put forward a three-part theory of the
syllogism. For example, a syllogism is "a form of words in which, when
certain assumptions are made, something other that what has been
assumed necessarily follows...." (24b19-21) So it does not seem,
granting the earlier general remark, that Aristotle intends to restrict
the concept of a syllogism to the form of a bipartite premise followed
by a conclusion.
But now, Aristotle seems to be saying that he is going to tell us how
every syllogism is "effected", to quote the Tredennick translation in
the Loeb edition of Aristotle's works. Indeed, beginning at (25b31 ff.)
Aristotle starts into the First Figure of the syllogism, and it seems
like we are going to show how the Figures effect each and every
syllogism. It's an easy interpretation, and one that many scholars have
made, yes, but I would like to urge some caution in taking it just like
that.
To begin with, on the side of cautious interpretaition, I would like to
point out that the original Greek goes as follows: "...kai pôs
gignetai pâs sullogismos...." (25b27). The crucial verb is 'gignetai',
which means a lot of things. To be sure, it can mean "happen" or
"amount to" or "occur". Its form is the 3rd person singular of the verb
'gignomai'. But it can also mean 'be born' or 'come into being'. Thus,
it could well be that Aristotle is *not* saying that every syllogism is
of the form of one of the three Figures, but that he has a more general
syllogistic form in mind, and all of the instances of the form of the
syllogism are born, or come into being, or arise from, the particular
forms that he next describes.
For a dictionary definition, you might check out the "Pocket Oxford
Classical Greek Dictionary," ed. J. Morwood & J. Taylor, Oxford Univ.
Press, 2002.
It took me a while to come around to this point of view. If you're
following along out there in cyberspace, I'd urge you to have a look at
Tredennick's translation as well as the original Attic Greek text. This
is an important point as far as modern interpretations of Aristotle's
theory of deduction are concerned.
Thanks,
--RLA
waveletter
"effected" as the Tredennick translation gives (in the Loeb edition of
Aristotle's "Prior Analytics") or "arises" or "is caused", as I
suggested might be the case.
If you happen to have Prof. Robin Smith's translation of the "Prior
Analytics," then you can see that he provides a slightly different
rendering of Aristotle's Greek text. Here it is:
"...let us now say through what premises, when, and how every deduction
comes about" (25b27, at the very beginning of Book A, Chapter 4). So
the more modern Smith translation allows for the case that the
syllogistic forms that follow this passage are not the only forms, but
just the fundamental forms in some sense.
The reference is Aristotle, "Prior Analytics," trans. Robin Smith,
Indianapolis, IN: Hackett, 1989.
--Ron
waveletter
I want to pick up the puzzle about the historical origin of the
mnemonics for Aristotelian syllogisms. By the way, I can't reply to the
specific post in the tree that noted the puzzle, because Google groups
has this awkward policy of closing Replies to messages older than 30
days. Oh well.
Here's what I wrote on 02-May-2006:
> Although Aristotle does introduce letter variables for the terms in his
> logical apparatus, he does not have a notation for these quantification
> elements. In medieval times--precisely when is unclear--notations were
> adopted for universal and particular, affirmative and negative. If S
> and P are terms, subject and predicate, then we have
>
> SaP : "all S are P" : universal affirmative
> SeP : "no S is P" : universal negative
> SiP : "some S is P : particular affirmative
> SoP : "some S is not P" : particular negative.
>
> Some historical notes on how these mnemonics arose are given by I.M.
> Bochenski, "A History of Formal Logic," trans. I. Thomas, Notre Dame,
> IN: Notre Dame University Press, 1961, pp. 210 ff. I hope to comment on
> this next time.
So, just a few notes on the origin of "a, e, i, o" and the "Barbara,
Celarent, Darii, Ferio" that come out of them.
(1) The mnemonics seem to originate in the early 13th century--that is,
around 1200. This is because Prof. Minio-Palulello of Oxford found the
word 'Festino' on a manuscript from no later than 1200 C.E.
(2) The mnemonics had become well-enough known by 1250 C.E. that Peter
of Spain could write them in verse (Bochenski, "History of Formal
Logic," p. 212).
(3) However, the water is muddied considerably by the theory of their
Byzantine origin, put forward by Prof. Prantl in the 19th century.
(4) Prof. Karl von Prantl (Prof. at the Munich Observatory) contended
that that the mnemonics were much older, and were devised by the
Byzantine scholar Michael Psellus (1018-1078). Prantl held that the
"Summulae" of Peter of Spain were a translation of Psellus's work,
"Sunopsis eis tên Aristotelous logikên epistêmên" or "Synopsis in
the logical knowledge of Aristotle."
(5) It turned out that Prof. Prantl was quite wrong, and that, in fact,
other scholars subsequently demonstrated that the "Sunopsis" was
written by George Scholarios (1400-1464 C.E.), thus exploding the myth
of the Byzantine origin of the "a,e,i,o" mnemonics.
(6) So, if this account is right, we can place the origin of our
familiar "a, e, i, o" mnemonics at around 1200 C.E., and say that they
were most prominently codified or, at least, popularized by Peter of
Spain.
Of course, in reading Bochenski's "History", it seems that the
discussion about the origin of the mnemonics was quite fluid at the
time of his text (1956, Freiburg edition). Perhaps some later
scholarship has directed better light onto the question. Does anyone on
the list know more about these questions? References?
Thanks!
--Ron
waveletter
So, I'm still thrashing around the Tredennick translation (Loeb edn.)
of the beginning of Book A, Ch. 4 of the "Prior Analytics" (Bekker
number 25b27). Tredennick translated the Attic Greek verb 'gignetai' as
'effected', suggesting that every syllogism has the form of those that
follows, namely with two premises and a conclusion, beginning with the
Barbara form (if A is predicated of all B, and B of all C, A must be
predicated of all C) which is introduced soon afterwards.
There are a number of reasons why we should think that Aristotle did
not mean that every syllogism had two premises and a conclusion, like
Barbara; for example:
1. His definition of syllogism in Book A, Ch. 1 is decidedly more
general (24b19).
2. He does explicitly consider long chains of term containment later in
the "Pr. An.", to wit, in Ch. 23, beginning at (40b37ff). Here he says
that "...when C is connected to another term, and this to another, and
this to yet another, and the series is not connected with B, in this
case too we shall have no syllogism with reference to B" (40b40-41a2).
So he's saying that the long chain is OK, but it has to chain with any
term (B) that appears in the conclusion.
3. The case is made clearer in (41a13ff): "Since, then, we must take
some common term which is related to both, and this may be done in
three ways, viz., by predicating A of C and C of B, or C of both, or
both of C, and these are the figures already described, it is evident
that every syllogism must be effected ("gignesthai") by means of one of
these figures; for the same principle will also hold good if A is
connected with B by more than one term; the figure will be the same
also in the case of several terms." I used the Tredennick translation
again, just to show the consistent, questionable choice of 'effect' for
'comes about' or 'arises' or 'is born'.
4. There are places in "Prior Analytics" Book B that also clearly allow
for more than two middle terms. For example: "...let it be required to
establish that A is predicated of all F, the middle terms being B, C,
D, and E" (66a39-40).
So, I think I've shown that Aristotle holds syllogisms to be chains
using the "a, e, i, o" quantifiers on categorical terms where there are
at least two premises. The longer chains arise from the two-premise
chains. This whole argument is important, because understanding
Aristotle's logic correctly on this point affects conflicting scholarly
interpretations of syllogistic in the mid-20th century.
Thanks!
--Ron