FOM: Reading Aristotle ...

[Walter Felscher]

It was most interesting to read Mr. Tait's note on Platon
and Aristoteles from January 21st. In particular, my eye was
caught by his formulation

  Just as Aristotle's epistemology is the epistemology of
  classification, the logic which supports it, the
  syllogistic, is the logic of classification.

Below, I shall collect in (A) some definitions as stated by
Aristoteles, and shall in (B) comment on the classificatory
character of syllogistic logic.  As it fits into the
occasion, I shall add in (C) some remarks on definitions.



(A)  Conclusions and Proofs


In Analytica Priora, [book] 1.[chapter] 1, 24b, Aristoteles
explains the  notion of "conclusion" (Schluss); it is
rephrased in a slightly clearer way in Topic 1.1, 100a :

    a conclusion is a speaking in which, under certain
    assumptions, something different than the assumed
    follows necessarily by reason of the assumptions.

In APri.1.4, 25b-26a, Aristoteles then begins to speak of
his three "figures", or schemata, of conclusions; the first of
which is
          every B is A   every C is B  :  every C is A
          no B is A      every C is B  :  no C is A
          every B is A   no C is B        -
          no B is A      no C is B        -

containing the two "complete" conclusions indicated. The
term B , disappearing in the concluded sentence, is called the
middle term of the conclusion.

While the notion of "proof" (demonstration) is used
occasionally in Analytica Priora, the first explanation for
it seems to appear only in Analytica Posteriora where, at
the start of 1.2 , 71b , it is first said that a conclusion
is called scientific if, by having it, we know; then

    A scientific conclusion is called a proof, and by it
    then we know 'through proof'.

A more fitting explanation is given at the start of Topic 1.1 ,
100 a :

    a demonstration (apodeixis) is a conclusion from true
    and first sentences or from such sentences, which arise
    from true and first sentences

and this is what is actually used by Aristoteles in
Analytica Posteriora : a demonstration is a chain, or rather
a tree, of successive conclusions.

In APost.1.2 , 72a , the maximal premisses of a deduction
(i.e.  those not arriving as conclusions from earlier ones)
are classified as "axioms" if they need to be known by
everyone in the field, and as "theses" otherwise. The theses
are classified as "hypotheses" if they are assumed as true or
false, and as "definitions" otherwise.  In APost.1.14 , 79a ,
it is said that apodictical conclusions - those in
apodictical demonstrations - use the 1st figure, and that,
in particular, this is the case for mathematical proofs.

Proofs employing Aristoteles' (figures of) conclusions use
the middle terms as cuts. Today's cut-free deduction systems
are, therefore, as a-Aristotelian as can be.



(B)  The classificatory character of Aristoteles' logic


Consider a classification of a basic concept A (a category
in Aristoteles' sense) into concepts

    Ai        :  A with specific difference i ,
    ...

    Ai,..,j,k :  Ai,..,j  with specific difference k ;

for instance one of the systems of Linnaeus or Jussieu
classifying the concept flora. If n is the length of the
index sequence i,..,j , then there are n definitions leading
from A to Ai,..,j ; hence by n-1 applications of barbara we
can conclude that every Ai,..,j is an A .

Conversely, a deduction employing only the two valid
conclusions from the 1st figure may relate premisses "every
B is A" and "no B is A" to a definition (by genus and
proprium) of the middle term B .

In so far, chains of definitions correspond to chains of
conclusions; in particular, first (i.e. undefinable)
concepts correspond to maximal premisses of a deduction. Of
later authors who have emphasized this parallelity, I
particularly wish to mention Pascal in his De l'‚sprit
Ge/ome/trique of 1658 .

A considerable part of the later chapters of Analytica
Posteriora, as well as the books 6 and 7 of the Topic, are
occupied by detailed discussions of the choice of
appropriate definitions of the middle terms also in the
cases of incomplete conclusions coming from the figures
different from the first.


(C)  Definitions in nomine and Definitions in re


For today's scientist, definitions are nothing but
attributions of names and, in so far, are arbitrary;
declarations to this effect can be found already in Hobbes
(De corpore 1655) and in Pascal l.c. The situation was
different for Aristoteles and for the scholastic
theologians.

Aristoteles distinguishes (in APost 1.13 , 78a ) between
conclusions and proofs showing THAT their result is the
case, e.g.

    the planets do not twinkle
    what not twinkles is near        [accidental observation]
    hence the planets are near ,

and conclusions showing WHY it is the case, e.g.

    the planets are near
    what is near does not twinkle    [necessary by nearness]
    hence the planets do not twinkle

Proofs of the first kind may work with the accidental, while
those of the second kind work with the necessary.

Investigating the r“le of middle terms, the distinction
THAT/WHY is reflected in their definitions: there are those
which explain an abbreviation by saying what is abbreviated,
and those which explain the possible WHY of a name (APost
2.7 , 92b , and 2.10 , 93b/94a ). The mediaeval termini were
explained e.g. by Occam (Summa totius logicae , I.26 ):

   Diffinitionum quaedam est diffinitio exprimens quid
   nominis, et quaedam est diffinitio quid rei. Diffinitio
   quid rei non est necessaria disputandi scienti
   significatum vocabuli ... quia talis diffinitio non
   tantum exprimit quid nomen significat, sed exprimit, quid
   res est. Aliae sunt diffinitiones importantis quid
   nominis, quae non sunt nisi orationes exprimentes quid
   nomina significant; et tales diffinitiones proprissime
   sunt nominum negativorum et connotativorum et
   respectivorum ... alia exprimens quid rei non est
   proprissima ... quia tale connotativum non habet nisi
   diffinitionem exprimentem quid nominis tantum.

In today's algebra, for instance, the definition of a 'ring'
is nominal, while that of a PID or a UFD may be viewed as a
definition in re. .

In so far, it should not be surprising that the older
literature is full of discussions whether a definition is
'right' in that it captures the essence of the concept to be
defined. For mystically inclined authors, moreover, the
'right' definition or name will have an additionally loaded
importance - from the divine "I am who I am" to the sad tale of
Rumpelstilzchen.


W.F.