Aristotle
[Footnote 11: Analyt. Prior. II. viii.-x. p. 59, b. 1-p. 61, a. 4.]
[Footnote 12: Ibid. x. p. 61, a. 7-15.]
Of this reversing process, one variety is what is called the
_Reductio ad Absurdum_; in which the conclusion is reversed by taking
its contradictory (never its contrary), and then joining this last
with one of the premisses, in order to prove the contradictory or
contrary of the other premiss.[13] The _Reductio ad Absurdum_ is
distinguished from the other modes of reversal by these
characteristics: (1) That it takes the contradictory, and not the
contrary, of the conclusion; (2) That it is destined to meet the case
where an opponent declines to admit the conclusion; whereas the other
cases of reversion are only intended as confirmatory evidence towards
a person who already admits the conclusion; (3) That it does not
appeal to or require any concession on the part of the opponent; for
if he declines to admit the conclusion, you presume, as a matter of
course, that he must adhere to the contradictory of the conclusion;
and you therefore take this contradictory for granted (without asking
his concurrence) as one of the bases of a new syllogism; (4) That it
presumes as follows:--When, by the contradictory of the conclusion
joined with one of the premisses, you have demonstrated the opposite
of the other premiss, the original conclusion itself is shown to be
beyond all impeachment on the score of form, _i.e._ beyond
impeachment by any one who admits the premisses. You assume to be
true, for the occasion, the very proposition which you mean finally
to prove false; your purpose in the new syllogism is, not to
demonstrate the original conclusion, but to prove it to be true by
demonstrating its contradictory to be false.[14]**
[Footnote 13: Analyt. Prior. II. xi. p. 61, a. 18, seq.]
[Footnote 14: Ibid. p. 62, a. 11: [Greek: phanero\n ou)=n o(/ti ou)
to\ e)nanti/on, a)lla\ to\ a)ntikei/menon, u(pothete/on e)n a(/pasi
toi=s sullogismoi=s. ou(/tô ga\r to\ a)nagkai=on e)/stai kai\ to\
a)xi/ôma e)/ndoxon. ei) ga\r kata\ panto\s ê)\ kata/phasis ê)\
a)po/phasis, deichthe/ntos o(/ti ou)ch ê( a)po/phasis, a)na/gkê tê\n
kata/phasin a)lêtheu/esthai.] See Scholia, p. 190, b. 40, seq.,
Brand.]
By the _Reductio ad Absurdum_ you can in all the three figures
demonstrate all the four varieties of conclusion, universal and
particular, affirmative and negative; with the single exception, that
you cannot by this method demonstrate in the First figure the
Universal Affirmative.[15] With this exception, every true conclusion
admits of being demonstrated by either of the two ways, either
directly and ostensively, or by reduction to the impossible.[16]
[Footnote 15: Ibid. p. 61, a. 35-p. 62, b. 10; xii. p. 62, a. 21.
Alexander, ap. Schol. p. 191, a. 17-36, Brand.]
[Footnote 16: Ibid. xiv. p. 63, b. 12-21.]
In the Second and Third figures, though not in the First, it is
possible to obtain conclusions even from two premisses which are
contradictory or contrary to each other; but the conclusion will, as
a matter of course, be a self-contradictory one. Thus if in the
Second figure you have the two premisses--All Science is good; No
Science is good--you get the conclusion (in _Camestres_), No Science
is Science. In opposed propositions, the same predicate must be
affirmed and denied of the same subject in one of the three different
forms--All and None, All and Not All, Some and None. This shows why
such conclusions cannot be obtained in the First figure; for it is
the characteristic of that figure that the middle term must be
predicate in one premiss, and subject in the other.[17] In dialectic
discussion it will hardly be possible to get contrary or
contradictory premisses conceded by the adversary immediately after
each other, because he will be sure to perceive the contradiction:
you must mask your purpose by asking the two questions not in
immediate succession, but by introducing other questions between the
two, or by other indirect means as suggested in the Topica.[18]
[Footnote 17: Analyt. Prior. II. xv. p. 63, b. 22-p. 64, a. 32.
Aristotle here declares _Subcontraries_ (as they were later
called),--Some men are wise, Some men are not wise,--to be opposed
only in expression or verbally ([Greek: kata\ tê\n le/xin mo/non]).]
[Footnote 18: Ibid. II. xv. p. 64, a. 33-37. See Topica, VIII. i. p.
155, a. 26; Julius Pacius, p. 372, note. In the Topica, Aristotle
suggests modes of concealing the purpose of the questioner and
driving the adversary to contradict himself: [Greek: e)n de\ tô=s
Topikoi=s paradi/dôsi metho/dous tô=n kru/pseôn di' a(\s tou=to
dothê/setai] (Schol. p. 192, a. 18, Br.). Compare also Analyt. Prior.
II. xix. p. 66, a. 33.]
Aristotle now passes to certain general heads of Fallacy, or general
liabilities to Error, with which the syllogizing process is beset.
What the reasoner undertakes is, to demonstrate the conclusion before
him, and to demonstrate it in the natural and appropriate way; that
is, from premisses both more evident in themselves and logically
prior to the conclusion. Whenever he fails thus to demonstrate, there
is error of some kind; but he may err in several ways: (1) He may
produce a defective or informal syllogism; (2) His premisses may be
more unknowable than his conclusion, or equally unknowable; (3) His
premisses, instead of being logically prior to the conclusion, may be
logically posterior to it.[19]
[Footnote 19: Ibid. II. xvi. p. 64, b. 30-35: [Greek: kai\ ga\r ei)
o(/lôs mê\ sullogi/zetai, kai\ ei) di' a)gnôstote/rôn ê)\ o(moi/ôs
a)gnô/stôn, kai\ ei) dia\ tô=n u(ste/rôn to\ pro/teron; ê( ga\r
a)po/deixis e)k pistote/rôn te kai\ prote/rôn e)stin.... ta\ _me\n
di' au(tô=n pe/phuke gnôri/zesthai, ta\ de\ di' a)/llôn_.]]
Distinct from all these three, however, Aristotle singles out and
dwells upon another mode of error, which he calls _Petitio
Principii_. Some truths, the _principia_, are by nature knowable
through or in themselves, others are knowable only through other
things. If you confound this distinction, and ask or assume something
of the latter class as if it belonged to the former, you commit a
_Petitio Principii_. You may commit it either by assuming at once
that which ought to be demonstrated, or by assuming, as if it were a
_principium_, something else among those matters which in natural
propriety would be demonstrated by means of a _principium_. Thus,
there is (let us suppose) a natural propriety that C shall be
demonstrated through A; but you, overlooking this, demonstrate B
through C, and A through B. By thus inverting the legitimate order,
you do what is tantamount to demonstrating A through itself; for your
demonstration will not hold unless you assume A at the beginning, in
order to arrive at C. This is a mistake made not unfrequently, and
especially by some who define parallel lines; for they give a
definition which cannot be understood unless parallel lines be
presupposed.[20]
[Footnote 20: Analyt. Prior. II. xvi. p. 64, b. 33-p. 65, a. 9.
_Petere principium_ is, in the phrase of Aristotle, not [Greek: tê\n
a)rchê\n ai)tei=sthai], but [Greek: to\ e)n a)rchê=| ai)tei=sthai] or
[Greek: to\ e)x a)rchê=s ai)tei=sthai] (xvi. p. 64, b. 28, 34).]
When the problem is such, that it is uncertain whether A can be
predicated either of C or of B, if you then assume that A is
predicable of B, you may perhaps not commit _Petitio Principii_, but
you certainly fail in demonstrating the problem; for no demonstration
will hold where the premiss is equally uncertain with the conclusion.
But if, besides, the case be such, that B is identical with C, that
is, either co-extensive and reciprocally convertible with C, or
related to C as genus or species,--in either of these cases you
commit _Petitio Principii_ by assuming that A may be predicated of
B.[21] For seeing that B reciprocates with C, you might just as well
demonstrate that A is predicable of B, because it is predicable of C;
that is, you might demonstrate the major premiss by means of the
minor and the conclusion, as well as you can demonstrate the
conclusion by means of the major and the minor premiss. If you cannot
so demonstrate the major premiss, this is not because the structure
of the syllogism forbids it, but because the predicate of the major
premiss is more extensive than the subject thereof. If it be
co-extensive and convertible with the subject, we shall have a
circular proof of three propositions in which each may be alternately
premiss and conclusion. The like will be the case, if the _Petitio
Principii_ is in the minor premiss and not in the major. In the First
syllogistic figure it may be in either of the premisses; in the
Second figure it can only be in the minor premiss, and that only in
one mode (_Camestres_) of the figure.[22] The essence of _Petitio
Principii_ consists in this, that you exhibit as true _per se_ that
which is not really true _per se_.[23] You may commit this fault
either in Demonstration, when you assume for true what is not really
true, or in Dialectic, when you assume as probable and conformable to
authoritative opinion what is not really so.[24]**
[Footnote 21: Ibid. p. 65, a. 1-10.]
[Footnote 22: Ibid. p. 65, a. 10: [Greek: ei) ou)=n tis, a)dê/lou
o)/ntos o(/ti to\ A u(pa/rchei tô=| G, o(moi/ôs de\ kai\ o(/ti tô=|
B, ai)toi=to tô=| B u(pa/rchein to\ A, ou(/pô dê=lon ei) to\ e)n
a)rchê=| ai)tei=tai, a)ll' o(/ti ou)k a)podei/knusi, dê=lon; ou) ga\r
a)rchê\ a)podei/xeôs to\ o(moi/ôs a)/dêlon. ei) me/ntoi to\ B pro\s
to\ G ou(/tôs e)/chei ô(/ste tau)to\n ei)=nai, ê)\ dê=lon o(/ti
a)ntistre/phousin, ê)\ u(pa/rchei tha/teron thate/rô|, to\ e)n
a)rchê=| ai)tei=tai. kai\ ga\r a)/n, o(/ti tô=| B to\ A u(pa/rchei,
di' e)kei/nôn deiknu/oi, ei) a)ntistre/phoi. nu=n de\ tou=to kôlu/ei,
a)ll' ou)ch o( tro/pos. ei) de\ tou=to poioi=, to\ ei)rême/non a)\n
poioi= kai\ a)ntistre/phoi ô(s dia\ triô=n.]
This chapter, in which Aristotle declares the nature of Petitio
Principii, is obscure and difficult to follow. It has been explained
at some length, first by Philoponus in the Scholia (p. 192, a. 35, b.
24), afterwards by Julius Pacius (p. 376, whose explanation is
followed by M. B. St. Hilaire, p. 288), and by Waitz, (I. p. 514).
But the translation and comment given by Mr. Poste appear to me the
best: "Assuming the conclusion to be affirmative, let us examine a
syllogism in Barbara:--
All B is A.
. All C is B.
. . All C is A.
And let us first suppose that the major premiss is a Petitio
Principii; _i.e._ that the proposition _All B is A_ is identical with
the proposition _All C is A_. This can only be because the terms B
and C are identical. Next, let us suppose that the minor premiss is a
Petitio Principii: _i.e._ that the proposition _All C is B_ is
identical with the proposition _All C is A_. This can only be because
B and A are identical. The identity of the terms is, their
convertibility or their sequence ([Greek: u(pa/rchei, e(/petai]).
This however requires some limitation; for as the major is always
predicated ([Greek: u(pa/rchei, e(/petai]) of the middle, and the
middle of the minor, if this were enough to constitute Petitio
Principii, every syllogism with a problematical premiss would be a
Petitio Principii." (See the Appendix A, pp. 178-183, attached to Mr.
Poste's edition of Aristotle's Sophistici Elenchi.)
Compare, about Petitio Principii, Aristot. Topic. VIII. xiii. p. 162,
b. 34, in which passage Aristotle gives to the fallacy called Petitio
Principii a still larger sweep than what he assigns to it in the
Analytica Priora. Mr. Poste's remark is perfectly just, that
according to the above passage in the Analytica, every syllogism with
a problematical (_i.e._ real as opposed to verbal) premiss would be a