And consequently the first is in the
middle, and the middle is in the third because of itself. For the first is either the
cause of the middle or vice versa. And in the same way the middle is either the cause
of the third or vice versa.Scriptum super libros Posteriorum
by Walter Burleigh
translated by John Longeway, from
the edition of Pamphilius de Montebononiensis, Venice 1514
Chapter 7
[I.6 75a28-37] “Now since it is necessary” (Quoniam autem est necesse etc.)
The seventh conclusion in this book is that the first term is in the middle term, and
the middle in the third term, because of itself. This conclusion is proven using the
sixth conclusion thus: A demonstration is from what inheres per se, therefore the
major premise of a demonstration receives in its definition the second of the
extremes <that is, the minor, or third term>.
And consequently the first is in the
middle, and the middle is in the third because of itself. For the first is either the
cause of the middle or vice versa. And in the same way the middle is either the cause
of the third or vice versa.
Note that something is said to be in another because of itself when what
precedes another is the cause of the other in such a way that it is not a partial
(diminuta) cause, nor has in itself a condition which is not a cause. For instance,
triangle is the cause of this that there is the having of three angles, etc. But plane
figure
is a partial cause in respect of this passion, so it is not its precise cause.
[I.7 75a37-b20] The eighth conclusion of this book is that it is not possible to demonstrate by descending from one genus to another. So it is to be understood that the principles of one science do not conclude anything in another science. And he proves this conclusion thus: In (41vb) a demonstration it is necessary for the middle and extremes to be of the same genus. But the subject in demonstration of which the passion must be concluded is of one genus and not another, and the consequent cannot be the middle in a demonstration from the consideration of one science, and the extremes, or one of the extremes, from the consideration of another. Consequently demonstrating by descending from one genus to another genus does not occur. But when the middle <is of one genus> and the extreme of another genus, since that which is of one genus is not proven of what is of another genus except accidentally, it follows that either the conclusion of the demonstration or one of its premisses is true accidentally, and so the demonstration is not from what inheres per se.
To make this conclusion clear Aristotle distinguishes those <things> that enter
into a demonstration. [I.7 75a 38-b1] There are three <things> that enter into a
demonstration, namely the subject, passion and axiom (dignitas).
And I hold that
different sciences might well have the same axioms, for instance, in geometry and
arithmetic this is a common axiom occurring in each: If equals be removed from equals
the remainders are equal. But different sciences do not have the same genus or the
same passion of the subject genus unless one is subalternate to the other, so that it
is a true subalternate science, and then they can, in a way, have the same subject. And
since no descent from genus to genus occurs, in every case there will be no
demonstrating that of contraries there is the same discipline; nor that by reducing
a cubical number to a cubical number there arises a bicubical number; nor that a
straight line is the most beautiful of lines, and contrary to the circle, for this is not
in the line as such, but is in it by reason of something common and through the form
of line.
[I.8 75b21-25] The ninth conclusion is that demonstration is from what is always (perpetuis). This is proven thus: Demonstration is from what is universal, but universals are always, therefore demonstration is from what is always.
[I.8 75b26-32] The tenth conclusion is that demonstration is from indestructibles. This conclusion is proven using the ninth conclusion, thus: Demonstration is from what is always, but what is always is incorruptible. And from these it follows that every definition is of incorruptibles, for definition is either a principle of demonstration or the conclusion of a demonstration or the whole demonstration itself, differing in arrangement . Then the genus of definition includes demonstration, but every demonstration is of indestructibles. Therefore every definition is of indestructibles.
[I.8 75b33-36] And since it has been said that does not occur except from
what is always there is a doubt concerning what is for the most part the case, how do
they make a demonstration, for instance, concerning the obscuration of the moon. For
can one demonstrate an eclipse of the moon when the moon is not always eclipsed?
Aristotle resolves this doubt by saying that since those things that are for the most
part the case are always the case insofar as they agree with demonstration. But this
ought not to be understood in such a way that from propositions that are for the most
part true there must arise those which are always true, and then from these there
must arise demonstrations. For example, this happens, that the moon is eclipsed. And
so the moon is obscured, and the Earth is placed between the sun and the moon, and
from these there can arise demonstrations in this way: Whenever the Moon is
obscured it is eclipsed, but whenever the Earth lies between the Moon and the Sun
it is obscured; therefore etc. And this conclusion is necessary since this moon is
eclipsed, which is, thus, the contingent part.
Chapter 8
[I.9 75b37-76a13] “Now since it is evident that etc.” (Quoniam autem
manifestum etc.) This conclusion is the eleventh, that not everything that is
syllogized
from true and indemonstrable propositions is known or demonstrated. This
conclusion is proven thus. In demonstration it is necessary for the first to be said of
the middle and the middle of the third because of itself. But not in every syllogism
from true and indemonstrable propositions is the first said of the middle and the
middle of the third because of itself; therefore not every such syllogism is a
demonstration, and consequently not everything that is syllogized from true and
indemonstrable <propositions> is demonstrated or known.
As to the assumption, that is, that something arises from true, indemonstrable
and immediate propositions in which the first is not said of the middle nor the middle
of the third because of itself, it is explained from the fact that the argument of
Briso by which he wants to square the circle is from true, indemonstrable and
immediate propositions and is not a demonstration. His argument is formed thus:
Where greater and less <than a given quantity> are found, there an equal is to be
found, but a square greater, and one less than a circle are found, therefore etc.
That
argument is not a demonstration, for the elements greater and less go beyond (42ra)
the circle and the square. Hence the middle is in it prior to being in that concerning
which the demonstration arises, and therefore it is not in it according to itself, and
without this there is no demonstration.
Then, to exclude a certain doubt, the Philosopher says that if the middle does not arise in the same genus as the conclusion of the demonstration, so that it happens that the conclusion is not at all demonstrated through the middle, then there is knowing by an inferior knowledge, since knowing propter quid is superior knowledge. For there is no knowledge that is knowing propter quid to which the middle pertains.
[I.9 76a13-15] The twelfth conclusion is that every demonstration is from principles made appropriate to the conclusion. And this follows from the eleventh conclusion of this book. For if it does not suffice to demonstration that it be from true and immediate and indemonstrable propositions, but is required that the first be in the middle and the middle in the third because of itself, then it is necessary that demonstration be from principles appropriate to the conclusion.
[I.9 76a16-36] The thirteenth conclusion is that it belongs to no science to demonstrate its own proper principles. This conclusion is proved from the present conclusion thus: Demonstration has being from proper principles, but proper principles do not have other proper principles prior to them; therefore it belongs to no science to prove its own proper principles.
But even though proper principles have been proved in this particular science, still proving them from common <principles> is <more a matter of> making them plain, and he is more a knower and understander who has knowledge from the more principal causes. Hence a superior science is greater than an inferior science.
[I.9 76a37-b11] The fourteenth conclusion of this book is that it is necessary,
if common principles are to agree with a demonstration in a special science, that they
be appropriated to the object of the science. This conclusion follows from the
preceding conclusion, for if demonstration arises from proper principles, then it is
necessary for common
principles, when they enter into the demonstration, to be made
proper. For example, there is one principle in the conclusion of geometry <and>
arithmetic to the effect that if equals be taken from equals what remains is equal.
And arithmetic uses this principle and makes it appropriate to its genus, about which
it considers. So that if from equals, that is, equal numbers, equal numbers be taken,
the remaining numbers are equal.
[I.10 76a37-b39] After this, the philosopher distinguishes
among those things
that are alone properties (proprietates) in demonstrations. Of this sort are axioms,
postulates (petitiones) <and> suppositions (suppositiones). And these can all easily be
accepted. Everything that is in a demonstration is either a principle or is principiated.
If it is principiated it is a passion. If it is a principle it is proper or common. If it is
common, it is an axiom. If it is proper it is complex or incomplex. If it is incomplex it
is a subject. If it is complex it is either demonstrable or indemonstrable. If it is
demonstrable it is either probable to the learner, and thus it is a supposition not
without qualification, but in a certain manner, or it is contrary to the opinion of the
learner, and thus it is a postulate (quaestio). But if it is an indemonstrable principle
it is a position, which is divided into supposition without qualification and definition.
For if it receives being or non-being, thus it is a supposition, but if it receives neither
being nor non-being, thus it is definition.
Chapter 9
[I.11 77a10-21] “Now to happen etc.” (Contingere autem etc.) The fifteenth
conclusion is that no demonstration receives affirmation and denial of something of
itself, unless the conclusion is such that it cannot be shown except through
affirmation of something of itself. This conclusion is proven only if, in an argument
(syllogismo) to the impossible, the conclusion cannot be shown otherwise than through
affirmation and denial of something of itself. Therefore an affirmation and denial of
something of itself is never found unless the conclusion is such that it cannot be
shown otherwise than through affirmation and denial of something of itself. Now how
it is that affirmation and denial of something of itself is received into an argument
to the impossible is clear, for if Callias is a non-animal it is to be argued thus: Callias
is an animal, therefore Callias is not a non-animal and <it was assumed that>
Callias is
a non-animal. It follows then that Callias is not an animal, therefore Callias is not
Callias. This conclusion that Callias is not Callias is not concluded except through
affirmation and denial of something of itself, and nonetheless, from the falsity of this
conclusion is concluded the falsity of the other premise.
[I.11 77a22-35] The sixteenth conclusion is that demonstration leading to the impossible receives this principle, that of everything the affirmation or the denial <of a predicate> is true <but not both>, but made appropriate to the genus of the subject. That an argument (syllogismus) leading to the impossible receives this principle is clear from the Prior Analytics, and that what is according to the principle <is received> in the same way, namely, <as it is> made appropriate to the genus of the subject, is clear from the preceding conclusion of the book at hand.
Chapter 10
[I.12 77a36-b15] (42rb) “Now if the same etc.” (Si autem idem etc.) This is the
seventeenth conclusion of this book: Every conclusion that a demonstrator seeks is
from proper <premisses>, but what is a principle in one demonstration is the
conclusion
in another. And then it is argued thus: Every proposition from which
something is demonstrated is from proper <principles>, but every conclusion a
demonstrator seeks is a proposition from which something is demonstrated; therefore
every conclusion that a demonstrator seeks is from proper <principles>. And from this
conclusion a corollary follows, namely, that different sciences do not coincide in their
conclusions, nor does any special artisan have it as his job to enumerate the
conclusions of another science. And thus
every science has its own proper
conclusions, since the question and the answer belong to the same science, and
therefore, just as a science has its own proper questions so it has its own proper
answers. And since disputations arise from question and answer, just as every science
has it proper questions and answers, so every science has its own proper disputations.
For instance, the geometer disputes as a geometer when geometry disputes those
things pertaining to geometry, it is clear that he proceeds well by disputing <thus>.
But if he does not dispute thus, although he disputes in geometry, but not about
geometry, the disputation does not proceed, and it is not suitable. Hence it is not to
be disputed in geometry about what is not geometrical.