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Posterior Analytics
By Aristotle


Translated by G. R. G. Mure

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BOOK I

Part 1 

All instruction given or received by way of argument proceeds from
pre-existent knowledge. This becomes evident upon a survey of all
the species of such instruction. The mathematical sciences and all
other speculative disciplines are acquired in this way, and so are
the two forms of dialectical reasoning, syllogistic and inductive;
for each of these latter make use of old knowledge to impart new,
the syllogism assuming an audience that accepts its premisses, induction
exhibiting the universal as implicit in the clearly known particular.
Again, the persuasion exerted by rhetorical arguments is in principle
the same, since they use either example, a kind of induction, or enthymeme,
a form of syllogism. 

The pre-existent knowledge required is of two kinds. In some cases
admission of the fact must be assumed, in others comprehension of
the meaning of the term used, and sometimes both assumptions are essential.
Thus, we assume that every predicate can be either truly affirmed
or truly denied of any subject, and that 'triangle' means so and so;
as regards 'unit' we have to make the double assumption of the meaning
of the word and the existence of the thing. The reason is that these
several objects are not equally obvious to us. Recognition of a truth
may in some cases contain as factors both previous knowledge and also
knowledge acquired simultaneously with that recognition-knowledge,
this latter, of the particulars actually falling under the universal
and therein already virtually known. For example, the student knew
beforehand that the angles of every triangle are equal to two right
angles; but it was only at the actual moment at which he was being
led on to recognize this as true in the instance before him that he
came to know 'this figure inscribed in the semicircle' to be a triangle.
For some things (viz. the singulars finally reached which are not
predicable of anything else as subject) are only learnt in this way,
i.e. there is here no recognition through a middle of a minor term
as subject to a major. Before he was led on to recognition or before
he actually drew a conclusion, we should perhaps say that in a manner
he knew, in a manner not. 

If he did not in an unqualified sense of the term know the existence
of this triangle, how could he know without qualification that its
angles were equal to two right angles? No: clearly he knows not without
qualification but only in the sense that he knows universally. If
this distinction is not drawn, we are faced with the dilemma in the
Meno: either a man will learn nothing or what he already knows; for
we cannot accept the solution which some people offer. A man is asked,
'Do you, or do you not, know that every pair is even?' He says he
does know it. The questioner then produces a particular pair, of the
existence, and so a fortiori of the evenness, of which he was unaware.
The solution which some people offer is to assert that they do not
know that every pair is even, but only that everything which they
know to be a pair is even: yet what they know to be even is that of
which they have demonstrated evenness, i.e. what they made the subject
of their premiss, viz. not merely every triangle or number which they
know to be such, but any and every number or triangle without reservation.
For no premiss is ever couched in the form 'every number which you
know to be such', or 'every rectilinear figure which you know to be
such': the predicate is always construed as applicable to any and
every instance of the thing. On the other hand, I imagine there is
nothing to prevent a man in one sense knowing what he is learning,
in another not knowing it. The strange thing would be, not if in some
sense he knew what he was learning, but if he were to know it in that
precise sense and manner in which he was learning it. 

Part 2

We suppose ourselves to possess unqualified scientific knowledge of
a thing, as opposed to knowing it in the accidental way in which the
sophist knows, when we think that we know the cause on which the fact
depends, as the cause of that fact and of no other, and, further,
that the fact could not be other than it is. Now that scientific knowing
is something of this sort is evident-witness both those who falsely
claim it and those who actually possess it, since the former merely
imagine themselves to be, while the latter are also actually, in the
condition described. Consequently the proper object of unqualified
scientific knowledge is something which cannot be other than it is.

There may be another manner of knowing as well-that will be discussed
later. What I now assert is that at all events we do know by demonstration.
By demonstration I mean a syllogism productive of scientific knowledge,
a syllogism, that is, the grasp of which is eo ipso such knowledge.
Assuming then that my thesis as to the nature of scientific knowing
is correct, the premisses of demonstrated knowledge must be true,
primary, immediate, better known than and prior to the conclusion,
which is further related to them as effect to cause. Unless these
conditions are satisfied, the basic truths will not be 'appropriate'
to the conclusion. Syllogism there may indeed be without these conditions,
but such syllogism, not being productive of scientific knowledge,
will not be demonstration. The premisses must be true: for that which
is non-existent cannot be known-we cannot know, e.g. that the diagonal
of a square is commensurate with its side. The premisses must be primary
and indemonstrable; otherwise they will require demonstration in order
to be known, since to have knowledge, if it be not accidental knowledge,
of things which are demonstrable, means precisely to have a demonstration
of them. The premisses must be the causes of the conclusion, better
known than it, and prior to it; its causes, since we possess scientific
knowledge of a thing only when we know its cause; prior, in order
to be causes; antecedently known, this antecedent knowledge being
not our mere understanding of the meaning, but knowledge of the fact
as well. Now 'prior' and 'better known' are ambiguous terms, for there
is a difference between what is prior and better known in the order
of being and what is prior and better known to man. I mean that objects
nearer to sense are prior and better known to man; objects without
qualification prior and better known are those further from sense.
Now the most universal causes are furthest from sense and particular
causes are nearest to sense, and they are thus exactly opposed to
one another. In saying that the premisses of demonstrated knowledge
must be primary, I mean that they must be the 'appropriate' basic
truths, for I identify primary premiss and basic truth. A 'basic truth'
in a demonstration is an immediate proposition. An immediate proposition
is one which has no other proposition prior to it. A proposition is
either part of an enunciation, i.e. it predicates a single attribute
of a single subject. If a proposition is dialectical, it assumes either
part indifferently; if it is demonstrative, it lays down one part
to the definite exclusion of the other because that part is true.
The term 'enunciation' denotes either part of a contradiction indifferently.
A contradiction is an opposition which of its own nature excludes
a middle. The part of a contradiction which conjoins a predicate with
a subject is an affirmation; the part disjoining them is a negation.
I call an immediate basic truth of syllogism a 'thesis' when, though
it is not susceptible of proof by the teacher, yet ignorance of it
does not constitute a total bar to progress on the part of the pupil:
one which the pupil must know if he is to learn anything whatever
is an axiom. I call it an axiom because there are such truths and
we give them the name of axioms par excellence. If a thesis assumes
one part or the other of an enunciation, i.e. asserts either the existence
or the non-existence of a subject, it is a hypothesis; if it does
not so assert, it is a definition. Definition is a 'thesis' or a 'laying
something down', since the arithmetician lays it down that to be a
unit is to be quantitatively indivisible; but it is not a hypothesis,
for to define what a unit is is not the same as to affirm its existence.

Now since the required ground of our knowledge-i.e. of our conviction-of
a fact is the possession of such a syllogism as we call demonstration,
and the ground of the syllogism is the facts constituting its premisses,
we must not only know the primary premisses-some if not all of them-beforehand,
but know them better than the conclusion: for the cause of an attribute's
inherence in a subject always itself inheres in the subject more firmly
than that attribute; e.g. the cause of our loving anything is dearer
to us than the object of our love. So since the primary premisses
are the cause of our knowledge-i.e. of our conviction-it follows that
we know them better-that is, are more convinced of them-than their
consequences, precisely because of our knowledge of the latter is
the effect of our knowledge of the premisses. Now a man cannot believe
in anything more than in the things he knows, unless he has either
actual knowledge of it or something better than actual knowledge.
But we are faced with this paradox if a student whose belief rests
on demonstration has not prior knowledge; a man must believe in some,
if not in all, of the basic truths more than in the conclusion. Moreover,
if a man sets out to acquire the scientific knowledge that comes through
demonstration, he must not only have a better knowledge of the basic
truths and a firmer conviction of them than of the connexion which
is being demonstrated: more than this, nothing must be more certain
or better known to him than these basic truths in their character
as contradicting the fundamental premisses which lead to the opposed
and erroneous conclusion. For indeed the conviction of pure science
must be unshakable. 

Part 3

Some hold that, owing to the necessity of knowing the primary premisses,
there is no scientific knowledge. Others think there is, but that
all truths are demonstrable. Neither doctrine is either true or a
necessary deduction from the premisses. The first school, assuming
that there is no way of knowing other than by demonstration, maintain
that an infinite regress is involved, on the ground that if behind
the prior stands no primary, we could not know the posterior through
the prior (wherein they are right, for one cannot traverse an infinite
series): if on the other hand-they say-the series terminates and there
are primary premisses, yet these are unknowable because incapable
of demonstration, which according to them is the only form of knowledge.
And since thus one cannot know the primary premisses, knowledge of
the conclusions which follow from them is not pure scientific knowledge
nor properly knowing at all, but rests on the mere supposition that
the premisses are true. The other party agree with them as regards
knowing, holding that it is only possible by demonstration, but they
see no difficulty in holding that all truths are demonstrated, on
the ground that demonstration may be circular and reciprocal.

Our own doctrine is that not all knowledge is demonstrative: on the
contrary, knowledge of the immediate premisses is independent of demonstration.
(The necessity of this is obvious; for since we must know the prior
premisses from which the demonstration is drawn, and since the regress
must end in immediate truths, those truths must be indemonstrable.)
Such, then, is our doctrine, and in addition we maintain that besides
scientific knowledge there is its originative source which enables
us to recognize the definitions. 

Now demonstration must be based on premisses prior to and better known
than the conclusion; and the same things cannot simultaneously be
both prior and posterior to one another: so circular demonstration
is clearly not possible in the unqualified sense of 'demonstration',
but only possible if 'demonstration' be extended to include that other
method of argument which rests on a distinction between truths prior
to us and truths without qualification prior, i.e. the method by which
induction produces knowledge. But if we accept this extension of its
meaning, our definition of unqualified knowledge will prove faulty;
for there seem to be two kinds of it. Perhaps, however, the second
form of demonstration, that which proceeds from truths better known
to us, is not demonstration in the unqualified sense of the term.

The advocates of circular demonstration are not only faced with the
difficulty we have just stated: in addition their theory reduces to
the mere statement that if a thing exists, then it does exist-an easy
way of proving anything. That this is so can be clearly shown by taking
three terms, for to constitute the circle it makes no difference whether
many terms or few or even only two are taken. Thus by direct proof,
if A is, B must be; if B is, C must be; therefore if A is, C must
be. Since then-by the circular proof-if A is, B must be, and if B
is, A must be, A may be substituted for C above. Then 'if B is, A
must be'='if B is, C must be', which above gave the conclusion 'if
A is, C must be': but C and A have been identified. Consequently the
upholders of circular demonstration are in the position of saying
that if A is, A must be-a simple way of proving anything. Moreover,
even such circular demonstration is impossible except in the case
of attributes that imply one another, viz. 'peculiar' properties.

Now, it has been shown that the positing of one thing-be it one term
or one premiss-never involves a necessary consequent: two premisses
constitute the first and smallest foundation for drawing a conclusion
at all and therefore a fortiori for the demonstrative syllogism of
science. If, then, A is implied in B and C, and B and C are reciprocally
implied in one another and in A, it is possible, as has been shown
in my writings on the syllogism, to prove all the assumptions on which
the original conclusion rested, by circular demonstration in the first
figure. But it has also been shown that in the other figures either
no conclusion is possible, or at least none which proves both the
original premisses. Propositions the terms of which are not convertible
cannot be circularly demonstrated at all, and since convertible terms
occur rarely in actual demonstrations, it is clearly frivolous and
impossible to say that demonstration is reciprocal and that therefore
everything can be demonstrated. 

Part 4

Since the object of pure scientific knowledge cannot be other than
it is, the truth obtained by demonstrative knowledge will be necessary.
And since demonstrative knowledge is only present when we have a demonstration,
it follows that demonstration is an inference from necessary premisses.
So we must consider what are the premisses of demonstration-i.e. what
is their character: and as a preliminary, let us define what we mean
by an attribute 'true in every instance of its subject', an 'essential'
attribute, and a 'commensurate and universal' attribute. I call 'true
in every instance' what is truly predicable of all instances-not of
one to the exclusion of others-and at all times, not at this or that
time only; e.g. if animal is truly predicable of every instance of
man, then if it be true to say 'this is a man', 'this is an animal'
is also true, and if the one be true now the other is true now. A
corresponding account holds if point is in every instance predicable
as contained in line. There is evidence for this in the fact that
the objection we raise against a proposition put to us as true in
every instance is either an instance in which, or an occasion on which,
it is not true. Essential attributes are (1) such as belong to their
subject as elements in its essential nature (e.g. line thus belongs
to triangle, point to line; for the very being or 'substance' of triangle
and line is composed of these elements, which are contained in the
formulae defining triangle and line): (2) such that, while they belong
to certain subjects, the subjects to which they belong are contained
in the attribute's own defining formula. Thus straight and curved
belong to line, odd and even, prime and compound, square and oblong,
to number; and also the formula defining any one of these attributes
contains its subject-e.g. line or number as the case may be.

Extending this classification to all other attributes, I distinguish
those that answer the above description as belonging essentially to
their respective subjects; whereas attributes related in neither of
these two ways to their subjects I call accidents or 'coincidents';
e.g. musical or white is a 'coincident' of animal. 

Further (a) that is essential which is not predicated of a subject
other than itself: e.g. 'the walking [thing]' walks and is white in
virtue of being something else besides; whereas substance, in the
sense of whatever signifies a 'this somewhat', is not what it is in
virtue of being something else besides. Things, then, not predicated
of a subject I call essential; things predicated of a subject I call
accidental or 'coincidental'. 

In another sense again (b) a thing consequentially connected with
anything is essential; one not so connected is 'coincidental'. An
example of the latter is 'While he was walking it lightened': the
lightning was not due to his walking; it was, we should say, a coincidence.
If, on the other hand, there is a consequential connexion, the predication
is essential; e.g. if a beast dies when its throat is being cut, then
its death is also essentially connected with the cutting, because
the cutting was the cause of death, not death a 'coincident' of the
cutting. 

So far then as concerns the sphere of connexions scientifically known
in the unqualified sense of that term, all attributes which (within
that sphere) are essential either in the sense that their subjects
are contained in them, or in the sense that they are contained in
their subjects, are necessary as well as consequentially connected
with their subjects. For it is impossible for them not to inhere in
their subjects either simply or in the qualified sense that one or
other of a pair of opposites must inhere in the subject; e.g. in line
must be either straightness or curvature, in number either oddness
or evenness. For within a single identical genus the contrary of a
given attribute is either its privative or its contradictory; e.g.
within number what is not odd is even, inasmuch as within this sphere
even is a necessary consequent of not-odd. So, since any given predicate
must be either affirmed or denied of any subject, essential attributes
must inhere in their subjects of necessity. 

Thus, then, we have established the distinction between the attribute
which is 'true in every instance' and the 'essential' attribute.

I term 'commensurately universal' an attribute which belongs to every
instance of its subject, and to every instance essentially and as
such; from which it clearly follows that all commensurate universals
inhere necessarily in their subjects. The essential attribute, and
the attribute that belongs to its subject as such, are identical.
E.g. point and straight belong to line essentially, for they belong
to line as such; and triangle as such has two right angles, for it
is essentially equal to two right angles. 

An attribute belongs commensurately and universally to a subject when
it can be shown to belong to any random instance of that subject and
when the subject is the first thing to which it can be shown to belong.
Thus, e.g. (1) the equality of its angles to two right angles is not
a commensurately universal attribute of figure. For though it is possible
to show that a figure has its angles equal to two right angles, this
attribute cannot be demonstrated of any figure selected at haphazard,
nor in demonstrating does one take a figure at random-a square is
a figure but its angles are not equal to two right angles. On the
other hand, any isosceles triangle has its angles equal to two right
angles, yet isosceles triangle is not the primary subject of this
attribute but triangle is prior. So whatever can be shown to have
its angles equal to two right angles, or to possess any other attribute,
in any random instance of itself and primarily-that is the first subject
to which the predicate in question belongs commensurately and universally,
and the demonstration, in the essential sense, of any predicate is
the proof of it as belonging to this first subject commensurately
and universally: while the proof of it as belonging to the other subjects
to which it attaches is demonstration only in a secondary and unessential
sense. Nor again (2) is equality to two right angles a commensurately
universal attribute of isosceles; it is of wider application.

Part 5

We must not fail to observe that we often fall into error because
our conclusion is not in fact primary and commensurately universal
in the sense in which we think we prove it so. We make this mistake
(1) when the subject is an individual or individuals above which there
is no universal to be found: (2) when the subjects belong to different
species and there is a higher universal, but it has no name: (3) when
the subject which the demonstrator takes as a whole is really only
a part of a larger whole; for then the demonstration will be true
of the individual instances within the part and will hold in every
instance of it, yet the demonstration will not be true of this subject
primarily and commensurately and universally. When a demonstration
is true of a subject primarily and commensurately and universally,
that is to be taken to mean that it is true of a given subject primarily
and as such. Case (3) may be thus exemplified. If a proof were given
that perpendiculars to the same line are parallel, it might be supposed
that lines thus perpendicular were the proper subject of the demonstration
because being parallel is true of every instance of them. But it is
not so, for the parallelism depends not on these angles being equal
to one another because each is a right angle, but simply on their
being equal to one another. An example of (1) would be as follows:
if isosceles were the only triangle, it would be thought to have its
angles equal to two right angles qua isosceles. An instance of (2)
would be the law that proportionals alternate. Alternation used to
be demonstrated separately of numbers, lines, solids, and durations,
though it could have been proved of them all by a single demonstration.
Because there was no single name to denote that in which numbers,
lengths, durations, and solids are identical, and because they differed
specifically from one another, this property was proved of each of
them separately. To-day, however, the proof is commensurately universal,
for they do not possess this attribute qua lines or qua numbers, but
qua manifesting this generic character which they are postulated as
possessing universally. Hence, even if one prove of each kind of triangle
that its angles are equal to two right angles, whether by means of
the same or different proofs; still, as long as one treats separately
equilateral, scalene, and isosceles, one does not yet know, except
sophistically, that triangle has its angles equal to two right angles,
nor does one yet know that triangle has this property commensurately
and universally, even if there is no other species of triangle but
these. For one does not know that triangle as such has this property,
nor even that 'all' triangles have it-unless 'all' means 'each taken
singly': if 'all' means 'as a whole class', then, though there be
none in which one does not recognize this property, one does not know
it of 'all triangles'. 

When, then, does our knowledge fail of commensurate universality,
and when it is unqualified knowledge? If triangle be identical in
essence with equilateral, i.e. with each or all equilaterals, then
clearly we have unqualified knowledge: if on the other hand it be
not, and the attribute belongs to equilateral qua triangle; then our
knowledge fails of commensurate universality. 'But', it will be asked,
'does this attribute belong to the subject of which it has been demonstrated
qua triangle or qua isosceles? What is the point at which the subject.
to which it belongs is primary? (i.e. to what subject can it be demonstrated
as belonging commensurately and universally?)' Clearly this point
is the first term in which it is found to inhere as the elimination
of inferior differentiae proceeds. Thus the angles of a brazen isosceles
triangle are equal to two right angles: but eliminate brazen and isosceles
and the attribute remains. 'But'-you may say-'eliminate figure or
limit, and the attribute vanishes.' True, but figure and limit are
not the first differentiae whose elimination destroys the attribute.
'Then what is the first?' If it is triangle, it will be in virtue
of triangle that the attribute belongs to all the other subjects of
which it is predicable, and triangle is the subject to which it can
be demonstrated as belonging commensurately and universally.

Part 6

Demonstrative knowledge must rest on necessary basic truths; for the
object of scientific knowledge cannot be other than it is. Now attributes
attaching essentially to their subjects attach necessarily to them:
for essential attributes are either elements in the essential nature
of their subjects, or contain their subjects as elements in their
own essential nature. (The pairs of opposites which the latter class
includes are necessary because one member or the other necessarily
inheres.) It follows from this that premisses of the demonstrative
syllogism must be connexions essential in the sense explained: for
all attributes must inhere essentially or else be accidental, and
accidental attributes are not necessary to their subjects.

We must either state the case thus, or else premise that the conclusion
of demonstration is necessary and that a demonstrated conclusion cannot
be other than it is, and then infer that the conclusion must be developed
from necessary premisses. For though you may reason from true premisses
without demonstrating, yet if your premisses are necessary you will
assuredly demonstrate-in such necessity you have at once a distinctive
character of demonstration. That demonstration proceeds from necessary
premisses is also indicated by the fact that the objection we raise
against a professed demonstration is that a premiss of it is not a
necessary truth-whether we think it altogether devoid of necessity,
or at any rate so far as our opponent's previous argument goes. This
shows how naive it is to suppose one's basic truths rightly chosen
if one starts with a proposition which is (1) popularly accepted and
(2) true, such as the sophists' assumption that to know is the same
as to possess knowledge. For (1) popular acceptance or rejection is
no criterion of a basic truth, which can only be the primary law of
the genus constituting the subject matter of the demonstration; and
(2) not all truth is 'appropriate'. 

A further proof that the conclusion must be the development of necessary
premisses is as follows. Where demonstration is possible, one who
can give no account which includes the cause has no scientific knowledge.
If, then, we suppose a syllogism in which, though A necessarily inheres
in C, yet B, the middle term of the demonstration, is not necessarily
connected with A and C, then the man who argues thus has no reasoned
knowledge of the conclusion, since this conclusion does not owe its
necessity to the middle term; for though the conclusion is necessary,
the mediating link is a contingent fact. Or again, if a man is without
knowledge now, though he still retains the steps of the argument,
though there is no change in himself or in the fact and no lapse of
memory on his part; then neither had he knowledge previously. But
the mediating link, not being necessary, may have perished in the
interval; and if so, though there be no change in him nor in the fact,
and though he will still retain the steps of the argument, yet he
has not knowledge, and therefore had not knowledge before. Even if
the link has not actually perished but is liable to perish, this situation
is possible and might occur. But such a condition cannot be knowledge.

When the conclusion is necessary, the middle through which it was
proved may yet quite easily be non-necessary. You can in fact infer
the necessary even from a non-necessary premiss, just as you can infer
the true from the not true. On the other hand, when the middle is
necessary the conclusion must be necessary; just as true premisses
always give a true conclusion. Thus, if A is necessarily predicated
of B and B of C, then A is necessarily predicated of C. But when the
conclusion is nonnecessary the middle cannot be necessary either.
Thus: let A be predicated non-necessarily of C but necessarily of
B, and let B be a necessary predicate of C; then A too will be a necessary
predicate of C, which by hypothesis it is not. 

To sum up, then: demonstrative knowledge must be knowledge of a necessary
nexus, and therefore must clearly be obtained through a necessary
middle term; otherwise its possessor will know neither the cause nor
the fact that his conclusion is a necessary connexion. Either he will
mistake the non-necessary for the necessary and believe the necessity
of the conclusion without knowing it, or else he will not even believe
it-in which case he will be equally ignorant, whether he actually
infers the mere fact through middle terms or the reasoned fact and
from immediate premisses. 

Of accidents that are not essential according to our definition of
essential there is no demonstrative knowledge; for since an accident,
in the sense in which I here speak of it, may also not inhere, it
is impossible to prove its inherence as a necessary conclusion. A
difficulty, however, might be raised as to why in dialectic, if the
conclusion is not a necessary connexion, such and such determinate
premisses should be proposed in order to deal with such and such determinate
problems. Would not the result be the same if one asked any questions
whatever and then merely stated one's conclusion? The solution is
that determinate questions have to be put, not because the replies
to them affirm facts which necessitate facts affirmed by the conclusion,
but because these answers are propositions which if the answerer affirm,
he must affirm the conclusion and affirm it with truth if they are
true. 

Since it is just those attributes within every genus which are essential
and possessed by their respective subjects as such that are necessary
it is clear that both the conclusions and the premisses of demonstrations
which produce scientific knowledge are essential. For accidents are
not necessary: and, further, since accidents are not necessary one
does not necessarily have reasoned knowledge of a conclusion drawn
from them (this is so even if the accidental premisses are invariable
but not essential, as in proofs through signs; for though the conclusion
be actually essential, one will not know it as essential nor know
its reason); but to have reasoned knowledge of a conclusion is to
know it through its cause. We may conclude that the middle must be
consequentially connected with the minor, and the major with the middle.

Part 7

It follows that we cannot in demonstrating pass from one genus to
another. We cannot, for instance, prove geometrical truths by arithmetic.
For there are three elements in demonstration: (1) what is proved,
the conclusion-an attribute inhering essentially in a genus; (2) the
axioms, i.e. axioms which are premisses of demonstration; (3) the
subject-genus whose attributes, i.e. essential properties, are revealed
by the demonstration. The axioms which are premisses of demonstration
may be identical in two or more sciences: but in the case of two different
genera such as arithmetic and geometry you cannot apply arithmetical
demonstration to the properties of magnitudes unless the magnitudes
in question are numbers. How in certain cases transference is possible
I will explain later. 

Arithmetical demonstration and the other sciences likewise possess,
each of them, their own genera; so that if the demonstration is to
pass from one sphere to another, the genus must be either absolutely
or to some extent the same. If this is not so, transference is clearly
impossible, because the extreme and the middle terms must be drawn
from the same genus: otherwise, as predicated, they will not be essential
and will thus be accidents. That is why it cannot be proved by geometry
that opposites fall under one science, nor even that the product of
two cubes is a cube. Nor can the theorem of any one science be demonstrated
by means of another science, unless these theorems are related as
subordinate to superior (e.g. as optical theorems to geometry or harmonic
theorems to arithmetic). Geometry again cannot prove of lines any
property which they do not possess qua lines, i.e. in virtue of the
fundamental truths of their peculiar genus: it cannot show, for example,
that the straight line is the most beautiful of lines or the contrary
of the circle; for these qualities do not belong to lines in virtue
of their peculiar genus, but through some property which it shares
with other genera. 

Part 8

It is also clear that if the premisses from which the syllogism proceeds
are commensurately universal, the conclusion of such i.e. in the unqualified
sense-must also be eternal. Therefore no attribute can be demonstrated
nor known by strictly scientific knowledge to inhere in perishable
things. The proof can only be accidental, because the attribute's
connexion with its perishable subject is not commensurately universal
but temporary and special. If such a demonstration is made, one premiss
must be perishable and not commensurately universal (perishable because
only if it is perishable will the conclusion be perishable; not commensurately
universal, because the predicate will be predicable of some instances
of the subject and not of others); so that the conclusion can only
be that a fact is true at the moment-not commensurately and universally.
The same is true of definitions, since a definition is either a primary
premiss or a conclusion of a demonstration, or else only differs from
a demonstration in the order of its terms. Demonstration and science
of merely frequent occurrences-e.g. of eclipse as happening to the
moon-are, as such, clearly eternal: whereas so far as they are not
eternal they are not fully commensurate. Other subjects too have properties
attaching to them in the same way as eclipse attaches to the moon.

Part 9

It is clear that if the conclusion is to show an attribute inhering
as such, nothing can be demonstrated except from its 'appropriate'
basic truths. Consequently a proof even from true, indemonstrable,
and immediate premisses does not constitute knowledge. Such proofs
are like Bryson's method of squaring the circle; for they operate
by taking as their middle a common character-a character, therefore,
which the subject may share with another-and consequently they apply
equally to subjects different in kind. They therefore afford knowledge
of an attribute only as inhering accidentally, not as belonging to
its subject as such: otherwise they would not have been applicable
to another genus. 

Our knowledge of any attribute's connexion with a subject is accidental
unless we know that connexion through the middle term in virtue of
which it inheres, and as an inference from basic premisses essential
and 'appropriate' to the subject-unless we know, e.g. the property
of possessing angles equal to two right angles as belonging to that
subject in which it inheres essentially, and as inferred from basic
premisses essential and 'appropriate' to that subject: so that if
that middle term also belongs essentially to the minor, the middle
must belong to the same kind as the major and minor terms. The only
exceptions to this rule are such cases as theorems in harmonics which
are demonstrable by arithmetic. Such theorems are proved by the same
middle terms as arithmetical properties, but with a qualification-the
fact falls under a separate science (for the subject genus is separate),
but the reasoned fact concerns the superior science, to which the
attributes essentially belong. Thus, even these apparent exceptions
show that no attribute is strictly demonstrable except from its 'appropriate'
basic truths, which, however, in the case of these sciences have the
requisite identity of character. 

It is no less evident that the peculiar basic truths of each inhering
attribute are indemonstrable; for basic truths from which they might
be deduced would be basic truths of all that is, and the science to
which they belonged would possess universal sovereignty. This is so
because he knows better whose knowledge is deduced from higher causes,
for his knowledge is from prior premisses when it derives from causes
themselves uncaused: hence, if he knows better than others or best
of all, his knowledge would be science in a higher or the highest
degree. But, as things are, demonstration is not transferable to another
genus, with such exceptions as we have mentioned of the application
of geometrical demonstrations to theorems in mechanics or optics,
or of arithmetical demonstrations to those of harmonics.

It is hard to be sure whether one knows or not; for it is hard to
be sure whether one's knowledge is based on the basic truths appropriate
to each attribute-the differentia of true knowledge. We think we have
scientific knowledge if we have reasoned from true and primary premisses.
But that is not so: the conclusion must be homogeneous with the basic
facts of the science. 

Part 10

I call the basic truths of every genus those clements in it the existence
of which cannot be proved. As regards both these primary truths and
the attributes dependent on them the meaning of the name is assumed.
The fact of their existence as regards the primary truths must be
assumed; but it has to be proved of the remainder, the attributes.
Thus we assume the meaning alike of unity, straight, and triangular;
but while as regards unity and magnitude we assume also the fact of
their existence, in the case of the remainder proof is required.

Of the basic truths used in the demonstrative sciences some are peculiar
to each science, and some are common, but common only in the sense
of analogous, being of use only in so far as they fall within the
genus constituting the province of the science in question.

Peculiar truths are, e.g. the definitions of line and straight; common
truths are such as 'take equals from equals and equals remain'. Only
so much of these common truths is required as falls within the genus
in question: for a truth of this kind will have the same force even
if not used generally but applied by the geometer only to magnitudes,
or by the arithmetician only to numbers. Also peculiar to a science
are the subjects the existence as well as the meaning of which it
assumes, and the essential attributes of which it investigates, e.g.
in arithmetic units, in geometry points and lines. Both the existence
and the meaning of the subjects are assumed by these sciences; but
of their essential attributes only the meaning is assumed. For example
arithmetic assumes the meaning of odd and even, square and cube, geometry
that of incommensurable, or of deflection or verging of lines, whereas
the existence of these attributes is demonstrated by means of the
axioms and from previous conclusions as premisses. Astronomy too proceeds
in the same way. For indeed every demonstrative science has three
elements: (1) that which it posits, the subject genus whose essential
attributes it examines; (2) the so-called axioms, which are primary
premisses of its demonstration; (3) the attributes, the meaning of
which it assumes. Yet some sciences may very well pass over some of
these elements; e.g. we might not expressly posit the existence of
the genus if its existence were obvious (for instance, the existence
of hot and cold is more evident than that of number); or we might
omit to assume expressly the meaning of the attributes if it were
well understood. In the way the meaning of axioms, such as 'Take equals
from equals and equals remain', is well known and so not expressly
assumed. Nevertheless in the nature of the case the essential elements
of demonstration are three: the subject, the attributes, and the basic
premisses. 

That which expresses necessary self-grounded fact, and which we must
necessarily believe, is distinct both from the hypotheses of a science
and from illegitimate postulate-I say 'must believe', because all
syllogism, and therefore a fortiori demonstration, is addressed not
to the spoken word, but to the discourse within the soul, and though
we can always raise objections to the spoken word, to the inward discourse
we cannot always object. That which is capable of proof but assumed
by the teacher without proof is, if the pupil believes and accepts
it, hypothesis, though only in a limited sense hypothesis-that is,
relatively to the pupil; if the pupil has no opinion or a contrary
opinion on the matter, the same assumption is an illegitimate postulate.
Therein lies the distinction between hypothesis and illegitimate postulate:
the latter is the contrary of the pupil's opinion, demonstrable, but
assumed and used without demonstration. 

The definition-viz. those which are not expressed as statements that
anything is or is not-are not hypotheses: but it is in the premisses
of a science that its hypotheses are contained. Definitions require
only to be understood, and this is not hypothesis-unless it be contended
that the pupil's hearing is also an hypothesis required by the teacher.
Hypotheses, on the contrary, postulate facts on the being of which
depends the being of the fact inferred. Nor are the geometer's hypotheses
false, as some have held, urging that one must not employ falsehood
and that the geometer is uttering falsehood in stating that the line
which he draws is a foot long or straight, when it is actually neither.
The truth is that the geometer does not draw any conclusion from the
being of the particular line of which he speaks, but from what his
diagrams symbolize. A further distinction is that all hypotheses and
illegitimate postulates are either universal or particular, whereas
a definition is neither. 

Part 11

So demonstration does not necessarily imply the being of Forms nor
a One beside a Many, but it does necessarily imply the possibility
of truly predicating one of many; since without this possibility we
cannot save the universal, and if the universal goes, the middle term
goes witb. it, and so demonstration becomes impossible. We conclude,
then, that there must be a single identical term unequivocally predicable
of a number of individuals. 

The law that it is impossible to affirm and deny simultaneously the
same predicate of the same subject is not expressly posited by any
demonstration except when the conclusion also has to be expressed
in that form; in which case the proof lays down as its major premiss
that the major is truly affirmed of the middle but falsely denied.
It makes no difference, however, if we add to the middle, or again
to the minor term, the corresponding negative. For grant a minor term
of which it is true to predicate man-even if it be also true to predicate
not-man of it--still grant simply that man is animal and not not-animal,
and the conclusion follows: for it will still be true to say that
Callias--even if it be also true to say that not-Callias--is animal
and not not-animal. The reason is that the major term is predicable
not only of the middle, but of something other than the middle as
well, being of wider application; so that the conclusion is not affected
even if the middle is extended to cover the original middle term and
also what is not the original middle term. 

The law that every predicate can be either truly affirmed or truly
denied of every subject is posited by such demonstration as uses reductio
ad impossibile, and then not always universally, but so far as it
is requisite; within the limits, that is, of the genus-the genus,
I mean (as I have already explained), to which the man of science
applies his demonstrations. In virtue of the common elements of demonstration-I
mean the common axioms which are used as premisses of demonstration,
not the subjects nor the attributes demonstrated as belonging to them-all
the sciences have communion with one another, and in communion with
them all is dialectic and any science which might attempt a universal
proof of axioms such as the law of excluded middle, the law that the
subtraction of equals from equals leaves equal remainders, or other
axioms of the same kind. Dialectic has no definite sphere of this
kind, not being confined to a single genus. Otherwise its method would
not be interrogative; for the interrogative method is barred to the
demonstrator, who cannot use the opposite facts to prove the same
nexus. This was shown in my work on the syllogism. 

Part 12

If a syllogistic question is equivalent to a proposition embodying
one of the two sides of a contradiction, and if each science has its
peculiar propositions from which its peculiar conclusion is developed,
then there is such a thing as a distinctively scientific question,
and it is the interrogative form of the premisses from which the 'appropriate'
conclusion of each science is developed. Hence it is clear that not
every question will be relevant to geometry, nor to medicine, nor
to any other science: only those questions will be geometrical which
form premisses for the proof of the theorems of geometry or of any
other science, such as optics, which uses the same basic truths as
geometry. Of the other sciences the like is true. Of these questions
the geometer is bound to give his account, using the basic truths
of geometry in conjunction with his previous conclusions; of the basic
truths the geometer, as such, is not bound to give any account. The
like is true of the other sciences. There is a limit, then, to the
questions which we may put to each man of science; nor is each man
of science bound to answer all inquiries on each several subject,
but only such as fall within the defined field of his own science.
If, then, in controversy with a geometer qua geometer the disputant
confines himself to geometry and proves anything from geometrical
premisses, he is clearly to be applauded; if he goes outside these
he will be at fault, and obviously cannot even refute the geometer
except accidentally. One should therefore not discuss geometry among
those who are not geometers, for in such a company an unsound argument
will pass unnoticed. This is correspondingly true in the other sciences.

Since there are 'geometrical' questions, does it follow that there
are also distinctively 'ungeometrical' questions? Further, in each
special science-geometry for instance-what kind of error is it that
may vitiate questions, and yet not exclude them from that science?
Again, is the erroneous conclusion one constructed from premisses
opposite to the true premisses, or is it formal fallacy though drawn
from geometrical premisses? Or, perhaps, the erroneous conclusion
is due to the drawing of premisses from another science; e.g. in a
geometrical controversy a musical question is distinctively ungeometrical,
whereas the notion that parallels meet is in one sense geometrical,
being ungeometrical in a different fashion: the reason being that
'ungeometrical', like 'unrhythmical', is equivocal, meaning in the
one case not geometry at all, in the other bad geometry? It is this
error, i.e. error based on premisses of this kind-'of' the science
but false-that is the contrary of science. In mathematics the formal
fallacy is not so common, because it is the middle term in which the
ambiguity lies, since the major is predicated of the whole of the
middle and the middle of the whole of the minor (the predicate of
course never has the prefix 'all'); and in mathematics one can, so
to speak, see these middle terms with an intellectual vision, while
in dialectic the ambiguity may escape detection. E.g. 'Is every circle
a figure?' A diagram shows that this is so, but the minor premiss
'Are epics circles?' is shown by the diagram to be false.

If a proof has an inductive minor premiss, one should not bring an
'objection' against it. For since every premiss must be applicable
to a number of cases (otherwise it will not be true in every instance,
which, since the syllogism proceeds from universals, it must be),
then assuredly the same is true of an 'objection'; since premisses
and 'objections' are so far the same that anything which can be validly
advanced as an 'objection' must be such that it could take the form
of a premiss, either demonstrative or dialectical. On the other hand,
arguments formally illogical do sometimes occur through taking as
middles mere attributes of the major and minor terms. An instance
of this is Caeneus' proof that fire increases in geometrical proportion:
'Fire', he argues, 'increases rapidly, and so does geometrical proportion'.
There is no syllogism so, but there is a syllogism if the most rapidly
increasing proportion is geometrical and the most rapidly increasing
proportion is attributable to fire in its motion. Sometimes, no doubt,
it is impossible to reason from premisses predicating mere attributes:
but sometimes it is possible, though the possibility is overlooked.
If false premisses could never give true conclusions 'resolution'
would be easy, for premisses and conclusion would in that case inevitably
reciprocate. I might then argue thus: let A be an existing fact; let
the existence of A imply such and such facts actually known to me
to exist, which we may call B. I can now, since they reciprocate,
infer A from B. 

Reciprocation of premisses and conclusion is more frequent in mathematics,
because mathematics takes definitions, but never an accident, for
its premisses-a second characteristic distinguishing mathematical
reasoning from dialectical disputations. 

A science expands not by the interposition of fresh middle terms,
but by the apposition of fresh extreme terms. E.g. A is predicated
of B, B of C, C of D, and so indefinitely. Or the expansion may be
lateral: e.g. one major A, may be proved of two minors, C and E. Thus
let A represent number-a number or number taken indeterminately; B
determinate odd number; C any particular odd number. We can then predicate
A of C. Next let D represent determinate even number, and E even number.
Then A is predicable of E. 

Part 13

Knowledge of the fact differs from knowledge of the reasoned fact.
To begin with, they differ within the same science and in two ways:
(1) when the premisses of the syllogism are not immediate (for then
the proximate cause is not contained in them-a necessary condition
of knowledge of the reasoned fact): (2) when the premisses are immediate,
but instead of the cause the better known of the two reciprocals is
taken as the middle; for of two reciprocally predicable terms the
one which is not the cause may quite easily be the better known and
so become the middle term of the demonstration. Thus (2, a) you might
prove as follows that the planets are near because they do not twinkle:
let C be the planets, B not twinkling, A proximity. Then B is predicable
of C; for the planets do not twinkle. But A is also predicable of
B, since that which does not twinkle is near--we must take this truth
as having been reached by induction or sense-perception. Therefore
A is a necessary predicate of C; so that we have demonstrated that
the planets are near. This syllogism, then, proves not the reasoned
fact but only the fact; since they are not near because they do not
twinkle, but, because they are near, do not twinkle. The major and
middle of the proof, however, may be reversed, and then the demonstration
will be of the reasoned fact. Thus: let C be the planets, B proximity,
A not twinkling. Then B is an attribute of C, and A-not twinkling-of
B. Consequently A is predicable of C, and the syllogism proves the
reasoned fact, since its middle term is the proximate cause. Another
example is the inference that the moon is spherical from its manner
of waxing. Thus: since that which so waxes is spherical, and since
the moon so waxes, clearly the moon is spherical. Put in this form,
the syllogism turns out to be proof of the fact, but if the middle
and major be reversed it is proof of the reasoned fact; since the
moon is not spherical because it waxes in a certain manner, but waxes
in such a manner because it is spherical. (Let C be the moon, B spherical,
and A waxing.) Again (b), in cases where the cause and the effect
are not reciprocal and the effect is the better known, the fact is
demonstrated but not the reasoned fact. This also occurs (1) when
the middle falls outside the major and minor, for here too the strict
cause is not given, and so the demonstration is of the fact, not of
the reasoned fact. For example, the question 'Why does not a wall
breathe?' might be answered, 'Because it is not an animal'; but that
answer would not give the strict cause, because if not being an animal
causes the absence of respiration, then being an animal should be
the cause of respiration, according to the rule that if the negation
of causes the non-inherence of y, the affirmation of x causes the
inherence of y; e.g. if the disproportion of the hot and cold elements
is the cause of ill health, their proportion is the cause of health;
and conversely, if the assertion of x causes the inherence of y, the
negation of x must cause y's non-inherence. But in the case given
this consequence does not result; for not every animal breathes. A
syllogism with this kind of cause takes place in the second figure.
Thus: let A be animal, B respiration, C wall. Then A is predicable
of all B (for all that breathes is animal), but of no C; and consequently
B is predicable of no C; that is, the wall does not breathe. Such
causes are like far-fetched explanations, which precisely consist
in making the cause too remote, as in Anacharsis' account of why the
Scythians have no flute-players; namely because they have no vines.

Thus, then, do the syllogism of the fact and the syllogism of the
reasoned fact differ within one science and according to the position
of the middle terms. But there is another way too in which the fact
and the reasoned fact differ, and that is when they are investigated
respectively by different sciences. This occurs in the case of problems
related to one another as subordinate and superior, as when optical
problems are subordinated to geometry, mechanical problems to stereometry,
harmonic problems to arithmetic, the data of observation to astronomy.
(Some of these sciences bear almost the same name; e.g. mathematical
and nautical astronomy, mathematical and acoustical harmonics.) Here
it is the business of the empirical observers to know the fact, of
the mathematicians to know the reasoned fact; for the latter are in
possession of the demonstrations giving the causes, and are often
ignorant of the fact: just as we have often a clear insight into a
universal, but through lack of observation are ignorant of some of
its particular instances. These connexions have a perceptible existence
though they are manifestations of forms. For the mathematical sciences
concern forms: they do not demonstrate properties of a substratum,
since, even though the geometrical subjects are predicable as properties
of a perceptible substratum, it is not as thus predicable that the
mathematician demonstrates properties of them. As optics is related
to geometry, so another science is related to optics, namely the theory
of the rainbow. Here knowledge of the fact is within the province
of the natural philosopher, knowledge of the reasoned fact within
that of the optician, either qua optician or qua mathematical optician.
Many sciences not standing in this mutual relation enter into it at
points; e.g. medicine and geometry: it is the physician's business
to know that circular wounds heal more slowly, the geometer's to know
the reason why. 

Part 14

Of all the figures the most scientific is the first. Thus, it is the
vehicle of the demonstrations of all the mathematical sciences, such
as arithmetic, geometry, and optics, and practically all of all sciences
that investigate causes: for the syllogism of the reasoned fact is
either exclusively or generally speaking and in most cases in this
figure-a second proof that this figure is the most scientific; for
grasp of a reasoned conclusion is the primary condition of knowledge.
Thirdly, the first is the only figure which enables us to pursue knowledge
of the essence of a thing. In the second figure no affirmative conclusion
is possible, and knowledge of a thing's essence must be affirmative;
while in the third figure the conclusion can be affirmative, but cannot
be universal, and essence must have a universal character: e.g. man
is not two-footed animal in any qualified sense, but universally.
Finally, the first figure has no need of the others, while it is by
means of the first that the other two figures are developed, and have
their intervals closepacked until immediate premisses are reached.

Clearly, therefore, the first figure is the primary condition of knowledge.

Part 15

Just as an attribute A may (as we saw) be atomically connected with
a subject B, so its disconnexion may be atomic. I call 'atomic' connexions
or disconnexions which involve no intermediate term; since in that
case the connexion or disconnexion will not be mediated by something
other than the terms themselves. It follows that if either A or B,
or both A and B, have a genus, their disconnexion cannot be primary.
Thus: let C be the genus of A. Then, if C is not the genus of B-for
A may well have a genus which is not the genus of B-there will be
a syllogism proving A's disconnexion from B thus: 

all A is C, no B is C, therefore no B is A. Or if it is B which has
a genus D, we have 

all B is D, no D is A, therefore no B is A, by syllogism; and the
proof will be similar if both A and B have a genus. That the genus
of A need not be the genus of B and vice versa, is shown by the existence
of mutually exclusive coordinate series of predication. If no term
in the series ACD...is predicable of any term in the series BEF...,and
if G-a term in the former series-is the genus of A, clearly G will
not be the genus of B; since, if it were, the series would not be
mutually exclusive. So also if B has a genus, it will not be the genus
of A. If, on the other hand, neither A nor B has a genus and A does
not inhere in B, this disconnexion must be atomic. If there be a middle
term, one or other of them is bound to have a genus, for the syllogism
will be either in the first or the second figure. If it is in the
first, B will have a genus-for the premiss containing it must be affirmative:
if in the second, either A or B indifferently, since syllogism is
possible if either is contained in a negative premiss, but not if
both premisses are negative. 

Hence it is clear that one thing may be atomically disconnected from
another, and we have stated when and how this is possible.

Part 16

Ignorance-defined not as the negation of knowledge but as a positive
state of mind-is error produced by inference. 

(1) Let us first consider propositions asserting a predicate's immediate
connexion with or disconnexion from a subject. Here, it is true, positive
error may befall one in alternative ways; for it may arise where one
directly believes a connexion or disconnexion as well as where one's
belief is acquired by inference. The error, however, that consists
in a direct belief is without complication; but the error resulting
from inference-which here concerns us-takes many forms. Thus, let
A be atomically disconnected from all B: then the conclusion inferred
through a middle term C, that all B is A, will be a case of error
produced by syllogism. Now, two cases are possible. Either (a) both
premisses, or (b) one premiss only, may be false. (a) If neither A
is an attribute of any C nor C of any B, whereas the contrary was
posited in both cases, both premisses will be false. (C may quite
well be so related to A and B that C is neither subordinate to A nor
a universal attribute of B: for B, since A was said to be primarily
disconnected from B, cannot have a genus, and A need not necessarily
be a universal attribute of all things. Consequently both premisses
may be false.) On the other hand, (b) one of the premisses may be
true, though not either indifferently but only the major A-C since,
B having no genus, the premiss C-B will always be false, while A-C
may be true. This is the case if, for example, A is related atomically
to both C and B; because when the same term is related atomically
to more terms than one, neither of those terms will belong to the
other. It is, of course, equally the case if A-C is not atomic.

Error of attribution, then, occurs through these causes and in this
form only-for we found that no syllogism of universal attribution
was possible in any figure but the first. On the other hand, an error
of non-attribution may occur either in the first or in the second
figure. Let us therefore first explain the various forms it takes
in the first figure and the character of the premisses in each case.

(c) It may occur when both premisses are false; e.g. supposing A atomically
connected with both C and B, if it be then assumed that no C is and
all B is C, both premisses are false. 

(d) It is also possible when one is false. This may be either premiss
indifferently. A-C may be true, C-B false-A-C true because A is not
an attribute of all things, C-B false because C, which never has the
attribute A, cannot be an attribute of B; for if C-B were true, the
premiss A-C would no longer be true, and besides if both premisses
were true, the conclusion would be true. Or again, C-B may be true
and A-C false; e.g. if both C and A contain B as genera, one of them
must be subordinate to the other, so that if the premiss takes the
form No C is A, it will be false. This makes it clear that whether
either or both premisses are false, the conclusion will equally be
false. 

In the second figure the premisses cannot both be wholly false; for
if all B is A, no middle term can be with truth universally affirmed
of one extreme and universally denied of the other: but premisses
in which the middle is affirmed of one extreme and denied of the other
are the necessary condition if one is to get a valid inference at
all. Therefore if, taken in this way, they are wholly false, their
contraries conversely should be wholly true. But this is impossible.
On the other hand, there is nothing to prevent both premisses being
partially false; e.g. if actually some A is C and some B is C, then
if it is premised that all A is C and no B is C, both premisses are
false, yet partially, not wholly, false. The same is true if the major
is made negative instead of the minor. Or one premiss may be wholly
false, and it may be either of them. Thus, supposing that actually
an attribute of all A must also be an attribute of all B, then if
C is yet taken to be a universal attribute of all but universally
non-attributable to B, C-A will be true but C-B false. Again, actually
that which is an attribute of no B will not be an attribute of all
A either; for if it be an attribute of all A, it will also be an attribute
of all B, which is contrary to supposition; but if C be nevertheless
assumed to be a universal attribute of A, but an attribute of no B,
then the premiss C-B is true but the major is false. The case is similar
if the major is made the negative premiss. For in fact what is an
attribute of no A will not be an attribute of any B either; and if
it be yet assumed that C is universally non-attributable to A, but
a universal attribute of B, the premiss C-A is true but the minor
wholly false. Again, in fact it is false to assume that that which
is an attribute of all B is an attribute of no A, for if it be an
attribute of all B, it must be an attribute of some A. If then C is
nevertheless assumed to be an attribute of all B but of no A, C-B
will be true but C-A false. 

It is thus clear that in the case of atomic propositions erroneous
inference will be possible not only when both premisses are false
but also when only one is false. 

Part 17

In the case of attributes not atomically connected with or disconnected
from their subjects, (a, i) as long as the false conclusion is inferred
through the 'appropriate' middle, only the major and not both premisses
can be false. By 'appropriate middle' I mean the middle term through
which the contradictory-i.e. the true-conclusion is inferrible. Thus,
let A be attributable to B through a middle term C: then, since to
produce a conclusion the premiss C-B must be taken affirmatively,
it is clear that this premiss must always be true, for its quality
is not changed. But the major A-C is false, for it is by a change
in the quality of A-C that the conclusion becomes its contradictory-i.e.
true. Similarly (ii) if the middle is taken from another series of
predication; e.g. suppose D to be not only contained within A as a
part within its whole but also predicable of all B. Then the premiss
D-B must remain unchanged, but the quality of A-D must be changed;
so that D-B is always true, A-D always false. Such error is practically
identical with that which is inferred through the 'appropriate' middle.
On the other hand, (b) if the conclusion is not inferred through the
'appropriate' middle-(i) when the middle is subordinate to A but is
predicable of no B, both premisses must be false, because if there
is to be a conclusion both must be posited as asserting the contrary
of what is actually the fact, and so posited both become false: e.g.
suppose that actually all D is A but no B is D; then if these premisses
are changed in quality, a conclusion will follow and both of the new
premisses will be false. When, however, (ii) the middle D is not subordinate
to A, A-D will be true, D-B false-A-D true because A was not subordinate
to D, D-B false because if it had been true, the conclusion too would
have been true; but it is ex hypothesi false. 

When the erroneous inference is in the second figure, both premisses
cannot be entirely false; since if B is subordinate to A, there can
be no middle predicable of all of one extreme and of none of the other,
as was stated before. One premiss, however, may be false, and it may
be either of them. Thus, if C is actually an attribute of both A and
B, but is assumed to be an attribute of A only and not of B, C-A will
be true, C-B false: or again if C be assumed to be attributable to
B but to no A, C-B will be true, C-A false. 

We have stated when and through what kinds of premisses error will
result in cases where the erroneous conclusion is negative. If the
conclusion is affirmative, (a, i) it may be inferred through the
'appropriate' middle term. In this case both premisses cannot be false
since, as we said before, C-B must remain unchanged if there is to
be a conclusion, and consequently A-C, the quality of which is changed,
will always be false. This is equally true if (ii) the middle is taken
from another series of predication, as was stated to be the case also
with regard to negative error; for D-B must remain unchanged, while
the quality of A-D must be converted, and the type of error is the
same as before. 

(b) The middle may be inappropriate. Then (i) if D is subordinate
to A, A-D will be true, but D-B false; since A may quite well be predicable
of several terms no one of which can be subordinated to another. If,
however, (ii) D is not subordinate to A, obviously A-D, since it is
affirmed, will always be false, while D-B may be either true or false;
for A may very well be an attribute of no D, whereas all B is D, e.g.
no science is animal, all music is science. Equally well A may be
an attribute of no D, and D of no B. It emerges, then, that if the
middle term is not subordinate to the major, not only both premisses
but either singly may be false. 

Thus we have made it clear how many varieties of erroneous inference
are liable to happen and through what kinds of premisses they occur,
in the case both of immediate and of demonstrable truths.

Part 18

It is also clear that the loss of any one of the senses entails the
loss of a corresponding portion of knowledge, and that, since we learn
either by induction or by demonstration, this knowledge cannot be
acquired. Thus demonstration develops from universals, induction from
particulars; but since it is possible to familiarize the pupil with
even the so-called mathematical abstractions only through induction-i.e.
only because each subject genus possesses, in virtue of a determinate
mathematical character, certain properties which can be treated as
separate even though they do not exist in isolation-it is consequently
impossible to come to grasp universals except through induction. But
induction is impossible for those who have not sense-perception. For
it is sense-perception alone which is adequate for grasping the particulars:
they cannot be objects of scientific knowledge, because neither can
universals give us knowledge of them without induction, nor can we
get it through induction without sense-perception. 

Part 19

Every syllogism is effected by means of three terms. One kind of syllogism
serves to prove that A inheres in C by showing that A inheres in B
and B in C; the other is negative and one of its premisses asserts
one term of another, while the other denies one term of another. It
is clear, then, that these are the fundamentals and so-called hypotheses
of syllogism. Assume them as they have been stated, and proof is bound
to follow-proof that A inheres in C through B, and again that A inheres
in B through some other middle term, and similarly that B inheres
in C. If our reasoning aims at gaining credence and so is merely dialectical,
it is obvious that we have only to see that our inference is based
on premisses as credible as possible: so that if a middle term between
A and B is credible though not real, one can reason through it and
complete a dialectical syllogism. If, however, one is aiming at truth,
one must be guided by the real connexions of subjects and attributes.
Thus: since there are attributes which are predicated of a subject
essentially or naturally and not coincidentally-not, that is, in the
sense in which we say 'That white (thing) is a man', which is not
the same mode of predication as when we say 'The man is white': the
man is white not because he is something else but because he is man,
but the white is man because 'being white' coincides with 'humanity'
within one substratum-therefore there are terms such as are naturally
subjects of predicates. Suppose, then, C such a term not itself attributable
to anything else as to a subject, but the proximate subject of the
attribute B--i.e. so that B-C is immediate; suppose further E related
immediately to F, and F to B. The first question is, must this series
terminate, or can it proceed to infinity? The second question is as
follows: Suppose nothing is essentially predicated of A, but A is
predicated primarily of H and of no intermediate prior term, and suppose
H similarly related to G and G to B; then must this series also terminate,
or can it too proceed to infinity? There is this much difference between
the questions: the first is, is it possible to start from that which
is not itself attributable to anything else but is the subject of
attributes, and ascend to infinity? The second is the problem whether
one can start from that which is a predicate but not itself a subject
of predicates, and descend to infinity? A third question is, if the
extreme terms are fixed, can there be an infinity of middles? I mean
this: suppose for example that A inheres in C and B is intermediate
between them, but between B and A there are other middles, and between
these again fresh middles; can these proceed to infinity or can they
not? This is the equivalent of inquiring, do demonstrations proceed
to infinity, i.e. is everything demonstrable? Or do ultimate subject
and primary attribute limit one another? 

I hold that the same questions arise with regard to negative conclusions
and premisses: viz. if A is attributable to no B, then either this
predication will be primary, or there will be an intermediate term
prior to B to which a is not attributable-G, let us say, which is
attributable to all B-and there may still be another term H prior
to G, which is attributable to all G. The same questions arise, I
say, because in these cases too either the series of prior terms to
which a is not attributable is infinite or it terminates.

One cannot ask the same questions in the case of reciprocating terms,
since when subject and predicate are convertible there is neither
primary nor ultimate subject, seeing that all the reciprocals qua
subjects stand in the same relation to one another, whether we say
that the subject has an infinity of attributes or that both subjects
and attributes-and we raised the question in both cases-are infinite
in number. These questions then cannot be asked-unless, indeed, the
terms can reciprocate by two different modes, by accidental predication
in one relation and natural predication in the other. 

Part 20

Now, it is clear that if the predications terminate in both the upward
and the downward direction (by 'upward' I mean the ascent to the more
universal, by 'downward' the descent to the more particular), the
middle terms cannot be infinite in number. For suppose that A is predicated
of F, and that the intermediates-call them BB'B"...-are infinite,
then clearly you might descend from and find one term predicated of
another ad infinitum, since you have an infinity of terms between
you and F; and equally, if you ascend from F, there are infinite terms
between you and A. It follows that if these processes are impossible
there cannot be an infinity of intermediates between A and F. Nor
is it of any effect to urge that some terms of the series AB...F are
contiguous so as to exclude intermediates, while others cannot be
taken into the argument at all: whichever terms of the series B...I
take, the number of intermediates in the direction either of A or
of F must be finite or infinite: where the infinite series starts,
whether from the first term or from a later one, is of no moment,
for the succeeding terms in any case are infinite in number.

Part 21

Further, if in affirmative demonstration the series terminates in
both directions, clearly it will terminate too in negative demonstration.
Let us assume that we cannot proceed to infinity either by ascending
from the ultimate term (by 'ultimate term' I mean a term such as was,
not itself attributable to a subject but itself the subject of attributes),
or by descending towards an ultimate from the primary term (by 'primary
term' I mean a term predicable of a subject but not itself a subject).
If this assumption is justified, the series will also terminate in
the case of negation. For a negative conclusion can be proved in all
three figures. In the first figure it is proved thus: no B is A, all
C is B. In packing the interval B-C we must reach immediate propositions--as
is always the case with the minor premiss--since B-C is affirmative.
As regards the other premiss it is plain that if the major term is
denied of a term D prior to B, D will have to be predicable of all
B, and if the major is denied of yet another term prior to D, this
term must be predicable of all D. Consequently, since the ascending
series is finite, the descent will also terminate and there will be
a subject of which A is primarily non-predicable. In the second figure
the syllogism is, all A is B, no C is B,..no C is A. If proof of this
is required, plainly it may be shown either in the first figure as
above, in the second as here, or in the third. The first figure has
been discussed, and we will proceed to display the second, proof by
which will be as follows: all B is D, no C is D..., since it is required
that B should be a subject of which a predicate is affirmed. Next,
since D is to be proved not to belong to C, then D has a further predicate
which is denied of C. Therefore, since the succession of predicates
affirmed of an ever higher universal terminates, the succession of
predicates denied terminates too. 

The third figure shows it as follows: all B is A, some B is not C.
Therefore some A is not C. This premiss, i.e. C-B, will be proved
either in the same figure or in one of the two figures discussed above.
In the first and second figures the series terminates. If we use the
third figure, we shall take as premisses, all E is B, some E is not
C, and this premiss again will be proved by a similar prosyllogism.
But since it is assumed that the series of descending subjects also
terminates, plainly the series of more universal non-predicables will
terminate also. Even supposing that the proof is not confined to one
method, but employs them all and is now in the first figure, now in
the second or third-even so the regress will terminate, for the methods
are finite in number, and if finite things are combined in a finite
number of ways, the result must be finite. 

Thus it is plain that the regress of middles terminates in the case
of negative demonstration, if it does so also in the case of affirmative
demonstration. That in fact the regress terminates in both these cases
may be made clear by the following dialectical considerations.

Part 22

In the case of predicates constituting the essential nature of a thing,
it clearly terminates, seeing that if definition is possible, or in
other words, if essential form is knowable, and an infinite series
cannot be traversed, predicates constituting a thing's essential nature
must be finite in number. But as regards predicates generally we have
the following prefatory remarks to make. (1) We can affirm without
falsehood 'the white (thing) is walking', and that big (thing) is
a log'; or again, 'the log is big', and 'the man walks'. But the affirmation
differs in the two cases. When I affirm 'the white is a log', I mean
that something which happens to be white is a log-not that white is
the substratum in which log inheres, for it was not qua white or qua
a species of white that the white (thing) came to be a log, and the
white (thing) is consequently not a log except incidentally. On the
other hand, when I affirm 'the log is white', I do not mean that something
else, which happens also to be a log, is white (as I should if I said
'the musician is white,' which would mean 'the man who happens also
to be a musician is white'); on the contrary, log is here the substratum-the
substratum which actually came to be white, and did so qua wood or
qua a species of wood and qua nothing else. 

If we must lay down a rule, let us entitle the latter kind of statement
predication, and the former not predication at all, or not strict
but accidental predication. 'White' and 'log' will thus serve as types
respectively of predicate and subject. 

We shall assume, then, that the predicate is invariably predicated
strictly and not accidentally of the subject, for on such predication
demonstrations depend for their force. It follows from this that when
a single attribute is predicated of a single subject, the predicate
must affirm of the subject either some element constituting its essential
nature, or that it is in some way qualified, quantified, essentially
related, active, passive, placed, or dated. 

(2) Predicates which signify substance signify that the subject is
identical with the predicate or with a species of the predicate. Predicates
not signifying substance which are predicated of a subject not identical
with themselves or with a species of themselves are accidental or
coincidental; e.g. white is a coincident of man, seeing that man is
not identical with white or a species of white, but rather with animal,
since man is identical with a species of animal. These predicates
which do not signify substance must be predicates of some other subject,
and nothing can be white which is not also other than white. The Forms
we can dispense with, for they are mere sound without sense; and even
if there are such things, they are not relevant to our discussion,
since demonstrations are concerned with predicates such as we have
defined. 

(3) If A is a quality of B, B cannot be a quality of A-a quality of
a quality. Therefore A and B cannot be predicated reciprocally of
one another in strict predication: they can be affirmed without falsehood
of one another, but not genuinely predicated of each other. For one
alternative is that they should be substantially predicated of one
another, i.e. B would become the genus or differentia of A-the predicate
now become subject. But it has been shown that in these substantial
predications neither the ascending predicates nor the descending subjects
form an infinite series; e.g. neither the series, man is biped, biped
is animal, &c., nor the series predicating animal of man, man of Callias,
Callias of a further. subject as an element of its essential nature,
is infinite. For all such substance is definable, and an infinite
series cannot be traversed in thought: consequently neither the ascent
nor the descent is infinite, since a substance whose predicates were
infinite would not be definable. Hence they will not be predicated
each as the genus of the other; for this would equate a genus with
one of its own species. Nor (the other alternative) can a quale be
reciprocally predicated of a quale, nor any term belonging to an adjectival
category of another such term, except by accidental predication; for
all such predicates are coincidents and are predicated of substances.
On the other hand-in proof of the impossibility of an infinite ascending
series-every predication displays the subject as somehow qualified
or quantified or as characterized under one of the other adjectival
categories, or else is an element in its substantial nature: these
latter are limited in number, and the number of the widest kinds under
which predications fall is also limited, for every predication must
exhibit its subject as somehow qualified, quantified, essentially
related, acting or suffering, or in some place or at some time.

I assume first that predication implies a single subject and a single
attribute, and secondly that predicates which are not substantial
are not predicated of one another. We assume this because such predicates
are all coincidents, and though some are essential coincidents, others
of a different type, yet we maintain that all of them alike are predicated
of some substratum and that a coincident is never a substratum-since
we do not class as a coincident anything which does not owe its designation
to its being something other than itself, but always hold that any
coincident is predicated of some substratum other than itself, and
that another group of coincidents may have a different substratum.
Subject to these assumptions then, neither the ascending nor the descending
series of predication in which a single attribute is predicated of
a single subject is infinite. For the subjects of which coincidents
are predicated are as many as the constitutive elements of each individual
substance, and these we have seen are not infinite in number, while
in the ascending series are contained those constitutive elements
with their coincidents-both of which are finite. We conclude that
there is a given subject (D) of which some attribute (C) is primarily
predicable; that there must be an attribute (B) primarily predicable
of the first attribute, and that the series must end with a term (A)
not predicable of any term prior to the last subject of which it was
predicated (B), and of which no term prior to it is predicable.

The argument we have given is one of the so-called proofs; an alternative
proof follows. Predicates so related to their subjects that there
are other predicates prior to them predicable of those subjects are
demonstrable; but of demonstrable propositions one cannot have something
better than knowledge, nor can one know them without demonstration.
Secondly, if a consequent is only known through an antecedent (viz.
premisses prior to it) and we neither know this antecedent nor have
something better than knowledge of it, then we shall not have scientific
knowledge of the consequent. Therefore, if it is possible through
demonstration to know anything without qualification and not merely
as dependent on the acceptance of certain premisses-i.e. hypothetically-the
series of intermediate predications must terminate. If it does not
terminate, and beyond any predicate taken as higher than another there
remains another still higher, then every predicate is demonstrable.
Consequently, since these demonstrable predicates are infinite in
number and therefore cannot be traversed, we shall not know them by
demonstration. If, therefore, we have not something better than knowledge
of them, we cannot through demonstration have unqualified but only
hypothetical science of anything. 

As dialectical proofs of our contention these may carry conviction,
but an analytic process will show more briefly that neither the ascent
nor the descent of predication can be infinite in the demonstrative
sciences which are the object of our investigation. Demonstration
proves the inherence of essential attributes in things. Now attributes
may be essential for two reasons: either because they are elements
in the essential nature of their subjects, or because their subjects
are elements in their essential nature. An example of the latter is
odd as an attribute of number-though it is number's attribute, yet
number itself is an element in the definition of odd; of the former,
multiplicity or the indivisible, which are elements in the definition
of number. In neither kind of attribution can the terms be infinite.
They are not infinite where each is related to the term below it as
odd is to number, for this would mean the inherence in odd of another
attribute of odd in whose nature odd was an essential element: but
then number will be an ultimate subject of the whole infinite chain
of attributes, and be an element in the definition of each of them.
Hence, since an infinity of attributes such as contain their subject
in their definition cannot inhere in a single thing, the ascending
series is equally finite. Note, moreover, that all such attributes
must so inhere in the ultimate subject-e.g. its attributes in number
and number in them-as to be commensurate with the subject and not
of wider extent. Attributes which are essential elements in the nature
of their subjects are equally finite: otherwise definition would be
impossible. Hence, if all the attributes predicated are essential
and these cannot be infinite, the ascending series will terminate,
and consequently the descending series too. 

If this is so, it follows that the intermediates between any two terms
are also always limited in number. An immediately obvious consequence
of this is that demonstrations necessarily involve basic truths, and
that the contention of some-referred to at the outset-that all truths
are demonstrable is mistaken. For if there are basic truths, (a) not
all truths are demonstrable, and (b) an infinite regress is impossible;
since if either (a) or (b) were not a fact, it would mean that no
interval was immediate and indivisible, but that all intervals were
divisible. This is true because a conclusion is demonstrated by the
interposition, not the apposition, of a fresh term. If such interposition
could continue to infinity there might be an infinite number of terms
between any two terms; but this is impossible if both the ascending
and descending series of predication terminate; and of this fact,
which before was shown dialectically, analytic proof has now been
given. 

Part 23

It is an evident corollary of these conclusions that if the same attribute
A inheres in two terms C and D predicable either not at all, or not
of all instances, of one another, it does not always belong to them
in virtue of a common middle term. Isosceles and scalene possess the
attribute of having their angles equal to two right angles in virtue
of a common middle; for they possess it in so far as they are both
a certain kind of figure, and not in so far as they differ from one
another. But this is not always the case: for, were it so, if we take
B as the common middle in virtue of which A inheres in C and D, clearly
B would inhere in C and D through a second common middle, and this
in turn would inhere in C and D through a third, so that between two
terms an infinity of intermediates would fall-an impossibility. Thus
it need not always be in virtue of a common middle term that a single
attribute inheres in several subjects, since there must be immediate
intervals. Yet if the attribute to be proved common to two subjects
is to be one of their essential attributes, the middle terms involved
must be within one subject genus and be derived from the same group
of immediate premisses; for we have seen that processes of proof cannot
pass from one genus to another. 

It is also clear that when A inheres in B, this can be demonstrated
if there is a middle term. Further, the 'elements' of such a conclusion
are the premisses containing the middle in question, and they are
identical in number with the middle terms, seeing that the immediate
propositions-or at least such immediate propositions as are universal-are
the 'elements'. If, on the other hand, there is no middle term, demonstration
ceases to be possible: we are on the way to the basic truths. Similarly
if A does not inhere in B, this can be demonstrated if there is a
middle term or a term prior to B in which A does not inhere: otherwise
there is no demonstration and a basic truth is reached. There are,
moreover, as many 'elements' of the demonstrated conclusion as there
are middle terms, since it is propositions containing these middle
terms that are the basic premisses on which the demonstration rests;
and as there are some indemonstrable basic truths asserting that 'this
is that' or that 'this inheres in that', so there are others denying
that 'this is that' or that 'this inheres in that'-in fact some basic
truths will affirm and some will deny being. 

When we are to prove a conclusion, we must take a primary essential
predicate-suppose it C-of the subject B, and then suppose A similarly
predicable of C. If we proceed in this manner, no proposition or attribute
which falls beyond A is admitted in the proof: the interval is constantly
condensed until subject and predicate become indivisible, i.e. one.
We have our unit when the premiss becomes immediate, since the immediate
premiss alone is a single premiss in the unqualified sense of 'single'.
And as in other spheres the basic element is simple but not identical
in all-in a system of weight it is the mina, in music the quarter-tone,
and so on--so in syllogism the unit is an immediate premiss, and in
the knowledge that demonstration gives it is an intuition. In syllogisms,
then, which prove the inherence of an attribute, nothing falls outside
the major term. In the case of negative syllogisms on the other hand,
(1) in the first figure nothing falls outside the major term whose
inherence is in question; e.g. to prove through a middle C that A
does not inhere in B the premisses required are, all B is C, no C
is A. Then if it has to be proved that no C is A, a middle must be
found between and C; and this procedure will never vary.

(2) If we have to show that E is not D by means of the premisses,
all D is C; no E, or not all E, is C; then the middle will never fall
beyond E, and E is the subject of which D is to be denied in the conclusion.

(3) In the third figure the middle will never fall beyond the limits
of the subject and the attribute denied of it. 

Part 24

Since demonstrations may be either commensurately universal or particular,
and either affirmative or negative; the question arises, which form
is the better? And the same question may be put in regard to so-called
'direct' demonstration and reductio ad impossibile. Let us first examine
the commensurately universal and the particular forms, and when we
have cleared up this problem proceed to discuss 'direct' demonstration
and reductio ad impossibile. 

The following considerations might lead some minds to prefer particular
demonstration. 

(1) The superior demonstration is the demonstration which gives us
greater knowledge (for this is the ideal of demonstration), and we
have greater knowledge of a particular individual when we know it
in itself than when we know it through something else; e.g. we know
Coriscus the musician better when we know that Coriscus is musical
than when we know only that man is musical, and a like argument holds
in all other cases. But commensurately universal demonstration, instead
of proving that the subject itself actually is x, proves only that
something else is x- e.g. in attempting to prove that isosceles is
x, it proves not that isosceles but only that triangle is x- whereas
particular demonstration proves that the subject itself is x. The
demonstration, then, that a subject, as such, possesses an attribute
is superior. If this is so, and if the particular rather than the
commensurately universal forms demonstrates, particular demonstration
is superior. 

(2) The universal has not a separate being over against groups of
singulars. Demonstration nevertheless creates the opinion that its
function is conditioned by something like this-some separate entity
belonging to the real world; that, for instance, of triangle or of
figure or number, over against particular triangles, figures, and
numbers. But demonstration which touches the real and will not mislead
is superior to that which moves among unrealities and is delusory.
Now commensurately universal demonstration is of the latter kind:
if we engage in it we find ourselves reasoning after a fashion well
illustrated by the argument that the proportionate is what answers
to the definition of some entity which is neither line, number, solid,
nor plane, but a proportionate apart from all these. Since, then,
such a proof is characteristically commensurate and universal, and
less touches reality than does particular demonstration, and creates
a false opinion, it will follow that commensurate and universal is
inferior to particular demonstration. 

We may retort thus. (1) The first argument applies no more to commensurate
and universal than to particular demonstration. If equality to two
right angles is attributable to its subject not qua isosceles but
qua triangle, he who knows that isosceles possesses that attribute
knows the subject as qua itself possessing the attribute, to a less
degree than he who knows that triangle has that attribute. To sum
up the whole matter: if a subject is proved to possess qua triangle
an attribute which it does not in fact possess qua triangle, that
is not demonstration: but if it does possess it qua triangle the rule
applies that the greater knowledge is his who knows the subject as
possessing its attribute qua that in virtue of which it actually does
possess it. Since, then, triangle is the wider term, and there is
one identical definition of triangle-i.e. the term is not equivocal-and
since equality to two right angles belongs to all triangles, it is
isosceles qua triangle and not triangle qua isosceles which has its
angles so related. It follows that he who knows a connexion universally
has greater knowledge of it as it in fact is than he who knows the
particular; and the inference is that commensurate and universal is
superior to particular demonstration. 

(2) If there is a single identical definition i.e. if the commensurate
universal is unequivocal-then the universal will possess being not
less but more than some of the particulars, inasmuch as it is universals
which comprise the imperishable, particulars that tend to perish.

(3) Because the universal has a single meaning, we are not therefore
compelled to suppose that in these examples it has being as a substance
apart from its particulars-any more than we need make a similar supposition
in the other cases of unequivocal universal predication, viz. where
the predicate signifies not substance but quality, essential relatedness,
or action. If such a supposition is entertained, the blame rests not
with the demonstration but with the hearer. 

(4) Demonstration is syllogism that proves the cause, i.e. the reasoned
fact, and it is rather the commensurate universal than the particular
which is causative (as may be shown thus: that which possesses an
attribute through its own essential nature is itself the cause of
the inherence, and the commensurate universal is primary; hence the
commensurate universal is the cause). Consequently commensurately
universal demonstration is superior as more especially proving the
cause, that is the reasoned fact. 

(5) Our search for the reason ceases, and we think that we know, when
the coming to be or existence of the fact before us is not due to
the coming to be or existence of some other fact, for the last step
of a search thus conducted is eo ipso the end and limit of the problem.
Thus: 'Why did he come?' 'To get the money-wherewith to pay a debt-that
he might thereby do what was right.' When in this regress we can no
longer find an efficient or final cause, we regard the last step of
it as the end of the coming-or being or coming to be-and we regard
ourselves as then only having full knowledge of the reason why he
came. 

If, then, all causes and reasons are alike in this respect, and if
this is the means to full knowledge in the case of final causes such
as we have exemplified, it follows that in the case of the other causes
also full knowledge is attained when an attribute no longer inheres
because of something else. Thus, when we learn that exterior angles
are equal to four right angles because they are the exterior angles
of an isosceles, there still remains the question 'Why has isosceles
this attribute?' and its answer 'Because it is a triangle, and a triangle
has it because a triangle is a rectilinear figure.' If rectilinear
figure possesses the property for no further reason, at this point
we have full knowledge-but at this point our knowledge has become
commensurately universal, and so we conclude that commensurately universal
demonstration is superior. 

(6) The more demonstration becomes particular the more it sinks into
an indeterminate manifold, while universal demonstration tends to
the simple and determinate. But objects so far as they are an indeterminate
manifold are unintelligible, so far as they are determinate, intelligible:
they are therefore intelligible rather in so far as they are universal
than in so far as they are particular. From this it follows that universals
are more demonstrable: but since relative and correlative increase
concomitantly, of the more demonstrable there will be fuller demonstration.
Hence the commensurate and universal form, being more truly demonstration,
is the superior. 

(7) Demonstration which teaches two things is preferable to demonstration
which teaches only one. He who possesses commensurately universal
demonstration knows the particular as well, but he who possesses particular
demonstration does not know the universal. So that this is an additional
reason for preferring commensurately universal demonstration. And
there is yet this further argument: 

(8) Proof becomes more and more proof of the commensurate universal
as its middle term approaches nearer to the basic truth, and nothing
is so near as the immediate premiss which is itself the basic truth.
If, then, proof from the basic truth is more accurate than proof not
so derived, demonstration which depends more closely on it is more
accurate than demonstration which is less closely dependent. But commensurately
universal demonstration is characterized by this closer dependence,
and is therefore superior. Thus, if A had to be proved to inhere in
D, and the middles were B and C, B being the higher term would render
the demonstration which it mediated the more universal. 

Some of these arguments, however, are dialectical. The clearest indication
of the precedence of commensurately universal demonstration is as
follows: if of two propositions, a prior and a posterior, we have
a grasp of the prior, we have a kind of knowledge-a potential grasp-of
the posterior as well. For example, if one knows that the angles of
all triangles are equal to two right angles, one knows in a sense-potentially-that
the isosceles' angles also are equal to two right angles, even if
one does not know that the isosceles is a triangle; but to grasp this
posterior proposition is by no means to know the commensurate universal
either potentially or actually. Moreover, commensurately universal
demonstration is through and through intelligible; particular demonstration
issues in sense-perception. 

Part 25

The preceding arguments constitute our defence of the superiority
of commensurately universal to particular demonstration. That affirmative
demonstration excels negative may be shown as follows. 

(1) We may assume the superiority ceteris paribus of the demonstration
which derives from fewer postulates or hypotheses-in short from fewer
premisses; for, given that all these are equally well known, where
they are