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BOOK I. - Aristotle, Posterior Analytics 
Aristotle’s Posterior Analytics, trans. E.S. Bouchier, B.A. (Oxford: Blackwell, 1901).
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Whether a Demonstrative Science exists
Previous knowledge is required for all scientific studies or methods of instruction. Examples from Mathematics, Dialectic and Rhetoric. Previous knowledge as variously expressed in theses concerning either the existence of a thing or the meaning of the word denoting it. Learning consists in the conversion of universal into particular knowledge.
All communications of knowledge from teacher to pupil by way of reasoning pre-suppose some pre-existing knowledge. The truth of this statement may be seen from a complete enumeration of instances:—it is thus that the mathematical sciences are attained and every art also. The same is the case with dialectical arguments whether proceeding by means of the syllogism or of induction, for the former kind makes such assumptions as people who understand the meaning admit, the latter uses the recognized clearness of the particular as an indication of the universal, so that both convey their information by means of things already known. So too orators produce conviction in a like manner, using either Example, which is equivalent to induction, or Enthymeme, which corresponds to syllogism.
Pre-existing knowledge of two kinds is required: one must either assume beforehand that something exists, or one must understand what the word means, while sometimes both sorts of knowledge are required. As an example of the first case we may take the necessity for previously knowing the proposition ‘everything must be either affirmed or denied.’ Of the second case an instance would be the knowledge of the meaning conveyed by the word ‘triangle’; of the combination of both kinds, the knowledge both of what ‘Unit’ means, and of the fact that ‘Unit’ exists. The distinction is necessary, since the grounds of certainty differ in the two cases.
Some facts become known as a result of previously acquired knowledge, while others are learned at the moment of perceiving the object. This latter happens in the case of all things comprised under a universal, with which one is already acquainted. It is known to the pupil, before perceiving any particular triangle, that the interior angles of every triangle are equal to two right angles; but it is only at the moment of sense-perception that he learns that this figure inscribed in the semi-circle is a triangle.
In some cases knowledge is only acquired in this latter way, and the particular is not learned by means of a middle term: that is to say, in the cases where we touch the concrete particular, that is in the case of things which are not predicable of any subject. We ought to admit that, even before arriving at particulars, and so obtaining a syllogism, we do, from one point of view perhaps, possess knowledge, although from another we do not. For how, it may be asked, when he did not know whether the thing existed at all or not, could he have known absolutely that it contains two right angles? The answer is that he knows it from a particular point of view, in that he knows the universal, but he does not know it absolutely. On any other view we shall have the dilemma of the Meno—a man will either learn nothing at all or only what he knows before. This difficulty must not be solved as some try to do. The question is asked, ‘Do you or do you not know every dyad to be even?’ On receiving an affirmative reply they bring forward some dyad of the existence of which the other was ignorant, and so could not have known it to be even. The solution suggested is to say that one does not know every dyad to be even, but only that which one knows to be a dyad. On the other hand one knows that of which one possesses or has received a demonstration, and no demonstration concerns merely (e.g.) every triangle, or number, one may happen to know, but every possible triangle or number. No demonstrative proposition is taken as referring to ‘any number you may know of,’ or ‘any straight line you may know of,’ but to the entire subject. Nothing, however, I should suppose, precludes our knowing already what we learn from one point of view and not knowing it from another. The absurdity would consist not in having some sort of knowledge of what one learns, but in having knowledge of it in a certain respect—I mean in the very same respect and manner in which one learns it.
What Knowing is, what Demonstration is, and of what it consists
Scientific knowledge of a thing consists in knowing its cause demonstratively. The principles required for Demonstration. Meaning of ‘Thesis,’ ‘Hypothesis,’ ‘Axiom,’ ‘Definition.’
We suppose ourselves to know anything absolutely and not accidentally after the manner of the sophists, when we consider ourselves to know that the ground from which the thing arises is the ground of it, and that the fact cannot be otherwise. Science must clearly consist in this, for those who suppose themselves to have scientific knowledge of anything without really having it imagine that they are in the position described above, while those who do possess such knowledge are actually in that position in relation to the object.
Hence it follows that everything which admits of absolute knowledge is necessary. We will discuss later the question as to whether there is any other manner of knowing a thing, but at any rate we hold that that ‘knowledge comes through demonstration.’ By ‘demonstration’ I mean a scientific syllogism, and by ‘scientific’ a syllogism the mere possession of which makes us know.
If then the definition of knowledge be such as we have stated, the premises of demonstrative knowledge must needs be true, primary, immediate, better known than, anterior to, and the cause of, the conclusion, for under these conditions the principles will also be appropriate to the conclusion. One may, indeed, have a syllogism without these conditions, but not demonstration, for it will not produce scientific knowledge. The premises must be true, because it is impossible to know that which is not, e.g. that the diagonal of a square is commensurate with the side. The conclusion must proceed from primary premises that are indemonstrable premises, for one cannot know things of which one can give no demonstration, since to know demonstrable things in any real sense is just to have a demonstration of them. The premises must be Causal, Better known and Anterior; Causal, because we only know a thing when we have learned its cause, Anterior because anteriority is implied by causation, previously known not only in our second sense, viz. that their meaning is understood, but that one knows that they exist.
Now the expressions ‘anterior’ and ‘better known’ have each a double meaning; things which are naturally anterior are not the same as things anterior to us, nor yet are things naturally better known better known to us. I mean by things anterior, or better known, ‘to us,’ such as are nearer our sense-perception, while things which are absolutely anterior or better known are such as are more removed from it. Those things are the furthest removed from it which are most Universal, nearest to it stands the Particular, and these two are diametrically opposed.
The phrase ‘the conclusion must result from primary principles’ means that it must come from elements appropriate to itself, (for I attach the same meaning to primary principle [πρω̂τον] and to element [ἀρχή]). Now the element of demonstration is an immediate proposition; ‘immediate’ meaning a proposition with no other proposition anterior to it. A premise is either of the two parts of a predication, wherein one predicate is asserted of one subject. A dialectical premise is one which offers an alternative between the two parts of the predication, a demonstrative premise is one which lays down definitely that one of them is true.
Predication is either part of a Contradiction. Contradiction is an opposition of propositions which excludes any intermediate proposition. That part of a Contradiction which affirms one thing of another is Affirmation, that which denies one thing of another is Negation.
I apply the name Thesis to an immediate syllogistic principle which cannot be proved, and the previous possession of which is a necessary condition for learning something, but not all. That which is an indispensable antecedent to the acquisition of any knowledge I call an Axiom; for there are some principles of this kind, and ‘axiom’ is the name generally applied to them.
A Thesis which embodies one or other part of a predication (that is that the subject does, or does not, exist) is a Hypothesis; one which makes no such assertion a Definition. Definition is really a kind of Thesis; e.g. the arithmetician ‘lays it down’ that Unity is indivisibility in respect to quantity, but this is not a Hypothesis, for the nature of unity and the fact of its existence are not one and the same question.
Since then belief and knowledge with regard to any subject result from the possession of a demonstrative syllogism, and since a syllogism is demonstrative when the principles from which it is drawn are true, we must not merely have a previous knowledge of some or all of these primary principles, but have a higher knowledge of them than of the conclusion.
The Cause always possesses the quality which it impresses on a subject in a higher degree than that subject; thus, that for which we love anything is dear in a higher degree than the actual object of our love. Hence if our knowledge and belief is due to its primary principles, we have a higher knowledge of these latter and believe more firmly in them, because the thing itself is a consequence of them. Now it is not possible to believe less in what one knows than in what one neither knows nor has attained to by some higher faculty than knowledge. But this will happen unless he whose belief is produced by demonstration has a previous knowledge of the primary principles, for it is more needful to believe in these principles, either all or some, than in the conclusion to which they lead.
Now in order to attain to that knowledge which comes by demonstration one must not only be better acquainted with and believe more firmly in the elementary principles than in the conclusion, but nothing must be better known nor more firmly believed in than the opposites of those principles from which a false conclusion contrary to the science itself can be educed; that is to say if he who possesses absolute knowledge is to be quite immovable in his opinions.
A refutation of the error into which some have fallen concerning Science and Demonstration
Certain objections met. (1) That first principles are hypothetical; (2) That their consequences establish one another by a circular proof.
Now some persons, because of the necessity of knowledge of the primary principles, infer that knowledge does not exist, while others suppose that it does exist and that everything whatever is capable of demonstration. Neither of these views is either true or necessary. Those who assume that knowledge is not possible at all, think that it would involve an infinite regress, since one cannot know the later terms of a series by means of the earlier when such a series has no primary terms. In this they are right, for it is impossible to complete the infinite. But if there be a limit to the regress, and primaries do exist, they say that these must be unknowable, supposing that they admit of no demonstration, which is the only way of knowing they allow to exist. But if it be impossible to learn these primary principles, one cannot know their results either absolutely or in any proper sense, but only hypothetically, viz. on the assumption that such principles do exist.
The other party agrees with them in holding that knowledge can only be attained by demonstration, but considers that there is nothing to prevent a demonstration of everything being given, maintaining that demonstration may proceed in a circle, all things being proved reciprocally.
We, on the other hand, hold that not every form of knowledge is demonstrative, but that the knowledge of ultimate principles is indemonstrable. The necessity of this fact is obvious, for if one must needs know the antecedent principles and those on which the demonstration rests, and if in this process we at last reach ultimates, these ultimates must necessarily be indemonstrable. Our view then is not only that knowledge exists, but that there is something prior to science by means of which we acquire knowledge of these ultimates. On the other hand it is clear that absolute demonstration cannot proceed in a circle if it be admitted that the demonstration must be drawn from anterior and better known principles than itself; for it is impossible for the same things to be both anterior and posterior in relation to the same objects, except from a different point of view, e.g. some things may be anterior relatively to us and others absolutely anterior, a distinction which inductive proof illustrates. If this be so the definition of absolute knowledge might be considered defective, since it really has a double sense; or that second kind of demonstration drawn from principles better known in relation to us is ambiguous.
Those who hold that demonstration proceeds in a circle not only meet with the difficulty already mentioned, but really say that ‘this is if this is,’—an easy method of proving anything whatsoever. This appears plainly when three terms are assumed (for it is immaterial whether one says that the proof passes through many or few terms before returning to the starting point, as also whether it be through a few or two only). For when:
(for that is how the circular proof proceeds). Let A be placed in the position C held before. Then to say that ‘If B is, A must be,’ is equivalent to saying that C must be, and this proves that ‘If A is, C must be’; and C is here identical with A.
Thus those who hold that the demonstration proceeds in a circle simply declare that if A is, A must be—an easy method of proving anything.
Nor is even this proof possible except in the case of reciprocals such as Properties. It has been already shewn (Prior An. II. 5) that it is never necessary that a conclusion should follow when only one thing is assumed (by ‘one thing’ I mean one term or one proposition); such can only happen when there are at least two antecedent propositions capable of producing a syllogism.
If then A be a consequence of B and of C, and these latter consequences of each other, and also of A, it is possible to prove reciprocally all the questions that can be raised, in the first figure, as has been shewn in the treatise on the Syllogism (Prior An. II. 5). But it has also been shewn that in the other figures no circular demonstration can be effected, or none concerning the premises in question.
Circular demonstration is never admissible in the case of terms not reciprocal. Hence, as few such terms occur in demonstrations, it is clearly useless and untrue to maintain that demonstration consists in proving each term of a series by means of the others, and that consequently everything is demonstrable.
The meaning of ‘Distributive,’ ‘Essential,’ ‘Universal’
Demonstration deals with necessary truths. The definition of ‘distributively true,’ ‘essential,’ ‘universal.’
Now since the object of absolute knowledge can never undergo change, the objects of demonstrative knowledge must be necessary. Knowledge becomes demonstrative when we possess a demonstration of it, and hence demonstration is a conclusion drawn from necessary premises. We must now then state from what premises and conclusions demonstrations may be drawn; and first let us define what we mean by ‘Distributively true,’ ‘Essential’ and ‘Universal.’
By ‘Distributively true,’ I mean a quality which is not merely present in some instances and absent in others, or present at some times and absent at others; e.g. if the quality ‘Animal’ be distributively predicable of man, if it be true to say ‘this is a man,’ it must also be true to say ‘this is an animal’; and if he be the one now, then he must be the other now; so too if ‘Point’ be true of every line. An empirical proof of this is the fact that when the question is raised whether one thing is true of another distributively, our objections take the form of asserting that it is not true of some particular instance or at some particular time.
I. ‘Essential’ qualities are all those which enter into the essence of a thing, (as ‘line’ does into that of ‘triangle,’ and ‘point’ into that of ‘line’; for ‘line’ and ‘point’ belong to the essence of ‘triangle’ and line respectively), and are mentioned in their definition.
II. Essential qualities are, further, attributes of subjects in the definition of which the subject is mentioned, thus ‘Straight’ or ‘Curved’ are essential attributes of ‘Line’; ‘Odd’ or ‘Even’ of ‘Number’; as also ‘Prime’ and ‘Compound,’ ‘Equilateral’ (as 3) and ‘Scalene’ (as 6); in all these cases the things form part of the definition of the real nature of the attributes mentioned, these things being in the first instance ‘Line,’ in the second ‘Number.’ So too in other instances I call attributes which inhere in either of these ways ‘essential,’ while attributes which do not belong to the subject in either of these ways I call ‘accidental’; e.g. ‘Musical’ or ‘White’ as applied to ‘Animal.’
III. Thirdly, essential is that which is not predicated of anything other than itself as attribute of subject; thus if I say, ‘the walking thing,’ some other independent thing is ‘walking’ or is ‘white.’ On the other hand substances and everything which denotes a particular object are not what they are in virtue of being anything else but what they are. Things then which are not predicable of any subject I call ‘essential,’ those which are so predicable ‘accidental’ [in the sense of dependent].
IV. In a fourth sense the attribute which exists in a subject as a result of itself is essential, while that which is not self-caused is accidental. E.g. Suppose lightning to appear while a person is walking. This is accidental, for the lightning is not caused by his walking, but, as we say, ‘it was a coincidence.’ If, however, the attribute be self-caused it is essential: e.g. if someone is wounded and dies, his death is an essential consequence of the wound, since it has been caused by it:—the wound and death are not an accidental coincidence. In the case then of the objects of absolute knowledge, that which is called ‘essential’ in the sense of inhering in the attributes or of having the latter inhering in it is self-caused and necessary; for it must inhere either unconditionally or as one of a pair of contraries, e.g. as either straight or curved inhere in line, odd or even in number. Contrariety consists in either the privation or the contradiction of a quality in the case of homogeneous subjects: e.g. in the case of numbers ‘even’ is that which is not ‘odd,’ in so far as one of these qualities is necessarily present in a subject. Hence, if one of these qualities must be either affirmed or denied, essential attributes are necessary. This then may suffice for the definition of Distributive and Essential.
By ‘Universal’ I mean that which is true of every case of the subject and of the subject essentially and as such. It is clear then that all universal attributes inhere in things necessarily. Now ‘essentially’ and ‘as such’ are identical expressions: e.g. Point and Straight are essential attributes of line, in that they are attributes of it as such. Or again the possession of two right angles is an attribute of triangle as such, for the angles of a triangle are essentially equal to two right angles. The condition of universality is satisfied only when it is proved to be predicable of any member that may be taken at random of the class in question, but of no higher class; e.g. the possession of two right angles is not a universal attribute of figure, for though one may demonstrate of a particular figure that it has two right angles, it cannot be done of any and every figure, nor does the demonstrator make use of any and every figure, for a square is a figure, but its angles are not equal to two right angles. Any and every isosceles triangle has its angles equal to two right angles, but it is not a primary, ‘triangle’ standing yet higher. Thus any primary taken at random which is shewn to have its angles equal to two right angles, or to possess any other quality, is the primary subject of the universal predicate, and it is to that demonstration primarily and essentially applies; to everything else it applies only in a sense. Nor is this quality of having its angles equal to two right angles a universal attribute of isosceles triangle, but is of a wider application.
From what causes mistakes arise with regard to the discovery of the Universal. How they may be avoided
Demonstration must disregard all accidental circumstances, and aim at the discovery of the essential and universal.
We must not fail to notice that mistakes frequently arise from the primary universal not being really demonstrated in the way in which it is thought to be demonstrated. We fall into this mistake firstly when no universal can be found above the particular or particulars: secondly, when such a universal is found applicable to specifically different subjects, but yet has no name; thirdly, when the universal to be demonstrated stands to the true universal in the relation of part to whole.
In this last case the demonstration is indeed applicable to all the particular parts, but will not contain a primary universal. I consider the demonstration to be primary and essential when it is a demonstration of a primary universal. If then it were to be proved that perpendiculars to the same line are parallel, it might be thought that this was the primary subject of the demonstration because it is true in the case of all right angles so formed. This, however, is not the whole truth. The lines are parallel not because each of the angles at their base is a right angle, and consequently equal to the other, but because such angles are in all cases equal to two right angles.
So, too, if there were no other kind of triangle than the isosceles it might be supposed that the quality of possessing angles equal to two right angles was true of the subject as isosceles. Again, the law that proportionals, whether numbers, lines, solids, or periods of time, may be permuted, would be a case, as it used to be proved, viz., of each case separately, though it may really be proved of all together by means of a single demonstration; but since no single designation included magnitudes, times and solids, and since these differ specifically, they were treated of separately. The law is now, however, proved universally. It does not apply to numbers or lines as such, but only because it belongs to the universal conception as such in which all are supposed to be. Hence even if it be proved of equilateral, scalene and isosceles triangles separately, whether by means of the same or by different proofs that every one has its angles equal to two right angles, one will not know except accidentally, that triangle possesses this quality nor will one know it of the universal triangle, even though there is no other sort of triangle than those mentioned. One does not in fact know it of triangle as such, nor yet of every individual triangle, except distributively, nor does one know it of every triangle ideally, even if there is no triangle of which one does not know it.
When, we may ask, is our knowledge not universal and when is it absolute? It is clear that our knowledge of the law would be universal if triangularity and equilateral triangularity were identical in conception. If, however, the two concepts be not identical but diverse, and if the quality in question belong to triangle as such, then a knowledge of the law as relating merely to a particular form of triangle is not universal. Now does this quality belong to triangle as such, or to isosceles triangle as such? Further, what is its essential primary subject? Also, when does the demonstration of this establish anything universal? Clearly when, after the elimination of accidental qualities, the quality to be demonstrated is found to belong to the subject and to no higher subject. For example, the quality of having its angles equal to two right angles will be found to belong to bronze isosceles triangle, but will still be present when the qualities ‘bronze’ and ‘isosceles’ are eliminated; so too, it may be said they will cease to be present when Form or Limit are eliminated. But they are not the first conditions of such disappearance. What then will first produce this result? If it is triangle, the quality of having two right angles belongs to the particular kinds of triangles as a result of its belonging essentially to triangle, and the demonstration in regard to triangle is a universal demonstration.
Demonstration is founded on Necessary and Essential Principles
For necessary conclusions necessary premises are required.
If then demonstrative knowledge be derived from necessary principles (and that which one knows is never contingent), and if the essential attributes of a subject be necessary (and essential attributes either inhere in the definition of the subject, or, in cases where one of a pair of opposites must necessarily be true, have the subjects inhering in their definition), then it is clear that the demonstrative syllogism must proceed from necessary premises Every attribute is predicable either in the way mentioned or accidentally, but accidental attributes are not necessary. We should then either express ourselves as above or lay it down as an elementary principle that demonstration is something necessary, and that if a thing has been demonstrated it can never be other than it is; and consequently that the demonstrative syllogism must proceed from necessary premises. It is indeed possible to syllogize from true premises without demonstrating anything, but not so if the premises be also necessary, for this very necessity is the characteristic of demonstration.
An empirical confirmation of the view that demonstration results from necessary premises is that when we bring forward objections against persons who imagine themselves to be producing a demonstration, we bring our objections in the form ‘There is no necessity.’ Whether we hold that the things in question are really contingent or only considered to be so for the sake of a particular argument. It is clear from this that it is folly to suppose oneself to have made a good choice of scientific principles so long as the premise be generally accepted and also true, after the manner of the sophists who assume that ‘Knowing is identical with possessing knowledge.’ It is not in fact that which is generally accepted or rejected which constitutes a principle, but the primary properties of the genus with which the demonstration deals; nor is everything which is true also appropriate to the conclusion to be demonstrated.
It is also clear from the following considerations that the syllogism can proceed from necessary premises only. If one who, in a case where demonstration is possible, is not acquainted with the cause, can have no real knowledge of the demonstration, then one who knows that A is necessarily predicable of C, whilst B, the middle term by means of which the demonstration has been effected, is not necessary, must be ignorant of the cause of the thing, for in this case the conclusion is not rendered necessary by the middle term; in fact the middle, since it is not necessary, may not exist at all, but the conclusion is necessary.
Moreover if one who now knows (accidentally) the cause of a necessary conclusion remain unchanged while the thing itself remains unchanged, and if, though he has not forgotten it, yet he has no real knowledge of it, then he can never have had any real knowledge of it before. When the middle term is not anything necessary, it may pass away. In such a case, if the man remain unchanged while the thing remains unchanged, he may hold fast the cause of the thing, but he has no real knowledge of the thing itself, nor has he ever had such knowledge. But if the thing denoted by the middle term has not passed away, but yet is capable of doing so, that which results from it is only the possible, not the necessary; and when one’s inference is derived only from the possible one cannot be said to have knowledge in the true sense of the word. When the conclusion is necessary there is nothing to prevent the middle term, by means of which the conclusion was proved from being necessary, for it is possible to infer the necessary from the not necessary, just as one may infer the true from the untrue.
But when the middle term is necessary the conclusion also is necessary, just as true premises always produce a true conclusion. Thus, suppose A to be necessarily predicable of B, and B of C; A then must be necessarily predicable of C. But when the conclusion is not necessary, it is impossible that the middle should be necessary.
Suppose that, Some C is A, and again that All B is A, and that All C is B. But then All C will be A, which is contrary to our original hypothesis.
Since then that which one knows demonstratively must be necessary, it is clear that one ought to obtain the demonstration by means of a necessary middle term. Otherwise one will neither know the cause of the thing demonstrated nor the necessity of its being what it is, but one will either think one knows it without doing so (that is if one suppose to be necessary that which is not necessary), or one will think one knows it in a different way if one knows the fact of the conclusion with the help of middle terms, and when one knows its cause without the help of middle terms. Now there is no demonstrative science of accidents (attributes) which are not essential according to our definition of ‘essential.’ It is not in this connection possible to prove that the conclusion is necessarily true, for the accidental may not be true; (it is of accidents of this kind that I am speaking).
A difficulty might perhaps be raised as to why accidental premises are asked for for the purposes of a conclusion, if the conclusion drawn from them be not necessary; for it might be maintained that it would make no difference if any sort of premise were brought forward and then the conclusion were subjoined. Premises should however be laid down not because the conclusion is necessarily true because of them, but because the person who admits the premises must necessarily admit the conclusion, and his admission will be correct if the premises are true.
Now since only the essential attributes of any genus and those belonging to it as such are necessary, it is clear that scientific demonstrations both deal with and are drawn from essential attributes. As accidental attributes are not necessary one does not require to know the cause of the conclusion, not even if this be an eternal attribute without being essential, as in the case of syllogisms based on universal concomitance. In this latter connection the essential will be known, but not the fact that it is essential, nor yet why it is so. (By ‘knowledge of why it is essential’ I mean ‘knowing its cause.’) In order then to possess knowledge of this sort the middle term must result from the nature of the minor, and the major from the nature of the middle.
The Premises and the Conclusion of a Demonstration must belong to the same genus
Premises must be homogeneous with the conclusion. No transference of premises from one genus to another is valid unless the one is subaltern to the other.
It is not possible to arrive at a demonstration by using for one’s proof a different genus from that of the subject in question; e.g. one cannot demonstrate a geometrical problem by means of arithmetic. There are three elements in demonstrations:—(1) the conclusion which is demonstrated, i.e., an essential attribute of some genus; (2) axioms or self-evident principles from which the proof proceeds; (3) the genus in question whose properties, i.e. essential attributes, are set forth by the demonstrations. Now the axioms which form the grounds of the demonstration may be identical for different genera; but in cases where the genera differ, as do arithmetic and geometry, it is not possible, e.g. to adapt an arithmetical demonstration to attributes of spatial magnitudes, unless such magnitudes happen to be numbers. That such transference is possible in certain connections I will explain later (cf. Chap. IX.).
Arithmetical demonstration is restricted to the genus with which it is properly concerned, and so with other sciences. Hence if a demonstration is to be transferred from one science to another the subjects must be the same either absolutely or in some respect. Otherwise such a transference is clearly impossible, for the extremes and the middle terms must necessarily belong to the same genus, for if not they would not be essentially but only accidentally predicable of the subject.
Hence one cannot shew by means of geometry that opposites are dealt with by a single science nor yet that two cubes when multiplied together produce another cube. Nor can one prove what belongs to one science by means of another except when one is subordinate to the other, as optics are to geometry and harmonics to arithmetic.
Neither is geometry concerned with the question of an attribute of line which does not inhere in it as such, and does not result from the special principles of geometry, as for instance the question whether the straight line is the most beautiful kind of line, or whether the straight line is the opposite of a circumference, for these qualities of beauty and opposition do not belong to line as a result of its particular genus, but because it has some qualities in common with other subjects.
Demonstration is concerned only with what is eternal
The conclusion of a demonstration must be of everlasting application. Perishable things are, strictly speaking, indemonstrable. This applies also to definitions, which are a partial demonstration.
It is clear that if the premises from which the syllogism proceeds are universal, the conclusion of such a demonstration and of demonstration in general must be eternal. There is then no knowledge properly speaking of perishable things, but only accidentally, because the knowledge of perishable things is not universal but under restrictions of time and manner. When this is the case, the minor premise at least must be other than universal and must be perishable:—perishable because then the conclusion will contain a similar element, other than universal because then the predication will apply to some and not others of the subjects in question; so that no universal conclusion can be drawn but only one referring to this or that definite time. The same holds good with regard to definitions, seeing that definition is either the starting point of a demonstration, or itself a demonstration which differs from definition only in the way in which it is expressed or, lastly, in form a conclusion of a demonstration.
Demonstrations and sciences concerning things which occur only frequently (e.g. lunar eclipses) are clearly of everlasting application in so far as they are demonstrations, while in so far as they are not of everlasting application they are particular. As in the case of eclipses so is it with other subjects of the kind.
Demonstration is founded not on general, but on special and indemonstrable principles; nor is it easy to know whether one really possesses knowledge drawn from these principles
All demonstration is derived from special principles, themselves indemonstrable, the knowledge of which, in each genus, is the supreme knowledge on which the whole deduction depends.
Since it is clear that nothing can be demonstrated except from its own elementary principles, that is to say when the thing demonstrated is an essential attribute of the subject, it does not suffice for the possession of knowledge that a thing shall have been demonstrated from true, indemonstrable and ultimate premises. Otherwise demonstrations would be admissible resembling that of Bryson demonstrating the squaring of the circle. Now such arguments demonstrate by means of a common principle which will apply to another science as well, so that the same arguments are of service in other sciences distinct in kind. Thus we have no essential but only an accidental knowledge of the thing, for otherwise the demonstration would not also be applicable to another kind of subjects.
We have more than an accidental knowledge of anything when we see it in the light of its essential nature, after starting from the elementary principles of the things as such. Thus we know the law that a triangle has two right angles when we know of what figure this is an essential attribute and know it after starting from the principles peculiar to Triangle. Hence if the attribute is essentially an attribute of the subject, the middle term of the demonstration must necessarily be included in the same genus, or, if not, one of the genera must be subordinate to the other, as when proportions in harmonics are proved by means of arithmetical premises. Such relations are proved in the same way as in arithmetic, but there is a difference between the two cases, for the question of the Fact falls under the one science (since the subjects of the two sciences differ generically) but the Cause is established by the superior science, to which the properties in question are essential. It is plain even from the case of the subordinate sciences that no absolute demonstration of a thing can be attained save by starting from its own elementary principles. In this case, however, the elementary principles of the sciences in question are not mutually exclusive.
If this be admitted it is also clear that it is impossible to demonstrate the special elementary principles of each science, for the principles of such a demonstration would be the elementary principles of everything, and the science formed by them would be the universal master science; seeing that one who learns a thing through the recognition of higher causes has a better knowledge of it, and the principles through which he learns the thing are anterior when they are causes not themselves produced by any higher cause. If then his knowledge be of this higher kind it must have attained to the highest possible degree, and if this subjective knowledge of his constitute a science, that science must be higher than any other, and in fact the highest science.
The demonstration of one thing is not applicable to another genus except in the case already mentioned, as illustrated by the application of geometrical demonstrations to mechanical or optical, or of arithmetical demonstrations to harmonic theorems.
Now it is hard to decide if we really know a thing or not, for it is hard to decide whether our knowledge is derived from the elementary principles of the subject or not, and it is in this that knowledge consists. We imagine that, if we possess a syllogism drawn from true and primary premises, we really possess knowledge. This, however, is not the case, for the conclusions should belong to the same genus as the primary principles.
The Definition and Division of Principles
Such indemonstrable principles may be either peculiar to each science or common to several sciences, though common only by analogy. All demonstration involves three things:—the object demonstrated, common axioms or principles, and the special modifications or properties of the subject genus. The distinction between Hypothesis and Petition.
I mean by the elementary principles in each genus those whose existence it is not possible to prove. Now the meaning of the primary principles and that of their consequences are assumed; the existence of the elementary principles must also be assumed, that of everything else proved. For instance the meaning of Unit, or Straight, or Triangle must be assumed, that Unit and Magnitude exist must also be assumed, everything else must be proved.
Of the principles employed in demonstrative science some are peculiar to each science, others are common to all, i.e. common in the sense of analogous, since their use is confined to each genus as comprehended by a particular science. Principles peculiar to one science are such as the proposition ‘Line, or Straight, is of such and such a nature;’ common principles are such as, ‘If one take equals from equals the remainders are equal.’ Each of these principles is taken as applicable to all cases belonging to the particular genus; for its results will be the same whether it be treated universally or only particularly, e.g. in geometry to spatial magnitudes or in arithmetic to numbers.
Those principles too are peculiar whose existence is assumed not demonstrated, namely those whose essential attributes are investigated by the science; as arithmetic investigates units, geometry points and lines, for these sciences assume that the thing in question exists, and that it is identical with some particular object. They likewise assume the meaning of the essential attributes of the thing, as arithmetic assumes the meaning of Odd, Even, Square or Cube, and geometry that of Incommensurable, and Inclined or at an Angle, while the existence of these qualities is shewn by means of the common principles and the conclusions already demonstrated. The same thing applies to astronomy.
In short in every demonstrative science there are three elements: (1) the things whose existence it assumes, namely the subject or genus, the essential attributes of which are investigated by the science; (2) what are called ‘Common Axioms’ which the demonstration uses as its primary principles; and (3) Properties, the meaning of which is assumed.
However nothing prevents some sciences from overlooking one or other of these elements; e.g. a science may not expressly assume the existence of the subject genus if this be self-evident (for the existence of Number is more obvious than that of Cold or Heat), or it may not assume the meaning of the properties if it is obvious, just as in the case of their common principles the sciences do not assume the meaning of ‘taking equals from equals,’ because this is known. None the less however there are naturally these three elements in a science:—the subject of proof, the things proved and the grounds of proof.
That which must needs exist and must necessarily be supposed to exist is neither Hypothesis nor Petition but Axiom. Demonstration is not concerned with the outward expression of an idea but with its inner significance, for that is the case with syllogism in general, and one may always raise objections to the external expression but not always to the inner significance.
Everything which, being capable of proof, is assumed without being proved, if admitted by the learner is a Hypothesis, which hypothesis is not an absolute hypothesis but only one with reference to the person who accepts it.
If however something be assumed with regard to which the learner has no opinion or a contrary one it is a Petition. This then is the difference between hypothesis and petition; petition being that which is somewhat opposed to the learners opinion, or, in a wider sense, whatever, though capable of demonstration, is assumed and employed without any proof.
Definitions are not hypotheses, since it is not asserted that their subjects do or do not exist. Hypotheses are formulated as propositions, Definitions require only to be understood, and no Hypothesis consists in that alone, unless it be maintained that mere Hearing is a Hypothesis. Hypotheses are the premises from the existence of which the conclusion is inferred.
The hypotheses of the geometrician are not, as some assert, false, saying that, though one ought not to make use of false propositions, yet the geometrician calls a line a foot long which is not a foot long, or declares that he has drawn a straight line, though the line is really not straight. The geometrician in reality draws no conclusion from the fact of the particular line that he draws actually possessing the quality which he names, but from the existence of the things which that line represents.
Moreover all postulates and hypotheses are universal or particular, definitions are neither.
On certain Principles which are common to all Sciences
[ The possibility of Demonstratiou presupposes the validity of universal predicates, but does not require Platonic ideas]. The ‘Common Axioms’ are expressly formulated in exceptional cases. They connect the sciences with one another, and with Dialectic and Metaphysics, thus giving unity to all forms of true Thought.
[It does not follow, if demonstration is to exist, that there must be Ideas or a Unity outside the many individual things, but it does follow that some unity must be truly predicable of the many. If no such unity existed we should have no universal; and without a universal there could be no middle term and consequently no demonstration. Since demonstration does exist there must be some self-identical unity, a real and no mere nominal unity, predicable of many individual things.] No demonstration lays down that it is impossible both to affirm and to deny a quality of a thing at the same time, unless it is necessary to present the conclusion in a corresponding form by the help of that axiom. In that event the conclusion is proved by our assuming that the major is predicable of the middle term, and that to deny the major of the middle is untrue. It makes no difference if the thing denoted by the middle be assumed to exist or to be non-existent, and the same applies to the thing denoted by the minor. If it be granted that Man is such and such; i.e. if, though Not-man be also such and such, it be simply granted that man is animal and not not-animal; then Callias [being man] will be animal and not not-animal, even though not-Callias be also man. The reason of this is that the major is not only predicated of the middle but of something else outside it, because it has a wider application, so that it makes no difference to the conclusion whether the middle be an affirmative or a negative expression.
Demonstration by means of reduction to absurdity assumes the truth of the law ‘everything may be either affirmed or denied of a subject,’ and this not always in a universal sense but simply to the extent required, namely so as to be applicable to the particular genus in question. I mean by ‘applying to the genus,’ that genus with which one’s demonstration is concerned, as has been remarked above. (Chap. X.).
All sciences overlap as far as their common principles are concerned. (By these I mean the principles used by them as the grounds of demonstration, not the subjects of the demonstration nor yet the thing demonstrated). Now dialectic is common to all the sciences, and if one were to try and give a universal proof of the common principles of science, such as ‘Everything can be either affirmed or denied,’ or ‘if equals be taken from equals,’ or some maxim of that kind [the resulting science would similarly be common to all sciences]. But dialectic does not deal with any definite objects of this sort nor with any single genus. Otherwise it would not have used the interrogative form, for this cannot be employed for purposes of demonstration; since the same thing cannot be proved from opposite propositions. This has been proved in the treatise on the syllogism. (Prior An. II. 15).
On Questions, and, in passing, on the way in which Sciences are extended
Corresponding to the special principles of a science are special questions which must not be transferred from one genus to another, so that no discussion of a science with persons ignorant of it can lead to valid results. Two kinds of opposites to a science exist:—questions or demonstrations entirely outside its range and those which involve a breach of some of its laws.
If a syllogistic question be the same as one of the members of an alternative, and if there be premises in each science from which the syllogism belonging specially to each science may be deduced, there must be some scientific question from which the special syllogism corresponding to each science is derived.
It is plain then that not every question can be a geometrical or a medical question, and similarly with all other special sciences, but only those questions can be geometrical proceeding from which some of the matters connected with geometry are proved, or something proved on the same principles as geometry; e.g. optical theorems. The same is the case with other sciences. Now with regard to these questions, in the case of geometry they must be explained in accordance with the principles and conclusions of geometry, but no account need be given of the principles themselves by the geometrician as such, and this applies to other sciences also.
One should not then ask every possible question of a person acquainted with a particular science, nor need he answer every question asked of him, but only a question concerning the definite subject of the particular science. If one enter into a discussion with a geometrician as such, it is clear that the proof he gives will be a sound one if drawn according to these principles, otherwise unsound. It is also clear that in such circumstances one cannot confute a geometrician except accidentally, so that we must not discuss geometry before persons ignorant of that science, for any unsound arguments put forward will remain unnoticed. The same is the case with other sciences.
Since then there are geometrical questions, it may be asked whether there are also ungeometrical, and what kind of ignorance in connection with each science causes certain questions to bear the same relation to that science as ungeometrical bear to geometrical questions. Further is a syllogism resting on ignorance a syllogism formed from premises which contradict the science it belongs to, or rather a fallacy which nevertheless does belong to the science in question, e.g. geometry? Or, again, is a question belonging to another pursuit, such as a musical question, ungeometrical as regards geometry? Again, is the supposition that parallel lines can meet in one sense geometrical and from another point of view ungeometrical? ‘Ungeometrical’ is in fact an ambiguous expression, as is ‘unrhythmical.’ One thing may be ungeometrical or unrhythmical from not possessing the quality in question at all, another from having it defectively. So too the form of ignorance resulting from bad or defective principles is contrary to Science. In mathematical sciences the fallacy is more easily perceived than in other sciences, because in them the middle term is always expressed twice, something being predicated distributively of the middle term, and the latter in turn predicated distributively of another subject. The predicate is not however used distributively. In mathematics one may, as it were, see by an immediate act of thought the relations of the middle term, while in words they remain unnoticed. E. g. as regards the question, ‘Is every circle a figure?’ If one describe a circle on paper it clearly is so. If the conclusion be drawn ‘then the epic cycle is a figure,’ this is clearly untrue.
No objection should be raised to a science on the ground that its premises are inductive, for just as nothing can be a premise which does not apply to several instances (otherwise it would not be universally predicable, and Syllogism is drawn from universals), so an objection must have a universal application. Premises and the objections to them correspond to one another, and any objection one urges against a premise should be capable of serving either as a demonstrative or as a dialectical premise.
The laws of the syllogism are violated when the common attribute of both major and minor terms is treated as their predicate. An instance is the syllogism of Caeneus that ‘fire increases in geometrical proportion’; ‘for,’ as he says, ‘fire increases rapidly and so does geometrical proportion.’ No syllogism can, however, be formed thus. The truth is: if the proportion which increases most quickly in respect to quantity be the geometrical, and if fire be that which increases most quickly in respect to motion . . . .
Thus it is sometimes impossible to draw a conclusion from two premises of this kind, at other times it is possible, though the possibility may not be observed. If it were impossible to draw any true conclusion from false premises, it would be easy to bring the syllogism to a conclusion, for it would necessarily be convertible. For instance let A exist by hypothesis, and when A exists let something else (B for instance) exist also, which one knows in this instance does exist. By conversion then it may be shewn from B that A exists. Conversion is more frequent in pure mathematics because these admit of no accidental qualities (and in this differ from dialectical arguments) but only of definitions.
Mathematical science is advanced not by the use of a number of middle terms, but by the subsumption of one term under another (as A under B, B under C, C under D, and so to infinity). The process may also take two directions, A being predicable both of C and E. Suppose A represents any number definite or indefinite.
B any odd number of definite magnitude.
C any odd number whatsoever.
(Then A will be seen to be predicable of C).
Again:— Let D be an even definite number.
E any even number whatsoever.
Then A is predicable of E.
The difference between the Demonstration and Science of a thing’s Nature and those of its Cause
There are two classes of demonstration, one giving the Fact, the other the Cause of the fact; such demonstrations being effected either by the same or separate sciences. If the former, the propositions may be immediate and convertible, when we have the demonstration of the cause, or mediate and inconvertible, when we have only the demonstration of the fact. If different sciences are employed, and one is subordinate to the other, the superior gives the Cause, the inferior the Fact.
A difference exists between knowing that a fact is and knowing its cause. This may be considered firstly in connection with the same science and from two points of view, viz. (1) in the case where the syllogism is not deduced from ultimate propositions (for here the primary cause is not expressed, while knowledge of the cause goes back to the primary cause). (2) The second aspect of the distinction is seen when the propositions from which the conclusion is drawn are ultimate and reciprocal, but the middle employed is not the cause but the better known effect. Nothing in fact prevents in the case of reciprocating terms, that term which is not the cause being better known to us, so that our demonstration will be through this as a middle. E. g. Planets are proved to be near the earth from the fact that they do not twinkle, as follows. Let
C designate Planets.
B Not twinkling.
A Being near.
Here B may rightly be predicated of C, for planets do not twinkle. Also A is true of B, for that which does not twinkle is near,—a truth to be arrived at by induction or observation. A then must be true of C, so that we have now demonstrated that the planets are near.
This syllogism then does not deal with the cause of the phenomenon but with the fact; for the planets are not really near because they do not twinkle, but do not twinkle because they are near. It is also possible to prove the first fact by means of the second, and the demonstration will then be of the cause. Thus:—
Let C be the Planets.
B Being near.
A Not twinkling.
Here B is true of C, and A (‘not twinkling’) of B. Therefore A is true of C. Thus the syllogism is a syllogism of the cause, for it comprehends the primary cause. Another instance is the method by which the moon is proved to be spherical by a reference to its regular increases. It proceeds thus:—If that which increases in this particular way be spherical, and if the moon do so increase, it is clear that the moon is spherical. As thus expressed the syllogism demonstrates only the fact, but when the middle term is transposed it is a demonstration of the cause. The moon is not spherical in consequence of its increases, but undergoes these particular increases because it is spherical. Let C be the Moon; B spherical form; A the method of increase. In cases, however, where the middle terms are not interchangeable, and where the effect is better known than the cause, the fact may be proved but not the cause. This is also the case when the middle term is wider than the other two terms. Here too the demonstration is of the fact, not of the cause, for the primary cause is not stated. E. g. To the question ‘why does not a wall breathe’? suppose the answer to be given ‘because it is not an animal.’ Now if this negative quality be the cause of its not breathing, the corresponding affirmative ‘is an animal’ ought to be the cause of this phenomenon, just as granting that a negation of a quality be the cause why something does not exist, the affirmation of it is the cause why it does exist. E. g. If a want of balance between heat and cold be the cause of the absence of health, a due balance between them must be the cause of its presence. So conversely, if the affirmation be the cause of the presence of a quality the negation is the cause of its absence. But in the first instance quoted this does not hold good. Not every animal in fact does breathe. The syllogism which demonstrates a cause of this kind belongs to the second figure. E. g. Let A be Animal; B Breathing; C Wall. Now A is true of all B (for everything which breathes is an animal), but of no C. Hence B is true of no C, and therefore no wall breathes. Such statements of cause resemble hyperbolical expressions, for one is guilty of a kind of hyperbole if one depart from the proximate cause and take the more remote as one’s middle term. Of such a nature is the inference of Anacharsis that the Scythians have no flute-players because they have no vines.
Such are the differences between the syllogism of the fact and that of the cause, as regards the same science and the position of the middle terms; but from another point of view the fact sometimes differs from the cause in that each is examined by a different science. This is the case when the sciences are of such a nature that one is subordinate to the other, as optics to geometry, mechanics to the measurement of solids, harmonics to arithmetic and records of observation to astronomy. Some of these subordinate sciences have almost similar names; e.g. mathematical and nautical astronomy, mathematical and acoustic harmonics. In these cases the fact depends on the observational sciences, the cause on the mathematical sciences; for the mathematician can demonstrate the causes though he often does not know the fact, just as those who are aware of a universal law, through want of observation, are often ignorant of some of the particular facts. These superior sciences will be such as differ in essence from the subordinate sciences, and deal merely with abstract forms. Thus mathematics are concerned with forms, and do not deal with any concrete subject; and even if the propositions of geometry happen to be true of a concrete subject they are true of it not as concrete. Now there is a science which bears the same relation to optics as optics to geometry; e.g. knowledge about the rainbow. The fact that there is such a thing falls within the province of the natural philosopher, the cause within that of the optician, either as such or in so far as he is a mathematician.
Many sciences which are not subordinate one to another, yet sometimes have similar interrelations: e.g. medicine and geometry. Thus the fact that circular wounds heal more slowly must be learned by the surgeon, the cause of it by the geometrician.
The figure proper to Demonstrate Syllogism
The first figure of the syllogism is the most scientific, being the most suitable for the attainment of the cause. Further it alone can examine into the simple fact which must be both affirmative and universal. The other two figures reinforce their demonstrations by an appeal to the first figure, the latter never makes use of them.
Of the figures of the syllogism the most proper for scientific demonstration is the first, for mathematical sciences, such as arithmetic, geometry and optics, and generally speaking all sciences which investigate the cause of things, effect their demonstration by its means. The demonstration of the cause is in fact carried out either exclusively or generally and in most cases by means of this figure, so that in this respect also it appears to be the most proper for science, seeing that the examination of the cause is the most important element in knowing. Further, the knowledge of what a thing is can only be attained by means of this figure, for in the second figure no affirmative conclusion is produced, and the knowledge of what a thing is involves affirmation. In the third figure there are indeed affirmative conclusions, but not universal ones, and the knowledge of what a thing is is of the character of a universal; thus, ‘two-footed’ is true of man universally and without restriction. Moreover the first figure has no need of the assistance of the two other figures, while these latter are strengthened and extended by means of the first until they arrive at ultimate principles. It is clear then that the first figure is the most important instrument of scientific knowledge.
On immediate negative propositions
Yet demonstration is possible in the other figures, and if of a negative character is as valid in the second figure as in the first.
Just as the quality A may inhere in B without the intervention of a middle term, so it may not inhere without such intervention. By these expressions I mean that there is no middle term connecting A and B. In that case inherence and non-inherence will no longer depend on the presence of a third term. When then either A or B, or both, are true of the whole of a third term, it is impossible that A should not be true of B immediately. We may suppose all C to be A. Then if all C is not B (for it is possible that all of a subject should be A, but none of it B) the conclusion will follow that B is not A. For if all A is C, and no B is C, then no B is A.
The same proof will be adopted if both terms are distributively predicable of a third. That B need not be predicable of a subject of which A is distributively predicable, and conversely that A need not be predicable of a third term of which B is distributively predicable may be seen clearly from a consideration of those series of terms wherein no term of the one series can be interchanged with one in the other series. Thus if none of the terms in the series A, C, D are predicable of any in the series B, E, F; if further A is distributively predicable of G, a term belonging to the same series, then it is clear that no G will be B, for otherwise these distinct series would have interchangeable terms. So too if B is distributively predicable of some other subject. If, however, neither A nor B is distributively predicable of any third term, and if A is not predicable of B, A must be not predicable of B immediately. This is so because if any middle term were present, one of the two terms named would have to be distributively predicable of a third term, since the syllogism must be either in the first or the second figure. Now if it be in the first, B will be distributively predicable of a third term, for in this case the premise must be affirmative; if it be in the second A or B may be distributively predicable of a third term, for when either premise is of a negative character a conclusion may be attained, though this is impossible when both premises are negative.
It is plain therefore that one term may be proved to be deniable of another immediately, and we have now shewn when and how this may happen.
On ignorance resulting from a defective arrangement of terms in mediate propositions
Concerning ignorance and error; firstly in the case where two terms are predicated of one another immediately.
That ignorance which results not from the simple absence of knowledge but from a faulty arrangement of terms is a logical deception which, in cases where one thing is predicable or not predicable of one another immediately, takes two forms, (1) an immediate supposition that one thing is or is not predicable of another, (2) a supposition to this effect arrived at through a syllogism. Now in the case of the simple or immediate supposition the mistake is simple, in the case of that which is produced by the syllogism it may assume several forms. Suppose it to be proved immediately that no B is A; then if one conclude, with the help of a middle term C, that B is A, one’s reasoning will have led one astray. Here it is possible for both premises to be false or else for only one. Thus if no C be A, and no B be C, and if each of these premises be transposed, both will be false. It is in fact possible for C to be so placed with regard to A and B that it is neither included in A nor is universally predicable of B. Now B cannot be true of another term distributively, since the hypothesis was that A was not immediately predicable of C, and there is no necessity why A should be universally predicable of all C, so that here both premises are false. Further one of the premises may be true, not however either of the two, but only AC; for the premise CB will be always false, because C is predicable of no part of B. The premise AC may however be true, as when both C and B are shewn to be immediately predicable of A. For when the same thing is predicated primarily of more than one term, no one of these latter will be predicable of another. Nor does it affect the case if A be shewn to be predicable of C not immediately (but by means of a term taken from a higher class). Only in the case of premises such as these and only in this manner can mistakes arise in connection with predicating one term of another, for no syllogism in another figure can prove universal predication.
Mistakes connected with the proof that one term is not predicable of another may however occur in either the first or the second figure.
We will first mention in how many ways this may happen in the first figure, and what the position of the premises must then be.
For instance suppose A to be immediately predicable of B and C. Then if one take as premises ‘No C is A,’ and ‘all B is C,’ the premises will be false. A mistake will also follow if only one of the premises, either of the two, be false. It is possible for the premise AC to be true, BC false, AC being true because A is not distributively predicable of C, BC false because it is impossible for C to be B when no A is C, for then the premise AC would no longer be true. When however both premises are true the conclusion also will be true. Further the premise BC may be true while the other is false; for instance in the case where both C and A are B; since one of these terms must be included in the other. Hence if one assert that no C is A, the premise will be false. It is clear then that the conclusion will be false if one or both of the premises be false.
In the second figure it is not possible for both the premises to be entirely false; for when all B is A no third term can be found which will be predicable of the whole of one and not predicable of any part of the other term. If one want a syllogism at all one ought to select the premises in such a way that the middle term will be affirmed of one of the other two terms and denied of the second. If then, when thus stated, the premises are false, it is clear that the contrary of them will be true. This however is impossible1 , though nothing prevents each of the premises from being partially false when the conclusion is false, as in the case where some of A and also of B are C, while it is asserted that all A is C and no B is C. Here the two premises are false, not however entirely but only partially false. The same thing will happen when the position of the negative premise is changed1 . It is also possible in the second figure, for one premise, either of the two, to be false. Suppose that what all A is, B will be also. If then it be asserted that all A is C, and no B is C, the premise AC will be true, BC false. Again that which is predicable of no B will be predicable of no A, for if a thing be true of A it will be true also of B, but the hypothesis was that it was not true of A. If then it be asserted that all A is C, and no B is C, the latter premise will be true, the former false. Similarly if the negative premise be reversed, that which is predicable of no A will be predicable of no B. If then it be asserted that no A is C and all B is C, the former premise will be true, the latter false. Again, to assert that what is predicable of all B is predicable of no A is false, for a term which is predicable of all B must be predicable of some A. If then it be asserted that all B is C and no A is C, the former premise will be true, the latter false. It is clear then that whether both the premises are false or only one of them, an atomic or elementary error will attach to the resulting conclusion.
On ignorance resulting from a defective arrangement of terms in immediate propositions
Secondly concerning logical errors arising when two terms are connected by a common middle term.
In cases where one term is predicated or denied of another not immediately but by means of a middle term, when the conclusion is attained by the help of the proper middle term wrongly expressed, both premises cannot be false, but only the premise containing the major term. By the ‘proper middle term’ I mean that by which the syllogism which contradicts the opposite conclusion may be attained. Suppose that it be shewn by means of the middle C that B is A. Here, since if a conclusion is to be attained at all the premise CB must be affirmative, it is clear that this same premise will always be true, that is it can never he converted into a negative; but the premise AC will be false, for when this is converted the opposite conclusion will prove true. The same is the case if the middle be taken from another series of terms. Let D be such a term. Now if D inhere in all of A and be distributely predicable of B the premise BD must remain unchanged, while the other, major, premise must be converted to a negative form. Hence the former premise will be always true, the latter, or major, false. Generally speaking this sort of fallacious argument will be the same as that already mentioned where the proper middle term is employed.
But if the conclusion be not attained by means of the proper middle term, when the middle term used is included in A but is not predicable of any of B, both the premises must be false. Here the premises must be converted into their contrary if any conclusion is to be drawn from them. If their form remain unaltered they must both be false. E.g. If all D be A, but no B be D.
If these premises be converted into their contrary a conclusion will follow and both premises will be false.
But when the middle term (e.g. D) is not included in A the premise AD will be true, BD false. For AD is true because D is not included in A, DB is false because otherwise the conclusion also would be true, and the hypothesis was that the conclusion is false.
When a fallacious argument occurs in the second figure it is not possible for both the premises to be false in their entirety. When B is included in A no term can be predicable of the whole of the one and of none of the other, as has been remarked above (Chap. XVI). On the other hand one of the premises, either of the two, may be false. For instance, supposing that both A and B are C, if it be asserted that C is A, but C is not B, the premise CA will be true, the other premise false. Again if it were asserted that B is C, but A is not C, the premise CB will be true, the other premise false. We have now shewn when and from what premises the fallacy is produced if the fallacious syllogism be negative. If it be affirmative it is impossible, when the proper middle term is used, for both premises to be false, since, as was said before, if a conclusion is to be attained the premise CB must remain unaltered. Consequently the premise CA will always be false, for that is the one which is converted into a negative. The like is the case if the middle be taken from a different series of terms, as was remarked in connection with the negative fallacy. Here the premise DB must remain unaltered, while AD must be converted, and the fallacy is the same as the preceding. When however the proper middle is not used, if D be included in A the major premise containing those terms will be true, the other will be false. It is in fact possible that A should be predicable of several terms, no one of which is included under another. But if D be not included in A the premise containing them must clearly be false, for it is expressed affirmatively. The premise BD on the contrary may be either true or false; for it is quite possible for no D to be A while all B is D:—thus ‘no science is animal,’ but ‘all music is science.’ So too no D may be A, and no B may be D.
It is plain then that, when the middle term is not included in A, both or either of the premises may be false. It is now therefore possible to see in how many ways and from what causes syllogistic fallacies may arise, both in the case of immediate assertions and of those attained mediately through demonstration.
On ignorance as resulting from defective sense perception
Ignorance is the result of a defect in sense. Universals can only be attained by the help of Induction. Induction however depends on Sensation, the objects of which are particulars, of which no science is possible. Consequently Induction is necessary for the conversion of Sensation into Scientific knowledge.
It is also clear that if some branch of our perceptive faculties prove deficient the corresponding branch of science, which cannot be attained without those faculties, must fail also; that is to say if it be agreed that we must acquire knowledge either through induction or demonstration. Now although demonstration proceeds from universals and induction from particulars, it is impossible to attain to the knowledge of universals except by means of induction. Even the matter of the abstract sciences may be established through induction, since some qualities belong peculiarly to each class of thing and make them what they are, even though these qualities are not really separable from the things themselves. Induction without the power of perception is impossible, for perception is concerned with particulars, which cannot be grasped at all by means of science. The reason of this is that we cannot attain to universals without induction, nor use induction without sense perception.
Whether the Principles of Demonstration are finite or infinite
Syllogisms being either affirmative or negative, are the attributes of a subject and the subjects of an attribute limited or unlimited in number? Further, can an infinity of middle terms exist between two given extremes?
Every syllogism proceeds by means of three terms. The aim of one, the affirmative, class is to shew that C is A, because B is A and C is B; the negative syllogism has as one of its premises the proposition stating that one term is true of another, as its second that one term is not true of another.
It is clear then that these premises constitute the principles of demonstration and are what are called its hypotheses. When the premises have been expressed in this form the conclusion must follow; e. g. C is proved to be A by means of B, or again B is proved to be A by means of some other middle term, and similarly C is proved to be B.
It is plain therefore that if inferences depend on opinion and are merely dialectical the only thing the logician need keep in view is that the premises of his syllogism should be as generally recognized as possible. Hence if a middle term between A and B really exist, but is thought not to be so, an inference drawn according to the received opinion will be a dialectical inference; but in order to draw universally true inferences one should look to that which really is, not that which is thought to be. Of the former character is a term predicated of other terms essentially not accidentally. By ‘accidentally’ I mean after the manner in which we sometimes say ‘that white thing is a man,’ which is not the same as when we say ‘the man is white.’ In the latter case the man is not white because he is something else, but simply because he is man; in the former proposition whiteness is predicated as an accidental attribute of the man.
Now some things are of such a nature that they may be predicated essentially. Suppose a term C, which is such that it is not predicable of any other term, while B is immediately predicable of it. Further let E be predicable of F, and F of B. Now must this process terminate or can it proceed indefinitely? Again, if nothing be predicable of A essentially, but A be immediately predicable of H and of no prior term, must this process also terminate or can it also continue indefinitely?
This case differs from the one last mentioned, inasmuch as that amounts to asking whether it is possible, when one begins with a term which cannot be predicated of anything else while another term may be predicated of it, to advance upwards along an illimitable series? The other signifies, ‘can one, when starting with a term which is predicated of another term while no other is predicated of it, proceed downwards along an infinite series’? Also, can the intervening terms be infinite when the major and minor are definite? Thus, if C be A, and the middle term between them be B, while other terms exist between B and A, and still more between these others, can these middle terms be continued to infinity, or is that impossible? This enquiry is identical with the question whether demonstrations are illimitable, whether everything is capable of demonstration or whether the process must terminate in both directions. The same questions may, I consider, be asked concerning negative syllogisms and premises. Suppose that no B is, at least immediately, A, will there be then any intervening term, of which A is also, not predicable, prior to B? Suppose such an intervening term to be G, which is predicable of all B, and suppose another term prior to this, as H, which is predicable of all G. In these cases there is either an infinite series of terms of which A is denied antecedently, or there is a limit at which the series terminates. This does not, however, apply to reciprocally predicable terms, for here all the terms bear the same relation to one another, whether only the attributes are limitless, or both attribute and subjects, except where the reciprocation is effected in a different manner, so that the attribute is now predicated as essential and again as accidental.
Middle terms are not infinite
Middle terms cannot be limitless; otherwise the subject and attribute could never be brought into the relation demanded by the syllogism. Attribution also is limited both in the direction of the general and of the particular.
That the intervening terms of a predication cannot be infinite if predications terminate both in an upward and a downward direction is obvious. [I mean by ‘upward’ that which is more in the direction of universal, by ‘downward’ that which is nearer to the particular]. For if, when A is predicated of F, the intervening terms (here designated as B) could be infinite, it is clear that if one proceeded from A in the direction of the particular one could continue to predicate one term of another to infinity, [the terms intervening between A and F are here regarded as infinite]; and similarly, if one proceeded from F in the direction of the more general, one would traverse an infinite number of terms before arriving at A. If, however, there can be no such infinite progress or regress, the terms intervening between A and F cannot be infinite.
It is of no avail to maintain that some of the intervening terms, say A, B, C, follow one another so closely as to admit of no further intervening term, while others of the series are not so closely connected. For whichever of the B’s I care to select must have a certain relation to A or to F, and the intervening terms must be finite or infinite. To enquire from what starting point one begins the process to infinity, and whether this process is mediate or immediate is not to the purpose, for everything which follows any given point must be looked on as limitless.
In Negations some final and ultimate point is reached where the series must cease
If the series terminate in the case of affirmative demonstration, it will do so in negative demonstration. It will be found that demonstration may be carried out in various figures, but that the methods are limited in number so that the demonstrations are limited also. In every figure a primary or ultimate is reached of which the attribute is predicable, though the ultimate is not predicable of the attribute.
The process will also clearly terminate in the case of negative demonstration, if it be admitted that an upward and a downward limit are reached in affirmative demonstration. Suppose it to be impossible to proceed to infinity when starting from the last term and advancing upwards, (by the ‘last term’ I mean that which is not predicable of any other term, though some other term, e.g. F, may be predicable of it), and impossible also to proceed from the first term to the last, (by the ‘first term’ I mean that which is predicable of another term though no other is predicable of it). If this supposition be correct then the process of negative demonstration will also terminate. Negation is proved in three ways: (1) According to the first figure: all C is B, but no B is A. Then from the premise CB and from any minor premise whatsoever one must proceed to ultimate knowledge, for such a premise as this is affirmative. As to the major premise it is clear that when the major term is not predicable of another term (such as D) prior to the middle, this term must be distributively predicable of B. Again, if the major term be not predicable of another term prior to D, that other term must be distributively predicable of D. Hence, since the process of demonstration terminates in the direction of the universal it will do so likewise in that of the particular, and there will be some primary term of which the major (A) is not predicable immediately. (2) In the second figure: if all A be B, and no C be B, then no C is A. If a demonstration of this be required it may clearly be proved either by the method just mentioned, or by our present method or by the third method. The method adopted in the first figure has already been explained, so I will now explain the second. The system of proof is as follows. Suppose that all B is D and no C is D, while something must be predicable of B. If it be proved that C is not D, some other term which is not predicable of C must be predicable of D. Hence, since predication, as it advances continually to the next highest term, must terminate at some point, negation will similarly terminate. (3) The third method is as follows. If all B be A, but no B be C, C will not be predicable of everything of which A is predicable. This, again, may be proved by the two methods already mentioned, or according to our present method. We have shewn that the process must terminate if the two former methods be adopted. If we use the third figure we will thus state the premises. All E is B, but some E is C. Here the major premise, some E is not C, may be proved in the same way as before. Since our hypothesis was that the process terminates in the direction of the particular, it is now clear that negative demonstration (in this case the negation of C) will also terminate. It is plain, too, that the process will terminate in every case, even if the proof adopt not one method alone, but all three, according to the first, the second, or the third figure. All these three methods are definite, and that which is brought to a definite end in a definite manner must itself be definite. Granting then that the process of affirmative demonstration terminates, that of negative demonstration must do so likewise.
In Affirmations some final and ultimate point is reached where the series must cease
In the case of essential attributes, the attributions may easily be seen to be limited in number, so that the demonstrations of them are limited also. The mind cannot traverse an infinity, and as Substance, for instance, is definable, its attributes must be limited. In other words demonstration is applicable only to Essentials (καθ’ αὑτά) which cannot be unlimited, for that would render definition impossible. As it is possible, the attributes are limited. Hence demonstration possesses certain principles which are not themselves capable of any demonstration.
That affirmative demonstration terminates at a certain point may be proved dialectically as follows. It clearly terminates in the case of predications concerning the essence of a thing, for if the essential attributes can be defined and are knowable, and if one cannot reach to the end of the infinite, predications of essential attributes must needs have some limit. To give a general turn to the statement we may express ourselves thus. It is equally possible to say with truth that ‘this white thing is walking’ and ‘that great thing is a stick,’ or again ‘the stick is great’ and ‘the man is walking,’ but there is a difference between the two pairs of expressions. In saying ‘the white thing is a stick,’ I mean ‘that which has the accidental quality of whiteness is a stick,’ not that ‘the white thing’ is the subject of which ‘stick’ is the predicate. It is in fact a stick not because it is white nor from being essentially white, so that ‘this white thing’ is only accidentally a stick. But when I say ‘the stick is white,’ I do not mean that another thing distinct from stick is white, and that stick is an accidental quality of it; (as e.g. when I say ‘the musician is white;’ for in that case I mean that the man, who has the accidental quality of being a musician, is white) but the stick is the subject which is white without being, as a result of that, anything else than the genus or a species of ‘stick.’ Thus if we are to provide separate designations for the two methods, the latter form of expression may be called the ‘predication of attributes,’ the former either not predication at all or accidental, not absolute, predication. In the first case ‘white’ is the attribute,’ ‘stick’ that of which the attribute is predicated.
We may now lay down the rule that the attribute is always predicated of its subject absolutely, not accidentally, for that is how demonstrations are able to effect proof. Hence when one thing is predicated as an attribute of another it concerns Substance, Quality, Quantity, Relation, Action, Passion, Place or Time. Moreover that which denotes a substance denotes either the Genus or the Species of the thing of which the attributes are predicated, but that which does not denote a substance, but is predicated of another subject without being either the Genus or the Species of that subject, is an accident: e.g. White as predicated of Man; for ‘man’ neither belongs to the genus ‘white,’ nor is he a species of it. He should rather be called ‘animal,’ for man is a species of animal.
Everything which does not denote substance must be affirmed of some subject as an attribute, and nothing can be (e.g.) white, in the sense that it is simply white, without being at the same time something else besides. We may at once dismiss Ideas; they are mere empty names, and if they do exist cannot concern our argument, for demonstrations deal only with subjects such as we have already mentioned.
Further if one thing be not an attribute of another nor yet the latter an attribute of the former, and if no attribute of an attribute can exist, the two terms in question cannot be reciprocally predicable as attributes. One of them may be correctly predicable of the other, but each cannot really be predicable of the other, for one would have to be predicated as a substance, as if it were a genus or differentia of the attribute. It has however been proved that these attributions cannot be continued to infinity, either in the direction of the universal or of the particular. Take the proposition ‘Man is a biped, this again an animal, while animal belongs to some other genus.’ Nor can the process be infinite when ‘animal’ is predicated of ‘man,’ ‘man’ of ‘Callias,’ and ‘Callias’ of an individual definite man who is Callias. It is indeed possible to define every substance of this sort, but one cannot even in thought complete the infinite. Hence one cannot arrive at the infinite, either in the direction of the universal or of the particular, for one cannot define that substance of which infinites are predicated.
Two terms, of which one is an accident, cannot be reciprocally predicable as genera are; otherwise each would be a species of itself. Neither can qualities or any other of the categories be so predicated, unless the predication be accidental, for all these categories are accidents and are predicated of substances.
It may also be shewn that this process of predication is not limitless in the direction of the universal, for that which is predicated of any subject must denote Quality, Quantity, or some such attribute of substance.
All these attributes are however limited, not less than the classes contained in the categories, namely Quality, Quantity, Relation, Action, Passion, Place or Time; and our hypothesis is that one thing should be predicated of one, and things should not be predicated of each other unless they denote substances, for all the categories, except substance, are accidents, some essential, others accidents in a different sense.
All these then are predicated of some substance. Accidents however are not subjects, for we hold none of those things to be subjects which are not called what they are called in virtue of their being already something else; one accident being predicated of one subject, another of another. Hence nothing indefinite will be predicated of any subject either in the direction of the universal or of the particular, for the terms of which accidents are predicated are those which constitute the substance of a thing, and such terms cannot be limitless. As we advance towards the universal we find that these substances and their accidents are neither of them limitless. There must then be some term of which an attribute is predicated as a primary attribute, while of this latter something further is predicated. The process must in time terminate, and there must be something which is not predicated of anything more primary, and of which nothing more primary is predicated.
This then is one method of demonstrating that the process of predication has limits. Another is as follows. The existence of antecedent predicates renders propositions demonstrable. One cannot grasp demonstrable things in any better way than by knowing them, nor can knowledge of them be obtained without demonstration. But if one thing can only be learned by means of others, and we are unacquainted with these latter, and do not know them by the help of any higher perception than knowledge, we shall have no real knowledge of these subjects which can only be learned mediately. If then it be possible to obtain absolute knowledge of anything by means of demonstration, not merely knowledge restricted by particular conditions or hypotheses, the intervening predications of attributes must necessarily terminate. Otherwise, if there were always some term higher than that actually employed, everything would be demonstrable.
Since however one cannot pass beyond the limitless, one cannot know by means of demonstration that which cannot be demonstrated. If then we have no higher perception of the demonstrable than knowledge, the result must be that we cannot know anything absolutely by means of demonstration, but only conditionally.
This proof may win a dialectic assent to our assertion, but the following argument, based on the real nature of things, will prove more shortly that predications of attributes in demonstrative sciences, such as we are now considering, cannot be limitless in either direction.
Demonstration deals with all the essential attributes of things; and Essential has two meanings, viz.: (1) Attributes forming part of the definition of the subject; (2) Things of the definition of which the subject forms part. For instance odd is essential to number, for odd is an attribute of number, while number itself forms part of the definition of odd. Again, multitude or discrete forms part of the definition of number. Neither of these processes can be unlimited. (1) The process by which e.g. odd is predicated of number, cannot be so, for if it were, there would be some other attribute included in odd, of which odd itself would be predicable as an attribute. If this were so number will be predicable as primary subject of all the attributes thus becoming predicable of it. (2) If, however, unlimited attributes cannot be predicated of a single term, predications in demonstration must reach a limit in the direction of the universal. Every attribute must be predicated of a primary subject, as in this example of number, while conversely number is an attribute of these others, so that both will be convertible and will not overlap. Neither are the attributes which form part of the definition unlimited, for in that case definition would be impossible. Hence if all the attributes are regarded as essential, and if that which is essential cannot be unlimited, a limit to predication must be reached in the direction of the universal, and consequently in that of the particular. If this be so, that which falls between the two limits of predication must always be limited, and this at once shews that demonstrations must necessarily have ultimate principles, and that not everything can be asserted, and that not everything is, as some have held, capable of demonstration. If ultimate principles do exist not everything can be demonstrable, nor can the process of demonstration continue to infinity. A necessary consequence of either of these conclusions would be that there can be no immediate and inseparable propositions, but that everything must be mediate and separable, for that which is demonstrated is demonstrated by the interposition of one term between two others, not by the addition of one from outside. Hence, if Deduction could go on to infinity, infinite means might exist between two terms. This, however, is impossible if attributes are limited in both directions; and that they are so has already been proved dialectically, and has now been demonstrated in accordance with the real nature of things.
Several terms may have only one thing in common, but one middle term uniting attribute and subject is necessary for demonstration; for immediate propositions are indemonstrable and serve as the basis for demonstrating other propositions. Such elementary principles need not be everywhere identical; for ‘Unit’ in different sciences is only analogously the same.
After this proof it is clear that if the same quality belong to two terms: e.g. A to C and D, when neither of these terms is predicable of the other, either universally or in some other way, A will not always be predicable of them in consequence of possessing a common quality. For instance it is a common quality of isosceles and scalene triangles to have their angles equal to two right angles, for it belongs to them because they are a particular kind of figure and not in any other connection. But this is not always the case. Suppose a common quality B which is the cause of A belonging to C and D. It is clear then that B belongs to D in consequence of some other common quality, and that other quality in consequence of a third. This process would involve the intervention of an infinite number of terms between two other terms, which is impossible. If then one term be common to two others it is not necessary that it should be common to several additional terms, since there are also ultimate propositions. It is, however, necessary for the terms which have something in common with one another to be in the same genus and derived from the same series, if there is to be any community of essential attributes, for demonstration cannot pass from one genus to another. It is also clear that when A is predicable of B, if there be any common middle term A may be shewn to be so predicable. The elements of demonstration are all things which are of the nature of middle terms, and correspond in number to the quantity of middle terms existing. Although immediate propositions, either all of them or only those which are universal, are the real elements of demonstration, yet if there be no such elements there can be no demonstration; but the stage is that of seeking the primary principles of demonstration (viz. Induction). Similarly, suppose A to be not predicable of B; if there be either a middle or a more comprehensive term of which neither is predicable, the fact that A is not predicable of B may be demonstrated; if not, that is impossible. The primary principles and elements are equal in number to the terms of a demonstration, for the premises formed by these terms are the principles of demonstration. Also, just as some of these principles are themselves indemonstrable, such as that ‘this is that’ or ‘this is predicable of that,’ or the corresponding negatives, so some of these immediate principles pronounce that a thing is, others that it is not. When a proof of anything is required a middle term must be found which is predicated of the minor B as a primary attribute. Let such a middle be C, and let A be similarly predicated of C. If the process be continued in this way, no premise is added from outside in the course of the proof, and no attribute is predicated of the subject A. Thus the middle terms are continually compressed, until they form a single proposition not divisible by any further middle term. Unity is attained when the proposition is immediate and simply forms one immediate premise. Just as in other subjects the primary element is simple, though not identical in all cases, being in Weight a Mina, in Music a Semitone, and elsewhere something different, so in Syllogism the Unit is Immediate Premise, in Demonstration and Science it is Reason. Now in affirmative demonstration the middle term never falls outside the attributes of the predicate, and the same is sometimes the case in negative syllogisms, as in the case where A is not predicable of B because of C; namely, when all B is C and no C is A. But if it be required to prove that no C is A, one must take a mean between A and C, and the process will go on for ever. But if one have to prove that D is not predicable of E because C is predicable of all D but of none or of not all of E, the middle term will never fall outside of E, and E is the term of which D was not to be predicable.
In the third figure the middle term will never fall outside that term which is denied of another or of which another is denied.
Whether Universal or Particular Demonstration is superior
It may be supposed that particular demonstration is superior to universal: Because (1) It gives knowledge of the things in themselves. (2) The universal is a nonentity, and has no existence outside the particulars. But knowledge of the universal is really more extensive than knowledge of the particular. The universal has not a separate existence, but resembles other abstractions like Quality or Relation. It alone gives the Cause; it cannot end in an unknowable infinity; it gives knowledge of more things than of the one under consideration. It contains the particular potentially, and ends in Understanding, not, like the particular, in Sensation.
Since one sort of demonstration is universal and another particular, one affirmative and the other negative, the question is raised as to which is superior. A similar doubt attaches to the method of direct demonstration and of that which proceeds by reduction to the impossible. First then let us consider the universal and particular demonstration, and when we have explained that point we may consider direct and indirect demonstration. Some may perhaps regard the particular method as superior in virtue of the following considerations. If that demonstration which gives us more scientific knowledge be superior (for to produce that is the function of demonstration), if further we have more scientific knowledge of each thing when we know it essentially than when we know it through something else (e.g. we know better about the musician Coriscus, when we know the fact that Coriscus is musical than when we know that ‘man’ is musical, and so in other instances); if, thirdly, universal demonstration prove that something else, not merely the thing in question, is what it is (e.g. prove that the angles of an isosceles triangle are equal to two right angles, not because it is isosceles but because it is a triangle), while particular demonstration shews that the thing itself and not something else possesses the quality in question; if, in short, essential demonstration be of a superior kind and particular demonstration be more essential than universal, then particular demonstration would seem to be the superior. Further, they would argue, no universal can exist outside the particulars, while universal demonstration produces the impression that there is some independent universal in connection with the thing demonstrated, and that a natural quality of this kind exists in real objects (e.g. that there is a universal triangle outside particular triangles, and a universal figure outside particular figures, and a universal number outside particular numbers); and demonstration which is concerned with the existing is superior to that which is concerned with the non-existing, and that which leads to no errors to that which does. Now universal demonstration is of the latter kind, since the method adopted is cumulative, as e.g. in the demonstration of analogy, that ‘what is not in line, number, solid or plane is the universal of analogy.’ Since then universal demonstration is of this character, and since it is less concerned with existence than is particular demonstration, and since it may produce wrong opinions, it would seem to be inferior to particular demonstration. But is not this last argument favourable rather to universal than to particular demonstration? If the quality of having its angles equal to two right angles belong to a figure, not because it is isosceles but because it is triangle, he who only knows that it is isosceles knows less than he who knows it to be a triangle. Strictly speaking when this quality is proved to inhere in isosceles triangle, but not as a result of that figure being a triangle, the proof is not a demonstration at all. If however the proof be effected in the manner mentioned, one who knows everything in the light of its particular essential qualities has superior knowledge of it. If triangle has a wider denotation than isosceles triangle, and if the word ‘triangle’ is not equivocal and the same idea underlies all triangles, and if further the quality of having its angles equal to two right angles belongs to every triangle, then an isosceles triangle does not possess this quality because it is isosceles but because it is a triangle.
Consequently one who knows the universal has a higher knowledge of the thing’s essential qualities than one who knows the particular. Thus universal demonstration is superior to particular. Further, if the universal be one and unambiguous, the universal will exist in no less a degree than the particulars, but actually in a greater degree, in that the universal possesses only imperishable qualities while the particulars are more liable to perish. Moreover there is no necessity for supposing that the universal is anything outside the particulars because it expresses a unity, any more than that those categories have independent existence which signify, not substances, but qualities, relations or actions. If, in fact, it be supposed that the universal has a separate existence, it is not the demonstration which is to be blamed, but the listener who misunderstands it.
Moreover if demonstration be a syllogism proving the cause and reason of a thing, the universal contains the cause to a higher degree, for that which is an essential attribute of a thing is its own cause. Now the universal is primary and is therefore the cause of the attribute. Hence universal demonstration is superior for it gives a better proof of the cause and reason.
Further we pursue our search for the cause of a thing until, and think that we have learned it when, we see nothing else which can be regarded as the cause, whether it be in the region of becoming or being. This last must be the end and goal of our enquiry. Take the question, ‘for what reason did he come?’ ‘To receive the money, and this in order to pay his debt, and that again in order not to act unjustly.’ If we proceed in this way, when we find that a thing has happened on no other account and for no other reason than the fact we have attained to, we say that ‘he came’ or ‘it is, or becomes owing to this ultimate cause,’ and that we have then learned most completely why he came. But if the same happens with regard to all causes and all reasons, and if our knowledge is most complete when we know the ultimate cause, then in other cases also we have most complete knowledge of a thing when its existence is not merely the result of the existence of something else. When therefore we know that the external angles of a figure are equal to four right angles because it is isosceles, there remains the question ‘why have isosceles figures this quality?’ The reason of this is that they are triangles, and the reason why triangles possess this quality is that they are rectlinear figures. If this latter fact be not caused by something else, we have then the most complete knowledge of it, and have then attained to the universal. Hence universal demonstration is superior.
Further, the more a demonstration partakes of the nature of the particular, the larger is the indefinite element which it contains. In so far as things are indefinite they are unknowable, in so far as they are definite they are knowable. Hence things are more knowable the greater the universal element they contain, less knowable the greater the particular element. Demonstration is applicable in a higher degree to things which are more capable of demonstration, and corresponds in definiteness to the definiteness of its objects. Consequently that demonstration which is the more universal is superior, since it is demonstration in a higher sense. Moreover, that demonstration which brings one knowledge of other things as well as of the single object of study is preferable to that which gives information about the latter alone; and one who has a universal demonstration knows the particular as well, while one who knows the particular does not know the universal. Hence universal demonstration is superior from this point of view also.
We may also consider the following point. To prove more universally is to use for the proof a middle term which is nearer to the elementary law; now that which is nearest to this law is the ultimate, and so the ultimate must be identical with the elementary principle. If then the demonstration which is derived from the elementary principle be more exact than that which is not, that which is more nearly derived from it must be more exact than that which is more remote. Now the former has the larger universal element. Hence from this point of view the universal is superior. E.g. If one had to prove that A is predicable of D; the middle terms are B and C, B being the more universal. Then the demonstration based on B is more universal.
Some, however, of the arguments here used are merely dialectical, and the best proof that the universal demonstration is the superior may be derived from the fact that when we possess the major premise we in a manner know the minor also and possess it potentially. E.g. If we know that every triangle has its angles equal to two right angles, we know in a manner, or potentially, that an isosceles figure has this property, even if we do not know that an isosceles figure is a triangle. On the other hand one who possesses this minor premise, does not in any way know the universal, either potentially or actually. The universal too belongs to pure thought, while the particular is finally referable to acts of sensation. This may suffice to shew that universal demonstration is superior to particular.
That Affirmative is superior to Negative Demonstration
Affirmative demonstration is superior to negative. It requires fewer propositions, is more persuasive and comprehensible, and also more immediate, for the negative is only proved through the medium of the affirmative.
That affirmative demonstration is superior to negative is plain from the following considerations. We may suppose that, other circumstances being similar, the demonstration which proceeds from fewer postulates, hypotheses, or premises is superior. If these fewer postulates are as well known as the more numerous, knowledge will be attained more quickly by their means: a desirable result. Now the reason for the assertion that the demonstration proceeding from fewer premises, so long as they are universal, is superior, is as follows. If the middle terms be equally well known, then the antecedent terms will likewise be better known. Firstly then let it be supposed that, by means of the middle terms B, C and D, the demonstration is arrived at that E is A, and then the same demonstration by means of the middle terms F and G. Here the fact that D is A is similar to the fact that E is A, but the fact that D is A is antecedent to and better known than the fact that E is A, for the latter is demonstrated by means of the former, and that by which a thing is demonstrated is more convincing than the thing demonstrated. Hence, other circumstances being similar, the demonstration proceeding by means of fewer propositions is superior. In both cases alike the proof is attained by means of three terms and two premises, but affirmative demonstration assumes that a certain thing exists, negative demonstration first that it does and then that it does not exist, so that the latter is inferior to the former. Further, since it has been proved that, when both premises are negative, no conclusion can be arrived at, a negative syllogism must have one negative and one affirmative premise. We should now add the following condition. When the demonstration is extended in application the number of affirmative premises must be increased, while the negative premises in each syllogism can never be more than one. Suppose that no B is A, but all C is B. If the premises are to be further enlarged a middle term must be interposed between each of these pairs. Let the middle between A and B be D, and that between BC be E. Now it is clear that the term E is affirmative, and D must be affirmative when joined to B, negative when joined to A, for all B must be D, and no D must be A. Thus one premise, DA, is negative.
The same method applies to other syllogisms. In affirmative syllogisms the middle term is always used affirmatively when joined with one of the other two terms, but in negative syllogisms the middle term must be negative in one premise. Thus one premise is negative but the others are affirmative. Also if that by which a thing is proved be more comprehensible and convincing than the thing itself, and the negative demonstration be proved by affirmative premises, but not vice versâ, the affirmative demonstration would seem to be prior to, and more comprehensible and convincing than the negative.
Moreover, since the first principle of syllogism is the universal immediate premise, and since in the affirmative syllogism the universal premise is affirmative, in the negative it is negative; since also the affirmative premise is prior to and more comprehensible than the negative (for the negation only becomes known by means of the affirmation, and affirmation is prior to negation, just as ‘being’ is prior to ‘not-being’); then the primary principle of the affirmative syllogism is superior to that of the negative, and that syllogism which uses superior principles must itself be superior. Moreover, the affirmative syllogism is more primary, because without it no negative syllogism can be formed.
Direct Demonstration is superior to Reduction per impossible
Negative demonstration is superior to demonstration by reduction to the impossible, for, though both are proved by means of Not-being, in the case of the negative demonstration this Not-being is anterior to the demonstration, in the case of the other it follows. This advantage of priority makes the Negative superior.
Since the affirmative argument is superior to the negative it is clearly superior to the reduction to the impossible. The difference between them should be noticed. Thus, let no B be A, and all B be C. It follows necessarily that no C can be A. When terms are thus placed the negative demonstration shewing that C is not A is direct. The reduction to the impossible on the other hand proceeds as follows. If one have to prove that B is not A one must assume that it is A, and also that C is B; whence it follows that C is A. This is already known and acknowledged to be impossible. Hence the conclusion follows that B cannot be A. If then C be acknowledged to be A, B cannot be A.
The terms then are arranged in a similar way in both methods, but a difference arises according to which of the two negative premises is the better known, whether that shewing that B is not A, or that C is not A. When the conclusion that C is not A is better known we have a demonstration by reduction to the impossible, when the other negative proposition in the syllogism itself (B is not A) is better known, the demonstration is direct. Now the proposition B is A is naturally prior to the proposition C is A, for that from which the conclusion is drawn is prior to the conclusion itself. But the proposition C is not A is the conclusion, the proposition B is not A is a premise from which the conclusion is drawn; and the refutation of any statement does not consist merely in the conclusion but in the premises from which it is drawn. Now that from which a conclusion is drawn is a syllogism so constituted that one premise bears to the other the relation of whole to part or part to whole. The premises CA and BA, however, have not this relation to one another. If then the demonstration from prior and better known premises be superior, and if further both methods of demonstration rest on the assumption that something does not exist, if thirdly one of these methods be derived from a more, another from a less primary source, then negative demonstration is, from this fact alone, superior to reduction to the impossible. Hence, if affirmative be superior to negative demonstration, it is plainly superior to reduction to the impossible.
What science is more certain and prior, and what less certain and inferior
The highest science is that which gives both the fact and the cause. The science which gives the cause only is superior to that which gives the fact only. One science may also be superior to another because it has immaterial objects or simpler principles.
A science is more exact than and prior to another when it gives the fact and the cause at the same time, and when there are not separate sciences for each. Further a science which has no material object is more exact than and prior to one which has (as in the case of arithmetic as contrasted with harmonics). Lastly a science with simpler principles is superior to one which requires a greater number. What I mean by this may be illustrated by the following example. Point is a substance in position, Unit a substance without position. Hence ‘point’ is possessed of additional qualities or principles.
What constitutes one or many Sciences
A science is one when it applies to a single genus, and when all the principles used belong to that science. Otherwise demonstration would be impossible (cf. Bk. I, c. 7).
Those sciences are one and the same which belong to the same genus, namely those which have the same primary principles and common parts or essential qualities. One science differs from another when their elementary principles are not drawn from the same source, and when the principles of one science are not derived from those of the other. A proof of this may be seen when one reaches the indemonstrable propositions of a science. These, if the sciences be one, must belong to the same genus as the things which are demonstrated. Another proof of this is that the things demonstrated are homogeneous to those indemonstrable propositions by which they are proved.
Concerning many Demonstrations of the same thing
Several demonstrations of the same conclusion may be given, and their middle terms may be taken from different series as well as from the same series. Such middle terms must however be reciprocally predicable.
It is possible to give several demonstrations of the same things, not only by taking a middle term from the same series of terms, and that too a middle term which is not logically proximate (as for instance by taking as middles between A and B not only the proximate term C, but also D and E) but by taking one from a different series of terms. As an instance of this last let A represent Changing, B Rejoicing, D Moving, and again H represent Being calm. Now D may be correctly predicated of B, and also A of D, for one who rejoices experiences movement, and that which moves undergoes a change. Again A may also be predicated of H, and H of B, for everyone who rejoices feels a calm, and one who feels a calm undergoes a change. Hence the syllogism is established by different middle terms not derived from the same series. It is not however allowable that neither of these middle terms should be predicable of the other, for both are necessarily predicable of a common third term. The other figures of the syllogism may also be examined in order to see in how many ways a syllogism with the same conclusion may be constructed.
On fortuitous occurrences
No demonstration can prove fortuitous circumstances, for demonstration deals only with the necessary or sometimes with the probable.
No knowledge of a fortuitous occurrence can be attained by demonstration. The fortuitous does not resemble either the necessary or the probable, but is that which falls outside both of these classes, while demonstration deals with one or other of them, since every syllogism is drawn from necessary or probable premises. If then the premises be necessary the conclusion is so likewise, if the premises apply, in most cases only, the conclusion has a similar application. Hence, if the fortuitous be neither probable nor necessary, it cannot be demonstrated.
Sense perception cannot give Demonstrative Science
No Science can be attained by means of Sensation, which can never prove a universal, though repeated sensations may in time produce a universal, and this a knowledge of the Cause.
Nor can scientific knowledge be gained by means of sense perception, for even though perception may give information concerning a thing’s quality as opposed to its concrete existence, yet an act of perception must indicate the existence of the object in a particular place and at the present time. The universal on the other hand and that which is present in every example of a subject cannot be perceived by the senses, for the universal is not a particular thing visible at the present moment, for then it would not be a universal at all, seeing that we mean by Universal that which is eternal and omnipresent. Since the demonstrations rest on the universal, and universals cannot be perceived by the senses, it is clear that one cannot acquire scientific knowledge by means of sense perception. Even if we could have perceived that a triangle has its angles equal to two right angles, we should certainly have gone on to search for a demonstration of it, and should not, as some assert, have already known the fact by means of perception alone. Perception as an act must deal with the particular alone, while scientific knowledge consists in learning the universal. Thus even if we were on the moon and saw the earth shutting out the light, we should nevertheless be ignorant of the cause of an eclipse. We should indeed see that the moon was being eclipsed at that particular moment, but we should not know the cause of an eclipse in general, for our perception would not be of the universal. I do not deny that after seeing the same phenomenon occur repeatedly we might search out the universal law, and thus attain to demonstration, seeing that knowledge of the universal results from repeated acts of sense perception. But the value of the universal lies in its shewing the cause of particular phenomena, and consequently the universal is more important than the perception of particular cases or the immediate apprehension of such things as have for cause something other than themselves. Of self-caused primaries we are not now speaking. It is then clearly impossible to acquire scientific knowledge of any demonstrable thing, unless the meaning of ‘scientific knowledge acquired through demonstration’ be attached to the phrase ‘act of sense-perception.’
Certain doubtful questions may be solved by a reference to the failure of the sense perceptions. Thus if we had seen certain things we should have made no further enquiry about them, not because we know them simply from seeing them, but because the mere sight of them would have sufficed to give us the universal. E.g. If we saw that the burning-glass was porous and that the light filtered through the apertures, it would be clear why it burns, because we should see the phenomenon occur in every separate glass; but we should yet have to form the abstract idea that this quality is universally true of every possible glass1 .
On the difference of Principles corresponding to the difference of Syllogisms
The principles of demonstration cannot be the same in all cases, for true conclusions may be drawn from false premises, and even in the case of true syllogisms the principles may differ generically. Further all principles may be divided into Common and Special, corresponding to the grounds and the subjects (ἐξ ὡ̂ν καὶ περὶ ὅ) of demonstration.
It is impossible that all syllogisms should have the same elementary principles, and this may be proved by purely dialectical considerations, Some syllogisms are true, others false, and it is also possible to deduce a true conclusion from false premises, though only in one particular class of circumstances. For instance the proposition C is A may be true, but the middle term B is false, since B is not A, nor yet is C B. But if the middle terms to these premises be expressed, the falsity of the premises will become obvious; since a false conclusion presupposes false premises, while true conclusions result from true premises, and false and true premises are different from one another. Nor do false conclusions follow only from premises which are false in the same manner as themselves, for things which are false may be both the contrary to and inconsistent with each other, as may be illustrated by the assertions ‘Justice is either injustice or cowardice’; ‘Man is either a horse or an ox’; ‘Equal is either greater or less.’ That all syllogisms have not the same principles may also be proved as follows from conclusions already arrived at. Even true conclusions are not invarably derived from the same elementary principles, for in many cases the principles differ and do not suit every kind of argument: e.g. the conception of ‘unit’ cannot be used as a principle when theorizing concerning points, since units, unlike points, have no special position. In order to make the same principles suit various forms of syllogism it is necessary to use them as predicates of the major term, as subjects of the minor or as intermediate between major and minor; or else they must be variously related, some being intermediate between major and minor, others superior to the major or inferior to the minor.
No common principles can exist from which everything may be demonstrated (by ‘common principles,’ I mean those resembling the proposition,—‘it is possible either to affirm or deny everything.’) Existing things differ generically; some predicates can only be assigned to the genus quantity, others to that of quality, and these subjects and predicates together with the common principles of science join in producing a demonstration. Moreover the principles are not much less numerous than the conclusions, since the principles constitute premises, and may become formal premises by inserting a term between major and minor or adding a term either superior to the major or inferior to the minor. Further the conclusions are unlimited, the terms limited. Again some principles are necessary, some contingent.
If we consider the matter in this way we see that these limited principles cannot be identical, since the conclusions are unlimited. If an objector were to assert that these are the principles of geometry, those of calculation, those again of medicine, his assertion would simply amount to saying that different sciences have different principles. It is however absurd to say that they are the same principles in all cases just because they are principles and not something else; for by that method all distinct things might be proved identical. Nor can it be meant that every premise will prove every conclusion, which would be equivalent to claiming that all sciences should have the same principles—a ridiculous assumption, for this is not the case with existing kinds of exact science, nor is it possible in logical analysis. The immediate premises are principles, and distinct from them is the conclusion which is attained by means of the addition of an immediate premise. If it be asserted that it is the primary immediate premises that constitute those principles which are identical in every science, we should answer that there is a unique premise in each branch of science. If then it be agreed that not everything can be proved from any principle whatsoever, and yet that the principles of various sciences are not so unlike one another as to fall into distinct classes, there remains the suggestion that the principles of every science are akin, while the conclusions drawn from them differ. This however is clearly untrue, for it has been proved that the principles of sciences which differ generically are themselves generically different. Principles are in fact of two kinds, being either the sources or the subject of science. The former are common, the latter, such as ‘number’ or ‘magnitude,’ are peculiar to each science.
The distinction between Science and Opinion
Science depends on the Necessary, Opinion on the Contingent. Opinion may attain to immediate propositions, but as these are not necessary, Opinion is uncertain and can never be applied to the same object as Science.
Scientific knowledge and its object differ from Opinion and its object, in that Science is universal and rests on the necessary, and the necessary is not contingent. Some things are true and do exist, but yet are contingent so that they cannot be the object of science, for that would involve the identification of the contingent with its opposite, the necessary. Neither is the contingent the object of Reason (by which I mean the elementary principle of Knowledge), nor again of indemonstrable knowledge, which consists in the assumption of immediate propositions. Yet Reason, Knowledge and Opinion, together with everything which they make known, are true, so that the object of Opinion is still the true or the false, but yet contingent; that is to say it involves the apprehension of an immediate but not necessary proposition.
This view is in harmony with ordinary experience, which makes us regard Opinion as unreliable, and the nature of the things about which opinion is held is likewise unreliable. Also when one thinks that something cannot but be what it is, one never supposes that one merely opines that thing, but that one knows it. On the other hand when one thinks that the thing is now some one particular thing, but yet that nothing prevents it from taking a different form, then one supposes oneself merely to opine, since opinion refers to objects of this latter kind, whereas knowledge relates to the necessary. Why then, it may be asked, is it impossible1 to opine and know the same thing, and why is not opinion the same as knowledge, if it be laid down that everything which one knows may also be the subject of opinion, and those who merely opine pass in company through the intervening middle terms until they arrive at ultimate principles? If the former possess scientific knowledge why do not the latter also? The object of Opinion may be the Cause of things just as much as the Fact of their existence, and it is the Cause which supplies the middle term.
The difficulty may be explained thus. One who has such a clear perception of the uncontingent objects as also to possess the definitions by means of which the demonstrations of them are arrived at, will know those objects and not merely opine them. If on the other hand he knows them as true, but yet he does not know that the attributes in question belong essentially and specificially to the subject, he will only have opinion not scientific knowledge both of the fact and the cause, that is to say if his opinion rest on immediate propositions. If his opinion do not so rest, he will opine only the fact, not the cause. Opinion and Knowledge have not absolutely the same object, but their objects are similar in the manner in which the objects of Truth and Falsity are similar. The assertion of some that true and false opinions are of the same kind involves many absurdities, such as that a false opinion is not an opinion at all, since all opinions are assumed to be true. But since ‘the same’ is used in many senses, false and true opinions are in one sense the same, in another different. For instance, a true opinion that the diagonal is commensurable with the side of a square would be absurd, but since the diagonal concerning which the opinions are held remains the same whether the opinion about it be right or wrong, the object of the two kinds of opinions is one and the same, while according to their essential nature and definition those opinions are different. In a similar way knowledge and opinion may be said to have the same object. The knowledge concerning the nature of animal is of such a kind that its object cannot be other than animal. Opinion concerning the same is such that its object may be other than animal. Thus knowledge concerning man contains a reference to his essential characteristics, opinion contains no such reference. In this case the objects of knowledge and opinion are the same but regarded from a different point of view.
It is clear from this that one cannot opine and know the same thing at the same time. Otherwise one would suppose simultaneously that a thing was both contingent and necessary, which is impossible. It is possible, as has been said, for knowledge and opinion concerning the same object to exist in different persons, but in the same person they cannot. Otherwise he would have to suppose simultaneously that, e. g. Man is essentially animal (for that is equivalent to saying ‘man cannot but be animal’) and also ‘man is not essentially animal’ (for that is the meaning of ‘capable of being something else’ or ‘contingent’). How to distinguish between Inference, Reason, Knowledge, Art, Prudence and Wisdom, are questions belonging partly to Natural Philosophy and partly to Ethics. (Cf. de An. I, 1. Eth. VI, 3, 4).
Sagacity is a rapid perception of the middle term, or cause, resulting from a consideration of the major and minor terms.
Sagacity is a faculty for hitting upon the middle term in an imperceptible moment of time. For instance, suppose some one, seeing that the moon always has its bright side turned towards the sun, quickly inferred that this was so because the moon receives its light from the sun; or again, seeing someone conversing with a rich man, inferred that he was doing so in order to borrow money; or again inferred that the reason why two persons were friends of one another was that both were enemies of a third person. On seeing the major and minor of the syllogism the sagacious man is able to perceive all the causes or middle terms. Thus: Let A represent ‘having its bright side towards the sun’; B ‘lighted from the sun;’ C ‘the moon.’ Now B, ‘lighted from the sun,’ is true of C, the moon; A, ‘having their bright side towards the body from which the light is received’ is true of all objects denoted by B. Hence A is true of C because it is true of B.
[1 ]Because if the conclusion be false, both the premises cannot be true.
[1 ]I.e., if the negative premise be treated as the major instead of the minor.
[1 ]This translation follows the reading διὰ τί καίει with the Clarendon Press Edition. Poste, Zell and other versions follow the old reading ϕωτίζει ‘transmits light,’ and make ὔελσς refer to all kinds of glass.
[1 ]Reading πω̂ς ον̓̂ν οὐκ ἔστι . . . The negative seems necessary as this passage is evidently attributed to an imaginary objector.