Updated: May 6
Watson, John. “Aristotle's Posterior Analytics: I. Demonstration.” The Philosophical Review 13, no. 1 (1904): 1. https://doi.org/10.2307/2176405.
LIKE other works of Aristotle, the Posterior Analytics has had an influence upon the history of human thought out of all proportion to its length. Within a comparatively short compass the author succeeds in giving a tolerably complete and systematic statement of the processes by which scientific truth is reached. The main object of the treatise, it is true, is to explain the conditions under which the necessary conclusions of science may be drawn, a fact which naturally gave countenance to the doctrine that truth is reached by a deductive process. A careful examination, however, shows that the preeminence assigned to deduction cannot be justified by the contents of the work itself, in which the necessity of induction as an indispensable preparation for the deductions of sciences is everywhere kept in view, and indeed expressly stated. The treatise is so interesting in itself, and so valuable for the light it throws upon the philosophy of Aristotle in general, and especially upon his Metaphysic, that it may not be superfluous to give a summary of its main contents, and to attempt some estimate of their value.
"All teaching and all learning of a reflective character," Aristotle tells us, "start from knowledge that we already have." As we learn from a passage in the Ethics, the "teaching and learning" here referred to proceed either by induction or by syllogism; for, it is by induction, as Aristotle goes on to explain, that we obtain universal propositions, and it is from these universal propositions as premises that syllogism draws its conclusions. Thus, both induction and syllogism start from knowledge that we already have; the former being evolved from the perceptions of sense, and the latter from the premises supplied by induction. These two processes, as Aristotle points out in the present work, are common to dialectic, rhetoric, and the sciences. The proper subject of the treatise, however, is the method of science, and hence the two former methods of "teaching and learning" are referred to merely in order to show that reflection follows the same path in all cases, bringing forward universal propositions derived by the mind from perception, and deducing conclusions from them. Aristotle therefore at once proceeds to ask what the character of the data with which science is starts, and how from them the truths which constitute it are derived.
The view just stated of the relation of science to induction is Aristotle's substitute for Plato. According to the doctrine suggested in the Meno, learning is not the acquisition of knowledge for the first time, but the recollection of what we already know. Aristotle, on the other hand, maintains that we have no knowledge whatever prior to sensible perception, no knowledge of the universal prior to induction, and no scientific truth prior to the deductions drawn from the premises supplied by induction. Thus, the difficulty raised by Plato, that we either learn nothing, or only what we knew beforehand, is solved, when we see that we may know universal principles, and may yet be ignorant of the conclusions involved in them, until these are brought to light by the deductions of science. that have come under our observation; on the contrary, the principles from which science draws its conclusions are universal, and so also are the conclusions derived from them. From arithmetic we learn, not that all the ' twos' we have observed are 'even,' but that every possible 'two' must be 'even.' Nothing less than this will satisfy the demands of science. In other words, 'science ' in the strict sense of the term, is the knowledge of the cause or of that which 'cannot be otherwise.' Now, knowledge of this kind is obtained by means of demonstration or scientific syllogism, the data of which are supplied by induction. The character of those data may be deduced from the conclusions which have to be reached. If the judgments of science predicate what is necessary, the premises must be such as by a valid logical process will yield judgments of that kind. In the first place, therefore, the premises must be true. And this means that they must state what belongs to the actual nature of things; for the test of a true judgment is never in Aristotle the mere impossibility of thinking the opposite, but its conformity to the object; a judgment is true when it combines in thought what is combined in the thing, or separates in thought what is separated in the thing. The reason why the judgment, "The diagonal is commensurable," is false, is that it affirms a connection of subject and predicate which contradicts the actual nature of the diagonal. In the second place, the premises of a demonstrative syllogism must be primary or indemonstrable. For, if this is not admitted, we either fall into an infinite series, and therefore never reach an absolute conclusion, or we are forced to hold the equally untenable doctrine that nothing is true except what can be demonstrated. There must, then, Aristotle contends, be certain immediate or primary truths, which by their very nature are indemonstrable, and without which no demonstration, and therefore no science, is possible. In the third place, our premises must contain the ground or cause. For, as we have seen, the judgments of science are in all cases necessary, or express the 'essence ' or 'ground' of a thing. Hence the premises must be 'better known ' than the conclusion and 'prior' to it. This does not mean that, in the order of our knowledge, we start from what is, in the sense indicated, 'better known'; on the contrary, we begin with particular perceptions of sense, and only at a later stage advance to the universal. What we mean by saying that the premises of science are ' better known' than the conclusion, is that they contain the determination of the 'necessary' characteristics of a thing, and therefore the 'cause' or ' ground ' why it is what it is. As such they are logically 'prior' to the conclusion, forming as they do the indispensable presupposition of the conclusions reached in science.
To understand more fully the nature of the premises from which the necessary truths of science are deduced, there are certain terms which must be defined, (I) When a proposition is said to be true 'without exception' we mean that it is true of every member of a class, and of every member of that class at all times. Thus, if it is true that " every man is an animal," it is also true that every person who can be called "man," may also be called "animal"; and if at any given moment he is the one, he must also at the same time be the other. (2) By 'essential' it is meant that a certain element is included in the very conception or definition of a thing. Aristotle distinguishes two cases in which this principle holds good; for either a certain property is ' essential ' to the definition of the subject, or the subject to the definition of the property. We cannot define a 'line' without including the 'point’ and we cannot define 'straight' without including the ' line.' Again, when a property is predicated of an individual, it is said to be predicated essentially whereas a property which is not predicated of an individual, but of something which presupposes an individual, is said to be predicated 'accidentally.' In the judgment, "Socrates walks," the predicate belongs to the subject; but in the judgment, "the white walks," the predicate does not belong to the subject, but to something else not expressed ultimately, an individual. In the one case we have 'essential' predication, in the other 'accidental.' Lastly, that is said to be 'essential' which involves a causal connection; as, e. g., when a victim dies by the stroke of the sacrificial knife. These two last cases do not satisfy the requirements of strict science. The former only yields judgments which predicate a property of the individual, and from singular judgments no universal conclusion, such as science demands, can be derived. The latter, again, only gives us judgments which are conditionally necessary, whereas strictly scientific judgments are true at all times. Thus, as Aristotle himself points out, there remains only the case in which a property belongs to the very conception of the subject, or the subject to the definition of the predicate. (3) There is a third term, the universal which introduces a further limitation. Any predicate is 'essential' which is involved in the definition of the subject or involves the subject in its definition. Thus, in the judgment, "man is an animal," the predicate 'animal' is part of the definition of man and as such is 'essential '; but the judgment is not in the strict sense ' universal.' To be ' universal,' a judgment must state that which is true (I) 'without exception,' (2) 'essentially,' (3) of a class 'as such.' Now, to be true of a class 'as such' is to be true of a primary subject, and therefore true 'essentially' and without exception; but a predicate may apply to every member of a species without exception, or it may be part of the definition of a species, and yet, it may not be true of the genus or class 'as such.' The judgment that "the isosceles triangle contains two right angles" is true of all isosceles triangles, and the predicate is part of the definition of the subject; but it is not 'universal,' because ' isosceles triangle' is not the 'primary subject' to which the property of having two right angles belongs. In short, we only obtain a truly universal ' judgment, such as is required in scientific demonstration, when subject and predicate are convertible; in other words, when we have assigned the 'cause' or 'ground' of a thing. Hence Aristotle refuses to admit that we can reach scientific truth per enumerationem simplicem. Even supposing it could be proved of each species of triangle separately equilateral, scalene, and isosceles that its angles are equal to two right angles, we should not know it to be true of the triangle 'universally,' and therefore we should not know whether there might not be some other kind of triangle of which it was not true. The necessary basis of a scientific syllogism is, therefore, a major premise which predicates an essential attribute belonging to the primary subject. When this is the case, subject and predicate must be coextensive. No doubt the special sciences make use of premises that are not 'universal'; but these, as we shall immediately see, are not in the strict sense 'scientific because they only prove the 'fact,' not the 'cause'; and proof of the 'fact' is only a step towards the end of science, which is demonstration of the 'cause.'
The basis of a demonstrative syllogism, then, must be a truly 'universal' principle. This is obvious, if we consider that science in the strict sense consists entirely of necessary conclusions. You can infer that an isosceles triangle contains two right angles, granting that "the triangle as such" has this property; but no such conclusion can be drawn, unless the major premise is, in the sense defined, 'universal.' In other words, the middle term must contain the real 'ground' or 'cause.' If the middle term is not necessary, it may cease to be predicable; hence what was true may cease to be true; and obviously, from what may not be true, no absolute conclusion can be drawn. On the other hand, when the middle term is necessary, the conclusion must also be necessary, and such necessary conclusions constitute science. Our result then is, that both the premises and the conclusion of a demonstrative syllogism must be necessary; while the middle term, on which the conclusion is based, must contain the real 'ground' or 'cause' of the subject, or, what is the same thing, the attributes which belong to it in itself, and therefore to every member of the primary genus under consideration.
The premises of a science, then, are true and primary, and they contain the 'cause' or 'ground' of a thing. But, while these are the characteristics of all the premises employed by science, there is a distinction in the character of the premises themselves, to which it is necessary to refer. From Aristotle's point of view, there is no single science which contains the whole body of scientific truth. It is true that first philosophy or metaphysic has as its object the highest principles of being but, on the other hand, those principles do not enable us to determine things in their concrete or specific character. Thus, metaphysic has, as one of its tasks, to show that the laws of contradiction and excluded middle admit of no possible exception, and therefore must be presupposed in every one of the special sciences. On the other hand, each of the special sciences employs these laws only in so far as they apply to the special 'class of being' with which it deals. While, therefore, Aristotle calls them 'common' principles, he is careful to add that they are not taken in their abstract generality by the special sciences, but only in their specific application to the subject under investigation. And the same remark holds good of another class of 'common principles' or 'axioms,' viz., those which apply, not indeed to all kinds of being, like the laws of contradiction and excluded middle, but to the objects of two or more sciences. Of this character is the axiom that "if equals be taken from equals, the remainders are equal," a principle which is common to arithmetic and geometry. But here again the axiom is not employed in its complete generality. In arithmetic it is interpreted to mean, that "if equal numbers be taken from equal numbers," etc., whereas in geometry it means that "if equal magnitudes be taken from equal magnitudes," etc. In actual use, therefore, the axioms are not really 'common' principles, but in their specific sense, as employed in a particular science, are special or determinate principles.
This view of the so-called 'common' principles is in accordance with Aristotle's whole view of things. For him there is not a single ' kind of being,' but various mutually exclusive spheres of being, each of which is the object of a particular science. Hence, when he is laying down the conditions of science, he tells us that it involves three things: (I) The class of being, which is the object of a particular science, (2) the axioms or principles from which we argue, (3) the conclusion, which states an essential determination of the class under investigation. It is therefore an illegitimate procedure for any science to pass out of its own proper sphere. There are certain absolutely irreducible 'kinds of being,' each of which has its own special determinations; and therefore, the geometer can no more apply to magnitudes the properties of numbers than the arithmetician can characterize numbers by the attributes essential to magnitudes. There must be no … on pain of illogical and unscientific reasoning. It is therefore natural for Aristotle to point out that the 'common ' principles are in practice really 'special.' The employment of so-called 'common' principles is therefore no real violation of the doctrine that science must contain only 'universal ' judgments; for the axioms, as interpreted by the special sciences, express what is 'essential' to magnitude or number 'as such,' and what is true of every magnitude or number at all times.
Each of the special sciences, then, assumes the truth of the common principles or axioms in the limited sense required for its special purpose. No doubt these principles may be called 'special’ since, in the meaning assigned to them by a given science, they are not applicable to any other science; but, as in the wider sense they express the principles common to all being, or at least to more than one 'kind of being,' Aristotle distinguishes from them the principles which are peculiar to a given science. These are 'theses,' they are 'posited' by the science. They state the primary characteristics of the 'class of being,' with which the science deals, and therefore at once define it and affirm the existence of the object defined. A principle of this sort is called a "postulate.” Thus, geometry not only presupposes the definition of ‘magnitude,' or 'point' and 'line,' but it postulates the actual existence of 'magnitudes' or 'points and lines.' The 'point,' e. g., is defined as 'that which has no extension,' or ' that which is indivisible;’ the 'line' as ' that which has only one dimension;’ and geometry assumes that there are real 'points' and 'lines' corresponding to these definitions. The special principles or postulates, therefore, agree with the common principles or axioms in presupposing the truth or reality of their object. Unless the truth of the special principles is assumed, the science to which they belong has no premises from which 'universal' conclusions may be drawn; for these principles, as primary determinations of a certain 'class of being,' do not admit of demonstration. Assuming them, however, it is possible to advance to the concrete determination of the genus; and the problem of a given science is just to deduce, by means of demonstrative syllogisms, by the aid of induction, the totality of the essential properties, modifications, and functions of the class of being with which it deals.
Besides these special principles or postulates, each science employs another species of 'theses' viz., those which agree with the 'postulates' in being definitions but differ from them in not being presupposed as data of the science under investigation. This class of definitions comes to light in the course of the demonstration, and therefore presupposes it. They are therefore merely verbal, and but serve to embody the results of demonstration, when those results are taken as the premises of a new demonstration. Thus, in geometry the essence of the 'point' and the 'line' is expressed by the 'postulates' in which they are defined, but the content of the conceptions 'straight,' 'commensurable,' 'diverging and converging,' is expressed in the definition of these properties, which states what belongs 'essentially' to the subject determined by them. Similarly, the definition and reality of the 'unit' is in arithmetic a 'postulate,' but the definitions of 'odd' and 'even,' 'square' and 'cube' numbers, express the properties of numbers which are established in the course of the demonstration. The definitions proper are therefore data of demonstration, not in the sense that they are presupposed as the basis of all the demonstrations of a particular science, but only in the sense that they are presupposed at certain stages in the process of demonstration. The truth of the primary determinations of the genus under investigation is 'postulated,' the truth of the properties which characterize the species falling under the genus is demonstrated. Geometry 'postulates' the reality of the 'point' and 'line'; it 'demonstrates' the truth that 'the triangle has two right angles' from these postulates, in combination with the common principles or axioms, employing the definition of 'right angle' which has been obtained in the course of prior demonstrations, and has been embodied in a verbal definition.
The demonstrations of science, as we have seen, enable us to make certain 'universal' judgments in regard to the 'kind of being' with which a particular science deals. No such judgments are intelligible, if we adopt the view of the Platonists, that there are ideas, or abstract unities, which have an existence apart from the many individuals. Nor is the assumption of such unities essential to the explanation of demonstrative science. No doubt we must be able to predicate unity of many individuals; under no other condition, indeed, can we have a truly 'universal' judgment; for no 'universal' conclusion can be drawn, unless we have a middle term, comprehending an attribute which is identical in a number of things, and identical not merely in name but in reality. Thus, if 'man' is a separate and independent idea, the proposition 'Socrates is a man' can only mean that the name 'man' is applied to Socrates, because he is found to resemble Plato and Aristotle, not because he is identical in nature with them. Only if there is absolute identity in nature can we have a universal and necessary judgment, a judgment which expresses the ' essential ' nature of Socrates as 'man.' The two terms … indicate the doctrine of Aristotle, that a 'universal' judgment must express the essential connection of subject and attribute, a connection which is not accidental, but is involved in the very nature of the object.
From what has been said it is obvious that, in Aristotle's view, no science is possible, unless there are certain fixed or unchangeable ‘kinds of being’ which can be grasped and defined by thought. It is indispensable to his doctrine that, though the accidental properties belonging to things are infinite, the properties which are inseparable from a given 'class of being' must be limited in number. This is the main argument by which he seeks to show that scientific demonstration must start from indemonstrable premises. In all predication, as he argues, there must be a primary subject which cannot be predicated of anything else. We can no doubt say either 'the white is wood' or 'wood is white,' but the second form of expression alone corresponds to the nature of things, since 'wood' is the subject of which 'white' is predicable, whereas 'white' is not the subject of which 'wood' can be predicated, though in a proposition it may occupy the position of subject. Now, in predicating in the category of ‘essence’ we predicate either the genus or the species, and whenever we predicate in any other category, we merely state what is ' accidental ' or separable from the subject. The judgment, "man is an animal," is a predication of 'essence,' because it is implied that there cannot exist a man who is not as man an animal. But predication in any other category, such as 'quality' or 'quantity,' is of a different character. Thus "man is white" is not ' ‘essential' predication, for it does not mean that 'white' is inseparable from 'man'; if it were, "man is white" would mean that there is a genus or specie 'white,' and that 'man' is part of it. Whatever the kind of predication, however, there must be a subject which cannot be predicated of anything else; in other words, the individual is the real, and all real predication is a determination of the individual, whether that determination is 'essential' or 'accidental.'
Now, it is easy to show that the predicates which express the 'essence' of a subject must be limited in number. It is essential predication to say, "Man is two-footed," for here we are predicating a species. And we can go on to say, "The two-footed is animal," because here we predicate a genus. But we cannot go on to say "Animal is something else," since here we have reached the summum genus, and any further advance in this upward direction carries us beyond the genus to which 'two-footed' and 'man' belong, and therefore destroys the ' essential 'character of ' man.' And, in the descending series, we can say "Man is animal," where the predication is of the genus; then "Callias is man," for here we predicate the species; but if we attempt to descend further, and say "Something else is Callias," we are stopped by the impossibility of predicating the individual in consistency with the nature of things. There is therefore a fixed limit, both upwards and downwards. Nor can we predicate genera interchangeably, for this would mean that a genus is predicated of itself. If, e. g. y number = 'magnitude,' and 'magnitude' = number; then number = 'number,' and 'magnitude' = 'magnitude'; which is no predication at all, or at most only verbal predication. For the same reason the categories cannot be predicated of one another. In short, each kind of being has its fixed limits, and equally each kind of category.
Now, demonstration is only possible when we can find a middle term. If we know that "man is two-footed," and that "the two-footed is animal, “we can reach the conclusion that " man is animal." But as there is a limit both upward and downward, there must be a limited number of middle terms. If this is denied, we must hold that everything is demonstrable, a view which really destroys the possibility of all demonstration, since it lands us in an infinite series.
This conclusion might be reached by a simple analysis of demonstration. The judgments of demonstration must contain nothing but 'essential' properties, since a necessary conclusion cannot be derived from what is 'accidental.' Now, essential predication, as we have seen above, either (a) states a property involved in the subject, or (b) a property which is limited to the subject. It is obvious that a property essential to the determination of the subject must be ' universal ' in the strict sense of the term; i. e., it must be a determination of a summum genus as such. Hence the judgment in which this property is predicated must be primary, and therefore indemonstrable. And a judgment which affirms a predicate that is meaningless apart from the subject must have a definite or limited application. Thus, 'odd' has no meaning except as an attribute of 'number,' and therefore nothing is 'odd ' except a 'number'; it is accordingly an ultimate determination. Hence, we cannot demonstrate that numbers are 'odd,' but must accept the determination as primary. If a demonstration were possible, we should have to find a conception which included 'odd ' numbers and other species of 'odd ' than that of number. As this is impossible, we cannot demonstrate that numbers are 'odd,' but must accept the determination as a first principle. Our general conclusion, then, is that the properties involved in the definition of a class, as well as its specific determinations, must as ultimate be assumed by demonstration, not proved by it.
The sphere of a given science is evident from these considerations. A science is one when it deals with a single class of being and with the essential properties of that class. When the first principles are different, the sciences are different. Now it may be shown, in the first place dialectically, that there cannot be principles common to all the sciences 'common' in the sense of having the same specific meaning. It will be admitted that there are false as well as true syllogisms. But false premises yield a false conclusion, true premises a true conclusion. And as the premises are the … from which the conclusion is derived, the … of false syllogisms must be generically different from the … of true syllogisms. And not only so, but there may be a generic distinction even in the case of false principles themselves. Thus, we may form false syllogisms, either by concluding that justice is injustice, or that it is cowardice. Here the two false conclusions contradict each other and must therefore be derived from generically different first principles. And what is true of false syllogisms is even more obvious in the case of true syllogisms. We cannot establish a true geometrical conclusion from arithmetic because arithmetic deals with points that have no position, whereas geometry deals with points that have position. If we attempt to pass from units to points, we must find a middle term expressing what is characteristic of the unit or the point, or a conception predicated of both as the genus of two species, or subsumed under both, or higher than the one, lower than the other. But (1) a specific principle cannot be a middle term, since a middle term must be common to the two extremes; (2) it cannot be related to the extremes as genus to species, for 'unit' and 'point' would then have the same 'essence'; (3) nor can it be subsumed under both, for then there would obviously be two genera; (4) nor can it be higher than the one, lower than the other, for then it would be the genus of, say, the 'point,' while the 'point' would be the genus of the 'unit.' As these are the only possible suppositions, the principles of two sciences cannot be the same in kind. It is no real objection to this view, that there are 'common' principles, for these must be specified before they can be employed in demonstration.
We are now in a position to distinguish between science and opinion. The conclusions of science are 'universal,' being based upon premises which are necessarily true or cannot possibly be otherwise. The object of opinion, on the other hand, is that which may be true, but is not necessarily true, or that which, though necessarily true, is not known to be necessarily true. But, strictly speaking, even in the latter case the object of opinion is different from the object of science. Both may relate to the same object, but the mode of conception is fundamentally different, and therefore the object is really different. The same person cannot at once have an opinion in regard to a thing, and a scientific knowledge of it; for this would mean that he could hold contradictory notions and believe both to be true.
From what has been said, it is obvious that we have scientific knowledge only when we have discovered the 'cause' or 'ground.' But as such knowledge must from its very nature be true of actual things, there can be no knowledge of the 'cause' unless there is a previous knowledge of the ' fact.’ In the progress of science towards its goal, it is not unusual to begin by demonstrating the 'fact,' as a preparatory step to the demonstration of the ' cause '; a procedure which is perfectly natural, because the fact is more readily accessible to us than the cause. Thus, we learn from induction that bodies whose light gradually increases are spherical, and we infer that, since the moon gradually increases in light, it is spherical. This gives us the syllogism:
Bodies which gradually increase in light are spherical.
The moon gradually increases in light.
Therefore, the moon is spherical.
The proof, however, does not satisfy the demands of scientific demonstration, for the major premise states a 'fact,' without assigning a 'cause.' It is true that bodies which gradually increase in light are spherical, but until we know that the increase in light is an 'essential' attribute of 'spherical' bodies, that only 'spherical' bodies possess the attribute in question, we cannot obtain a really 'universal ' conclusion. Hence the proper form of the demonstrative syllogism states the cause,' and assumes the form:
Spherical bodies gradually increase in light.
The moon is a spherical body.
Therefore, the moon gradually increases in light.
The major premise is a 'universal 'judgment, in the sense defined above, because it states what is true of 'all' spherical bodies, what is ' essential ' to the class, and what is true of the class 'as such.' Aristotle's general view is, that we never have a premise expressing the 'cause,' except when subject and predicate are convertible. The demonstrative syllogism, therefore, naturally falls into the first figure the favorite figure of the mathematical sciences because this is the only figure in which we have a universal affirmative conclusion. It may be added that, while there can be no science, in the strict sense of the term, until a knowledge of the 'cause' has been obtained, it sometimes happens that the 'fact' is the object of a subordinate science, while the 'cause' is brought to light by another science. Thus, optics deals with the 'fact' in the case of visible phenomena, while geometry assigns the 'cause.' But this division of labor is obviously merely a matter of convenience and does not affect the general principle that scientific truth consists in the knowledge of causes.
In this article a summary of Aristotle's general view as to the nature of science has been given; a subsequent article will deal with his view of induction, as the method by which science is supplied with the premises from which its conclusions are drawn.