PART II — COMMENTARY ON THE POSTERIOR ANALYTICS¶
On the Most Certain Mode of Demonstration, Axioms, Hypotheses, and Geometric Postulates
Source Work: Expositio libri Posteriorum Analyticorum (c. 1270–1272) · Switch to Diplomatic Latin · Summary · GEOMETOR Notes
Book I, Lecture 1 — On the Necessity of Logic and the Most Certain Mode of Demonstration in the Mathematical Sciences¶
Latin Locus: Lib. I, lect. 1 · Commentary: Notes on Lib. I, lect. 1
[lib. 1 l. 1 n. 1] As \ Aristotle says at the beginning of the Metaphysics, the human race lives by art and reasonings: in saying which the Philosopher seems to touch upon a certain property of man whereby he differs from the other animals. For other animals are impelled to their acts by a certain natural instinct, whereas man is directed in his actions by the judgment of reason. And hence it is that various arts serve to bring human acts to completion easily and in an orderly manner. For art seems to be nothing other than a certain ordering of reason as to how human acts may arrive at their due end through determinate means. Now reason is not only able to direct the acts of the lower powers, but is also directive of its own act. For this is proper to the intellective part, that it reflect upon itself: for the intellect understands itself, and similarly reason can reason about its own act. If, therefore, from the fact that reason reasons about the act of the hand there was devised the art of building or of carpentry, through which man can exercise acts of this kind easily and in an orderly manner, by the same reason a certain art is necessary that is directive of the very act of reason itself, through which, namely, man may proceed in the very act of reason in an orderly manner, easily, and without error.
[lib. 1 l. 1 n. 2] And this art is logic, that is, rational science. Which is rational not only from the fact that it is according to reason (which is common to all arts), but also from the fact that it is concerned with the very act of reason itself as with its own proper matter.
[lib. 1 l. 1 n. 3] And therefore it seems to be the art of arts, because it directs us in the act of reason, from which all arts proceed. One must therefore take the parts of logic according to the diversity of the acts of reason.
[lib. 1 l. 1 n. 4] Now there are three acts of reason, the first two of which belong to reason insofar as it is a kind of intellect. For one action of the intellect is the understanding of indivisibles or incomplex things, according to which it conceives what a thing is. And this operation is called by some the informing of the intellect or imagination through the intellect. And to this operation of reason is ordered the doctrine that Aristotle hands down in the book of the Categories. The second operation of the intellect, however, is the composition or division of the intellect, in which truth or falsity is already found. And this act of reason is served by the doctrine that Aristotle hands down in the book Peri Hermeneias [On Interpretation]. The third act of reason, however, is according to that which is proper to reason, namely, to discourse from one thing to another, so that through what is known one may arrive at the knowledge of what is unknown. And the remaining books of logic serve this act.
[lib. 1 l. 1 n. 5] Now it must be observed that the acts of reason are similar, in a certain respect, to the acts of nature. Hence art too imitates nature insofar as it can. Now in the acts of nature a threefold diversity is found. For in certain things nature acts of necessity, in such a way that it cannot fail. In certain other things, however, nature operates for the most part, although at times it can fail of its proper act. Hence in these things there must necessarily be a twofold act: one that occurs in the majority of cases, as when a complete animal is generated from seed; the other when nature falls short of what is fitting to it, as when some monster is generated from seed owing to the corruption of some principle. And these three are also found in the acts of reason. For there is a certain process of reason that induces necessity, in which it is impossible for there to be a failure of truth; and through such a process of reason the certitude of science is acquired. There is another process of reason in which what is true is concluded in the majority of cases, yet without having necessity. The third process of reason, however, is that in which reason falls short of the true owing to the defect of some principle that ought to have been observed in reasoning.
[lib. 1 l. 1 n. 6] Now the part of logic that serves the first process is called the judicative part, inasmuch as judgment possesses the certitude of science. And because certain judgment concerning effects cannot be had except by resolving back into first principles, this part is called analytics, that is, resolutory. Now the certitude of judgment that is obtained through resolution comes either from the very form of the syllogism alone—and to this is ordered the book of the Prior Analytics, which treats of the syllogism unqualifiedly—or else, together with this, from the matter, because propositions that are per se and necessary are assumed—and to this is ordered the book of the Posterior Analytics, which treats of the demonstrative syllogism. The second process of reason, however, is served by another part of logic, which is called inventive. For discovery does not always possess certitude. Hence concerning those things that have been discovered, judgment is required in order that certitude may be had. Now just as in natural things, among those that act in the majority of cases, a certain gradation is observed (because the stronger the power of nature is, the more rarely does it fail of its effect), so too in the process of reason that is not accompanied by complete certitude, a certain gradation is found according as one approaches more or less closely to complete certitude. For through a process of this kind, sometimes, although science is not produced, belief or opinion is nevertheless produced on account of the probability of the propositions from which one proceeds: because reason leans wholly toward one part of a contradiction, albeit with fear of the other; and to this is ordered topics or dialectic. For the dialectical syllogism proceeds from probable premises, concerning which Aristotle treats in the book of the Topics. At other times, however, belief or opinion is not completely formed, but rather a certain suspicion, because reason does not lean wholly toward one part of the contradiction, although it is inclined more to this side than to that. And to this is ordered rhetoric. At still other times, estimation alone leans toward one part of a contradiction on account of some representation, after the manner in which a man conceives a loathing for some food if it is represented to him under the likeness of something loathsome. And to this is ordered poetics; for it belongs to the poet to lead one toward something virtuous through some fitting representation. Now all of these pertain to rational philosophy, for to lead from one thing to another belongs to reason. Finally, the third process of reason is served by the part of logic called sophistic, concerning which Aristotle treats in the book of Sophistical Refutations.
[lib. 1 l. 1 n. 7] Setting aside the other parts of logic, therefore, our present attention must be directed to the judicative part, insofar as it is handed down in the book of the Posterior Analytics. This book is divided into two parts: in the first, he shows the necessity of the demonstrative syllogism, with which this book is concerned; in the second, he determines the nature of the demonstrative syllogism itself, at We suppose ourselves to know, et cetera.
[lib. 1 l. 1 n. 8] Now the necessity of anything ordered to an end is taken from its end; and the end of the demonstrative syllogism is the acquisition of scientific knowledge; hence, if scientific knowledge could not be acquired through syllogism or argument, there would be no necessity for the demonstrative syllogism. Now \ Plato held that scientific knowledge in us is not caused by syllogism, but by the impression of ideal Forms upon our souls, from which he also said that natural forms flow out into natural things, holding these natural forms to be certain participations of Forms separated from matter. From this it followed that natural agents did not cause forms in lower things, but merely prepared matter to participate in the separated Forms. And similarly he held that scientific knowledge is not caused in us through study and exercise, but that impediments are merely removed, and man is brought back, as it were, to the recollection of things that he naturally knows from the impression of separated Forms. The teaching of Aristotle, however, is contrary on both counts. For he holds that natural forms are reduced to act by forms that exist in matter, namely, by the forms of natural agents. And similarly he holds that scientific knowledge comes to be in us in act through some knowledge pre-existing in us. And for scientific knowledge to come to be in us in this way is for it to come about through a syllogism or an argument of whatever kind. For in arguing we proceed from one thing to another.
[lib. 1 l. 1 n. 9] In order to show the necessity of the demonstrative syllogism, therefore, Aristotle sets down at the outset that knowledge is acquired in us from some pre-existing knowledge. Accordingly, he does two things: first, he establishes his proposition; second, he teaches the mode of foreknowledge, at Now it is necessary in two ways, et cetera. Concerning the first he does two things. First, he introduces a universal proposition containing what is proposed, namely, that the reception of knowledge in us comes about from some pre-existing knowledge. And therefore he says, Every teaching and every learning, and not every knowledge, because not all knowledge depends upon prior knowledge: for that would be to proceed to infinity. Every reception of learning, however, comes about from pre-existing knowledge. Now the names teaching (doctrina) and learning (disciplina) pertain to the acquisition of knowledge. For teaching is the action of one who makes something to be known, whereas learning is the reception of knowledge from another. Yet teaching and learning are not taken here insofar as they relate solely to the acquisition of scientific knowledge, but with respect to the acquisition of knowledge of whatever kind. This is clear because he manifests this proposition even in disputative and rhetorical arguments, through which scientific knowledge is not acquired. On this account, too, he does not say from pre-existing scientific knowledge or understanding, but universally from knowledge. Moreover, he adds intellective in order to exclude the reception of sensitive or imaginative knowledge. For to proceed from one thing to another belongs to reason alone.
[lib. 1 l. 1 n. 10] Second, when he says, For the mathematical sciences, et cetera, he manifests the aforesaid proposition by induction. And first, in demonstrative matters, in which scientific knowledge is acquired. Now among these the mathematical sciences hold the principal place, on account of their most certain mode of demonstration. And consequently all the other arts follow as well, because in all of them there is some mode of demonstration; otherwise they would not be sciences.
[lib. 1 l. 1 n. 11] Second, when he says, And similarly, et cetera, he manifests the same thing in disputative or dialectical arguments, which employ syllogism and induction, in both of which one proceeds from something foreknown. For in a syllogism the knowledge of some universal conclusion is received from other known universals; whereas in induction a universal is concluded from singulars, which are manifest with respect to sense.
[lib. 1 l. 1 n. 12] Third, when he says, And similarly rhetorical arguments, et cetera, he manifests the same thing in rhetorical matters, in which persuasion is effected through an enthymeme or through an example, and not through a syllogism or a complete induction, on account of the uncertainty of the matter with which rhetoric deals, namely, the singular acts of men, in which universal propositions cannot be assumed with truth. And therefore in place of the syllogism, in which there must necessarily be some universal proposition, it uses an enthymeme; and similarly in place of induction, in which a universal is concluded, it uses an example, in which one proceeds from a singular not to a universal, but to a singular. Hence it is clear that just as an enthymeme is a kind of truncated syllogism, so an example is a kind of imperfect induction. If, therefore, in syllogism and induction one proceeds from something foreknown, the same must be understood in the enthymeme and the example.
Book I, Lecture 2 — On the Twofold Foreknowledge (That It Is and What It Is) of Subject and Attribute: Unit and Triangle¶
Latin Locus: Lib. I, lect. 2 · Commentary: Notes on Lib. I, lect. 2
[lib. 1 l. 2 n. 1] After the Philosopher has shown that all learning comes about from pre-existing knowledge, he now shows what the mode of foreknowledge is. And concerning this he does two things: first, he determines the mode of foreknowledge with respect to those things that must be foreknown in order to obtain knowledge of the conclusion whose scientific knowledge is sought; second, he determines the mode of foreknowledge of the conclusion itself whose scientific knowledge is sought through demonstration, at Before one is led by induction, et cetera. Now in foreknowledge two things are included, namely, knowledge and the order of knowledge. First, therefore, he determines the mode of foreknowledge with respect to the knowledge itself; second, with respect to the order of knowledge, at Now it is possible to know, et cetera.
[lib. 1 l. 2 n. 2] Concerning the first, it must be known that that of which scientific knowledge is sought through demonstration is some conclusion in which a proper attribute is predicated of some subject, which conclusion is inferred from certain principles. And because the knowledge of simple things precedes the knowledge of composite things, it is necessary that before knowledge of the conclusion is had, the subject and the proper attribute be known in some manner. And similarly the principle from which the conclusion is inferred must be foreknown, since the conclusion becomes known from the knowledge of the principle.
[lib. 1 l. 2 n. 3] Now of these three—namely, the principle, the subject, and the proper attribute—there is a twofold mode of foreknowledge, namely, that it is (quia est) and what it is (quid est). Now it was shown in Book VII of the Metaphysics that complex expressions are not defined. For there is no definition of white man, and much less of any enunciation. Hence, since a principle is a certain enunciation, one cannot foreknow of it what it is, but only that it is true. Concerning the proper attribute, however, it can indeed be known what it is, because, as is shown in the same book, accidents in a certain manner have a definition. Yet the being (esse) of a proper attribute, as of any accident, is to inhere in a subject—which is precisely what is concluded by demonstration. Therefore one does not foreknow of a proper attribute that it is, but what it is alone. The subject, however, both has a definition and does not depend upon the proper attribute for its being; rather, its own proper being is understood prior to the very being of the proper attribute in it. And therefore concerning the subject it is necessary to foreknow both what it is and that it is, especially since the middle term of a demonstration is taken from the definition of the subject and of the proper attribute.
[lib. 1 l. 2 n. 4] On this account, therefore, the Philosopher says that it is necessary to have foreknowledge in two ways: because there are two things that are foreknown concerning those matters of which we possess foreknowledge, namely, that it is and what it is. And there are some things concerning which it is necessary first to know that they are, such as principles, of which he afterward gives an example by setting down the first of all principles, namely, that of everything either the affirmation or the negation is true.
[lib. 1 l. 2 n. 5] There are other things, however, concerning which it is necessary to understand beforehand what it is that is spoken, that is, what is signified by the name—namely, concerning proper attributes. And he does not say what it is unqualifiedly, but what it is that is spoken, because before it is known of something whether it is, one cannot properly know of it what it is: for of non-beings there are no definitions. Hence the question whether it is precedes the question what it is. Yet it cannot be shown of anything whether it is unless one first understands what is signified by the name. On this account, too, the Philosopher in Book IV of the Metaphysics, in disputing against those who deny the principles, teaches that one should begin from the signification of names. Now he gives the example of the triangle, concerning which one must foreknow that its name signifies this, namely, what is contained in its definition. For since accidents are referred to substance in a certain order, there is nothing unfitting in that which is an accident with respect to one thing being also a subject with respect to another—just as surface is an accident of bodily substance, and yet surface is the primary subject of color. That, however, which is a subject in such a way that it is an accident of nothing is substance. Hence in those sciences whose subject is some substance, that which is the subject can in no way be a proper attribute, as is the case in first philosophy and in natural science, which treats of the mobile subject. In those sciences, however, which treat of certain accidents, nothing prevents that which is taken as a subject with respect to some proper attribute from being taken also as a proper attribute with respect to a prior subject. Yet this does not proceed to infinity. For one must arrive at something primary in that science which is taken as a subject in such a way as in no way to be a proper attribute, as is clear in the mathematical sciences, which treat of continuous or discrete quantity. For in these sciences those things that are primary in the genus of quantity are supposed, such as the unit, the line, the surface, and other such things. Once these have been supposed, certain other things are sought through demonstration, such as the equilateral triangle and the square in geometry, and other such things. Now demonstrations of this sort are called, as it were, operative, such as Upon a given finite straight line to construct an equilateral triangle. Once this has been found, further proper attributes are in turn proved of it, such as that its angles are equal, or something of this sort. It is clear, therefore, that in the first mode of demonstration the triangle stands as a proper attribute, whereas in the second it stands as a subject. Hence the Philosopher here gives the example of the triangle insofar as it is a proper attribute, not insofar as it is a subject, when he says that concerning the triangle it is necessary to foreknow that it signifies this.
[lib. 1 l. 2 n. 6] He also says that there are certain things concerning which one must foreknow both, namely, what it is and that it is. And he gives the example of the unit, which is the principle in every genus of quantity. For although in some manner it is an accident with respect to substance, nevertheless in the mathematical sciences, which treat of quantity, it cannot be taken as a proper attribute, but only as a subject, since in this genus it has nothing prior to it.
[lib. 1 l. 2 n. 7] Now he shows the reason for this diversity, because there is not a like mode of manifestation for the aforesaid three—namely, principle, proper attribute, and subject. For the nature of knowledge is not the same in them: for principles are known through the act of the intellect composing and dividing, whereas the subject and the proper attribute are known through the act of apprehending what a thing is. Yet even this does not belong in the same way to the subject and to the proper attribute: since the subject is defined absolutely, because nothing outside its own essence is placed in its definition, whereas the proper attribute is defined with dependence upon the subject, which is placed in its definition. Hence, since they are not known in the same way, it is no wonder that the foreknowledge of them is diverse.
[lib. 1 l. 2 n. 8] Next, when he says, Now it is possible to know, et cetera, he determines the mode of foreknowledge on the part of the very order that foreknowledge implies. For one thing is prior to another both according to time and according to nature. And this twofold order must be considered in foreknowledge. For something is foreknown as being known prior in time; and of such things he says that some things it happens that one knows by knowing them prior in time to those things with respect to which they are said to be foreknown. Certain other things, however, are known simultaneously in time, though prior in nature; and of these he says that of certain foreknown things one receives knowledge simultaneously in time with those things with respect to which they are foreknown.
[lib. 1 l. 2 n. 9] What these latter things are he manifests by adding that of this sort are whatever things are contained under certain universals of which one possesses knowledge, that is, things clearly known to be contained under such universals. And he manifests this further by an example. For since two propositions are required in order to infer a conclusion, namely, a major and a minor, once the major proposition is known, knowledge of the conclusion is not yet had. The major proposition is therefore foreknown prior to the conclusion not only in nature, but also in time. Again, if in the minor proposition something contained under the universal proposition that is the major is introduced or assumed, concerning which it is not manifest that it is contained under this universal, knowledge of the conclusion is not yet had, because the truth of the minor proposition will not yet be certain. If, however, a term is assumed in the minor proposition concerning which it is manifest that it is contained under the universal in the major proposition, the truth of the minor proposition is evident, because what is taken under the universal carries with it the recognition of that universal, and thus knowledge of the conclusion is had immediately. As if someone were to demonstrate thus: Every triangle has three angles equal to two right angles—once this is known, knowledge of the conclusion is not yet had; but when afterward it is assumed, This figure inscribed in a semicircle is a triangle, immediately it is known that it has three angles equal to two right angles. If, however, it were not manifest that this figure inscribed in a semicircle is a triangle, the conclusion would not immediately be known once the assumption was introduced; rather, it would be necessary to seek a further middle term through which this figure would be demonstrated to be a triangle. Giving an example, therefore, of those things that are known before the conclusion prior in time, the Philosopher says that someone receiving knowledge of a conclusion through demonstration foreknew this proposition even according to time, namely, that every triangle has three angles equal to two right angles; yet in introducing this assumption, namely, that what is in the semicircle is a triangle, he knew the conclusion simultaneously—namely, in time—because what was introduced carries with it the recognition of the universal under which it is contained, so that there is no need to seek a further middle term. And therefore he adds that of certain things learning takes place in this manner, that is, knowledge of them is received per se, and there is no need to know them through some other middle term that would be ultimate in the resolution whereby mediated truths are reduced to immediate ones. Or it may be read thus: that the ultimate, that is, the extreme term that is taken under the universal middle, need not be known to fall under that universal through some other middle term. And what those things are that always carry with them the recognition of their universal he manifests by adding that of this sort are singulars, which are not predicated of any subject, since no middle term can be found between singulars and their species.
Book I, Lecture 3 — On Universal Knowledge and Recognizing the Singular Triangle Inscribed in a Semicircle¶
Latin Locus: Lib. I, lect. 3 · Commentary: Notes on Lib. I, lect. 3
[lib. 1 l. 3 n. 1] After the Philosopher has shown how certain other things must be foreknown before knowledge of the conclusion is acquired, he now wishes to show how it happens that the conclusion itself is foreknown before knowledge of it is acquired through syllogism or induction. And concerning this he does two things. First, he determines the truth of the matter, saying that before induction or syllogism is introduced to produce knowledge of some conclusion, that conclusion is in a certain manner known, and in a certain manner not known: for unqualifiedly it is unknown, yet it is known only in a qualified sense. For instance, if this conclusion is to be proved, A triangle has three angles equal to two right angles, before it was demonstrated, he who receives scientific knowledge of it through demonstration did not know it unqualifiedly, yet knew it in a qualified sense. Hence in a certain manner he foreknew it, though not unqualifiedly. Now the reason for this is that, as has already been shown, the principles must be foreknown prior to the conclusion. And principles stand to conclusions in demonstrative matters just as active causes in natural things stand to their effects (whence in Book II of the Physics the propositions of a syllogism are placed in the genus of efficient cause). Now before an effect is produced in act, it pre-exists in its active causes virtually, though not in act, which is to exist unqualifiedly. And similarly, before a conclusion is deduced from demonstrative principles, in those very foreknown principles the conclusion is foreknown virtually, though not in act: for in this way it pre-exists in them. And thus it is clear that it is not foreknown unqualifiedly, but in a qualified sense.
[lib. 1 l. 3 n. 2] Second, at But if not, et cetera, from the truth thus determined he excludes a certain difficulty that \ Plato raises in the book Meno, so entitled after the name of his disciple. Now the difficulty is as follows. He introduces someone altogether unskilled in the art of geometry who is questioned in an orderly sequence concerning the principles from which a certain geometrical conclusion is inferred, beginning from principles known through themselves (per se nota); to all of these the man ignorant of geometry answered what is true, and as the questions were carried forward step by step down to the conclusions, at every point he answered what is true. From this, therefore, Plato wishes to conclude that even those who appear unskilled in certain arts possess knowledge of them before they are instructed in them. And thus it follows that either a man learns nothing, or he learns the very things he already knew beforehand.
[lib. 1 l. 3 n. 3] Concerning this, therefore, he does four things. First, he sets forth that the aforesaid difficulty cannot be avoided unless the previously determined truth is presupposed, namely, that the conclusion which someone learns through demonstration or induction was known prior thereto not unqualifiedly, but insofar as it exists virtually in its principles—concerning which someone ignorant of the science, upon being questioned, is able to answer the truth. According to Plato’s teaching, however, the conclusion was foreknown unqualifiedly; hence it was not learned anew, but was rather brought back to memory through a certain deduction of reason—just as Anaxagoras too holds concerning natural forms that prior to generation they pre-existed in matter unqualifiedly, whereas \ Aristotle holds that they pre-exist in potency and not unqualifiedly.
[lib. 1 l. 3 n. 4] Second, when he says, For clearly not as some, et cetera, he sets down the false reply of certain thinkers to Plato’s difficulty—those, namely, who said that before a conclusion was demonstrated or learned in whatever manner, it was in no way known. For according to Plato’s difficulty an objection could be raised against them in this way: if someone were to ask an unlearned person, ‘Do you know that every dyad is even?’ and upon his saying so, that is, granting that he knows it, the questioner were to bring forward a certain dyad which the person questioned did not suppose to exist—say, that dyad which is the third part of six—it would be concluded that he knew the third part of six to be an even number, which was in fact unknown to him, though he learns it through the demonstration introduced. And thus it seems to follow that either he would not learn this, or he would learn what he already knew beforehand. In order to evade this difficulty, therefore, they solved it by saying that he who upon being questioned answered that he knew every dyad to be an even number did not assert that he knew every dyad unqualifiedly, but only every one that he knew to be a dyad. Hence, since that dyad which was brought forward was altogether unknown to him, he in no way knew that this dyad was an even number. And thus it follows that in one who knows the principles, the conclusion is in no way foreknown, neither unqualifiedly nor in a qualified sense.
[lib. 1 l. 3 n. 5] Third, at Yet they know, et cetera, he refutes this solution in the following manner. That is scientifically known of which a demonstration is possessed, or of which a demonstration is newly received—and this latter phrase is added on account of the learner, who begins to know. Now learners do not receive a demonstration concerning every dyad of which they have knowledge, but concerning every dyad unqualifiedly, and similarly concerning every number or every triangle. Therefore it is not true that one knows this concerning every number which he knows to be a number, or concerning every dyad which he knows to be a dyad, but rather concerning every one unqualifiedly. And that one does not know it concerning every number which he knows to be a number, but concerning every one unqualifiedly, he proves at For no proposition, et cetera: the conclusion agrees in its terms with the premises, for the subject and predicate of the conclusion are the major and minor extremes in the premises; yet in the premises no proposition is assumed concerning number or concerning a straight line with the qualification which you know, but rather concerning every one unqualifiedly; neither, therefore, does the conclusion of the demonstration carry the aforesaid qualification, but holds of every one unqualifiedly.
[lib. 1 l. 3 n. 6] Fourth, at Yet nothing prevents, et cetera, he sets down the true solution to the aforesaid difficulty in accordance with the truth previously determined, saying that nothing prevents what someone learns from being at first in a certain manner known and in a certain manner unknown. For there is nothing absurd if someone in a certain manner foreknows what he learns; what would be absurd is if he foreknew it in the very same manner in which he learns it. For to learn, properly speaking, is for scientific knowledge to be generated in someone. Now what is generated, prior to its generation, was neither altogether non-being nor altogether being, but in a certain manner being and in a certain manner non-being: being in potency, indeed, yet non-being in act; and this is what it is to be generated, namely, to be reduced from potency to act. Hence what someone learns was neither altogether known beforehand, as Plato held, nor altogether unknown, as was held according to the solution refuted above; rather, it was known potentially or virtually in the foreknown universal principles, while unknown in act according to proper knowledge. And this is what it is to learn: to be brought from potential, or virtual, or universal knowledge into proper and actual knowledge.
Book I, Lecture 4 — What Unqualified Scientific Knowledge Is and How Demonstration Proceeds from Primary, True, Immediate Causes¶
Latin Locus: Lib. I, lect. 4 · Commentary: Notes on Lib. I, lect. 4
[lib. 1 l. 4 n. 1] After the Philosopher has shown the necessity of the demonstrative syllogism, here he now begins to determine the nature of the demonstrative syllogism itself. And this is divided into two parts: in the first, he determines the nature of the demonstrative syllogism; in the second, he treats of the middle term from which the demonstrative syllogism proceeds, and this in the second book, at The things we seek are equal in number, et cetera. The first part is divided into two: in the first, he treats of the demonstrative syllogism absolutely; in the second, by comparing one demonstration to another, at Since demonstration is in one case universal, et cetera. The first is again divided into two: in the first, he treats of the demonstrative syllogism; in the second, he shows that in demonstrations one does not proceed to infinity, at Now every syllogism is through three terms, et cetera. The first is divided into two: in the first, he treats of the demonstrative syllogism through which we acquire scientific knowledge; in the second, he shows how a certain ignorance is also acquired in us through syllogism, at Now ignorance not according to negation, et cetera. Concerning the first he does three things: first, he treats of the demonstrative syllogism by showing what it is; second, he treats of the matter of the demonstrative syllogism by showing what and of what sort the premises are from which it is formed, at Since it is impossible, et cetera; third, he treats of its form by showing in which figure it is chiefly constructed, at Of the figures, the most scientific, et cetera. Concerning the first he does three things: first, he shows concerning the demonstrative syllogism what it is; second, he explains certain things that are placed in the definition of the demonstrative syllogism, at Now a principle of demonstration is, et cetera; third, he excludes certain errors concerning demonstration that might arise from what has been set down, at Now to some indeed, et cetera.
[lib. 1 l. 4 n. 2] Concerning the first, it must be known that in all things that exist for the sake of an end, the definition taken from the final cause is the reason for the definition taken from the material cause, and the middle term proving it: for it is on this account that a house must be made of stones and timber, namely, because it is a shelter protecting us from cold and heat. Accordingly, \ Aristotle here gives two definitions of demonstration: one of which is taken from the end of demonstration, which is to know scientifically (scire); and from this the other is concluded, which is taken from the matter of demonstration.
[lib. 1 l. 4 n. 3] Hence concerning this he does three things: first, he defines knowing scientifically itself; second, he defines demonstration through its end, which is knowing scientifically itself, at Now we say that we know, et cetera; third, from both definitions he concludes the definition of demonstration that is taken by reference to the matter of demonstration, at If, therefore, to know is as we have posited, et cetera. Concerning the first he does five things. First, he determines what sort of knowing it is that he intends to define.
[lib. 1 l. 4 n. 4] Concerning this, it must be known that we are said to know something unqualifiedly when we know it in itself. We are said to know something in a qualified sense when we know it in another thing in which it exists—either as a part in a whole, as if, knowing a house, we were said to know the wall; or as an accident in a subject, as if, knowing Coriscus, we were said to know the one approaching; or as an effect in its cause, as was said above when we noted that we foreknow the conclusion in the principles; or in any similar manner. And this is to know per accidens, because when something is known per se, we are said to know that which happens to belong to it in whatever manner. The Philosopher therefore intends to define knowing unqualifiedly, and not knowing per accidens. For this latter mode of knowing is sophistical; indeed, the sophists employ such a mode of arguing: ‘I know Coriscus; Coriscus is the one approaching; therefore I know the one approaching.’
[lib. 1 l. 4 n. 5] Second, when he says, When we judge that we know the cause, et cetera, he sets down the definition of knowing unqualifiedly itself. Concerning this it must be considered that to know something scientifically is to know it completely, and this is to apprehend its truth completely: for the principles of a thing’s being and of its truth are the same, as is clear from Book II of the Metaphysics. It is necessary, therefore, that one who knows scientifically, if he knows completely, should know the cause of the thing known. Yet if he knew the cause alone, he would not yet know the effect in act—which is to know unqualifiedly—but only virtually, which is to know in a qualified sense and, as it were, per accidens. And therefore one who knows unqualifiedly must also know the application of the cause to the effect. Moreover, because scientific knowledge is also certain knowledge of a thing, whereas what can be otherwise cannot be known by anyone with certitude, it is further necessary that what is scientifically known cannot be otherwise. Because, therefore, scientific knowledge is complete knowledge, he says: when we judge that we know the cause; because it is actual knowledge through which we know unqualifiedly, he adds: and that it is the cause of that thing; and because it is certain knowledge, he subjoins: and that it is not possible for it to be otherwise.
[lib. 1 l. 4 n. 6] Third, at It is manifest, therefore, et cetera, he manifests the definition set down from the fact that both those who know and those who do not know, yet suppose themselves to know, take knowing in the manner just stated: for those who do not know, yet suppose themselves to know, opine that they stand thus in their knowing, as has been said, whereas those who truly know genuinely do stand thus. Now this is a right manifestation of a definition. For a definition is the account (ratio) that the name signifies, as is said in Book IV of the Metaphysics; and the signification of a name must be taken from what speakers commonly intend to signify by that name—whence too in Book II of the Topics it is said that names are to be used as the many use them. Moreover, if anyone considers rightly, by this explanation what the name signifies is shown rather than something being directly defined: for he does not explain science (scientia), to which a definition in the strict sense could be assigned inasmuch as it is a species of some genus, but rather explains the act to know (scire) itself. Hence from the very start he said, We suppose ourselves to know, et cetera, and did not say, ‘To know is such and such a thing.’
[lib. 1 l. 4 n. 7] Fourth, at Wherefore, that of which, et cetera, he concludes a certain corollary from the definition set down, namely, that that of which scientific knowledge is had unqualifiedly must be necessary, that is, incapable of being otherwise.
[lib. 1 l. 4 n. 8] Fifth, at Whether indeed, et cetera, he replies to a tacit question, namely, whether there is any other mode of knowing scientifically besides the aforesaid. This he promises to state in what follows: for there is also knowing through an effect, as will become clear below; and we are also said in some manner to know the indemonstrable principles themselves, of which no cause can be taken. Nevertheless, the proper and complete mode of knowing scientifically is the one stated above.
[lib. 1 l. 4 n. 9] Next, when he says, Now we say that we know, et cetera, he defines the demonstrative syllogism by reference to its end, which is to know scientifically. Concerning this he does three things. First, he sets down that to know is the end of the demonstrative syllogism or its effect, since to know seems to be nothing other than to understand the truth of some conclusion through demonstration. Second, at By demonstration I mean, et cetera, he defines the demonstrative syllogism through this end, saying that demonstration is a scientific syllogism, that is, one producing scientific knowledge. Third, he explains what he had called scientific, at And by scientific I mean, et cetera, saying that a scientific syllogism is one according to which we know scientifically simply by possessing it—lest perhaps someone should understand a scientific syllogism to mean any syllogism that some science might happen to employ.
[lib. 1 l. 4 n. 10] Next, when he says, If, therefore, to know is, et cetera, he concludes from what has been said the definition of the demonstrative syllogism taken from its matter. And concerning this he does two things: first, he concludes it; second, he manifests it, at Now they must indeed be true, et cetera. Concerning the first he does three things. First, he sets down the consequence whereby the material definition of demonstration is concluded from what precedes, saying that if to know signifies what we have stated—namely, to know the cause of a thing, et cetera—it is necessary that demonstrative science, that is, knowledge acquired through demonstration, proceed from propositions that are true, primary, and immediate, that is, not demonstrated through any middle term, but manifest through themselves (which are indeed called immediate insofar as they lack a demonstrating middle term, and primary in relation to other propositions that are proved through them); and furthermore from propositions better known than, prior to, and causes of the conclusion.
[lib. 1 l. 4 n. 11] Second, at For thus the principles will also be, et cetera, he excuses himself from adding another clause that seemed to need appending, namely, that demonstration proceeds from proper principles. For he says that this is already understood through what has been stated: since if the propositions of a demonstration are causes of the conclusion, they must necessarily be its proper principles, for causes must be proportioned to their effects.
[lib. 1 l. 4 n. 12] Third, at For a syllogism indeed, et cetera, he manifests the necessity of the aforesaid consequence, saying that although a syllogism as such does not require the aforesaid conditions in the propositions from which it proceeds, demonstration nevertheless requires them; otherwise it would not produce scientific knowledge.
[lib. 1 l. 4 n. 13] Next, when he says, Now they must indeed be true, et cetera, he manifests the definition set down, manifesting at the same time what he had just stated, namely, that unless the aforesaid conditions belonged to demonstration, it could not produce scientific knowledge. First, therefore, he shows that demonstration always proceeds from true premises in order to produce scientific knowledge: because what is not cannot be known—for example, that the diagonal is symmetrical, that is, commensurable with the side of the square (for quantities are called incommensurable when no common measure of them can be taken, and such quantities are those that do not have to one another the ratio of a number to a number, which necessarily occurs in the case of the diagonal of a square and its side, as is clear from Book X of \ Euclid). Now what is not true, is not: for to be and to be true are convertible. Therefore what is scientifically known must be true. And thus the conclusion of a demonstration, which produces scientific knowledge, must be true, and consequently its premises as well: for what is true cannot be scientifically known from false premises, even though it can be syllogistically concluded from them, as he will show below.
[lib. 1 l. 4 n. 14] Second, at And from primary and indemonstrable, et cetera, he shows that demonstration proceeds from primary and immediate, or indemonstrable, premises. For no one can possess scientific knowledge unless he possesses a demonstration of those things of which there can be a demonstration—and I say this per se, and not per accidens. He adds this qualification because it would be possible for someone to know some conclusion without having a demonstration of those premises even though they were demonstrable, inasmuch as he might know that conclusion through other principles; and that would be per accidens. Grant, therefore, that some demonstrator syllogizes from demonstrable, or mediated, premises: either he possesses a demonstration of them, or he does not. If he does not have one, then he does not scientifically know the premises, and thus neither does he know the conclusion on account of the premises. If, however, he does have one, then since in demonstrations one cannot proceed to infinity, as he will show below, one will eventually have to arrive at certain immediate and indemonstrable principles. And thus demonstration must proceed from immediate principles, either directly or through certain intermediates. Hence in the first book of the Topics, too, it is said that demonstration proceeds from primary and true things, or from things that have derived their credence through them.
[lib. 1 l. 4 n. 15] Third, at And they must be causes, et cetera, he proves that the propositions of a demonstration are causes of the conclusion, because we know scientifically precisely when we know the causes. And from this he concludes further that they are prior and better known, because every cause is naturally prior to and better known than its effect. Moreover, the cause of a demonstrative conclusion must be better known not only with respect to the knowledge of what it is, but also with respect to the knowledge that it is: for in order to demonstrate that there is an eclipse of the sun, it does not suffice to know what the interposition of the moon is, but one must also know that the moon is interposed between the sun and the earth. And because prior and better known are spoken of in two ways—namely, in relation to us (quoad nos) and according to nature—he consequently says that the things from which demonstration proceeds are prior and better known unqualifiedly and according to nature, and not in relation to us. And in explanation of this he says that things prior and better known unqualifiedly are those that are remote from sense, such as universals, whereas things prior and better known in relation to us are those closest to sense, namely, singulars, which are opposed to universals either by the opposition of prior and posterior or by the opposition of near and remote.
[lib. 1 l. 4 n. 16] Now the contrary of this seems to be found in Book I of the Physics, where it is said that universals are prior in relation to us and posterior according to nature. But it must be replied that here he is speaking of the order of the singular to the universal unqualifiedly, whose order must be taken according to the order of sensitive and intellective knowledge in us. Now sensitive knowledge is prior in us to intellective knowledge, because intellectual knowledge proceeds in us from sense; hence the singular is prior and better known in relation to us than the universal. In Book I of the Physics, however, the order set down is not that of the universal to the singular unqualifiedly, but that of the more universal to the less universal—for instance, of animal to man—and thus in relation to us the more universal must be prior and better known. For in every generation, what is in potency is prior in time and posterior in nature, whereas what is complete in act is prior in nature and posterior in time; and knowledge of a genus is, as it were, potential in comparison to knowledge of the species, in which all the essentials of a thing are known in act. Hence in the generation of our scientific knowledge, knowing the more common comes prior to knowing the less common. Again, in the book of the Physics it is said that the natural road for us is from things better known to us; therefore demonstration does not proceed from things that are prior unqualifiedly, but from those prior in relation to us. To this it must be replied that here he speaks according as what is in sense is better known in relation to us than what is in the intellect, whereas there he speaks according as what is better known in relation to us is also in the intellect; and demonstrations do not proceed from singulars that are in sense, but from universals alone, which are in the intellect. Or else it must be said that in every demonstration one must indeed proceed from things that are better known in relation to us—not singulars, however, but universals—for nothing can be made known to us except through what is better known to us. Now sometimes what is better known in relation to us is also better known unqualifiedly and according to nature, as happens in mathematics, in which, on account of abstraction from matter, demonstrations are made solely from formal principles; and in such sciences demonstrations proceed from things that are better known unqualifiedly. At other times, however, what is better known in relation to us is not better known unqualifiedly, as happens in natural things, in which the essences and powers of things, because they exist in matter, are hidden from us, yet become known to us through those things that outwardly appear concerning them. Hence in such matters demonstrations are made for the most part through effects, which are better known in relation to us and not unqualifiedly. At present, however, he is speaking not of this latter mode of demonstration, but of the first. Finally, because in this explanation he had also omitted to manifest that demonstration proceeds from proper principles, he consequently adds that this too is obtained from what has already been set down. For by saying that demonstration is from primary things, it is already established that it is from proper principles, just as was also said above. For first and principle appear to be the same: since what is first and greatest in each genus is the cause of all the things that come after, as is said in Book II of the Metaphysics.
Book I, Lecture 5 — On the Division of Immediate Principles into Axioms, Hypotheses, and Postulates¶
Latin Locus: Lib. I, lect. 5 · Commentary: Notes on Lib. I, lect. 5
[lib. 1 l. 5 n. 1] Because the Philosopher had said above that demonstration proceeds from primary and immediate things, and these had not yet been explained by him, he therefore intends to make them clear. And this part is divided into three sections: in the first, he shows what an immediate proposition is; in the second, he shows that propositions of this sort must be better known than the conclusion, at Since one must believe and know, et cetera; in the third, he excludes certain errors that took occasion from what had been said, at Now to some indeed, et cetera. Concerning the first he does two things: first, he shows what an immediate proposition is; second, he divides it, at Of an immediate principle, et cetera. Concerning the first he proceeds as follows. First, he resumes what had been said above, namely, that a principle of demonstration is an immediate proposition: for he had also said above that demonstration proceeds from primary and immediate premises.
[lib. 1 l. 5 n. 2] Second, at An immediate proposition is, et cetera, he defines the immediate proposition, saying that an immediate proposition is one to which no other is prior. The reason for this explanation is evident from what has gone before. For it was said above that demonstration proceeds from things that are prior. Whenever, therefore, some proposition is mediated—that is, possesses a middle term through which the predicate is demonstrated of the subject—the propositions from which it is demonstrated must be prior to it: for the predicate of the conclusion inheres prior in the middle term than in the subject, and the middle term likewise inheres prior in the subject than the predicate does. It remains, therefore, that a proposition to which no other is prior is immediate.
[lib. 1 l. 5 n. 3] Third, at Now a proposition is, et cetera, he shows what a proposition is, since it is placed in the definition of an immediate proposition. And concerning this he does three things. First, he defines a proposition unqualifiedly, saying that a proposition is either part of an enunciation, in which one thing is predicated of one thing. For an enunciation has two parts, namely, affirmation and negation. Now everyone who syllogizes must put forward one of the two, and not both: for putting forward both belongs properly to one who raises a question at the outset; and thus a proposition is distinguished from a problem. Likewise, just as in a single syllogism only one thing is concluded, so too the proposition that serves as a principle of the syllogism must be one; and that proposition is one in which one thing is predicated of one thing. Hence by saying one of one, the Philosopher distinguishes a proposition from an enunciation called multiple, in which several things are predicated of one or one thing of several.
[lib. 1 l. 5 n. 4] Second, at Dialectical, et cetera, he sets down the difference between a dialectical proposition and a demonstrative proposition, saying that while every proposition takes one part of an enunciation, a dialectical proposition takes either of them indifferently. For dialectic has a path to either part of a contradiction, inasmuch as it proceeds from probable premises; hence in putting forward a premise it takes whichever part of the contradiction is granted, and puts it forward by way of a question. A demonstrative proposition, however, takes one part determinately, because the demonstrator never has a path except to demonstrating what is true; hence in putting forward a premise he always takes the true part of the contradiction. On this account, too, he who demonstrates does not ask, but assumes his premise as something known.
[lib. 1 l. 5 n. 5] Third, at Now an enunciation, et cetera, he defines the enunciation that is placed in the definition of a proposition, saying that an enunciation comprises either part of a contradiction, as is clear from what has been said. What a contradiction is he consequently shows, saying that contradiction is an opposition of which there is no intermediate in itself (secundum se). For although in privation and habit, and in immediate contraries, there is no intermediate with respect to a determinate subject, nevertheless there is an intermediate unqualifiedly: for a stone is neither blind nor sighted, and whiteness is neither even nor odd. Moreover, whatever immediacy they possess with respect to a determinate subject they possess insofar as they participate somewhat in contradiction: for privation is negation in a determinate subject, and one of two contraries likewise contains something of privation. Contradiction, however, lacks an intermediate unqualifiedly in all things, and it does not have this from another, but of itself; and on this account he says that there is no intermediate of it in itself. Consequently he also explains what each part of a contradiction is: for contradiction is the opposition of affirmation and negation; hence one part of it is affirmation, which predicates something of something, and the other is negation, which removes something from something.
[lib. 1 l. 5 n. 6] Next, when he says, Of an immediate principle, et cetera, he divides the immediate principle. And concerning this he does two things: first, he divides it; second, he subdivides it, at Of positions, one kind, et cetera. First, therefore, he says that the immediate principle of a syllogism is twofold. One is what is called a position [or thesis], which cannot be demonstrated—and on this account is called immediate—yet it is not necessary for one who is to be taught, that is, someone who is to be instructed in a demonstrative science, to possess it, that is, to conceive it in his mind or assent to it beforehand. The other is what is called a dignity [or axiom] or maximal proposition, which anyone who is to be taught must necessarily possess in his mind and assent to. And it is manifest that certain principles are of this sort, as is proved in Book IV of the Metaphysics concerning the principle that affirmation and negation are not simultaneously true, the contrary of which no one can inwardly believe in his mind even if he utters it with his mouth. And in the case of such principles we employ the aforesaid name, namely, dignity [or axiom] or maximal proposition, on account of the certitude of principles of this sort for manifesting other things.
[lib. 1 l. 5 n. 7] For the understanding of this division, it must be known that every proposition whose predicate is contained in the definition (ratio) of its subject is immediate and known through itself (per se nota), so far as it is in itself. Yet the terms of certain propositions are such that they are known to everyone, such as being, and one, and the other attributes that belong to being insofar as it is being: for being is the first conception of the intellect. Hence propositions of this sort must be held as known through themselves not only in themselves, but also in relation to everyone—for instance, that it is impossible for the same thing both to be and not to be, and that the whole is greater than its part, and the like. Consequently, all sciences receive principles of this sort from metaphysics, to which it belongs to consider being unqualifiedly and the attributes of being. Certain other propositions, however, are immediate, yet their terms are not known to everyone. Hence, although the predicate belongs to the definition of the subject, nevertheless, because the definition of the subject is not known to everyone, it is not necessary that such propositions be granted by everyone. For example, the proposition All right angles are equal, so far as it is in itself, is known through itself or immediate, because equality enters into the definition of a right angle; for a right angle is that which a straight line falling upon another straight line makes in such a way that the angles on either side are rendered equal. And therefore principles of this sort are received with a certain position. There is also another way in which certain propositions are called hypotheses (suppositiones). For there are certain propositions that cannot be proved except through the principles of another science, and therefore they must be supposed in a given science even though they are proved through the principles of another science: just as to draw a straight line from any point to any point is supposed by the geometer and proved by the natural philosopher, who shows that between any two points there lies an intervening line.
[lib. 1 l. 5 n. 8] Next, when he says, Of position, one kind is, et cetera, he subdivides the other member of the first division, namely, position: saying that one kind of position is that which assumes either part of an enunciation, namely, affirmation or negation—which he signifies when he says, such as I mean that something is or is not. And this kind of position is called a hypothesis (suppositio), because it is supposed as possessing truth. Another kind of position, however, is that which does not signify to be or not to be, namely, a definition, which is simply called a position. For the definition of the unit is posited by the arithmetician as a certain principle, namely, that the unit is that which is indivisible according to quantity. Yet a definition is not called a hypothesis (suppositio): for that alone is properly supposed which signifies what is true or false. And therefore he adds that what the unit is—which signifies neither truth nor falsity—is not the same as that the unit is, which does signify truth or falsity.
[lib. 1 l. 5 n. 9] Yet one might ask: since a definition is not a proposition signifying to be or not to be, how is it placed under the subdivision of the immediate proposition? To this it must be replied that in making the subdivision he does not resume the immediate proposition to subdivide it, but rather the immediate principle; and not only a proposition, but also a definition, can be called a principle of a syllogism. Or else it may be said that although a definition in itself is not a proposition in act, it is nevertheless virtually a proposition, because once the definition is known, it is evident that the definition is truly predicated of its subject.
Book I, Lecture 6 — Against Those Who Deny Demonstration or Posit Circular Demonstration¶
Latin Locus: Lib. I, lect. 6 · Commentary: Notes on Lib. I, lect. 6
[lib. 1 l. 6 n. 1] After the Philosopher has shown what the immediate principles are, here he determines the nature of our knowledge of them. And concerning this he does two things: first, he shows that the immediate principles are better known than the conclusion; second, he shows that the falsity of their contraries must also be most certain, at Moreover, not only, et cetera. Concerning the first he does three things.
[lib. 1 l. 6 n. 2] First, he states his intention, saying that because we believe a concluded truth and scientifically know it by possessing a demonstrative syllogism—and this insofar as we scientifically know the demonstrative syllogism—it is necessary not only to foreknow the first principles prior to the conclusion, but also to know them better than the conclusion. Now he adds either all or some, because certain proximate principles require proof in order to become known, and prior to being proved they are no better known than the conclusion: just as the proposition that the exterior angle of a triangle is equal to the two opposite interior angles, prior to being proved, is just as unknown as the proposition that a triangle has three angles equal to two right angles; whereas certain other principles, as soon as they are proposed, are better known than the conclusion. Or in another way: certain conclusions are exceedingly well known inasmuch as they are grasped through sense, such as that the sun is eclipsed; hence the principle through which this is proved—namely, that the moon is interposed between the sun and the earth—is not unqualifiedly better known to us, although it is better known in the order of reason proceeding from cause to effect. Or in yet another way: he says this because he had also stated above that certain principles are known prior in time to the conclusion, whereas certain others become known simultaneously in time with the conclusion.
[lib. 1 l. 6 n. 3] Second, at For always that on account of which, et cetera, he proves his proposition in two ways. First, by a direct argument, as follows: Always that on account of which each thing is such, is itself more so—for example, if we love someone on account of another, as when we love a teacher on account of a student, we love the student more. Now we know conclusions and believe them on account of the principles. Therefore we know the principles more and believe them more than we do the conclusion.
[lib. 1 l. 6 n. 4] Now with regard to this argument it must be observed that a cause is always more potent than its effect. Whenever, therefore, cause and effect share a common name, that name is predicated more of the cause than of the effect, just as fire is hotter than the things heated by fire. Sometimes, however, cause and effect do not share a common name, and then, although the name of the effect does not belong to the cause, something worthier belongs to it instead: just as, although heat does not formally exist in the sun, there is nevertheless in it a certain power that is the principle of heat.
[lib. 1 l. 6 n. 5] Next, when he says, Now one cannot believe, et cetera, he proves the same thing by an argument leading to the impossible, which runs as follows. The principles are foreknown prior to the conclusion, as was established above; and thus at the time when the principles are known, the conclusion is not yet known. If, therefore, the principles were not better known than the conclusion, it would follow that a man knew the things he does not know either more than or equally with the things he does know. Yet this is impossible. Therefore it is impossible that the principles should not be better known than the conclusion. The text is expounded thus: neither one who knows scientifically, nor anyone better disposed in knowing than one who knows scientifically, should there happen to be anyone such (which he adds on account of one who has understanding of the principles, concerning whom this has not yet been made manifest), can believe more in things that do not happen—namely, to be known by him—than in the things he already knows. Yet this will happen unless one among those who believe a conclusion through demonstration foreknows, that is, knows better, the principles. In the Greek text this reads more plainly thus: For it is impossible to believe more than the things one knows those things which one neither happens to know scientifically nor is better disposed toward than if one happened to know them scientifically.
[lib. 1 l. 6 n. 6] Third, at For since it is necessary to believe more, et cetera, he explains what he had said, stating that the assertion that it is necessary to believe either all or some of the principles more than the conclusion must be understood of one who is to acquire scientific knowledge of it through demonstration. For if the conclusion were known from elsewhere, such as through sense, nothing would prevent the principles from not being better known than the conclusion in that respect.
[lib. 1 l. 6 n. 7] Next, when he says, Moreover, one must not only, et cetera, he shows that one must not only know the principles better than the conclusion to be demonstrated, but also that nothing may be more certain to him than that the opposites of the principles are false. And this is because one who possesses scientific knowledge must be unshakeable with respect to the principles, assenting to them most firmly. For whoever doubts the falsity of one of two opposites cannot cleave firmly to its opposite, because he always fears lest the other opposite be true.
Book I, Lecture 17 — That No Particular Science Demonstrates Its Own Proper Principles, Which Belong to First Philosophy¶
Latin Locus: Lib. I, lect. 17 · Commentary: Notes on Lib. I, lect. 17
[lib. 1 l. 17 n. 1] Above, the Philosopher had shown that demonstration does not proceed from extraneous principles; here, however, he shows that it does not proceed from common principles alone. And concerning this he does two things: first, he proves his proposition; second, he draws a certain conclusion from what has been said, at Now if this is manifest, et cetera. Concerning the first he does three things. First, he states his intention, saying that because it is manifest that it is impossible to demonstrate each thing through anything whatsoever, rather demonstration must proceed from the principles of each thing in such a way that what is demonstrated belongs to it insofar as it is that thing—that is, the principles of demonstration must belong per se to that which is demonstrated; if, I say, this is so, then in order for something to be scientifically known it does not suffice that it be demonstrated from true and immediate premises, but it must furthermore be demonstrated from proper principles.
[lib. 1 l. 17 n. 2] Second, at For it is possible to demonstrate thus, et cetera, he proves his proposition, namely, that it does not suffice to demonstrate something from true and immediate premises alone, because in that case it would be possible to demonstrate something in the manner in which Bryson demonstrated the tetragonism, that is, the squaring of the circle, showing that some square is equal to a circle through certain common principles in the following way: in whatever genus one can find something greater than and something less than a given thing, in that same genus one can also find something equal to it; now in the genus of squares one can find some square less than a circle—namely, one inscribed within the circle—and some square greater than the circle, within which the circle is inscribed; therefore one can find some square equal to the circle. Now this proof proceeds according to something common: for equal, greater, and less extend beyond the genus of quadrangle and circle. Hence it is clear that arguments of this sort demonstrate according to something common, because the middle term belongs to something wider than that of which the demonstration is made; and therefore arguments of this sort apply to other things as well, and do not belong to those things concerning which they are given as proximate principles. Consequently it is clear that one who knows through arguments of this sort does not know a thing insofar as it is what it is—that is, per se—but only per accidens: for if the demonstration were per se, it would not apply to another genus. For we know each thing per accidens whenever we do not know it insofar as it arises from its own principles, that is, insofar as it arises from per se principles—just as having three angles equal to two right angles belongs per se to a triangle, that is, insofar as it arises from its own principles. Wherefore, if the middle term assumed belonged per se to the conclusion, it would necessarily stand in the same proximity—that is, be proximate in genus—to the conclusion.
[lib. 1 l. 17 n. 3] Third, at Even if it is not so, et cetera, he removes a certain doubt. For sometimes it happens that the middle term of a demonstration is not in the same genus as the conclusion. How this happens he shows by saying: if the middle term is not in the same proximity to the conclusion, but stands as when something in harmonics, that is, in music, is demonstrated through arithmetic, it is indeed true that things of this sort are likewise demonstrated. For a demonstration is made in a lower science through the principles of a superior science, as has been shown, just as in the superior science through the principles of the superior science. Yet there is this difference: it belongs to the one science—namely, the lower—to know the that (quia) alone, for the \ subject-genus of the lower science is other than the subject-genus of the superior science from which the principles are taken; whereas to know the reason why (propter quid) belongs to the superior science, to whose subject those proper attributes belong per se. For since the proper attribute inheres in the subject on account of the middle term, that science will consider the reason why to which the middle term pertains, of which the proper attribute demonstrated holds per se. If, however, the subject pertains to another science, that science will not possess the reason why, but only the that, nor will the proper attribute demonstrated of such a subject belong to it per se, but through an extraneous middle term. If, on the other hand, both the middle term and the subject pertain to the same science, then it will belong to that science to know both the that and the reason why. Once this doubt has been removed, he goes on to draw the conclusion chiefly intended, saying that from what has been said it is clear that one cannot demonstrate each thing unqualifiedly—that is, in just any manner—but only insofar as it is demonstrated from the proper principles of each thing; even though the proper principles of the individual sciences have something common that is prior to them.
[lib. 1 l. 17 n. 4] Next, when he says, Now if this is manifest, et cetera, he draws a certain conclusion following from what has been said. And concerning this he does three things. First, he introduces the conclusion, saying that if this is true—namely, that demonstrations in the individual sciences are not made from common principles alone, and moreover that the principles of the sciences have something prior to them that is common—it is manifest that it does not belong to each particular science to demonstrate its own proper principles. For those prior principles through which the proper principles of the individual sciences could be proved are the common principles of all things; and that science which considers common principles of this sort is proper with respect to all—that is, it stands to the things common to all just as the other, particular sciences stand to the things proper to each. For just as arithmetic, since its subject is number, considers the attributes proper to number, so too first philosophy, which considers all principles, has for its subject being, which is common to all things, and therefore considers the attributes proper to being—which are common to all—as proper to itself.
[lib. 1 l. 17 n. 5] Second, when he says, For indeed he who knows from higher causes, et cetera, he shows the pre-eminence of this science which considers common principles—namely, first philosophy—over the others. For that through which something is proved must always be more scientifically known or better known. For he who knows something from higher causes must understand those causes more deeply, because he knows from things that are prior unqualifiedly, inasmuch as he does not know causes from their effects: for when someone knows causes from effects, he does not understand from things prior and better known unqualifiedly, but from things better known and prior in relation to us. When, however, the principles of a lower science are proved from the principles of a higher science, one does not proceed from effects to causes, but conversely. Hence such a process must be from things prior and better known unqualifiedly. Therefore what belongs to a higher science, from which what belongs to a lower science is proved, must be more scientifically known; and that through which all other things are proved while it itself is not proved from anything prior must be most scientifically known of all. Consequently, a higher science will be science to a greater degree than a lower one, and the supreme science, namely, first philosophy, will be science in the highest degree.
[lib. 1 l. 17 n. 6] Third, at Yet demonstration, et cetera, he returns to the principal conclusion, saying that demonstration does not cross over into another genus except in the way already stated, namely, as geometric demonstration extends to lower sciences—whether mechanical arts, which employ measurements, or speculative sciences, such as the sciences that treat of vision, like optics, which treats of the visual line—and similarly with arithmetic in comparison to harmonics, that is, music.
Book I, Lecture 18 — On the Three Elements Required in Every Demonstrative Science: Subject-Genus, Proper Attributes, and Common Axioms¶
Latin Locus: Lib. I, lect. 18 · Commentary: Notes on Lib. I, lect. 18
[lib. 1 l. 18 n. 1] After he has shown that demonstration proceeds not from common principles alone, but from proper principles, here, in order to make clear what has preceded, he determines the nature of proper and common principles. And concerning this he does two things. First, he shows the necessity of this determination, saying that it is difficult to recognize whether we know from proper principles (which alone is truly to know scientifically) or not from proper principles. For many suppose themselves to know scientifically whenever they possess a syllogism drawn from certain premises that are true and primary. Yet this is not true; rather, in order for us to know scientifically, the principles must be proximate to the things that are to be demonstrated (which are here called primary, just as above they were called extremes), or else proximate to the first indemonstrable principles.
[lib. 1 l. 18 n. 2] Second, at Now I call principles, et cetera, he determines the nature of proper and common principles. And concerning this he does two things: first, he treats of proper and common principles; second, he shows how the demonstrative sciences stand in relation to principles of this sort, at That it is impossible, et cetera. Concerning the first he does two things: first, he distinguishes principles from non-principles; second, he distinguishes principles from one another, at Now among the things used, et cetera.
[lib. 1 l. 18 n. 3] Concerning the first he does two things. First, he shows what principles are, saying that the principles in each genus are those things which, though true, cannot be demonstrated—either unqualifiedly, if they are first principles, or at least not within that science in which they are assumed as principles. And he adds though true in order to distinguish them from falsehoods, which are not demonstrated in any science.
[lib. 1 l. 18 n. 4] Second, at Now what both the primary things, et cetera, he shows the agreement and the difference between principles and non-principles. For principles agree with non-principles in this, that concerning both one must assume, as it were by supposing, what they signify—both the primary things, that is, the principles, and the things derived from them, that is, the things proved from the principles: because the what-it-is belongs properly to the science that treats of substance, namely, first philosophy, from which all other sciences receive this. Yet principles and the things derived from principles differ in this, that concerning the principles one must assume by supposing that they are, whereas concerning the other things, which are derived from the principles, one must demonstrate that they are. Thus in mathematics one assumes by supposing both what the unit is, which is a principle, and what the straight is, and what the triangle is, which are not principles, but proper attributes; yet that the unit exists, or that magnitude exists, the mathematician assumes as principles, whereas he demonstrates the others, namely, the things derived from the principles—for he demonstrates the existence of the equilateral triangle, the right angle, and also that this line is straight.
[lib. 1 l. 18 n. 5] Next, when he says, Now among the things used, et cetera, he distinguishes principles from one another: and first, proper principles from common principles; second, common principles from one another, at Now neither is a hypothesis, et cetera. The first part is divided into two: in the first, he divides proper and common principles; in the second, he clarifies a point that could occasion doubt, at Yet nothing prevents certain sciences, et cetera.
[lib. 1 l. 18 n. 6] Concerning the first he does three things. First, he sets down the division, saying that of the principles we use in the demonstrative sciences, some are proper to each individual science, while others are common. And because this might appear contrary to what was shown above—namely, that the demonstrative sciences do not proceed from common principles—he therefore adds that common principles are assumed in each demonstrative science according to analogy, that is, insofar as they are proportioned to that science. And this is what he goes on to explain by adding that it is useful to assume principles of this sort in the sciences to the extent that they pertain to the \ subject-genus contained under that science.
[lib. 1 l. 18 n. 7] Second, at Proper principles are, et cetera, he gives examples of both, saying that proper principles are, for instance, that a line is of such a sort, or what the straight is—for the definition of both the subject and the proper attribute is held as a principle in the sciences. Common principles, on the other hand, are, for instance, If equals be subtracted from equals, the remainders are equal, and other \ common conceptions of the mind.
[lib. 1 l. 18 n. 8] Third, at Now it is sufficient, et cetera, he shows how the demonstrative sciences employ the aforesaid principles. And first, concerning the common principles, he says that it is sufficient to assume each of these common principles to the extent that it pertains to the subject-genus with which the science deals. For geometry will accomplish the very same thing if it does not assume the aforesaid common principle in its full generality, but only in magnitudes, and arithmetic in numbers alone. For geometry will be able to draw its conclusion just as well if it says, If equal magnitudes be subtracted from equal magnitudes, the remaining magnitudes are equal, as if it said, If equals be subtracted from equals, the remainders are equal. And the same must be said concerning numbers.
[lib. 1 l. 18 n. 9] Second, at Now proper principles are, et cetera, he shows how the demonstrative sciences employ proper principles, saying that proper principles are the things that are supposed to exist in the sciences—namely, the subjects concerning which the science investigates the attributes that belong to them per se. Thus arithmetic considers units, and geometry considers signs, that is, points and lines: for the aforesaid sciences suppose that these exist and that they are such and such—that is, they suppose concerning them both that they are and what they are. Concerning the proper attributes, however, the aforesaid sciences suppose what each signifies: thus arithmetic supposes what the even is, and what the odd is, or what a square or cubic number is; and geometry supposes what the rational is in lines—for a line is called rational concerning which we can reckon by reference to a given line, and every line commensurable with a given line is of this sort, whereas one not commensurable with it is called irrational or surd; similarly geometry supposes what the deflected or the curved is. Yet concerning all the aforesaid proper attributes, the aforesaid sciences demonstrate that they exist through the common principles and from those principles that have been demonstrated from them. And what has been said concerning geometry and arithmetic must also be understood concerning astronomy. For every demonstrative science is concerned with three things: one of which is the subject-genus, whose per se proper attributes are investigated; another is the common axioms (communes dignitates), from which as from primary premises it demonstrates; and the third is the proper attributes (passiones), concerning which each science assumes what they signify.
[lib. 1 l. 18 n. 10] Next, when he says, Yet nothing prevents certain sciences, et cetera, he clarifies something that might occasion doubt. For because he had said that sciences suppose concerning principles that they are, concerning proper attributes what they are, and concerning subjects both, someone might believe that explicit mention must always be made of all of these. Hence he removes this doubt, saying that nothing prevents certain sciences from passing over some of the aforesaid, that is, omitting express mention of them: as when a science makes no explicit mention of supposing that its subject-genus exists if its existence is manifest—since it is not equally manifest of all things that they exist, as for instance that number exists and that hot or cold exists, the one being close to reason, the other to sense. Similarly, certain sciences do not set down by express mention what their proper attributes signify; just as it is not always necessary for sciences to make explicit mention of the common principles, because they are already known. Nevertheless, the aforesaid three are by nature presupposed in every science.
Book I, Lecture 19 — On Common Conceptions of the Mind, Hypotheses, Postulates, and Why the Geometer Does Not Demonstrate of Lines Drawn in Dust¶
Latin Locus: Lib. I, lect. 19 · Commentary: Notes on Lib. I, lect. 19
[lib. 1 l. 19 n. 1] After \ Aristotle has divided common principles from proper principles, here he distinguishes common principles from one another. And this is divided into three parts: in the first, he sets down the distinction of common principles from one another; in the second, he shows the difference between a definition and a certain kind of common principles, at Terms, therefore, are not, et cetera; in the third, he excludes a certain error, at For Forms to exist, therefore, et cetera. Concerning the first he does two things: first, he distinguishes \ common conceptions of the mind from postulates or hypotheses; second, he distinguishes postulates and hypotheses from one another, at Whatever things, therefore, et cetera.
[lib. 1 l. 19 n. 2] Concerning the first, it must be considered that common conceptions of the mind have something in common with the other principles of demonstration, and something proper to themselves. What they have in common is that both they and the other principles must necessarily be true per se. What is proper to these principles, however, is that not only must they necessarily be true per se, but it must also necessarily be seen that they are true per se. For no one can inwardly hold the contrary of them in opinion.
[lib. 1 l. 19 n. 3] He says, therefore, that a principle which not only must necessarily hold through itself, but must furthermore necessarily be seen to do so—namely, a common conception of the mind or axiom (dignitas)—is neither a postulate (petitio) nor a hypothesis (suppositio). And he proves this as follows. A postulate and a hypothesis can be confirmed by outward discourse, that is, by some argumentation. A common conception of the mind, however, is not directed to outward discourse, because it cannot be proved through any argumentation, but is directed to that discourse which is within the soul, because it immediately becomes known by the light of natural reason. That it is not directed to outward discourse is clear from the fact that no syllogism is constructed to prove common conceptions of the mind of this sort. And that principles of this sort are known not through outward discourse, but through inward reason, he proves from the fact that objection can always be raised against outward discourse, whether truly or apparently, whereas against inward reason it is not always possible to raise an objection. And the reason for this is that nothing is so true that it cannot be denied in spoken words: for even that most certain principle, that the same thing cannot simultaneously be and not be, some have denied with their mouths. Yet certain truths are so evident that their opposites cannot be grasped by the intellect; and therefore they cannot be opposed by inward reason, but only by outward discourse, which consists in spoken words. And of this sort are the common conceptions of the mind.
[lib. 1 l. 19 n. 4] Next, when he says, Whatever things, therefore, et cetera, he distinguishes hypotheses (suppositiones) and postulates (petitiones) from one another. Now it must be known that they have something in common, and in something they differ. What they have in common is that, although they are demonstrable, the demonstrator nevertheless assumes them without demonstrating them—and chiefly because they are not demonstrable through his own science, but through another, as was said above. Hence they are also reckoned among immediate principles, because the demonstrator employs them without a middle term, inasmuch as they have no middle term within that science. Yet they differ from one another: for if such a proposition already seems probable to the learner to whom the demonstration is addressed, it is called a hypothesis (suppositio); and thus a hypothesis is spoken of not unqualifiedly, but relatively to a given learner. If, however, the learner holds neither the same opinion nor the contrary opinion concerning it, the demonstrator must ask him to grant it, and then it is called a postulate (petitio). And if the learner holds the contrary opinion, then it will be a question, concerning which they must first dispute. Nevertheless, it is common to them all that the demonstrator uses each of them without demonstrating it, even though it is demonstrable.
[lib. 1 l. 19 n. 5] Next, when he says, Terms, therefore, are not, et cetera, he distinguishes definitions from hypotheses (suppositiones) by two arguments, the second of which begins at Furthermore, every postulate, et cetera. Concerning the first he does two things. First, he sets down the argument, which runs as follows: every postulate or hypothesis asserts that something is or is not; terms, however—that is, definitions—do not assert that anything is or is not; therefore terms, taken by themselves, are neither hypotheses nor postulates. When assumed within propositions, however, they become hypotheses, as when it is said, Man is a mortal rational animal. Yet terms taken by themselves need only be understood; and to understand is not to suppose, any more than to hear is. Rather, those things are supposed from whose being—that is, from whose truth as premises—the conclusion comes about in virtue of the fact that they are, that is, on account of the premises.
[lib. 1 l. 19 n. 6] Second, at Nor does the geometer, et cetera, he removes a certain objection. For some claimed that the geometer employs a false hypothesis whenever he says that a line is one foot long when it is not one foot long, or that a line drawn in the dust is straight when it is not straight. Yet Aristotle replies that the geometer does not suppose anything false on this account. For since the geometer demonstrates nothing concerning particulars, but only concerning universals, as was said above, whereas these drawn lines are certain particulars, it is manifest that he demonstrates nothing either concerning these lines or from them; rather, he employs them as sensible examples of the universals that are understood through these examples, concerning which and from which he demonstrates.
[lib. 1 l. 19 n. 7] Next, when he says, Furthermore, every postulate, et cetera, he sets down the second argument, which runs as follows: every hypothesis or postulate holds either of the whole or in part, that is, is either a universal or a particular proposition; definitions, however, are neither of these, because in them nothing is posited or predicated either universally or particularly; therefore definitions are neither hypotheses nor postulates.
[lib. 1 l. 19 n. 8] Next, when he says, For Forms to exist, therefore, et cetera, he shows from what has gone before that it is not necessary to posit Ideas, as \ Plato did. For it was shown above that demonstrations are concerned with universals, and in this way are concerned with eternal truths. Therefore, in order for demonstration to exist, it is not necessary that there be Forms, that is, Ideas, or any one thing existing apart from the many, just as the Platonists posited separated mathematical objects alongside the Ideas so that demonstrations might concern eternal things. Rather, if there is to be demonstration, it is necessary that there be one in many and predicated of many: because there will be no universal unless there is one predicated of many, and if there is no universal, there will be no middle term of demonstration, and hence no demonstration. And that the middle term of demonstration must be universal is evident from the fact that the middle term of demonstration must be one and the same thing predicated of several things not equivocally, but according to the very same intelligible nature (ratio)—which is the essence of the universal; whereas if it were equivocal, a fallacy could arise in arguing.
Book I, Lecture 42 — How One Science Is More Certain and Prior to Another: The Unit Without Position and the Point Having Position¶
Latin Locus: Lib. I, lect. 42 · Commentary: Notes on Lib. I, lect. 42
[lib. 1 l. 42 n. 1] After the Philosopher has assigned the reason for the unity and diversity of the sciences on the part of the \ subject-genus and on the part of the principles, here he pursues both further: and first, concerning the subjects of which scientific knowledge is had; second, concerning the principles, at Now whether the principles are the same, et cetera. Concerning the first, it must be considered that above he had laid down two conditions for the genus that serves as the subject of a science: one of which is that it be composed of primary elements, and the other is that its parts and proper attributes belong to it per se. Now one of these conditions is lacking in things that happen by fortune, because they do not come about per se, but per accidens and beside intention, as is proved in Book II of the Physics. The other condition is lacking in the things known through sense, even though they come first in our cognition. And therefore, first, he shows that there is no scientific knowledge of things that happen by fortune; second, he shows that there is no scientific knowledge of things insofar as they are known through sense, at Nor is it possible to know scientifically through sense, et cetera.
[lib. 1 l. 42 n. 2] First, therefore, he says that demonstrative science cannot be concerned with what happens by fortune, which comes about per accidens. And he proves this as follows. Every demonstrative syllogism proceeds either from necessary propositions or from propositions that are true for the most part. Now from necessary propositions a necessary conclusion follows, as was proved above; and similarly from propositions that are true for the most part there follows a conclusion that is true for the most part—or perhaps even necessary, insofar as what is necessary can follow from what is contingent just as what is true can follow from what is false. Never, however, from propositions that are true for the most part does there follow a proposition that is true only in the minority of cases, because otherwise it would follow that at times the premises would be true and the conclusion false, which is impossible, as was shown above. It is necessary, therefore, that the conclusion of a demonstrative syllogism be either necessary or true for the most part. Yet what happens by fortune is neither necessary nor for the most part, but comes to pass in the minority of cases, as is proved in Book II of the Physics. Therefore demonstrative science cannot be concerned with what happens by fortune.
[lib. 1 l. 42 n. 3] Now it must be considered that there can indeed be demonstration concerning things that happen for the most part, insofar as there is something of necessity in them. Yet the necessary, as is said in Book II of the Physics, exists differently in natural things, which are true for the most part and fail in a minority of cases, and in the disciplines, that is, in mathematical matters, which are always true. For in the mathematical disciplines necessity is a priori; whereas in natural things it is a posteriori (from what is nevertheless prior according to nature), namely, from the end and the form. Hence \ Aristotle there teaches that one should show the reason why (propter quid) thus: if this is to come to be—for example, that an olive tree be generated—it is necessary that this pre-exist, namely, olive seed; yet from olive seed an olive tree is not generated of necessity, because generation can be hindered through some corruption. Consequently, if a demonstration is framed from what is prior in the order of generation, it will not conclude of necessity—unless perhaps we take this very truth as necessary, that olive seed is generative of an olive tree for the most part, because it accomplishes this in accordance with the property of its nature unless it is impeded.
[lib. 1 l. 42 n. 4] Next, when he says, Nor through sense, et cetera, he shows that scientific knowledge is not of things insofar as they are known according to sense. And concerning this he does two things: first, he shows that scientific knowledge does not consist in sense; second, he shows how sense is ordered to scientific knowledge, at Nevertheless certain omissions are reduced, et cetera. Concerning the first he does two things: first, he shows that scientific knowledge is not through sense; second, he sets scientific knowledge above sense, at Now the universal is precious, et cetera. Concerning the first he does two things: first, he establishes the truth; second, he excludes the error of certain thinkers, at Yet it is manifest, et cetera.
[lib. 1 l. 42 n. 5] First, therefore, he says that just as there is no scientific knowledge of things that happen by fortune, so too scientific knowledge does not consist in the cognition that takes place through sense. And he proves this as follows. It is manifest that sense knows something of such a quality, and not this [substance]: for the per se object of sense is not substance and the what-it-is, but some sensible quality, such as hot, cold, white, black, and other things of this sort. Now qualities of this sort affect certain singular substances existing in a determinate place and time; hence it is necessary that what is sensed be a this-something, namely, a singular substance, and exist somewhere and now, that is, in a determinate place and time. From this it is clear that what is universal cannot fall under sense: for what is universal is not restricted to here and now, since otherwise it would no longer be universal; for we call that universal which is always and everywhere.
[lib. 1 l. 42 n. 6] Now this must not be understood by way of affirmation, as though it belonged to the essence (ratio) of the universal, or of that which is universal, to exist positively always and everywhere. For if it belonged to the essence of that nature which is universal—say, to the essence of man or animal—to exist always and everywhere, then every singular man or animal would have to exist always and everywhere, because the essence of man and of animal is found in every one of the singulars. And if it belonged to the very notion of universality itself, in the way that it belongs to the notion of a genus to contain species under itself, it would follow that nothing would be universal that was not actually found always and everywhere; and according to this, olive tree would not be a universal, because it cannot be found in every land. Rather, this must be understood by way of negation or abstraction, namely, in that the universal abstracts from every determinate time and place. Hence, so far as lies in itself, just as it is found in any given instance at one place or time, so it is suited by nature to be found in all. Thus, therefore, it is clear that the universal does not fall under sense. Since demonstrations, therefore, are chiefly universal, as was shown above, it is manifest that scientific knowledge acquired through demonstration does not consist in the cognition of sense.
[lib. 1 l. 42 n. 7] Next, when he says, Yet it is manifest that even if, et cetera, he excludes the error of certain thinkers who believed that scientific knowledge consists in the very perception of sense itself—and this argument appears to be directed against those who held that the intellect does not differ from sense, and consequently that there is no cognition other than sensitive cognition, as is recounted in Book III of De Anima and in Book IV of the Metaphysics. To exclude this, he says that even if we could perceive by sense that a triangle has three angles equal to two right angles, we would still need to seek a demonstration in order to possess scientific knowledge, nor would we scientifically know it through the perception of the senses: because sense is of singulars, whereas scientific knowledge consists in our knowing the universal, as has been shown. And because he had given an example of something that cannot in fact be perceived by sense, for greater clarity he sets down a further example drawn from things that are sensible, namely, an eclipse of the moon, which comes about from the opposition of the earth interposed between the sun and the moon so that the brightness of the sun cannot reach the moon on account of the earth’s shadow, upon entering which the moon is eclipsed. Suppose, therefore, that someone were standing upon the moon itself, and perceived by sense the interposition of the earth through its shadow: he would indeed perceive by sense that the moon was at that moment losing its light because of the earth’s shadow, yet he would not on that account wholly know the cause of the eclipse. For the per se cause of an eclipse is that which causes the eclipse universally. Now the universal is not known by sense; rather, from several singulars that have been seen, in which, after being considered many times, the same thing is found to occur, we receive universal knowledge. And thus through the universal cause we demonstrate something universally, of which there is scientific knowledge.
[lib. 1 l. 42 n. 8] Next, when he says, Now the universal is more precious, et cetera, he shows that scientific knowledge excels sense. For it is manifest that knowledge through a cause is nobler; now a per se cause is a universal cause, as has just been said, and therefore knowledge through a universal cause, such as scientific knowledge is, is precious. And because it is impossible to apprehend such a universal cause through sense, it follows that scientific knowledge, which manifests the universal cause, is more precious not only than all sensitive cognition, but also than every other intellective cognition—provided it concerns things that have a cause—because to know something scientifically through its universal cause is nobler than to apprehend in any other manner whatever has a cause without knowing its cause. Concerning primary truths, however, which do not have a cause, the case is different: for they are understood through themselves, and such knowledge of them (intelligentia) is more certain than all scientific knowledge, because it is from such understanding that scientific knowledge derives its certitude.
[lib. 1 l. 42 n. 9] Finally, he concludes his principal proposition, namely, that it is impossible to know anything demonstrable scientifically through sense—unless perhaps someone should use the name sense equivocally, calling demonstrative science sense because demonstrative science is determinately of one thing with certitude, just as sense is; on account of which certain estimations are also sometimes called sciences.
[lib. 1 l. 42 n. 10] Next, when he says, Nevertheless certain omissions in problems are reduced, et cetera, he shows how sense is ordered to scientific knowledge. For certain problematic doubts are traced back, as to their cause, to a failing of sense. For there are certain things about which we would no longer inquire in doubt if we had seen them—not indeed because scientific knowledge consists in seeing, but insofar as from the things seen we grasp, by way of experience (experimentum), the universal with which scientific knowledge is concerned. For instance, if we saw glass perforated throughout, and saw how light passes through the perforations of the glass, we would know the reason why glass is transparent. Here he employs this example in accordance with the opinion of those who held that light is a body and that certain bodies are transparent on account of minute passages called pores, which we cannot discern by sight owing to their smallness, and so we wonder why glass is transparent; and a similar example could be given for any things whatsoever that have a hidden sensible cause. And because he had said that scientific knowledge of such a thing does not consist in the act of seeing itself, he makes clear that this is true: for in seeing we cognize each of the singulars separately, whereas in knowing scientifically we must understand all of them together in the universal, namely, by understanding that the matter stands thus in every instance. For we see one by one in diverse pieces of glass, whereas we acquire scientific knowledge concerning all glass, that it is such.
Book II, Lecture 20 — On the Origin of the Habit of First Principles (Intellectus) from Sense, Memory, and Experience¶
Latin Locus: Lib. II, lect. 20 · Commentary: Notes on Lib. II, lect. 20
[lib. 2 l. 20 n. 1] After the Philosopher has shown how that which serves as the principle of demonstration as its middle term—namely, what a thing is and the reason why—becomes known, here he shows how the first common principles of demonstration become known. And first, he states what his intention concerns; second, he carries out what is proposed, at That indeed, therefore, it is impossible, et cetera. First, therefore, he says that from what follows it will become manifest, concerning the first indemonstrable principles, how the knowledge of them comes about in us and by what habit they are known—observing this order, however, that we first raise the difficulties surrounding this matter.
[lib. 2 l. 20 n. 2] Next, when he says, That indeed, therefore, it is impossible to know scientifically, et cetera, he carries out his purpose. And concerning this he does two things: first, he raises the difficulties; second, he solves them, at It is necessary, therefore, that we possess, et cetera. Concerning the first he does three things: first, he premises a truth from which the necessity of this inquiry is shown; second, he raises the questions, at Concerning the immediate principles, et cetera; third, he brings forward objections bearing on the question, at Now if indeed, et cetera. First, therefore, he says that it has already been stated above that it is impossible to know anything scientifically through demonstration unless the first immediate principles are foreknown; and therefore, for the science that treats of demonstration, it is useful that it be known how the first principles become known.
[lib. 2 l. 20 n. 3] Next, when he says, Now regarding the knowledge of the immediates, et cetera, he raises three difficulties concerning the aforesaid knowledge of principles. The first of these is whether the knowledge of all immediate principles is the same or not. The second difficulty is whether there is scientific knowledge (scientia) of all immediate principles, or of none, or scientific knowledge of some and another kind of cognition of others. The third question is whether the habits of these principles come to be in us after not having previously existed in us, or whether they have always been in us while nevertheless remaining hidden from us.
[lib. 2 l. 20 n. 4] Next, when he says, Now if indeed we possess them, et cetera, he raises objections regarding the last question, to which the others are ordered. And first he objects against the one side, saying that it is absurd to assert that we possess the habits of these principles while this remains hidden from us. For it is manifest that those who possess knowledge of the principles possess a knowledge more certain than the knowledge acquired through demonstration. Yet knowledge acquired through demonstration cannot be possessed in such a way as to escape the notice of its possessor: for it was stated at the beginning of this work that he who possesses scientific knowledge knows that it is impossible for the thing to be otherwise. Much less, therefore, can it happen that someone possesses knowledge of the immediate principles while this escapes his notice—an absurdity that would nevertheless follow if habits of this sort were innate in us and yet hidden from us.
[lib. 2 l. 20 n. 5] Second, at If, on the other hand, we acquire them, et cetera, he objects against the contrary side. For if someone says that we acquire the habits of the principles anew, not having possessed them previously, there remains the further difficulty of how we could come to know and learn principles of this sort anew without doing so from some pre-existing knowledge: for it is impossible to learn anything except from pre-existing knowledge, just as we said above regarding demonstration. Yet we cannot learn immediate principles from pre-existing knowledge, because pre-existing knowledge is more certain, inasmuch as it is the cause of certitude for the things that become known through it, whereas no knowledge is more certain than the knowledge of principles of this sort. Hence it does not seem possible that we should come to know them without having known them previously.
[lib. 2 l. 20 n. 6] Third, at It is manifest, therefore, et cetera, he concludes from the two preceding arguments that it is neither possible for us always to possess knowledge of these principles in a way that escapes our notice, nor possible for such knowledge to be generated in us anew when complete ignorance has preceded and no other cognitive habit or power is possessed.
[lib. 2 l. 20 n. 7] Next, when he says, It is necessary, therefore, et cetera, he solves the aforesaid questions: and first, he solves the last one; second, he solves the first two, at Now since among the habits of thought, et cetera. Concerning the first he does two things: first, he solves the difficulty; second, he clarifies the solution set down, at Now let us state again what was just said, et cetera. Concerning the first he does three things: first, he proposes that some cognitive power must pre-exist in us; second, he shows what this power is, at Now this clearly seems, et cetera; third, he shows how from this pre-existing cognitive principle the knowledge of principles comes to be in us, at From sense, therefore, et cetera. First, therefore, he says that from the outset there must necessarily exist in us a certain cognitive power that pre-exists the knowledge of the principles—yet not one that excels the knowledge of the principles in certitude. Hence the knowledge of principles does not come about in us from pre-existing knowledge in the same manner as happens in things that are known through demonstration.
[lib. 2 l. 20 n. 8] Next, when he says, Now this clearly seems to belong, et cetera, he shows what that pre-existing cognitive principle is; and with respect to this he sets down three grades among animals. The first of these is that which clearly belongs in common to all animals: for all of them possess a certain connatural power for judging sensible things, which is called sense, and which is not acquired anew, but follows upon their very nature.
[lib. 2 l. 20 n. 9] He sets down the second grade at Now although sense belongs to them, et cetera, saying that while sense exists in all animals, in certain of them some sensible impression remains after the sensible object has departed, as happens in all complete animals; whereas in certain others this does not happen, as in certain incomplete animals, such as those that do not move by progressive locomotion. And perhaps it happens in some animals that an impression remains with respect to certain more vehement sensibles, though not with respect to other, weaker ones. In whatever animals, therefore, no impression of sensible things remains at all, such animals possess no cognition except while they are actually sensing; and similarly, those animals in which such an impression is naturally suited to remain cannot possess any cognition beyond the moment of sensing with respect to those sensibles of which no impression remains in them. Animals in which such a retention of the impression does reside, however, are able still to possess a certain cognition in the soul beyond actual sensation; and these are the animals that possess memory.
[lib. 2 l. 20 n. 10] He sets down the third grade at When many such memories have arisen, therefore, et cetera, saying that while there are many such animals possessing memory, among them there is a further difference: for in some of them a certain reasoning takes place concerning the things retained in memory, as in men, whereas in others it does not, as in brute animals.
[lib. 2 l. 20 n. 11] Next, when he says, From sense, therefore, comes memory, et cetera, he shows in accordance with what has preceded how the knowledge of first principles comes about in us, concluding from the premises that from sense comes memory in those animals in which a sensible impression remains, as was said above. From memory arising many times concerning the same thing—yet in diverse singulars—comes experience (experimentum); for experience seems to be nothing other than grasping something from many instances retained in memory. Nevertheless, experience requires a certain reasoning about particulars, whereby one instance is compared with another, which belongs properly to reason: for example, when someone remembers that such an herb has many times cured many people of fever, there is said to be experience that such an herb is curative of fever. Reason, however, does not stop at the experience of particulars, but from the many particulars in which one is experienced it grasps one common thing that becomes firmly established in the soul, and considers it apart from the consideration of any of the singulars; and this common thing it takes as a principle of art and of scientific knowledge. For example, so long as the physician considered that this herb cured Socrates when feverish, and \ Plato, and many other individual men, it is experience; when, however, his consideration rises to the point that this species of herb cures a feverish person unqualifiedly, this is taken as a rule of the art of medicine. This, therefore, is what \ Aristotle says: that just as experience comes from memory, so too from experience, or still further from the universal resting as a whole in the soul (which is taken as holding in all instances just as has been experienced in certain ones; and this universal is indeed said to be resting in the soul insofar as it is considered apart from the singulars, in which there is motion; and he also calls it one apart from the many—not indeed according to real existence, but according to the consideration of the intellect, which considers some nature, say of man, without looking to Socrates and Plato; and although according to the consideration of the intellect it is one apart from the many, nevertheless according to existence it is one and the same in all the singulars—not indeed numerically one, as though numerically the same humanity belonged to all men, but one according to the intelligible nature of the species: for just as this white thing is like that white thing in whiteness, not as though numerically one whiteness existed in both, so too Socrates is like Plato in humanity, not as though numerically one humanity existed in both)—from this experience, therefore, and from such a universal received through experience, there comes to be in the soul that which is the principle of art and of scientific knowledge. And he distinguishes between art and science just as in Book VI of the Ethics, where it is said that art is the right reason of things to be made; and therefore he says here that if from experience some universal is grasped concerning generation—that is, concerning things that can be made or brought about, such as healing or agriculture—this pertains to art. Science, however, as is said in the same place, concerns necessary things; and therefore if the universal is considered concerning things that always exist in the same way, it pertains to science—for instance, concerning numbers or geometrical figures. And the mode just described holds for the principles of all sciences and arts. Hence he concludes that the habits of principles neither pre-exist in us as determinate and complete from the outset, nor come to be anew from other better-known pre-existing habits in the way that the habit of science is generated in us from the foreknowledge of principles; rather, the habits of principles come to be in us from pre-existing sense. And he illustrates this by the simile of a battle-line reformed when a defeated and routed army turns back to stand: for when one soldier makes a stand—that is, begins to stand fast immovably and ceases to flee—another halts and joins him, and afterward another, until enough gather together to form the beginning of a fresh battle. In the same way, from the sensation and memory of one particular, and again of another and another, one eventually arrives at that which is the principle of art and of scientific knowledge, as has been said.
[lib. 2 l. 20 n. 12] Now someone might suppose that sense alone, or the memory of singulars, suffices to cause the intelligible knowledge of principles, as certain of the ancients held who did not distinguish between sense and intellect. Therefore, in order to exclude this, the Philosopher adds that together with sense one must presuppose a nature of the soul of such a kind that it can undergo this—that is, is receptive of universal knowledge, which indeed comes about through the possible intellect—and furthermore that can actively bring this about through the \ agent intellect, which renders things intelligible in act by abstracting universals from singulars.
[lib. 2 l. 20 n. 13] Next, when he says, Now let us state again what was just said, et cetera, he clarifies what was stated in the preceding solution regarding how the universal is grasped from the experience of singulars, saying that what was stated above—though not with full plainness—namely, how the universal comes to be in the soul from the experience of singulars, must be stated again so that it may be made more plain. For if many singulars are taken which are undifferentiated with respect to some one feature existing in them, that one feature in respect of which they do not differ, once received in the soul, is the first universal—whatever that feature may be, whether it pertains to the essence of the singulars or not. For because we find Socrates and Plato and many others to be undifferentiated with respect to whiteness, we grasp this one thing, namely, white, as a universal that is an accident; and similarly, because we find Socrates and Plato and others to be undifferentiated with respect to rationality, we grasp this one thing in which they do not differ, namely, rational, as a universal that is a specific difference.
[lib. 2 l. 20 n. 14] How this one feature can be grasped he consequently makes clear. For it is manifest that it is the singular that is sensed properly and per se, and yet sense is in a certain manner also of the universal itself: for it cognizes Callias not only insofar as he is Callias, but also insofar as he is this man, and similarly Socrates insofar as he is this man. And hence it is that, once such a grasp by sense pre-exists, the intellective soul can consider man in both. If, however, it were the case that sense apprehended solely what belongs to particularity, and in no way apprehended along with it the universal nature existing in the particular, it would not be possible for universal knowledge to be caused in us from the apprehension of sense. And he subsequently manifests this very same thing in the ascent from species to genus. Hence he adds that once again among these—namely, in man and horse—the soul halts in its consideration until it arrives at something indivisible in them, which is a higher universal: for instance, we consider such-and-such an animal and such-and-such another, say man and horse, until we arrive at the common nature animal, which is the genus; and in this we do likewise until we arrive at some still higher genus. Since, therefore, we receive our knowledge of universals from singulars, he concludes that it is manifestly necessary for us to come to know the first universal principles through induction: for in this way—namely, by way of induction—sense brings about the universal within the soul, insofar as the singulars are considered.
[lib. 2 l. 20 n. 15] Next, when he says, Now since among the habits of thought, et cetera, he solves the first two questions, namely, whether there is scientific knowledge (scientia) of the first principles or some other habit. Regarding this, he takes from what has gone before that the knowledge of principles pertains to the intellect, to which it belongs to know the universal, since he calls the universal the principle of science. Now connected with the intellect there are two classes of habits bearing some relation to truth: some are always true, whereas others at times admit of falsity, as is clear in the case of opinion and calculative reasoning, which can be either true or false (and there are also erroneous habits, directed toward what is false). Since the principles, however, are supremely true, it is manifest that they pertain neither to habits that are always false nor to habits that sometimes admit of falsity, but solely to the habits that are always true; and of this sort are scientific knowledge (scientia) and understanding (intellectus, the habit of principles). In Book VI of the Ethics a third is added, namely, wisdom (sapientia); yet because wisdom, as is stated there, includes within itself both scientific knowledge and understanding (for it is a kind of science and the head of the sciences), he passes over it here. Setting wisdom aside, therefore, no other kind of cognition except understanding (intellectus) is more certain than scientific knowledge. Yet it is manifest that the principles of demonstrations are better known than the demonstrated conclusions, as was established in Book I; nor can there be scientific knowledge of the principles themselves, because all scientific knowledge comes about through some reasoning—namely, demonstrative reasoning—of which the very principles under discussion are the starting points. Since, therefore, nothing can be truer than scientific knowledge and understanding (for wisdom is included within these), it follows from a consideration of the premises that understanding (intellectus) is properly the habit of first principles. He also proves this by a second argument: namely, that demonstration is not of necessity the principle of demonstration, for otherwise one would proceed to infinity in demonstrations, which was refuted in Book I. Since demonstration causes scientific knowledge, therefore, it follows that neither can scientific knowledge be the principle of scientific knowledge in such a way that the principles of the sciences would be known through scientific knowledge. If, then, apart from scientific knowledge we possess no other kind of cognition that is always true, it remains that understanding (intellectus) will be the principle of scientific knowledge, inasmuch as the principles of the sciences are known through understanding—in such a way, namely, that this understanding, which is the principle of science, is cognitive of the principle from which science proceeds, whereas science as a whole stands in the same manner toward the entire subject matter with which science deals as understanding stands toward the principle of science.