Book 1
Chap. 1. Upon the Nature of Demonstration.
Chap. 2. Of Knowledge, and Demonstration, and its Elements.
Chap. 3. Refutation of certain opinions as to Science and Demonstration.
Chap. 4. Upon the terms "every," "per se," and "universal."
Chap. 5. Of Errors about the primary Universal.
Chap. 6. Demonstration consists of Principles per se; and of a necessary Medium.
Chap. 7. That we may not demonstrate by passing from one Genus to another.
Chap. 8. Things which are subject to Change are incapable of Demonstration per se.
Chap. 9. That the Demonstration of a thing ought to proceed from its own appropriate Principles: these last indemonstrable.
Chap. 10. Of the Definition and Division of Principles.
Chap. 11. Of certain Common Principles of all Sciences.
Chap. 12. Of Syllogistic Interrogation.
Chap. 13. The difference between Science, "that" a thing is, and "why" it is.
Chap. 14. The first Figure most suitable to Science.
Chap. 15. Of immediate negative Propositions.
Chap. 16. Of Ignorance, according to corrupt position of the Terms, where there are no Media.
Chap. 17. Continuation of the same with Media.
Chap. 18. Of the Dependence of Universals upon Induction, and of the latter upon Sense.
Chap. 19. Of the Principles of Demonstration, whether they are Finite or Infinite.
Chap. 20. Of Finite Media.
Chap. 21. It is shown that there are no Infinite Media in Negative Demonstration.
Chap. 22. That there are no Infinite Media in Affirmative Demonstration.
Chap. 23. Certain Corollaries
Chap. 24. The superiority of Universal to Particular Demonstration proved.
Chap. 25. The Superiority of Affirmative to Negative Demonstration proved.
Chap. 26. The Superiority of the same to Demonstration ad impossible proved.
Chap. 27. Upon the Nature of more Accurate Science.
Chap. 28. What constitutes one, and what different Sciences.
Chap. 29. That there may be several Demonstrations of the same thing.
Chap. 30. That there is no Science of the Fortuitous.
Chap. 31. That we do not possess Scientific Knowledge through Sensation.
Chap. 32. On the Difference of Priniciples according to the Diversity of Syllogisms.
Chap. 33. Upon the Difference between Science and Opinion.
Chap. 34. Of Sagacity.
Book 2
Chap. 1. That the subjects of Scientific Investigation are four.
Chap. 2. That all Investigation has reference to the Discovery of the Middle Term.
Chap. 3. Upon the Difference between Demonstration and Definition.
Chap. 4. That the Definition of a thing cannot be demonstrated.
Chap. 5. That there is no Conclusion by Divisions proved.
Chap. 6. Case of one Proposition defining the Definition itself.
Chap. 7. That what a thing is can neither be known by Demonstration nor by Definition.
Chap. 8. Of the logical Syllogism of what a thing is.
Chap. 9. Of certain Natures or Principles incapable of Demonstration.
Chap. 10. Upon Definition and its kinds.
Chap. 11. Of Causes and their Demonstration.
Chap. 12. Upon the causes of the Present, Past, and Future.
Chap. 13. Upon the Method of investigating Definition.
Chap. 14. Rules for Problems.
Chap. 15. Of Identical Problems.
Chap. 16. Of Causes and Effects.
Chap. 17. Extension of the same subject.
Chap. 18. Observation upon Cause to Singulars.
Chap. 19. Upon the Method and Habit necessary to the ascertainment of Principles.
All doctrine, and all intellectual discipline, arise from pre-existent knowledge. Now this is evident, if we survey them all, for both mathematical sciences are obtained in this manner, and also each of the other arts. It is the same also with arguments, as well those which result through syllogisms, as those which are formed through induction, for both teach through things previously known, the one assuming as if from those who understood them, the other demonstrating the universal by that which is evident as to the singular. Likewise also do rhetoricians persuade, for they do so either through examples, which is induction, or through enthymems, which is syllogism. It is necessary however to possess previous knowledge in a twofold respect; for with some things we must pre-suppose that they are, but with others we must understand what that is which is spoken of; and with others both must be known, as for instance, (we must pre-assume,) that of every thing it is true to affirm or deny that it is, but of a triangle, that it signifies so and so, and of the monad (we must know) both, viz. what it signifies and that it is, for each of these is not manifest to us in a similar manner. It is possible how ever to know from knowing some things previously, and receiving the knowledge of others at the same time, as of things which are contained under universals, and of which a man possesses knowledge. For he knew before that every triangle has angles equal to two right angles, but that this which is in a semi-circle is a triangle, he knew by induction at the same time. For of some things knowledge is acquired after this manner, nor is the extreme known through the middle, as such things as are singulars, and are not predicated of any subject. Perhaps however we must confess that we possess knowledge after a certain manner before induction or the assumption of a syllogism, but in another manner not. For what a man is ignorant about its existence at all, how could he know at all that it has two right angles? But it is evident that he thus knows because he knows the universal, but singly he does not know it. Still if this be not admitted, the doubt which is mentioned in the Meno will occur, either he will learn nothing, or those things which he knows, for he must not say, as some endeavour to solve the doubt, "Do you know that every duad is an even number or not?" for since if some one says that he does, they would bring forward a certain duad which he did not think existed, as therefore not even; and they solve the ambiguity, not by saying that he knew every duad to be even, but that he was ignorant as to what they know is a duad. Nevertheless they know that of which they possess and have received the demonstration, but they have received it not of every thing which they know to be a triangle or a number, but of every number and triangle singly, for no proposition is assumed of such a kind as the number which you know, or the rectilinear figure which you know, but universally. Still there is nothing (I think) to prevent a man who learns, in a certain respect knowing and in a certain respect being ignorant, for it is absurd, not that he should in some way know what he learns, but that he should thus know it, as he does when he learns it, and in the same manner.
WE think that we know each thing singly, (and not in a sophistical manner, according to accident,) when we think that we know the cause on account of which a thing is, that it is the cause of that thing, and that the latter cannot subsist otherwise; wherefore it is evident that knowledge is a thing of this kind, for both those who do not, and those who do know, fancy, the former, that they in this manner possess knowledge, but those who know, possess it in reality, so that it is impossible that a thing of which there is knowledge simply should subsist in any other way. Whether therefore there is any other mode of knowing we shall tell hereafter, but we say also that we obtain knowledge through demonstration, but I call demonstration a scientific syllogism, and I mean by scientific that according to which, from our possessing it, we know. If then to know is what we have laid down, it is necessary that demonstrative science should be from things true, first, immediate, more known than, prior to, and the causes of the conclusion, for thus there will be the appropriate first principles of whatever is demonstrated. Now syllogism will subsist even without these, but demonstration will not, since it will not produce knowledge. It is necessary then that they should be true, since we cannot know that which does not subsist, for instance, that the diameter of a square is commensurate with its side. But it must be from things first and indemonstrable, or otherwise a man will not know them, because he does not possess the demonstration of them, for to know those things of which there is demonstration not accidentally is to possess demonstration. But they must be causes, and more known, and prior; causes indeed, because we then know scientifically when we know the cause; and prior, since they are causes; previously known also, not only according to the other mode by understanding (what they signify), but by knowing that they are. Moreover they are prior and more known in two ways, for what is prior in nature, is not the same as that which is prior in regard to us, nor what is more known (simply) the same as what is more known to us. Now I call things prior and more known to us, those which are nearer to sense, and things prior and more known simply, those which are more remote from sense; and those things are most remote which are especially universal, and those nearest which are singular, and these are mutually opposed. That again is from things first, which is from peculiar principles, and I mean by first, the same thing as the principle, but the principle of demonstration is an immediate proposition, and that is immediate to which there is no other prior. Now a proposition is one part of enunciation, one of one, dialectic indeed, which similarly assumes either (part of contradiction), but demonstrative which definitely (assumes) that one (part) is true. Enunciation is either part of contradiction, and contradiction is an opposition which has no medium in respect to itself. But that part of contradiction (which declares) something, of somewhat, is affirmation, and that (which signifies) something from somewhat is negation. Of an immediate syllogistic principle, I call that the thesis, which it is not possible to demonstrate, nor is it necessary that he should possess it, who intends to learn any thing; but what he who intends to learn any thing must necessarily possess, that I call an axiom, for there are certain things of this kind, and in denominating these, we are accustomed generally to use this name. But of thesis, that which receives either part of contradiction, as for instance, I mean that a certain thing is, or that it is not, is hypothesis, but that which is without this, is definition. For definition is a thesis, since the arithmetician lays down unity to be that which is indivisible, according to quantity, yet it is not hypothesis, since what unity is, and that unity is, are not the same thing.
Notwithstanding, since we must believe in and know a thing from possessing such a syllogism as we call demonstration, and this is, because these are so, of which syllogism consists—it is necessary not only to have a previous knowledge of the first, or all, or some things, but that they should be more known, for that on account of which any thing exists, always exists itself in a greater degree; for example, that on account of which we love is itself more beloved. Hence if we know and believe on account of things first, we also know and believe those first things in a greater degree, because through them (we know and believe) things posterior. A man however cannot believe more than what he knows, those things which he does not know, nor with respect to which he is better disposed than if he knew. This however will happen, unless some one should previously know of those who give credence through demonstration, since it is more necessary to believe either in all or in certain first principles, than in the conclusion. It is not only however requisite that he who is to possess knowledge through demonstration, should know in a greater degree first principles, and believe rather in them than in the thing demonstrated, but also that nothing else should be more credible or more known to him than the opposites of the principles, from which a syllogism of contra-deception may consist, since it behoves him who possesses knowledge singly to be unchangeable.
To some, because it is necessary that first things should be known, science does not appear to exist, but to others to exist indeed, yet (they think) there are demonstrations of all things, neither of which opinions is true or necessary. For those who suppose that knowledge does not subsist at all, these think that we are to proceed to infinity as if we may not know things subsequent by things prior, of which there are no first, reasoning rightly, since it is impossible to penetrate infinites. And if (they say) we are to stop, and there are principles, these are unknown, since there is no demonstration of them, which alone they say is to know scientifically; but if it is not possible to know first things, neither can we know either simply or properly things which result from these, but by hypothesis, if these exist. Others however assent with respect to knowledge, for (they assert) that it is only through demonstration, but that nothing prevents there being a demonstration of all things, for demonstration may be effected in a circle, and (things be proved) from each other. We on the contrary assert, that neither is all science demonstrative, but that the science of things immediate is indemonstrable. And this is evidently necessary, for if it is requisite to know things prior, and from which demonstration subsists, but some time or other there is a stand made at things immediate, these must of necessity be indemonstrable. This therefore we thus assert, and we say that there is not only science, but also a certain principle of science, by which we know terms. But that it is impossible to demonstrate in a circle simply is evident, since demonstration must consist of things prior and more known, as it is impossible that the same should be prior and posterior to the same, unless in a different way, as for instance, some things with reference to us, but others simply in the manner in which induction makes known. If however this be so, to know simply will not be well defined, but it is two-fold, or the other demonstration is not simply so which is produced from things more known to us. Still there happens to those who assert there is demonstration in a circle, not only what has now been declared, but that they say nothing else than this is if it is, and in this manner we may easily demonstrate all things. Nevertheless it is evident that this occurs, when three terms are laid down, for to assert that demonstration recurs through many or through few terms, or whether through few or through two, makes no difference. For when A existing, B necessarily is, and from this last C, if A exists C will exist, if then, when A is, it is necessary that B should be, but this existing, A exists, (for this were to demonstrate in a circle,) let A be laid down in the place of C. To say therefore that because B is A is, is equivalent to saying that C is, and this is to say that A existing C is, but C is the same as A, so that it happens that they who assert there is demonstration in a circle, say nothing else than that A is because A is, and thus we may easily demonstrate all things. Neither however is this possible, except in those things which follow each other as properties: from one thing however being laid down, it has been proved that there will never necessarily result something else, (I mean by one thing, neither one term, nor one thesis being laid down,) but from two first and least theses, it is possible (to infer necessarily something else), since we may syllogize. If then A is consequent to B and to C, and these to each other, and to A, thus indeed it is possible to demonstrate all those things which are required from each other in the first figure, as we have shown in the books on Syllogism. It has also been shown that in the other figures there is either not a syllogism, or not one concerning the subjects assumed; but it is by no means possible to demonstrate in a circle those which do not reciprocate. Hence, since there are but few such in demonstrations, it is evidently vain and impossible to say, that there is demonstration of things from each other, and that on this account universal demonstration is possible.
Since