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observed, That facts are unfavourable to this opinion: for it does not appear, that Euclid, or Apollonius, or Archimedes, or Hugens, or Newton, ever made the least use of this art; and I am even of opinion, that no use can be made of it in mathematics. I would not wish to advance this rashly, since Aristotle has said, that mathematicians reason for the most part in the first figure. What led him to think so was, that the first figure only yields conclusions that are universal and affirmative, and the conclusions of mathematics are commonly of that kind. But it is to be observed, that the propositions of mathematics are not categorical propositions, consisting of one subject and one predicate. They express some relation which one quantity bears to another, and on that account must have three terms. The quantities compared make two, and the relation between them is a third. Now to such propositions we can neither apply the rules concerning the conversion of propositions, nor can they enter into a syllogism of any of the figures or modes. We observed before, that this conversion, A is greater than B, therefore B is less than A, does not fall within the rules of conversion given by Aristotle or the logicians; and we now add, that this simple reasoning, A is equal to B, and B to C; therefore A is equal to C, cannot be brought into any syllogism in figure and mode. There are indeed syllogisms into which mathematical propositions may enter, and
and of such we shall afterwards speak; but they have nothing to do with the system of figure and mode.
When we go without the circle of the mathematical sciences, I know nothing in which there seems to be so much demonstration as in that part of logic which treats of the figures and modes of syllogism; but the few remarks we have made, shew, that it has some weak places: and besides, this system cannot be used as an engine to rear itself...
The compass of the syllogistic system as an engine of science, may be discerned by a compen, dious and general view of the conclusion drawn, and the argument used to prove it, in each of the three figures. Inf
In the first figure, the conclusion affirms or denies something of a certain species or individual; and the argument to prove this conclusion is, That the same thing may be affirmed or denied of the whole genus to which that species or individual belongs.
In the second figure, the conclusion is, That some species or individual does not belong to such a genus; and the argument is, That some attribute common to the whole genus does not belong to that species or individual.
In the third figure, the conclusion is, That such an attribute belongs to part of a genus; and the argument is, That the attribute in question belongs E
to a species or individual which is part of that
I apprehend, that in this short view, every conclusion that falls within the compass of the three figures, as well as the mean of proof, is comprehended. The rules of all the figures might be easily deduced from it; and it appears, that there is only one principle of reasoning in all the three; so that it is not strange, that a syllogism of one figure should be reduced to one of another figure.
The general principle in which the whole terminates, and of which every categorical syllogism is only a particular application, is this, That what is affirmed or denied of the whole genus, may be affirmed or denied of every species and individual belonging to it. This is a principle of undoubted certainty indeed, but of no great depth. Aristotle and all the logicians assume it as an axiom or first principle, from which the syllogistic system, as it were, takes its departure: and after a tedious voyage, and great expence of demonstration, it lands at last in this principle as its ultimate conclusion. O curas hominum! O quantum est in rebus inane!
SECT. 6. On Modal Syllogisms.
Categorical propositions, besides their quantity and quality, have another affection, by which they are divided into pure and modal. In a pure pro
position, the predicate is barely affirmed or denied of the subject; but in a modal proposition, the affirmation or negation is modified, by being declared to be necessary, or contingent, or possible, or impossible. These are the four modes observed by Aristotle, from which he denominates a proposition modal. His genuine disciples maintain, that these are all the modes that can affect an affirmation or negation, and that the enumeration is complete. Others maintain, that this enumeration is incomplete; and that when an affirmation or negation is said to be certain or uncertain, probable or improbable, this makes a modal proposition, no less than the four modes of Aristotle. We shall not enter into this dispute; but proceed to observe that the epithets of pure and modal are applied to syllogisms as well as to propositions. A pure syllogism is that in which both premises are pure propositions. A modal syllogism is that in which either of the premises is a modal proposi tion.
The syllogisms, of which we have already said so much, are those only which are pure as well as categorical. But when we consider, that through all the figures and modes, a syllogism may have one premise modal of any of the four modes, while the other is pure, or it may have both premises modal, and that they may be either of the same mode or of different modes; what prodigious variety arises from all these combinations? Now it
is the business of a logician, to shew how the conclusion is affected in all this variety of cases. Aristotle has done this in his First Analytics, with immense labour; and it will not be thought strange, that when he had employed only four chapters in discussing one hundred and ninety-two modes, true and false, of pure syllogisms, he should employ fifteen upon modal syllogisms.
I am very willing to excuse myself from entering upon this great branch of logic, by the judgment and example of those who cannot be charged either with want of respect to Aristotle, or with a low esteem of the syllogistic art.
Keckerman, a famous Dantzican professor, who spent his life in teaching and writing logic, in his huge folio system of that science, published anno 1600, calls the doctrine of the modals the crux logicorum. With regard to the scholastic doctors, among whom this was a proyerb, De modalibus non gustabit asinus, he thinks it very dubious, whether they tortured most the modal syllogisms, or were most tortured by them. But those crabbed geniuses, says he, made this doctrine so very thorny, that it is fitter to tear a man's wits in pieces than to give them solidity. He desires it to be observed, that the doctrine of the modals is adapted to the Greek language. The modal terms were frequently used by the Greeks in their disputations; and, on that account, are so fully handled by Aristotle but in the Latin tongue you shall hardly