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the agreement was less complete, because the Earth is only one of many Planets, and the Planets only a small portion of all the heavenly bodies, such as Satellites, Comets, Meteors, and Double-Stars which revolve in such orbits.
The second of the Canons applies to cases where there is disagreement or difference, as in the following example :
Venus is a planet.
Therefore Venus is not self-luminous. The first of these propositions states a certain agreement to exist between Venus and planet, just as in the previous case of the Earth, but the second proposition states a disagreement between Planet and self-luminous bodies; hence we infer a disagreement between Venus and self-luminous body. But the reader will carefully observe that from two disagreements we can never infer anything If the following were put forth as an argument it would be evidently absurd :
Sirius is not a planet.
Therefore Sirius is not self-luminous. Both the premises or propositions given are true, and yet the conclusion is false, for all the fixed stars are self-luminous, or shine by their own light. We may, in fact, state as a third Canon that,
3. Two terms both disagreeing with one and the same third term may or may not agree with each other.
Self-evident rules, of an exactly similar nature to these three Canons, are the basis of all mathematical reasoning, and are usually called axioms. Euclid's first axiom is ihat “Things which are equal to the same thing are equal to one another;" and whether we apply it to the length of lines, the magnitude of angles, areas, solids, numbers,
degrees, or anything else which admits of being equal or unequal, it holds true. Thus if the lines A and are each equal to C it is evident that each is equal to the other.
Euclid does not give axioms corresponding to the second and third Canons, but they are really used in Geometry. Thus if A is equal to B, but D is not equal to B, it follows that A is not equal to D, or things of which one is equal, but the other unequal to the same third thing, are unequal to each other. Lastly, A and E are two lines both unequal to D and unequal to each other, whereas A and B are two lines both uncqual to D but equal to each other; thus we plainly see that “two things unequal to the same thing may or may not be equal to each other."
From what precedes it will be apparent that all reasoning requires that there should be one agreement at least; if there be two agreements we may reason to a third agreement; if there be one agreement and one difference we may reason to a second difference; but if there be two differences only we cannot reason to any conclusion whatever. These self-evident principles will in the next Lesson serve to explain some of the rules of the Syllogism.
Logicians however have not confined themselves to the use of these Canons, but have often put the same truth into a different form in axioms known as the Dicta de omni et nullo of Aristotle. This celebrated Latin phrase means “Statements concerning all and none,” and the axiom, or rather pair of axioms, is usually given in the following words:
Whatever is predicated of a term distributed whether affirmatively or negatively, may be predicated in like manner of everything contained under it. Or more briefly :
What pertains to the higher class pertains also to the lower.
This merely means, in untechnical language, that what may be said of all the things of any sort or kind may be said of any one or any part of those things; and, secondly, what may be denied of all the things in a class may be denied of any one or any part of them. Whatever may be said of “All planets” may be said of Venus, the Earth, Jupiter, or any other planet; and, as they may all be said to revolve in elliptic orbits, it follows that this may be asserted of Venus, the Earth, Jupiter, or any otlrer planet. Similarly, according to the negative part of the Dicta, we may deny that the planets are selfluminous, and knowing that Jupiter is a planet may deny that Jupiter is self-luminous. A little reflection would show that the affirmative Dictum is really the first of the Canons in a less complete and general form, and that the negative Dictum is similarly the second Canon. These Dicta in fact only apply to such cases of agreement between terms as consist in one being the name of a smaller class, and another of the larger class containing it. Logicians have for the most part strangely overlooked the important cases in which one term agrees with another to the extent of being identical with it; but this is a subject which we cannot fitly discuss here at any length. It is treated in my little work called The Substitution of Similars*.
Some logicians have held that in addition to the three laws which are called the Primary Laws of Thought,
there is a fourth called “The Principle or Law of Sufficient Reason.” It was stated by Leibnitz in the following words:
Nothing happens without a reason why it should be so rather than otherwise. For instance, if there be a pair of scales in every respect exactly alike on each side and with exactly equal weights in each scale, it must remain motionless and in equilibrium, because there is no reason why one side should go down more than the other. It is certainly a fundamental assumption in mechanical science that if a body is acted upon by two perfectly equal forces in different directions it will move equally between them, because there is no reason why it should move more to one side than the other. Mr Mansel, Sir W. Hamilton and others consider however that this law has no place in logic, even if it can be held self-evident at all ; and the question which appears open to doubt need not be discussed here.
I have so freely used the word axiom in this lesson that it is desirable to clear up its meaning as far as possible. Philosophers do not perfectly agree about its derivation or exact meaning, but it certainly comes from the verb åčiów, which is rendered, to think worthy. It generally denotes a self-evident truth of so simple a character that it must be assumed to be true, and, as it cannot be proved by any simpler proposition, must itself be taken as the basis of reasoning. In mathematics it is clearly used in this sense.
See Hamilton's Lectures on Logic, Lectures 5 and 6.
THE RULES OF THE SYLLOGISM.
SYLLOGISM is the common name for Mediate Inference, or inference by a medium or middle term, and is to be distinguished from the process of Immediate Inference, or inference which is performed without the use of any third or middle term.
We are in the habit of employing a middle term or medium whenever we are prevented from comparing two things together directly, but can compare each of them with a certain third thing. We cannot compare the sizes of two halls by placing one in the other, but we can measure each by a foot rule or other suitable measure, which forms a common measure, and enables us to ascertain with any necessary degree of accuracy their relative dimensions. If we have two quantities of cotton goods and want to compare them, it is not necessary to bring the whole of one portion to the other, but a sample is cut off, which represents exactly the quality of one portion, and, according as this sample does or does not agree with the other portion, so must the two portions of goods agree or differ.
The use of a middle term in syllogism is closely parallel to what it is in the above instances, but not exactly the same. Suppose, as an example, that we wish to ascertain whether or not “Whales are viviparous," and that we had not an opportunity of observing the fact directly; we could yet show it to be so if we knew that “whales are mammalian animals," and that “all mam