Every X is A, Some X is B, therefore Some A is BSome X is A, Every X is B, therefore Some A is B
Every X is A, Some X is B, therefore Some A is BSome X is A, Every X is B, therefore Some A is B
Every X is A, Some X is B, therefore Some A is BSome X is A, Every X is B, therefore Some A is B
Every X is A, Some X is B, therefore Some A is B
Some X is A, Every X is B, therefore Some A is B
In this tract, syllogisms have been divided into two classes: first,those which prove a universal conclusion; secondly, those which prove a partial conclusion, and which are (all but one) derived from the first by weakening one of the premises, in such manner as to produce a legitimate but weakened conclusion. Those of the first class are placed in the first column, and the other in the second.
In all works on logic, it is customary to write that premiss first which contains the predicate of the conclusion. Thus,
The premises thus arranged are called major and minor; the predicate of the conclusion being called the major term, and its subject the minor. Again, in the preceding case we see the various subjects coming in the order X, B; A, X; A, B: and the number of different orders which can appear is four, namely—
which are called the four figures, and every kind of syllogism in each figure is called a mood. I now put down the various moods of each figure, the letters of which will be a guide to find out those of the preceding list from which they are derived. Co means that a premiss of the preceding list has been converted; + that it has been strengthened; Co +, that both changes have taken place. Thus,
And Co + abbreviates the following: If some A be X, then some X is A (Co); and all that is true when Some X is A, is true when Every X is A (+); therefore the second is legitimate, if the first be so.
The above is the ancient method of dividing syllogisms; but, for the present purpose, it will be sufficient to consider the six from which the rest can be obtained. And since some of the six have A in the predicate of the conclusion, and not B, we shall join to them the six other syllogisms which are found by transposing B and A. The complete list, therefore, of syllogisms with the weakest premises and the strongest conclusions, in which a comparison of A and B is obtained by comparison of both with X, is as follows:
In the list of page19, there was nothing but recapitulation of forms, each form admitting a variation by interchanging A and B. This interchange having been made, and the results collected as above, if we take every case in which B is the predicate, or can be made the predicate by allowable conversion, we have a collection of all possibleweakestforms in which the result is one of the four ‘Every A is B,’ ‘No A is B,’ ‘Some A is B,’ ‘Some A is not B’; as follows. The premises are written in what appeared the most natural order, without distinction of major or minor; and the letters prefixed are according to the forms of the several premises, as in page10.
Every assertion which can be made upon two things by comparison with any third, that is, every simple inference, can be reduced to one of the preceding forms. Generally speaking, one of the premises is omitted, as obvious from the conclusion; that is, one premiss being named and the conclusion, that premiss is implied which is necessary to make the conclusion good. Thus, if I say, “That race must have possessed some of the arts of life, for they came from Asia,” it is obviously meant to be asserted, that all races coming from Asia must have possessed some of the arts of life. The preceding is then a syllogism, as follows:
‘That race’ is ‘a race of Asiatic origin:’Every ‘race of Asiatic origin’ is ‘a race which must have possessed some of the arts of life:’Therefore, That raceisa race which must have possessed some of the arts of life.
‘That race’ is ‘a race of Asiatic origin:’Every ‘race of Asiatic origin’ is ‘a race which must have possessed some of the arts of life:’Therefore, That raceisa race which must have possessed some of the arts of life.
‘That race’ is ‘a race of Asiatic origin:’Every ‘race of Asiatic origin’ is ‘a race which must have possessed some of the arts of life:’Therefore, That raceisa race which must have possessed some of the arts of life.
‘That race’ is ‘a race of Asiatic origin:’
Every ‘race of Asiatic origin’ is ‘a race which must have possessed some of the arts of life:’
Therefore, That raceisa race which must have possessed some of the arts of life.
A person who makes the preceding assertion either means to imply, antecedently to the conclusion, that all Asiatic races must have possessed arts, or he talks nonsense if he asserts the conclusion positively. ‘A must be B, for it is X,’ can only be true when ‘Every X is B.’ This latter proposition may be called the suppressed premiss; and it is in such suppressed propositions that the greatest danger of error lies. It is also in such propositions that men convey opinions which they would not willingly express. Thus, the honest witness who said, ‘I always thought him a respectable man—he kept his gig,’ would probably not have admitted in direct terms, ‘Every man who keeps a gig must be respectable.’
I shall now give a few detached illustrations of what precedes.
“His imbecility of character might have been inferred from his proneness to favourites; for all weak princes have this failing.” The preceding would stand very well in a history, and many would pass it over as containing very good inference. Written, however, in the form of a syllogism, it is,
which is palpably wrong. (Rule 1.) The writer of such a sentence as the preceding might have meant to say, ‘for all who have this failing are weak princes;’ in which case he would have inferred rightly. Every one should be aware that there is much false inference arising out of badness of style, which is just as injurious to the habits of the untrained reader as if the errors were mistakes of logic in the mind of the writer.
‘A is less than B; B is less than C: therefore A is less than C.’ This, at first sight, appears to be a syllogism; but, on reducing it to the usual form, we find it to be,
which is not a syllogism, since there is no middle term. Evident as the preceding is, the following additional proposition must be formed before it can be made explicitly logical. ‘If B be a magnitude less than C, then every magnitude less than B is also less than C.’ There is, then, before the preceding can be reduced to a syllogistic form, the necessity of a deduction from the second premiss, and the substitution of the result instead of that premiss. Thus,
But, if the additional argument be examined—namely, if B be less than C, then that which is less than B is less than C—it will be found to require precisely the same considerations repeated; for the original inference was nothing more. In fact, it may easily be seen as follows, that the proposition before us involves more than any simple syllogismcan express. When we say that A is less than B, we say that if A were applied to B, every part of A would match a part of B, and there would be parts of B remaining over. But when we say, ‘Every A is B,’ meaning the premiss of a common syllogism, we say that every instance of A is an instance of B, without saying any thing as to whether there are or are not instances of B still left, after those which are also A are taken away. If, then, we wish to write an ordinary syllogism in a manner which shall correspond with ‘A is less than B, B is less than C, therefore A is less than C,’ we must introduce a more definite amount of assertion than was made in the preceding forms. Thus,
Or thus:
The most technical form, however, is,
This sort of argument is calledà fortioriargument, because the premises are more than sufficient to prove the conclusion, and the extent of the conclusion is thereby greater than its mere form would indicate. Thus, ‘A is less than B, B is less than C, therefore,à fortiori, A is less than C,’ means that the extent to which A is less than C must be greater than that to which A is less than B, or B than C. In the syllogism last written, either of the bracketed premises might be struck out without destroying the conclusion; which last would, however, be weakened. As it stands, then, the part of the conclusion, ‘Some C is not A,’ follows ità fortiori.
The argumentà fortiori, may then be defined as a universallyaffirmative syllogism, in which both of the premises are shewn to be less than the whole truth, or greater. Thus, in ‘Every A is X, Every X is B, therefore Every A is B,’ we do not certainly imply that there are more Xs than As, or more Bs than Xs, so that we do not know that there are more Bs than As. But if we are at liberty to state the syllogism as follows,
All the As make up part (and part only) of the XsEvery X is B;
All the As make up part (and part only) of the XsEvery X is B;
All the As make up part (and part only) of the XsEvery X is B;
All the As make up part (and part only) of the Xs
Every X is B;
then we are certain that
All the As make up part (and part only) of the Bs.
All the As make up part (and part only) of the Bs.
All the As make up part (and part only) of the Bs.
All the As make up part (and part only) of the Bs.
But if we are at liberty further to say that
All the As make up part (and part only) of the XsAll the Xs make up part (and part only) of the Bs
All the As make up part (and part only) of the XsAll the Xs make up part (and part only) of the Bs
All the As make up part (and part only) of the XsAll the Xs make up part (and part only) of the Bs
All the As make up part (and part only) of the Xs
All the Xs make up part (and part only) of the Bs
then we conclude that
All the As make uppart of part(only) of the Bs
All the As make uppart of part(only) of the Bs
All the As make uppart of part(only) of the Bs
All the As make uppart of part(only) of the Bs
and the words in Italics mark that quality of the conclusion from which the argument is calledà fortiori.
Most syllogisms which give an affirmative conclusion are generally meant to implyà fortioriarguments, except only in mathematics. It is seldom, except in the exact sciences, that we meet with a proposition, ‘Every A is B,’ which we cannot immediately couple with ‘Some Bs are not As.’
When an argument is completely established, with the exception of one assertion only, so that the inference may be drawn as soon as that one assertion is established, the result is stated in a form which bears the name of anhypotheticalsyllogism. The word hypothesis means nothing but supposition; and the species of syllogism just mentioned first lays down the assertion that a consequence will be true if a certain condition be fulfilled, and then either asserts the fulfilment of the condition, and thence the consequence, or else denies the consequence, and thence denies the fulfilment of the condition. Thus, if we know that
When A is B, it follows that P is Q;
When A is B, it follows that P is Q;
When A is B, it follows that P is Q;
When A is B, it follows that P is Q;
then, as soon as we can ascertain that A is B, we can conclude that P is Q; or, if we can shew that P is not Q, we know that A is not B. But if we find that A is not B, we can infer nothing; for the preceding does not assert that P is Qonlywhen A is B. And if we find out that P is Q, we can infer nothing. This conditional syllogism may be converted into an ordinary syllogism, as follows. Let K be any ‘case in which A is B,’ and Z a ‘case in which P is Q’; then the preceding assertion amounts to ‘Every K is Z.’ Let L be a particular instance, the A of which may or may not be B. If A be B in the instance under discussion, or if A be not B, we have, in the one case and the other,
Similarly, according as a particular case (M) is or is not Z, we have
That is to say: The assertion of an hypothesis is the assertion of its necessary consequence, and the denial of the necessary consequence is the denial of the hypothesis; but the assertion of the necessary consequence gives no right to assert the hypothesis, nor does the denial of the hypothesis give any right to deny the truth of that which would (were the hypothesis true) be its necessary consequence.
Demonstration is of two kinds: which arises from this, that every proposition has a contradictory; and of these two, one must be true and the other must be false. We may then either prove a proposition to be true, or its contradictory to be false. ‘It is true that Every A is B,’ and, ‘it is false that there are some As which are not Bs,’ are the same proposition; and the proof of either is called the indirect proof of the other.
But how is any proposition to be proved false, except by proving acontradiction to be true? By proving a necessary consequence of the proposition to be false. But this is not a complete answer, since it involves the necessity of doing the same thing; or, so far as this answer goes, one proposition cannot be proved false unless by proving another to be false. But it may happen, that a necessary consequence can be obtained which is obviously and self-evidently false, in which case no further proof of the falsehood of the hypothesis is necessary. Thus the proof which Euclid gives that all equiangular triangles are equilateral is of the following structure, logically considered.
(1.) If there be an equiangular triangle not equilateral, it follows that a whole can be found which is not greater than its part.[1]
1. This is the proposition in proof of which nearly the whole of the demonstration of Euclid is spent.
1. This is the proposition in proof of which nearly the whole of the demonstration of Euclid is spent.
(2.) It is false that there can be any whole which is not greater than its part (self evident).
(3.) Therefore it is false that there is any equiangular triangle which is not equilateral; or all equiangular triangles are equilateral.
When a proposition is established by proving the truth of the matters it contains, the demonstration is calleddirect; when by proving the falsehood of every contradictory proposition, it is calledindirect. The latter species of demonstration is as logical as the former, but not of so simple a kind; whence it is desirable to use the former whenever it can be obtained.
The use of indirect demonstration in the Elements of Euclid is almost entirely confined to those propositions in which the converses of simple propositions are proved. It frequently happens that an established assertion of the form
may be easily made the means of deducing,
which last gives
The conversion of the second proposition into the third is usually made by an indirect demonstration, in the following manner. If possible, let there be one B which is not A, (2) being true. Then there is one thing which is not A and is B; but every thing not A is not B; therefore there is one thing which is B and is not B: which is absurd. It is then absurd that there should be one single B which is not A; or, Every B is A.
The following proposition contains a method which is of frequent use.
Hypothesis.—Let there be any number of propositions or assertions,—three for instance, A, B, and C,—of which it is the property that one or the other must be true,and one only. Let there be three other propositions, P, Q, and R, of which it is also the property that one, and one only, must be true. Let it also be a connexion of those assertions, that
When A is true, P is trueWhen B is true, Q is trueWhen C is true, R is true
When A is true, P is trueWhen B is true, Q is trueWhen C is true, R is true
When A is true, P is trueWhen B is true, Q is trueWhen C is true, R is true
When A is true, P is true
When B is true, Q is true
When C is true, R is true
Consequence: then it follows that
When P is true, A is trueWhen Q is true, B is trueWhen R is true, C is true
When P is true, A is trueWhen Q is true, B is trueWhen R is true, C is true
When P is true, A is trueWhen Q is true, B is trueWhen R is true, C is true
When P is true, A is true
When Q is true, B is true
When R is true, C is true
For, when P is true, then Q and R must be false; consequently, neither B nor C can be true, for then Q or R would be true. But either A, B, or C must be true, therefore A must be true; or, when P is true, A is true. In a similar way the remaining assertions may be proved.
We have hitherto supposed that the premises are actually true; and, in such a case, the logical conclusion is as certain as the premises. It remains to say a few words upon the case in which the premises are probably, but not certainly, true.
The probability of an event being about to happen, and that of an argument being true, may be so connected that the usual method of measuring the first may be made to give an easy method of expressing the second. Suppose an urn, or lottery, with a large number of balls, black or white; then, if there be twelve white balls to one black, we say it is twelve to one that a white ball will be drawn, or that a white ball is twelve times as probable as a black one. A certain assertion may be in the same condition as to the force of probability with which it strikes the mind: that is, the questions
Is the assertion true?Will a white ball be drawn?
Is the assertion true?Will a white ball be drawn?
Is the assertion true?Will a white ball be drawn?
Is the assertion true?
Will a white ball be drawn?
may be such that the answer, ‘most probably,’ expresses the same degree of likelihood in both cases.
We have before explained that logic has nothing to do with the truth or falsehood of assertions, but only professes, supposing them true, to collect and classify the legitimate methods of drawing inferences. Similarly, in this part of the subject, we do not trouble ourselves with the question, How are we to find the probability due to premises? but we ask: Supposing (happen how it may) that wehavefound the probability of the premises, required the probability of the conclusion. When the odds in favour of a conclusion are, say 6 to 1, there are, out of every 7 possible chances, 6 in favour of the conclusion, and 1 against it. Hence ⁶⁄₇ and ⅐ will represent the proportions, for and against, of all the possible cases which exist.
Thus we have the succession of such results as in the following table:—
Let the probability of a conclusion, as derived from the premises (that is on the supposition that it was never imagined to be possible till the argument was heard), be called theintrinsic probabilityof the argument. This is found by multiplying together the probabilities of all the assertions which are necessary to the argument. Thus, suppose that a conclusion was held to be impossible until an argument of a single syllogism was produced, the premises of which have severally five to one and eight to one in their favour. Then ⅚ × ⁸⁄₉, or ⁴⁰⁄₅₄, is the intrinsic probability of the argument, and the odds in its favour are 40 to 14, or 20 to 7.
But this intrinsic probability is not always that of the conclusion; the latter, of course, depending in some degree on the likelihood which the conclusion was supposed to have before the argument was produced. A syllogism of 20 to 7 in its favour, advanced in favour of a conclusion which was beforehand as likely as not, produces a much more probable result than if the conclusion had been thought absolutely false until the argument produced a certain belief in the possibility of its being true.The change made in the probability of a conclusion by the introduction of an argument (or of a new argument, if some have already preceded) is found by the following rule.
From the sum of the existing probability of the conclusion and the intrinsic probability of the new argument, take their product; the remainder is the probability of the conclusion, as reinforced by the argument. Thus,a + b − abis the probability of the truth of a conclusion after the introduction of an argument of the intrinsic probabilityb, the previous probability of the said conclusion having beena.
Thus, a conclusion which has at present the chance ⅔ in its favour, when reinforced by an argument whose intrinsic probability is ¾, acquires the probability ⅔ + ¾ − ⅔ × ¾ or, ⅔ + ¾ − ½, or ¹¹⁄₁₂; or, having 2 to 1 in its favour before, it has 11 to 1 in its favour after, the argument.
When the conclusion was neither likely nor unlikely beforehand (or had the probability ½), the shortest way of applying the preceding rule (in whicha + b − abbecomes ½ + ½b) is to divide the sum of the numerator and denominator of the intrinsic probability of the argument by twice the denominator. Thus, an argument of which the intrinsic probability is ¾, gives to a conclusion on which no bias previously existed, the probability ⅞ or3 + 42 × 4.