Page:The American Cyclopædia (1879) Volume X.djvu/585

 LOGIC 579 ange is round," &c. Or, forming a generic con- ception, we may say, "An orange is a fruit;" "Men are animals." We may thus predicate P of M, and M of S, and then, dropping the common or middle term M, may predicate P of S, a proposition derived by deduction from the two" premises or primary judgments. The formula, " M is P, S is M, therefore S is P," is called a syllogism, a term which includes any possible combination of two propositions from which is deduced a third, which is hence called a conclusion. The conclusions of preceding syllogisms may become the premises of others ad in/ftnitum. The premises may be negative as well as affirmative S are not P, as well as S are P ; they may also include only a part of the subject, as some S are P, some S are not P. Hence there are four cardinal propositions : Universal affirmative : All S are P. " negative : No S are P. Particular affirmative: Some S are P. " negative : Some S are not P. For convenience these propositions are desig- nated by the first four vowels ; thus : A, uni- versal affirmative ; E, universal negative ; I, particular affirmative ; O, particular negative. Combining these four propositions in all pos- sible ways of three in a set, we obtain 64 sets, which are called moods. Of these moods, however, only 11 are found to give valid con- clusions, viz.: AAA, AAI, AEE, AEO, All, AOO, EAE, EAO, EIO, IAI, and OAO. It is found also that the position of the middle term is of essential importance; for let the mood AAA be written thus : " All M are P ; all S are M; therefore all S are P;" and it is evi- dent at once that if M is included in the class P, and S is included in the class M, then S must be included in P also. But if the same mood be written, "All P are M; all S are M," then it does not follow that S is included in P ; for men are animals, and horses are animals, but men are not therefore horses. Every mood of the syllogism thus has what are termed fig- ures, of which there are four. In the first figure, the middle term is the subject of the major premise and the predicate of the minor ; in the second, the middle term is the predicate of both premises ; in the third, it is the sub- ject of both premises ; and in the fourth, it is the predicate of the major premise and the subject of the minor. The 11 moods each hav- ing 4 figures would give 44 syllogisms, of which, however, only 19 are found by exami- nation to be distinct and valid. These are des- ignated by the capital vowels in the following mnemonic hexameters : BArbArA, cElArEnt, DArll, fErWque, prioris: <?EsArE, cAmEstrEs, fEstlnO, bArOkO, seoimdce: Tertia dArAptl, dlsAmls, dAtlsI, fElAptOn, EOkArdO, fErlsOn, habet: quarto, insuper addit, BrAmAntlp, cAmEnsEs, dlmArls, fEsApO, frEsIsOn. When one of the premises is understood, but not expressed, in the statement, the syllogism is called an enthymeme. When several prem- ises are employed for the same conclusion, several syllogisms are in fact abridged into one formula, which is called a sorites. When one premise is assumed as hypothetically true, and the conclusion is stated as depending upon the truth of the other alone, we have what is called a conditional judgment ; and if the con- clusion is stated as depending upon the falsity of the other, we have a disjunctive judgment. A conditional or disjunctive proposition may be made the major premise, and then the syl- logism be completed as follows : " If A is B, C is D ; but A is B ; therefore is D." In this case the syllogism is called a conditional syllo- gism, or sometimes a hypothetical syllogism. "Either A is B or C is D ; but A is not B; therefore C is D." In this case the syllogism is called disjunctive. The major premise may affirm only a comparison or relation between the terms, as : " Where the boy is, there the father is ; but the boy is at home ; therefore, the father is at home." Besides the fulfilment of all the conditions of the formulas in syllo- gisms, there are found to be also certain con- ditions and laws in regard to the use of words, which are necessary to the validity of the rea- soning. The violation of these laws gives rise to fallacies, of which there are reckoned 13, 6 in dictione and 7 extra dictionem. 1. Equivo- cation occurs when a word is used in the same formula in two different senses. 2. Amphibo- logy when a word is so used as to leave it doubtful whether it be a subject or predicate, or when the reference of a pronoun is ambig- uous. 3 and 4. Composition and division are caused by using the same term both collective- ly and distributively in the same formula, thus : " 3 and 2 are two numbers ; but 5 is 3 and 2 ; therefore, 5 is two numbers." Here 3 and 2 are used distributively in the major and collec- tively in the minor premise. The reverse is true of the word Romans in the following : "The Romans conquered Carthage; Brutus and Caesar were Romans; therefore Brutus and Caesar conquered Carthage." 5. Accent may occasion a fallacy by varying the mean- ing of a proposition. Thus the purport of the question, "Do you ride to town to-day?" may be changed five times by changing the accented word, or omitting the emphatic ac- cent. 6. The form of the expression (figura dictionis) may lead to a fallacy, as when we infer from the fact that one word ending in #, as mensa, is of the feminine gender, that there- fore another word with a like termination, as poeta, is feminine also. 7. The fallacy of acci- dents arises when we affirm of something de- scribed 1)y some accidental property or circum- stance what is true only of its substance, as : " We buy raw meat in the market ; what we buy in the market, we eat ; therefore, we eat raw meat." Here we do not buy meat because it is raw, but because it is meat, for its essence and not for its accidents, and only its essential quality is common to the different members of the argument. 8. Mistaken application con- sists in giving to a statement a universal appli-