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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..d7b82bc --- /dev/null +++ b/.gitattributes @@ -0,0 +1,4 @@ +*.txt text eol=lf +*.htm text eol=lf +*.html text eol=lf +*.md text eol=lf diff --git a/LICENSE.txt b/LICENSE.txt new file mode 100644 index 0000000..6312041 --- /dev/null +++ b/LICENSE.txt @@ -0,0 +1,11 @@ +This eBook, including all associated images, markup, improvements, +metadata, and any other content or labor, has been confirmed to be +in the PUBLIC DOMAIN IN THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..96af02a --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #67017 (https://www.gutenberg.org/ebooks/67017) diff --git a/old/67017-0.txt b/old/67017-0.txt deleted file mode 100644 index 434bcc7..0000000 --- a/old/67017-0.txt +++ /dev/null @@ -1,1659 +0,0 @@ -The Project Gutenberg eBook of First notions of logic, by Augustus -De Morgan - -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you -will have to check the laws of the country where you are located before -using this eBook. - -Title: First notions of logic - (preparatory to the study of geometry) - -Author: Augustus De Morgan - -Release Date: December 26, 2021 [eBook #67017] - -Language: English - -Produced by: Richard Tonsing and the Online Distributed Proofreading - Team at https://www.pgdp.net (This file was produced from - images generously made available by The Internet Archive) - -*** START OF THE PROJECT GUTENBERG EBOOK FIRST NOTIONS OF LOGIC *** - - - - - - FIRST NOTIONS - OF - LOGIC - (PREPARATORY TO THE STUDY OF GEOMETRY) - - - BY - - AUGUSTUS DE MORGAN, - - OF TRINITY COLLEGE, CAMBRIDGE, - PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON. - - The root of all the mischief in the sciences, is this; that falsely - magnifying and admiring the powers of the mind, we seek not its real - helps.—BACON. - - - LONDON: - - PRINTED FOR TAYLOR AND WALTON, - - BOOKSELLERS AND PUBLISHERS TO UNIVERSITY COLLEGE. - - 28 UPPER GOWER STREET. - - M.DCCC.XXXIX. - - - - -⁂ This Tract contains no more than the author has found, from -experience, to be much wanted by students who are commencing with -Euclid. It will ultimately form an Appendix to his Treatise on -Arithmetic. - -The author would not, by any means, in presenting the minimum necessary -for a particular purpose, be held to imply that he has given enough of -the subject for all the ends of education. He has long regretted the -neglect of logic; a science, the study of which would shew many of its -opponents that the light esteem in which they hold it arises from those -habits of inference which thrive best in its absence. He strongly -recommends any student to whom this tract may be the first introduction -of the subject, to pursue it to a much greater extent. - -_University College, Jan, 8, 1839._ - - - LONDON:—PRINTED BY JAMES MOYES, - Castle Street, Leicester Square. - - - - - FIRST NOTIONS - OF - LOGIC. - - -What we here mean by Logic is the examination of that part of reasoning -which depends upon the manner in which inferences are formed, and the -investigation of general maxims and rules for constructing arguments, so -that the conclusion may contain no inaccuracy which was not previously -asserted in the premises. It has nothing to do with the truth of the -facts, opinions, or presumptions, from which an inference is derived; -but simply takes care that the inference shall certainly be true, if the -premises be true. Thus, when we say that all men will die, and that all -men are rational beings, and thence infer that some rational beings will -die, the _logical_ truth of this sentence is the same whether it be true -or false that men are mortal and rational. This logical truth depends -upon the structure of the sentence, and not on the particular matters -spoken of. Thus, - - Instead of, Write, - All men will die. Every A is B. - All men are rational beings. Every A is C. - Therefore some rational beings will die. Therefore some Cs are Bs. - -The second of these is the same proposition, logically considered, as -the first; the consequence in both is virtually contained in, and -rightly inferred from, the premises. Whether the premises be true or -false, is not a question of logic, but of morals, philosophy, history, -or any other knowledge to which their subject-matter belongs: the -question of logic is, does the conclusion certainly follow if the -premises be true? - -Every act of reasoning must mainly consist in comparing together -different things, and either finding out, or recalling from previous -knowledge, the points in which they resemble or differ from each other. -That particular part of reasoning which is called _inference_, consists -in the comparison of several and different things with one and the same -other thing; and ascertaining the resemblances, or differences, of the -several things, by means of the points in which they resemble, or differ -from, the thing with which all are compared. - -There must then be some propositions already obtained before any -inference can be drawn. All propositions are either assertions or -denials, and are thus divided into _affirmative_ and _negative_. Thus, A -is B, and A is not B, are the two forms to which all propositions may be -reduced. These are, for our present purpose, the most simple forms; -though it will frequently happen that much circumlocution is needed to -reduce propositions to them. Thus, suppose the following assertion, ‘If -he should come to-morrow, he will probably stay till Monday’; how is -this to be reduced to the form A is B? There is evidently something -spoken of, something said of it, and an affirmative connexion between -them. Something, if it happen, that is, the happening of something, -makes the happening of another something probable; or is one of the -things which render the happening of the second thing probable. - - A │is│ B - - - The happening of his arrival │ │an event from which it may be - to-morrow │is│ inferred as probable that he - │ │ will stay till Monday. - -The forms of language will allow the manner of asserting to be varied in -a great number of ways; but the reduction to the preceding form is -always possible. Thus, ‘so he said’ is an affirmation, reducible as -follows: - - What you have just said (or │is│the thing which he said. - whatever else ‘so’ refers to) │ │ - -By changing ‘is’ into ‘is not,’ we make a negative proposition; but care -must always be taken to ascertain whether a proposition which appears -negative is really so. The principal danger is that of confounding a -proposition which is negative with another which is affirmative of -something requiring a negative to describe it. Thus ‘he resembles the -man who was not in the room,’ is affirmative, and must not be confounded -with ‘he does not resemble the man who was in the room.’ Again, ‘if he -should come to-morrow, it is probable he will not stay till Monday,’ -does not mean the simple denial of the preceding proposition, but the -affirmation of the directly opposite proposition. It is, - - A │is│ B - - - The happening of his arrival │ │an event from which it may be - to-morrow, │is│ inferred to be _im_probable - │ │ that he will stay till Monday, - -whereas the following, - - The happening of his arrival │ │an event from which it may be - to-morrow, │is _not_│ inferred as probable that he - │ │ will stay till Monday, - -would be expressed thus: ‘If he should come to-morrow, that is no reason -why he should stay till Monday.’ - -Moreover, the negative words not, no, &c., have two kinds of meaning -which must be carefully distinguished. Sometimes they deny, and nothing -more: sometimes they are used to affirm the direct contrary. In cases -which offer but two alternatives, one of which is necessary, these -amount to the same thing, since the denial of one, and the affirmation -of the other, are obviously equivalent propositions. In many idioms of -conversation, the negative implies affirmation of the contrary in cases -which offer not only alternatives, but degrees of alternatives. Thus, to -the question, ‘Is he tall?’ the simple answer, ‘No,’ most frequently -means that he is the contrary of tall, or considerably under the -average. But it must be remembered, that, in all logical reasoning, the -negation is simply negation, and nothing more, never implying -affirmation of the contrary. - -The common proposition that two negatives make an affirmative, is true -only upon the supposition that there are but two possible things, one of -which is denied. Grant that a man must be either able or unable to do a -particular thing, and then _not unable_ and able are the same things. -But if we suppose various degrees of performance, and therefore degrees -of ability, it is false, in the common sense of the words, that two -negatives make an affirmative. Thus, it would be erroneous to say, ‘John -is able to translate Virgil, and Thomas is not unable; therefore, what -John can do Thomas can do,’ for it is evident that the premises mean -that John is so near to the best sort of translation that an affirmation -of his ability may be made, while Thomas is considerably lower than -John, but not so near to absolute deficiency that his ability may be -altogether denied. It will generally be found that two negatives imply -an affirmative of a weaker degree than the positive affirmation. - -Each of the propositions, ‘A is B,’ and ‘A is not B,’ may be subdivided -into two species: the _universal_, in which every possible case is -included; and the _particular_, in which it is not meant to be asserted -that the affirmation or negation is universal. The four species of -propositions are then as follows, each being marked with the letter by -which writers on logic have always distinguished it. - - A _Universal Affirmative_ Every A is B - E _Universal Negative_ No A is B - I _Particular Affirmative_ Some A is B - O _Particular Negative_ Some A is not B - -In common conversation the affirmation of a part is meant to imply the -denial of the remainder. Thus, by ‘some of the apples are ripe,’ it is -always intended to signify that some are not ripe. This is not the case -in logical language, but every proposition is intended to make its -amount of affirmation or denial, and no more. When we say, ‘Some A is -B,’ or, more grammatically, ‘Some As are Bs,’ we do not mean to imply -that some are not: this may or may not be. Again, the word some means, -‘one or more, possibly all.’ The following table will shew the bearing -of each proposition on the rest. - - _Every A is B_ affirms and contains _Some A is B_ and│_No A is B_ - denies │_Some A is not B_ - - _No A is B_ affirms and contains _Some A is not B_ │_Every A is B_ - and denies │_Some A is B_ - - _Some A is B_ does not │_Every A is B_ │but denies _No A is B_ - contradict │_Some A is not B_│ - - _Some A is not B_ does not │_No A is B_ │but denies _Every A is B_ - contradict │_Some A is B_ │ - -_Contradictory_ propositions are those in which one denies _any thing_ -that the other affirms; _contrary_ propositions are those in which one -denies _every thing_ which the other affirms, or affirms every thing -which the other denies. The following pair are contraries. - - Every A is B and No A is B - -and the following are contradictories, - - Every A is B to Some A is not B - No A is B to Some A is B - -A contrary, therefore, is a complete and total contradictory; and a -little consideration will make it appear that the decisive distinction -between contraries and contradictories lies in this, that contraries may -both be false, but of contradictories, one must be true and the other -false. We may say, ‘Either P is true, or _something_ in contradiction of -it is true;’ but we cannot say, ‘Either P is true, or _every thing_ in -contradiction of it is true.’ It is a very common mistake to imagine -that the _denial_ of a proposition gives a right to _affirm_ the -contrary; whereas it should be, that the _affirmation_ of a proposition -gives a right to _deny_ the contrary. Thus, if we deny that Every A is -B, we do not affirm that No A is B, but only that Some A is not B; -while, if we affirm that Every A is B, we deny No A is B, and also Some -A is not B. - -But, as to contradictories, affirmation of one is a denial of the other, -and denial of one is affirmation of the other. Thus, either Every A is -B, or Some A is not B: affirmation of either is denial of the other, and -_vice versá_. - -Let the student now endeavour to satisfy himself of the following. -Taking the four preceding propositions, A, E, I, O, let the simple -letter signify the affirmation, the same letter in parentheses the -denial, and the absence of the letter, that there is neither affirmation -nor denial. - - From A follow│(E), I, (O)│From (A) follow O - From E │(A), (I), O│From (E) I - From I │(E) │From (I) (A), E, O - From O │(A) │From (O) A, (E), I - -These may be thus summed up: The affirmation of a universal proposition, -and the denial of a particular one, enable us to affirm or deny all the -other three; but the denial of a universal proposition, and the -affirmation of a particular one, leave us unable to affirm or deny two -of the others. - -In such propositions as ‘Every A is B,’ ‘Some A is not B,’ &c., A is -called the _subject_, and B the _predicate_, while the verb ‘is’ or ‘is -not,’ is called the _copula_. It is obvious that the words of the -proposition point out whether the subject is spoken of universally or -partially, but not so of the predicate, which it is therefore important -to examine. Logical writers generally give the name of _distributed_ -subjects or predicates to those which are spoken of universally; but as -this word is rather technical, I shall say that a subject or predicate -enters wholly or partially, according as it is universally or -particularly spoken of. - -1. In A, or ‘Every A is B,’ the subject enters wholly, but the predicate -only partially. For it obviously says, ‘Among the Bs are all the As,’ -‘Every A is part of the collection of Bs, so that all the As make a part -of the Bs, the whole it _may_ be.’ Thus, ‘Every horse is an animal,’ -does not speak of all animals, but states that all the horses make up a -portion of the animals. - -2. In E, or ‘No A is B,’ both subject and predicate enter wholly. ‘No A -whatsoever is any one out of all the Bs;’ ‘search the whole collection -of Bs, and _every_ B shall be found to be something which is not A.’ - -3. In I, or ‘Some A is B,’ both subject and predicate enter partially. -‘Some of the As are found among the Bs, or make up a part (the whole -possibly, but not known from the preceding) of the Bs.’ - -4. In O, or ‘Some A is not B,’ the subject enters partially, and the -predicate wholly. ‘Some As are none of them any whatsoever of the Bs; -every B will be found to be no one out of a certain portion of the As.’ - -It appears then that, - -In affirmatives, the predicate enters partially. - -In negatives, the predicate enters wholly. - -In contradictory propositions, both subject and predicate enter -differently in the two. - -The _converse_ of a proposition is that which is made by interchanging -the subject and predicate, as follows: - - The proposition. Its converse. - A Every A is B Every B is A - E No A is B No B is A - I Some A is B Some B is A - O Some A is not B Some B is not A - -Now, it is a fundamental and self-evident proposition, that no -consequence must be allowed to assert more widely than its premises; so -that, for instance, an assertion which is only of some Bs can never lead -to a result which is true of all Bs. But if a proposition assert -agreement or disagreement, any other proposition which asserts the same, -to the same extent and no further, must be a legitimate consequence; or, -if you please, must amount to the whole, or part, of the original -assertion in another form. Thus, the converse of A is not true: for, in -‘Every A is B,’ the predicate enters partially; while in ‘Every B is A,’ -the subject enters wholly. ‘All the As make up a part of the Bs, then a -part of the Bs are among the As, or some B is A.’ Hence, the only -_legitimate_ converse of ‘Every A is B’ is, ‘Some B is A.’ But in ‘No A -is B,’ both subject and predicate enter wholly, and ‘No B is A’ is, in -fact, the same proposition as ‘No A is B.’ And ‘Some A is B’ is also the -same as its converse ‘Some B is A;’ here both terms enter partially. But -‘Some A is not B’ admits of no converse whatever; it is perfectly -consistent with all assertions upon B and A in which B is the subject. -Thus neither of the four following lines is inconsistent with itself. - - Some A is not B and Every B is A - Some A is not B and No B is A - Some A is not B and Some B is A - Some A is not B and Some B is not A. - -We find then, including converses, which are not identical with their -direct propositions, _six_ different ways of asserting or denying, with -respect to agreement or non-agreement, total or partial, between A and, -say X: these we write down, designating the additional assertions by U -and Y. - - │Identical. │ Identical. │ - A Every A is X│E No A is X│I Some A is X│O Some A is not X - U Every X is A│„ No X is A│„ Some X is A│Y Some X is not A - -We shall now repeat and extend the table of page 8 (A), &c., meaning, as -before, the denial of A, &c. - - From A or (O) follow A, (E), I (O) - From E or (I) (A), E, (I), O, (U), Y - From I or (E) (E) I - From O or (A) (A), O - From U or (Y) (E) I, U (Y) - From Y or (U) (U) Y - -Having thus discussed the principal points connected with the simple -assertion, we pass to the manner of making two assertions give a third. -Every instance of this is called a syllogism, the two assertions which -form the basis of the third are called premises, and the third itself -the conclusion. - -If two things both agree with a third in any particular, they agree with -each other in the same; as, if A be of the same colour as X, and B of -the same colour as X, then A is of the same colour as B. Again, if A -differ from X in any particular in which B agrees with X, then A and B -differ in that particular. If A be not of the same colour as X, and B be -of the same colour as X, then A is not of the colour of B. But if A and -B both differ from X in any particular, nothing can be inferred; they -may either differ in the same way and to the same extent, or not. Thus, -if A and B be both of different colours from X, it neither follows that -they agree, nor differ, in their own colours. - -The paragraph preceding contains the essential parts of all inference, -which consists in comparing two things with a third, and finding from -their agreement or difference with that third, their agreement or -difference with one another. Thus, Every A is X, every B is X, allows us -to infer that A and B have all those qualities in common which are -necessary to X. Again, from Every A is X, and ‘No B is X,’ we infer that -A and B differ from one another in all particulars which are essential -to X. The preceding forms, however, though they represent common -reasoning better than the ordinary syllogism, to which we are now -coming, do not constitute the ultimate forms of inference. Simple -_identity_ or _non-identity_ is the ultimate state to which every -assertion may be reduced; and we shall, therefore, first ask, from what -identities, &c., can other identities, &c., be produced? Again, since we -name objects in species, each species consisting of a number of -individuals, and since our assertion may include all or only part of a -species, it is further necessary to ask, in every instance, to what -extent the conclusion drawn is true, whether of all, or only of part? - -Let us take the simple assertion, ‘Every living man respires;’ or, every -living man is one of the things (however varied they may be) which -respire. If we were to inclose all living men in a large triangle, and -all respiring objects in a large circle, the preceding assertion, if -true, would require that the whole of the triangle should be contained -in the circle. And in the same way we may reduce any assertion to the -expression of a coincidence, total or partial, between two figures. -Thus, a point in a circle may represent an individual of one species, -and a point in a triangle an individual of another species: and we may -express that the whole of one species is asserted to be contained or not -contained in the other by such forms as, ‘All the △ is in the ○’; ‘None -of the △ is in the ○’. - -Any two assertions about A and B, each expressing agreement or -disagreement, total or partial, with or from X, and leading to a -conclusion with respect to A or B, is called a syllogism, of which X is -called the _middle term_. The plainest syllogism is the following:— - - Every A is X│ All the △ is in the ○ - Every X is B│ All the ○ is in the □ - Therefore Every A is B│Therefore All the △ is in the □ - -In order to find all the possible forms of syllogism, we must make a -table of all the elements of which they can consist; namely— - - A and X B and X - Every A is X A Every B is X - No A is X E No B is X - Some A is X I Some B is X - Some A is not X O Some B is not X - Every X is A U Every X is B - Some X is not A Y Some X is not B - -Or their synonymes, - - △ and ○ □ and ○ - All the △ is in the ○ A All the □ is in the ○ - None of the △ is in the ○ E None of the □ is in the ○ - Some of the △ is in the ○ I Some of the □ is in the ○ - Some of the △ is not in the ○ O Some of the □ is not in the ○ - All the ○ is in the △ U All the ○ is in the □ - Some of the ○ is not in the △ Y Some of the ○ is not in the □ - -Now, taking any one of the six relations between A and X, and combining -it with either of those between B and X, we have six pairs of premises, -and the same number repeated for every different relation of A and X. We -have then thirty-six forms to consider: but, thirty of these (namely, -all but (A, A) (E, E), &c.) are half of them repetitions of the other -half. Thus, ‘Every A is X, no B is X,’ and ‘Every B is X, no A is X,’ -are of the same form, and only differ by changing A into B and B into A. -There are then only 15 + 6, or 21 distinct forms, some of which give a -necessary conclusion, while others do not. We shall select the former of -these, classifying them by their conclusions; that is, according as the -inference is of the form A, E, I, or O. - -I. In what manner can a universal affirmative conclusion be drawn; -namely, that one figure is entirely contained in the other? This we can -only assert when we know that one figure is entirely contained in the -circle, which itself is entirely contained in the other figure. Thus, - - Every A is X│ All the △ is in the ○ A - Every X is B│ All the ○ is in the □ A - ∴ Every A is B│∴ All the △ is in the □ A - -is the only way in which a universal affirmative conclusion can be -drawn. - -II. In what manner can a universal negative conclusion be drawn; namely, -that one figure is entirely exterior to the other? Only when we are able -to assert that one figure is entirely within, and the other entirely -without, the circle. Thus, - - Every A is X│All the △ is in the ○ A - - No B is X│None of the □ is in E - │the ○ - - ∴ No A is B│None of the △ is in E - │the □ - -is the only way in which a universal negative conclusion can be drawn. - -III. In what manner can a particular affirmative conclusion be drawn; -namely, that part or all of one figure is contained in the other? Only -when we are able to assert that the whole circle is part of one of the -figures, and that the whole, or part of the circle, is part of the other -figure. We have then two forms. - - Every X is A│ All the ○ is in the △ A - - Every X is B│ All the ○ is in the □ A - - ∴ Some A is B│∴ Some of the △ is in I - │ the □ - - - Every X is A│ All the ○ is in the △ A - - Some X is B│ Some of the ○ is in I - │ the □ - - Some A is B│ Some of the △ is in I - │ the □ - -The second of these contains all that is strictly necessary to the -conclusion, and the first may be omitted. That which follows when an -assertion can be made as to some, must follow when the same assertion -can be made of all. - -IV. How can a particular negative proposition be inferred; namely, that -part, or all of one figure, is not contained in the other? It would seem -at first sight, whenever we are able to assert that part or all of one -figure is in the circle, and that part or all of the other figure is -not. The weakest syllogism from which such an inference can be drawn -would then seem to be as follows. - - Some A is X │ Some of the △ is in the ○ - Some B is not X│ Some of the □ is not in the ○ - ∴ Some B is not A│∴ Some of the △ is not in the □ - -But here it will appear, on a little consideration, that the conclusion -is only thus far true; that those As which are Xs cannot be _those_ Bs -which are not Xs; but they may be _other_ Bs, about which nothing is -asserted when we say that _some_ Bs are not Xs. And further -consideration will make it evident, that a conclusion of this form can -only be arrived at when one of the figures is entirely within the -circle, and the whole or part of the other without; or else when the -whole of one of the figures is without the circle, and the whole or part -of the other within; or lastly, when the circle lies entirely within one -of the figures, and not entirely within the other. That is, the -following are the distinct forms which allow of a particular negative -conclusion, in which it should be remembered that a particular -proposition in the premises may always be changed into a universal one, -without affecting the conclusion. For that which necessarily follows -from “some,” follows from “all.” - - Every A is X │ All the △ is in the ○ A - Some B is not X│ Some of the □ is not in the ○ O - ∴ Some B is not A│ Some of the □ is not in the △ O - - No A Is X │ None of the △ is in the ○ E - Some B is X │ Some of the □ is in the ○ I - ∴ Some B is not A│ Some of the □ is not in the △ O - - Every X is A │ All the ○ is in the △ A - Some X is not B│ Some of the ○ is not in the □ O - ∴ Some A is not B│ Some of the △ is not in the □ O - -It appears, then, that there are but six distinct syllogisms. All others -are made from them by strengthening one of the premises, or converting -one or both of the premises, where such conversion is allowable; or else -by first making the conversion, and then strengthening one of the -premises. And the following arrangement will shew that two of them are -universal, three of the others being derived from them by weakening one -of the premises in a manner which does not destroy, but only weakens, -the conclusion. - - 1. Every A is X 3. Every A is X - Every X is B No B is X ......... - ———————————— ———————————— - Every A is B No A is B - │ │ - │ ┌─────────┴─────────┐ - │ │ │ - 2. Some A is X 4. Some A is X 5. Every A is X 6. Every X is A - Every X is B No B is X Some B is not X Some X is not B - ———————————— ——————————————— ———————————————— ——————————————— - Some A is B Some A is not B Some B is not A Some A is not B - -We may see how it arises that one of the partial syllogisms is not -immediately derived, like the others, from a universal one. In the -preceding, AEE may be considered as derived from AAA, by changing the -term in which X enters universally into its contrary. If this be done -with the other term instead, we have - - No A is X│from which universal premises we cannot deduce a universal - │ conclusion, but only Some B is not A. - - Every X is B│ „ - -If we weaken one and the other of these premises, as they stand, we -obtain - - Some A is not X No A is X - Every X is B and Some X is B - ———————————————— ——————————————— - No conclusion Some B is not A - -equivalent to the fourth of the preceding: but if we convert the first -premiss, and proceed in the same manner, - - No X is A Some X is not A - From Every X is B we obtain Every X is B - ———————————————— ———————————————— - Some B is not A Some B is not A - -which is legitimate, and is the same as the last of the preceding list, -with A and B interchanged. - -Before proceeding to shew that all the usual forms are contained in the -preceding, let the reader remark the following rules, which may be -proved either by collecting them from the preceding cases, or by -independent reasoning. - -1. The middle term must enter universally into one or the other premiss. -If it were not so, the one premiss might speak of one part of the middle -term, and the other of the other; so that there would, in fact, be no -middle term. Thus, ‘Every A is X, Every B is X,’ gives no conclusion: it -may be thus stated; - - All the As make up _a part_ of the Xs - All the Bs make up _a part_ of the Xs - -And, before we can know that there is any common term of comparison at -all, we must have some means of shewing that the two parts are the same; -or the preceding premises by themselves are inconclusive. - -2. No term must enter the conclusion more generally than it is found in -the premises; thus, if A be spoken of partially in the premises, it must -enter partially into the conclusion. This is obvious, since the -conclusion must assert no more than the premises imply. - -3. From premises both negative no conclusion can be drawn. For it is -obvious, that the mere assertion of disagreement between each of two -things and a third, can be no reason for inferring either agreement or -disagreement between these two things. It will not be difficult to -reduce any case which falls under this rule to a breach of the first -rule: thus, No A is X, No B is X, gives - - Every A is (something which is not X) - Every B is (something which is not X) - -in which the middle term is not spoken of universally in either. Again, -‘No X is A, Some X is not B,’ may be converted into - - Every A is (a thing which is not X) - Some (thing which is not B) is X - -in which there is no middle term. - -4. From premises both particular no conclusion can be drawn. This is -sufficiently obvious when the first or second rule is broken, as in -‘Some A is X, Some B is X.’ But it is not immediately obvious when the -middle term enters one of the premises universally. The following -reasoning will serve for exercise in the preceding results. Since both -premises are particular in form, the middle term can only enter one of -them universally by being the predicate of a negative proposition; -consequently (Rule 3) the other premiss must be affirmative, and, being -particular, neither of its terms is universal. Consequently both the -terms as to which the conclusion is to be drawn enter partially, and the -conclusion (Rule 2) can only be a particular _affirmative_ proposition. -But if one of the premises be negative, the conclusion must be -_negative_ (as we shall immediately see). This contradiction shews that -the supposition of particular premises producing a legitimate result is -inadmissible. - -5. If one premiss be negative, the conclusion, if any, must be negative. -If one term agree with a second and disagree with a third, no agreement -can be inferred between the second and third. - -6. If one premiss be particular, the conclusion must be particular. This -is not very obvious, since the middle term may be universally spoken of -in a particular proposition, as in Some B is not X. But this requires -one negative proposition, whence (Rule 3) the other must be affirmative. -Again, since the conclusion must be negative (Rule 5) its predicate is -spoken of universally, and, therefore, must enter universally; the other -term A must enter, then, in a universal affirmative proposition, which -is against the supposition. - -In the preceding set of syllogisms we observe one form only which -produces A, or E, or I, but three which produce O. - -Let an assertion be said to be weakened when it is reduced from -universal to particular, and strengthened in the contrary case. Thus, -‘Every A is B’ is called stronger than ‘Some A is B.’ - -Every form of syllogism which can give a legitimate result is either one -of the preceding six, or another formed from one of the six, either by -changing one of the assertions into its converse, if that be allowable, -or by strengthening one of the premises without altering the conclusion, -or both. Thus, - - Some A is X may be written Some X is A - Every X is B „ Every X is B - - What follows will still follow from _Every_ X is A - „ Every X is B - -for all which is true when ‘Some X is A,’ is not less true when ‘Every X -is A.’ - -It would be possible also to form a legitimate syllogism by weakening -the conclusion, when it is universal, since that which is true of all is -true of some. Thus, ‘Every A is X, Every X is B,’ which yields ‘Every A -is B,’ also yields ‘Some A is B.’ But writers on logic have always -considered these syllogisms as useless, conceiving it better to draw -from any premises their strongest conclusion. In this they were -undoubtedly right; and the only question is, whether it would not have -been advisable to make the premises as weak as possible, and not to -admit any syllogisms in which more appeared than was absolutely -necessary to the conclusion. If such had been the practice, then - - Every X is A, Every X is B, therefore Some A is B - -would have been considered as formed by a spurious and unnecessary -excess of assertion. The minimum of assertion would be contained in -either of the following, - - 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. - - Universal. Particular. - - A Every A is X Some A is X I - A Every X is B ────── Every X is B A - ———————————— ————————————— - A Every A is B Some A is B I - - Some A is X I - No X is B E - - ┌ ——————————————— - A Every A is X │ Some A is not B O - E No X is B ─────┼ Every A is X A - ———————————— │ - E No A is B │ Some B is not X O - └ ——————————————— - Some B is not A O - - Every X is A A - ...... Some X is not B O - ——————————————— - Some A is not B O - -In all works on logic, it is customary to write that premiss first which -contains the predicate of the conclusion. Thus, - - Every X is B Every A is X - Every A is X would be written, and not Every X is B - ———————————— ———————————— - Every A is B Every A is B - -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— - - XB BX XB BX - AX AX XA XA - —— —— —— —— - AB AB AB AB - -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, - - A Every X is B A Every X is B - I Some A is X becomes A Every X is A: (Co +) - ———————————— ———————————— - I Some A is B I Some A is B - -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. - - _First Figure._ - - A Every X is B A Every X is B - A Every A is X I Some A is X - ———————————— —————————————— - A Every A is B I Some A is B - - E No X is B E No X is B - A Every A is X I Some A is X - ———————————— ———————————————— - E No A is B O Some A is not B - - - _Second Figure._ - - E No B is X (Co) E No B is X (Co) - A Every A is X I Some A is X - ———————————— ————————————————— - E No A is B O Some A is not B - - A Every B is X A Every B is X - E No A is X (Co) O Some A is not X - ———————————— ———————————————— - E No A is B O Some A is not B - - - _Third Figure._ - - A Every X is B E No X is B - A Every X is A (Co+) A Every X is A (Co+) - ———————————— —————————————————— - I Some A is B O Some A is not B - - I Some X is B (Co) O Some X is not B - A Every X is A A Every X is A - ———————————— ———————————————— - I Some A is B O Some A is not B - - A Every X is B E No X is B - I Some X is A (Co) I Some X is A (Co) - ———————————— ————————————————— - I Some A is B O Some A is not B - - - _Fourth Figure._ - - A Every B is X (+) I Some B is X - A Every X is A A Every X is A - ———————————— ———————————— - I Some A is B I Some B is A - - A Every B is X E No B is X (Co) - E No X is A A Every X is A (Co+) - ———————————— —————————————————— - E No A is B O Some A is not B - - E No B is X (Co) - I Some X is A (Co) - ———————————————— - O Some A is not B - -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: - - Every A is X Every B is X │Some A is X Some B is X - Every X is B Every X is A │No X is B No X is A - ———————————— ———————————— │———————————————— ———————————————— - Every A is B Every B is A │Some A is not B Some B is not A - │ - Every A is X Every B is X │Every A is X Every B is X - No X is B No X is A │Some B is not X Some A is not X - ———————————— ———————————— │———————————————— ———————————————— - No A is B No B is A │Some B is not A Some A is not B - │ - Some A is X Some B is X │Every X is A Every X is B - Every X is B Every X is A │Some X is not B Some X is not A - ———————————— ———————————— │———————————————— ———————————————— - Some A is B Some B is A │Some A is not B Some B is not A - -In the list of page 19, 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 possible -_weakest_ forms 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 page 10. - - A Every A is X - U Every X is B - ———————————— - A Every A is B - - I Some A is X I Some B is X - U Every X is B U Every X is A - ———————————— ———————————— - I Some A is B I Some A is B - - A Every A is X A Every B is X - E No B is X E No A is X - ———————————— ———————————— - E No A is B E No A is B - - I Some A is X A Every B is X U Every X is A - E No B is X O Some A is not X Y Some X is not B - ———————————————— ———————————————— ———————————————— - O Some A is not B O Some A is not B O Some A is not B - -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 race _is_ a 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, - - All weak princes are prone to favourites - He was prone to favourites - ———————————————— ——————————————————————— - Therefore He was a weak prince - -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, - - A is (a magnitude less than B) - B is (a magnitude less than C) - Therefore A is (a magnitude less than C) - -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, - - A is less than B - Less than B is less than C: following from B is less than C. - ————————— ————————————————— - Therefore A is less than C - -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 syllogism -can 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, - - Every A is B, and there are Bs which are not As - Every B is C, and there are Cs which are not Bs - ——————————————————————————————————————————————— - Therefore Every A is C, and there are Cs which are not As - -Or thus: - - The Bs contain all the As, and more - The Cs contain all the Bs, and more - ——————————————————————————————————— - The Cs contain all the As, and more - -The most technical form, however, is, - - From Every A is B; [Some B is not A] - Every B is C; [Some C is not B] - Follows Every A is C; [Some C is not A] - -This sort of argument is called _à fortiori_ argument, 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 it _à fortiori_. - -The argument _à fortiori_, may then be defined as a universally -affirmative 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 Xs - Every X is B; - -then we are certain that - - 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 Xs - All the Xs make up part (and part only) of the Bs - -then we conclude that - - All the As make up _part of part_ (only) of the Bs - -and the words in Italics mark that quality of the conclusion from which -the argument is called _à fortiori_. - -Most syllogisms which give an affirmative conclusion are generally meant -to imply _à fortiori_ arguments, 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 an _hypothetical_ syllogism. 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; - -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 Q _only_ when 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, - - Every K is Z Every K is Z - L is a K L is not a K - ———————— ————————————— - Therefore L is a Z No conclusion - -Similarly, according as a particular case (M) is or is not Z, we have - - Every K is Z Every K is Z - M is a Z M is not a Z - ————————————— ———————————— - No conclusion M is not a K - -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 a -contradiction 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] - -Footnote 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 called _direct_; when by proving the -falsehood of every contradictory proposition, it is called _indirect_. -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 - - Every A is B (1) - -may be easily made the means of deducing, - - Every (thing not A) is not B (2) - -which last gives - - Every B is A (3) - -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 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 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. - - Case 1. If │When P is Q, A is B - │When P is not Q, A is not B - It follows that│When A is B, P is Q - │When A is not B, P is not Q - - Case 2. If │When A is greater than B, P is greater than Q - „ │When A is equal to B, P is equal to Q - „ │When A is less than B, P is less than Q - - It follows that│When P is greater than Q, A is greater than B - „ │When P is equal to Q, A is equal to B - „ │When P is less than Q, A is less than B - - * * * * * - -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? - -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 we _have_ found 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:— - - Odds in favour of an event Probability for Probability against - - 1 to 1 ½ ½ - 2 to 1 ⅔ ⅓ - 3 to 1 ¾ ¼ - 3 to 2 ⅗ ⅖ - 4 to 1 ⅘ ⅕ - 4 to 3 ⁴⁄₇ ³⁄₇ - 5 to 1 ⅚ ⅙ - &c. &c. &c. - -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 the _intrinsic probability_ of 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 − ab_ is the probability of the truth of a -conclusion after the introduction of an argument of the intrinsic -probability _b_, the previous probability of the said conclusion having -been _a_. - -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 -which _a + b − ab_ becomes ½ + ½_b_) 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 or (3 + 4)/(2 × 4). - - - THE END. - - - LONDON:—PRINTED BY JAMES MOYES, - Castle Street, Leicester Square. - ------------------------------------------------------------------------- - - - - - TRANSCRIBER’S NOTES - - - 1. Silently corrected obvious typographical errors and variations in - spelling. - 2. Retained archaic, non-standard, and uncertain spellings as printed. - 3. Enclosed italics font in _underscores_. - -*** END OF THE PROJECT GUTENBERG EBOOK FIRST NOTIONS OF LOGIC *** - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the -United States without permission and without paying copyright -royalties. 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If you -are not located in the United States, you will have to check the laws of the -country where you are located before using this eBook. -</div> - -<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Title: First notions of logic</p> -<p style='display:block; margin-left:2em; text-indent:0; margin-top:0; margin-bottom:1em;'>(preparatory to the study of geometry)</p> - <p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Author: Augustus De Morgan</p> -<p style='display:block; text-indent:0; margin:1em 0'>Release Date: December 26, 2021 [eBook #67017]</p> -<p style='display:block; text-indent:0; margin:1em 0'>Language: English</p> - <p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em; text-align:left'>Produced by: Richard Tonsing and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive)</p> -<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK FIRST NOTIONS OF LOGIC ***</div> - -<div class='tnotes covernote'> - -<p class='c000'><strong>Transcriber’s Note:</strong></p> - -<p class='c000'>The cover image was created by the transcriber and is placed in the public domain.</p> - -</div> - -<div class='titlepage'> - -<div> - <h1 class='c001'><span class='xlarge'>FIRST NOTIONS</span><br /> <span class='small'>OF</span><br /> LOGIC<br /> <span class='large'>(PREPARATORY TO THE STUDY OF GEOMETRY)</span></h1> -</div> - -<div class='nf-center-c0'> -<div class='nf-center c002'> - <div>BY</div> - <div class='c003'><span class='large'>AUGUSTUS DE MORGAN,</span></div> - <div class='c003'>OF TRINITY COLLEGE, CAMBRIDGE,</div> - <div>PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON.</div> - <div class='c003'>The root of all the mischief in the sciences, is this; that falsely magnifying and admiring the powers of the mind, we seek not its real helps.—<span class='sc'>Bacon.</span></div> - <div class='c002'>LONDON:</div> - <div class='c003'>PRINTED FOR TAYLOR AND WALTON,</div> - <div class='c003'>BOOKSELLERS AND PUBLISHERS TO UNIVERSITY COLLEGE.</div> - <div class='c003'>28 UPPER GOWER STREET.</div> - <div class='c003'>M.DCCC.XXXIX.</div> - </div> -</div> - -</div> - -<p class='c004'>⁂ This Tract contains no more than the author has found, from -experience, to be much wanted by students who are commencing -with Euclid. It will ultimately form an Appendix to his Treatise on -Arithmetic.</p> - -<p class='c005'>The author would not, by any means, in presenting the minimum -necessary for a particular purpose, be held to imply that he has given -enough of the subject for all the ends of education. He has long regretted -the neglect of logic; a science, the study of which would shew -many of its opponents that the light esteem in which they hold it arises -from those habits of inference which thrive best in its absence. He -strongly recommends any student to whom this tract may be the first -introduction of the subject, to pursue it to a much greater extent.</p> - -<p class='c005'><em>University College, Jan, 8, 1839.</em></p> - -<div class='nf-center-c0'> -<div class='nf-center c002'> - <div><span class='small'>LONDON:—PRINTED BY JAMES MOYES,</span></div> - <div><span class='small'>Castle Street, Leicester Square.</span></div> - </div> -</div> - -<div class='chapter'> - <span class='pageno' id='Page_3'>3</span> - <h2 class='c006'><span class='large'>FIRST NOTIONS</span><br /> <span class='small'>OF</span><br /> LOGIC.</h2> -</div> - -<p class='c007'>What we here mean by Logic is the examination of that part of -reasoning which depends upon the manner in which inferences are -formed, and the investigation of general maxims and rules for constructing -arguments, so that the conclusion may contain no inaccuracy -which was not previously asserted in the premises. It has nothing to -do with the truth of the facts, opinions, or presumptions, from which an -inference is derived; but simply takes care that the inference shall -certainly be true, if the premises be true. Thus, when we say that all -men will die, and that all men are rational beings, and thence infer that -some rational beings will die, the <em>logical</em> truth of this sentence is the -same whether it be true or false that men are mortal and rational. This -logical truth depends upon the structure of the sentence, and not on the -particular matters spoken of. Thus,</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'>Instead of,</td> - <td class='c009'>Write,</td> - </tr> - <tr> - <td class='c008'>All men will die.</td> - <td class='c009'>Every A is B.</td> - </tr> - <tr> - <td class='c008'>All men are rational beings.</td> - <td class='c009'>Every A is C.</td> - </tr> - <tr> - <td class='c008'>Therefore some rational beings will die.</td> - <td class='c009'>Therefore some Cs are Bs.</td> - </tr> -</table> - -<p class='c010'>The second of these is the same proposition, logically considered, as the -first; the consequence in both is virtually contained in, and rightly -inferred from, the premises. Whether the premises be true or false, is -not a question of logic, but of morals, philosophy, history, or any other -knowledge to which their subject-matter belongs: the question of logic -is, does the conclusion certainly follow if the premises be true?</p> - -<p class='c005'><span class='pageno' id='Page_4'>4</span>Every act of reasoning must mainly consist in comparing together -different things, and either finding out, or recalling from previous -knowledge, the points in which they resemble or differ from each other. -That particular part of reasoning which is called <em>inference</em>, consists in -the comparison of several and different things with one and the same -other thing; and ascertaining the resemblances, or differences, of the -several things, by means of the points in which they resemble, or differ -from, the thing with which all are compared.</p> - -<p class='c005'>There must then be some propositions already obtained before any -inference can be drawn. All propositions are either assertions or denials, -and are thus divided into <em>affirmative</em> and <em>negative</em>. Thus, A is B, and -A is not B, are the two forms to which all propositions may be reduced. -These are, for our present purpose, the most simple forms; though it -will frequently happen that much circumlocution is needed to reduce -propositions to them. Thus, suppose the following assertion, ‘If he -should come to-morrow, he will probably stay till Monday’; how is -this to be reduced to the form A is B? There is evidently something -spoken of, something said of it, and an affirmative connexion between -them. Something, if it happen, that is, the happening of something, -makes the happening of another something probable; or is one of the -things which render the happening of the second thing probable.</p> - -<table class='table1' summary=''> - <tr> - <td class='brt c011'>A</td> - <td class='brt c011'>is</td> - <td class='c011'>B</td> - </tr> - <tr> - <td class='c012' colspan='3'> </td> - </tr> - <tr> - <td class='brt c012'>The happening of his arrival to-morrow</td> - <td class='brt c011'>is</td> - <td class='c012'>an event from which it may be inferred as probable that he will stay till Monday.</td> - </tr> -</table> - -<p class='c005'>The forms of language will allow the manner of asserting to be varied -in a great number of ways; but the reduction to the preceding form -is always possible. Thus, ‘so he said’ is an affirmation, reducible -as follows:</p> - -<table class='table1' summary=''> - <tr> - <td class='brt c012'>What you have just said (or whatever else ‘so’ refers to)</td> - <td class='brt c011'>is</td> - <td class='c012'>the thing which he said.</td> - </tr> -</table> - -<p class='c005'>By changing ‘is’ into ‘is not,’ we make a negative proposition; -<span class='pageno' id='Page_5'>5</span>but care must always be taken to ascertain whether a proposition -which appears negative is really so. The principal danger is that of -confounding a proposition which is negative with another which is -affirmative of something requiring a negative to describe it. Thus -‘he resembles the man who was not in the room,’ is affirmative, and -must not be confounded with ‘he does not resemble the man who was -in the room.’ Again, ‘if he should come to-morrow, it is probable he -will not stay till Monday,’ does not mean the simple denial of the preceding -proposition, but the affirmation of the directly opposite proposition. -It is,</p> - -<table class='table1' summary=''> - <tr> - <td class='brt c011'>A</td> - <td class='brt c011'>is</td> - <td class='c011'>B</td> - </tr> - <tr> - <td class='c012' colspan='3'> </td> - </tr> - <tr> - <td class='brt c012'>The happening of his arrival to-morrow,</td> - <td class='brt c011'>is</td> - <td class='c012'>an event from which it may be inferred to be <em>im</em>probable that he will stay till Monday,</td> - </tr> -</table> - -<p class='c010'>whereas the following,</p> - -<table class='table1' summary=''> - <tr> - <td class='brt c012'>The happening of his arrival to-morrow,</td> - <td class='brt c011'>is <em>not</em></td> - <td class='c012'>an event from which it may be inferred as probable that he will stay till Monday,</td> - </tr> -</table> - -<p class='c010'>would be expressed thus: ‘If he should come to-morrow, that is no -reason why he should stay till Monday.’</p> - -<p class='c005'>Moreover, the negative words not, no, &c., have two kinds of meaning -which must be carefully distinguished. Sometimes they deny, and -nothing more: sometimes they are used to affirm the direct contrary. -In cases which offer but two alternatives, one of which is necessary, -these amount to the same thing, since the denial of one, and the affirmation -of the other, are obviously equivalent propositions. In many -idioms of conversation, the negative implies affirmation of the contrary -in cases which offer not only alternatives, but degrees of alternatives. -Thus, to the question, ‘Is he tall?’ the simple answer, ‘No,’ most -frequently means that he is the contrary of tall, or considerably under -the average. But it must be remembered, that, in all logical reasoning, -the negation is simply negation, and nothing more, never implying -affirmation of the contrary.</p> - -<p class='c005'>The common proposition that two negatives make an affirmative, is -<span class='pageno' id='Page_6'>6</span>true only upon the supposition that there are but two possible things, -one of which is denied. Grant that a man must be either able or unable -to do a particular thing, and then <em>not unable</em> and able are the same -things. But if we suppose various degrees of performance, and therefore -degrees of ability, it is false, in the common sense of the words, -that two negatives make an affirmative. Thus, it would be erroneous -to say, ‘John is able to translate Virgil, and Thomas is not unable; -therefore, what John can do Thomas can do,’ for it is evident that the -premises mean that John is so near to the best sort of translation -that an affirmation of his ability may be made, while Thomas is considerably -lower than John, but not so near to absolute deficiency that -his ability may be altogether denied. It will generally be found that -two negatives imply an affirmative of a weaker degree than the positive -affirmation.</p> - -<p class='c005'>Each of the propositions, ‘A is B,’ and ‘A is not B,’ may be subdivided -into two species: the <em>universal</em>, in which every possible case is -included; and the <em>particular</em>, in which it is not meant to be asserted -that the affirmation or negation is universal. The four species of propositions -are then as follows, each being marked with the letter by -which writers on logic have always distinguished it.</p> - -<table class='table0' summary=''> - <tr> - <td class='c013'>A <em>Universal Affirmative</em></td> - <td class='c013'>Every A is</td> - <td class='c014'>B</td> - </tr> - <tr> - <td class='c013'>E <em>Universal Negative</em></td> - <td class='c013'>No A is</td> - <td class='c014'>B</td> - </tr> - <tr> - <td class='c013'>I <em>Particular Affirmative</em></td> - <td class='c013'>Some A is</td> - <td class='c014'>B</td> - </tr> - <tr> - <td class='c013'>O <em>Particular Negative</em></td> - <td class='c013'>Some A is not</td> - <td class='c014'>B</td> - </tr> -</table> - -<p class='c005'>In common conversation the affirmation of a part is meant to imply -the denial of the remainder. Thus, by ‘some of the apples are ripe,’ -it is always intended to signify that some are not ripe. This is not the -case in logical language, but every proposition is intended to make its -amount of affirmation or denial, and no more. When we say, ‘Some -A is B,’ or, more grammatically, ‘Some As are Bs,’ we do not mean to -imply that some are not: this may or may not be. Again, the word -some means, ‘one or more, possibly all.’ The following table will shew -the bearing of each proposition on the rest.</p> - -<table class='table1' summary=''> - <tr><td class='c015' colspan='2'><span class='pageno' id='Page_7'>7</span></td></tr> - <tr> - <td class='brt c012'><em>Every A is B</em> affirms and contains <em>Some A is B</em> and denies</td> - <td class='c016'><em>No A is B</em><br /><em>Some A is not B</em></td> - </tr> - <tr> - <td class='brt c012'><em>No A is B</em> affirms and contains <em>Some A is not B</em> and denies</td> - <td class='c016'><em>Every A is B</em><br /><em>Some A is B</em></td> - </tr> -</table> - -<table class='table1' summary=''> - <tr> - <td class='brt c012'><em>Some A is B</em> does not contradict</td> - <td class='brt c016'><em>Every A is B</em><br /><em>Some A is not B</em></td> - <td class='c012'>but denies <em>No A is B</em></td> - </tr> - <tr> - <td class='brt c012'><em>Some A is not B</em> does not contradict</td> - <td class='brt c016'><em>No A is B</em><br /><em>Some A is B</em></td> - <td class='c012'>but denies <em>Every A is B</em></td> - </tr> -</table> - -<p class='c005'><em>Contradictory</em> propositions are those in which one denies <em>any thing</em> -that the other affirms; <em>contrary</em> propositions are those in which one -denies <em>every thing</em> which the other affirms, or affirms every thing which -the other denies. The following pair are contraries.</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'>Every A is B</td> - <td class='c008'>and</td> - <td class='c009'>No A is B</td> - </tr> -</table> - -<p class='c010'>and the following are contradictories,</p> - -<table class='table0' summary=''> - <tr> - <td class='c013'>Every A is B</td> - <td class='c008'>to</td> - <td class='c017'>Some A is not B</td> - </tr> - <tr> - <td class='c013'>No A is B</td> - <td class='c008'>to</td> - <td class='c017'>Some A is B</td> - </tr> -</table> - -<p class='c010'>A contrary, therefore, is a complete and total contradictory; and a little -consideration will make it appear that the decisive distinction between -contraries and contradictories lies in this, that contraries may both be -false, but of contradictories, one must be true and the other false. We -may say, ‘Either P is true, or <em>something</em> in contradiction of it is true;’ -but we cannot say, ‘Either P is true, or <em>every thing</em> in contradiction of -it is true.’ It is a very common mistake to imagine that the <em>denial</em> of a -proposition gives a right to <em>affirm</em> the contrary; whereas it should be, -that the <em>affirmation</em> of a proposition gives a right to <em>deny</em> the contrary. -Thus, if we deny that Every A is B, we do not affirm that No A is B, -but only that Some A is not B; while, if we affirm that Every A is B, -we deny No A is B, and also Some A is not B.</p> - -<p class='c005'>But, as to contradictories, affirmation of one is a denial of the other, -and denial of one is affirmation of the other. Thus, either Every A -is B, or Some A is not B: affirmation of either is denial of the other, -and <i><span lang="fr" xml:lang="fr">vice versá</span></i>.</p> - -<p class='c005'>Let the student now endeavour to satisfy himself of the following. -Taking the four preceding propositions, A, E, I, O, let the simple letter -<span class='pageno' id='Page_8'>8</span>signify the affirmation, the same letter in parentheses the denial, and -the absence of the letter, that there is neither affirmation nor denial.</p> - -<table class='table1' summary=''> - <tr> - <td class='c018'>From A</td> - <td class='brt c019'>follow</td> - <td class='brt c018'>(E), I, (O)</td> - <td class='c018'>From (A)</td> - <td class='c019'>follow</td> - <td class='c020'>O</td> - </tr> - <tr> - <td class='c018'>From E</td> - <td class='brt c019'> </td> - <td class='brt c018'>(A), (I), O</td> - <td class='c018'>From (E)</td> - <td class='c019'> </td> - <td class='c020'>I</td> - </tr> - <tr> - <td class='c018'>From I</td> - <td class='brt c019'> </td> - <td class='brt c018'>(E)</td> - <td class='c018'>From (I)</td> - <td class='c019'> </td> - <td class='c020'>(A), E, O</td> - </tr> - <tr> - <td class='c018'>From O</td> - <td class='brt c019'> </td> - <td class='brt c018'>(A)</td> - <td class='c018'>From (O)</td> - <td class='c019'> </td> - <td class='c020'>A, (E), I</td> - </tr> -</table> - -<p class='c010'>These may be thus summed up: The affirmation of a universal proposition, -and the denial of a particular one, enable us to affirm or deny -all the other three; but the denial of a universal proposition, and the -affirmation of a particular one, leave us unable to affirm or deny two -of the others.</p> - -<p class='c005'>In such propositions as ‘Every A is B,’ ‘Some A is not B,’ &c., -A is called the <em>subject</em>, and B the <em>predicate</em>, while the verb ‘is’ or -‘is not,’ is called the <em>copula</em>. It is obvious that the words of the -proposition point out whether the subject is spoken of universally or -partially, but not so of the predicate, which it is therefore important to -examine. Logical writers generally give the name of <em>distributed</em> subjects -or predicates to those which are spoken of universally; but as this -word is rather technical, I shall say that a subject or predicate enters -wholly or partially, according as it is universally or particularly spoken of.</p> - -<p class='c005'>1. In A, or ‘Every A is B,’ the subject enters wholly, but the -predicate only partially. For it obviously says, ‘Among the Bs are -all the As,’ ‘Every A is part of the collection of Bs, so that all the As -make a part of the Bs, the whole it <em>may</em> be.’ Thus, ‘Every horse is -an animal,’ does not speak of all animals, but states that all the horses -make up a portion of the animals.</p> - -<p class='c005'>2. In E, or ‘No A is B,’ both subject and predicate enter wholly. -‘No A whatsoever is any one out of all the Bs;’ ‘search the whole -collection of Bs, and <em>every</em> B shall be found to be something which is -not A.’</p> - -<p class='c005'>3. In I, or ‘Some A is B,’ both subject and predicate enter partially. -‘Some of the As are found among the Bs, or make up a part (the -whole possibly, but not known from the preceding) of the Bs.’</p> - -<p class='c005'><span class='pageno' id='Page_9'>9</span>4. In O, or ‘Some A is not B,’ the subject enters partially, and the -predicate wholly. ‘Some As are none of them any whatsoever of the -Bs; every B will be found to be no one out of a certain portion of -the As.’</p> - -<p class='c005'>It appears then that,</p> - -<p class='c005'>In affirmatives, the predicate enters partially.</p> - -<p class='c005'>In negatives, the predicate enters wholly.</p> - -<p class='c005'>In contradictory propositions, both subject and predicate enter -differently in the two.</p> - -<p class='c005'>The <em>converse</em> of a proposition is that which is made by interchanging -the subject and predicate, as follows:</p> - -<table class='table0' summary=''> - <tr> - <th class='c008'>The proposition.</th> - <th class='c009'>Its converse.</th> - </tr> - <tr> - <td class='c021'>A Every A is B</td> - <td class='c014'>Every B is A</td> - </tr> - <tr> - <td class='c021'>E No A is B</td> - <td class='c014'>No B is A</td> - </tr> - <tr> - <td class='c021'>I Some A is B</td> - <td class='c014'>Some B is A</td> - </tr> - <tr> - <td class='c021'>O Some A is not B</td> - <td class='c014'>Some B is not A</td> - </tr> -</table> - -<p class='c005'>Now, it is a fundamental and self-evident proposition, that no consequence -must be allowed to assert more widely than its premises; so -that, for instance, an assertion which is only of some Bs can never -lead to a result which is true of all Bs. But if a proposition assert -agreement or disagreement, any other proposition which asserts the -same, to the same extent and no further, must be a legitimate consequence; -or, if you please, must amount to the whole, or part, of the -original assertion in another form. Thus, the converse of A is not -true: for, in ‘Every A is B,’ the predicate enters partially; while in -‘Every B is A,’ the subject enters wholly. ‘All the As make up a -part of the Bs, then a part of the Bs are among the As, or some B is -A.’ Hence, the only <em>legitimate</em> converse of ‘Every A is B’ is, ‘Some -B is A.’ But in ‘No A is B,’ both subject and predicate enter wholly, -and ‘No B is A’ is, in fact, the same proposition as ‘No A is B.’ And -‘Some A is B’ is also the same as its converse ‘Some B is A;’ here -both terms enter partially. But ‘Some A is not B’ admits of no -converse whatever; it is perfectly consistent with all assertions upon B -<span class='pageno' id='Page_10'>10</span>and A in which B is the subject. Thus neither of the four following -lines is inconsistent with itself.</p> - -<table class='table0' summary=''> - <tr> - <td class='c021'>Some A is not B and Every</td> - <td class='c014'>B is A</td> - </tr> - <tr> - <td class='c021'>Some A is not B and No</td> - <td class='c014'>B is A</td> - </tr> - <tr> - <td class='c021'>Some A is not B and Some</td> - <td class='c014'>B is A</td> - </tr> - <tr> - <td class='c021'>Some A is not B and Some</td> - <td class='c014'>B is not A.</td> - </tr> -</table> - -<p class='c005'>We find then, including converses, which are not identical with their -direct propositions, <em>six</em> different ways of asserting or denying, with -respect to agreement or non-agreement, total or partial, between A and, -say X: these we write down, designating the additional assertions by -U and Y.</p> - -<table class='table1' summary=''> - <tr> - <th class='brt c018'></th> - <th class='brt c011' colspan='2'>Identical.</th> - <th class='brt c011' colspan='2'>Identical.</th> - <th class='c018'> </th> - </tr> - <tr> - <td class='brt c018'>A Every A is X</td> - <td class='c011' rowspan='2'>E</td> - <td class='brt c018'>No A is X</td> - <td class='c011' rowspan='2'>I</td> - <td class='brt c018'>Some A is X</td> - <td class='c018'>O Some A is not X</td> - </tr> - <tr> - <td class='brt c018'>U Every X is A</td> - - <td class='brt c018'>No X is A</td> - - <td class='brt c018'>Some X is A</td> - <td class='c018'>Y Some X is not A</td> - </tr> -</table> - -<p class='c005'>We shall now repeat and extend the table of page <a href='#Page_8'>8</a> (A), &c., -meaning, as before, the denial of A, &c.</p> - -<table class='table0' summary=''> - <tr> - <td class='c021'>From A or (O)</td> - <td class='c008'>follow</td> - <td class='c008'>A,</td> - <td class='c008'>(E),</td> - <td class='c008'>I</td> - <td class='c008'>(O)</td> - <td class='c008'> </td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c021'>From E or (I)</td> - <td class='c008'> </td> - <td class='c008'>(A),</td> - <td class='c008'>E,</td> - <td class='c008'>(I),</td> - <td class='c008'>O,</td> - <td class='c008'>(U),</td> - <td class='c009'>Y</td> - </tr> - <tr> - <td class='c021'>From I or (E)</td> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c008'>(E)</td> - <td class='c008'>I</td> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c021'>From O or (A)</td> - <td class='c008'> </td> - <td class='c008'>(A),</td> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c008'>O</td> - <td class='c008'> </td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c021'>From U or (Y)</td> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c008'>(E)</td> - <td class='c008'>I,</td> - <td class='c008'> </td> - <td class='c008'>U</td> - <td class='c009'>(Y)</td> - </tr> - <tr> - <td class='c021'>From Y or (U)</td> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c008'>(U)</td> - <td class='c009'>Y</td> - </tr> -</table> - -<p class='c005'>Having thus discussed the principal points connected with the simple -assertion, we pass to the manner of making two assertions give a third. -Every instance of this is called a syllogism, the two assertions which -form the basis of the third are called premises, and the third itself the -conclusion.</p> - -<p class='c005'>If two things both agree with a third in any particular, they agree -with each other in the same; as, if A be of the same colour as X, and -B of the same colour as X, then A is of the same colour as B. Again, -if A differ from X in any particular in which B agrees with X, then -A and B differ in that particular. If A be not of the same colour as X, -<span class='pageno' id='Page_11'>11</span>and B be of the same colour as X, then A is not of the colour of B. -But if A and B both differ from X in any particular, nothing can be -inferred; they may either differ in the same way and to the same -extent, or not. Thus, if A and B be both of different colours from X, -it neither follows that they agree, nor differ, in their own colours.</p> - -<p class='c005'>The paragraph preceding contains the essential parts of all inference, -which consists in comparing two things with a third, and finding from -their agreement or difference with that third, their agreement or difference -with one another. Thus, Every A is X, every B is X, allows us -to infer that A and B have all those qualities in common which are -necessary to X. Again, from Every A is X, and ‘No B is X,’ we -infer that A and B differ from one another in all particulars which -are essential to X. The preceding forms, however, though they represent -common reasoning better than the ordinary syllogism, to which we -are now coming, do not constitute the ultimate forms of inference. -Simple <em>identity</em> or <em>non-identity</em> is the ultimate state to which every -assertion may be reduced; and we shall, therefore, first ask, from what -identities, &c., can other identities, &c., be produced? Again, since we -name objects in species, each species consisting of a number of individuals, -and since our assertion may include all or only part of a species, -it is further necessary to ask, in every instance, to what extent the -conclusion drawn is true, whether of all, or only of part?</p> - -<p class='c005'>Let us take the simple assertion, ‘Every living man respires;’ or, -every living man is one of the things (however varied they may be) -which respire. If we were to inclose all living men in a large triangle, -and all respiring objects in a large circle, the preceding assertion, if true, -would require that the whole of the triangle should be contained in the -circle. And in the same way we may reduce any assertion to the -expression of a coincidence, total or partial, between two figures. Thus, -a point in a circle may represent an individual of one species, and a -point in a triangle an individual of another species: and we may express -that the whole of one species is asserted to be contained or not contained -in the other by such forms as, ‘All the △ is in the ○’; -‘None of the △ is in the ○’.</p> - -<p class='c005'><span class='pageno' id='Page_12'>12</span>Any two assertions about A and B, each expressing agreement or -disagreement, total or partial, with or from X, and leading to a conclusion -with respect to A or B, is called a syllogism, of which X is called -the <em>middle term</em>. The plainest syllogism is the following:—</p> - -<table class='table1' summary=''> - <tr> - <td class='c019'> </td> - <td class='brt c018'>Every A is X</td> - <td class='c019'> </td> - <td class='c018'>All the △ is in the ○</td> - </tr> - <tr> - <td class='c019'> </td> - <td class='brt c018'>Every X is B</td> - <td class='c019'> </td> - <td class='c018'>All the ○ is in the □</td> - </tr> - <tr> - <td class='c019'>Therefore</td> - <td class='brt c018'>Every A is B</td> - <td class='c019'>Therefore</td> - <td class='c018'>All the △ is in the □</td> - </tr> -</table> - -<p class='c005'>In order to find all the possible forms of syllogism, we must make a -table of all the elements of which they can consist; namely—</p> - -<table class='table0' summary=''> - <tr> - <th class='c008'>A and X</th> - <th class='c008'> </th> - <th class='c009'>B and X</th> - </tr> - <tr> - <td class='c021'>Every A is X</td> - <td class='c008'>A</td> - <td class='c014'>Every B is X</td> - </tr> - <tr> - <td class='c021'>No A is X</td> - <td class='c008'>E</td> - <td class='c014'>No B is X</td> - </tr> - <tr> - <td class='c021'>Some A is X</td> - <td class='c008'>I</td> - <td class='c014'>Some B is X</td> - </tr> - <tr> - <td class='c021'>Some A is not X</td> - <td class='c008'>O</td> - <td class='c014'>Some B is not X</td> - </tr> - <tr> - <td class='c021'>Every X is A</td> - <td class='c008'>U</td> - <td class='c014'>Every X is B</td> - </tr> - <tr> - <td class='c021'>Some X is not A</td> - <td class='c008'>Y</td> - <td class='c014'>Some X is not B</td> - </tr> -</table> - -<p class='c005'>Or their synonymes,</p> - -<table class='table0' summary=''> - <tr> - <th class='c008'>△ and ○</th> - <th class='c008'> </th> - <th class='c009'>□ and ○</th> - </tr> - <tr> - <td class='c021'>All the △ is in the ○</td> - <td class='c008'>A</td> - <td class='c014'>All the □ is in the ○</td> - </tr> - <tr> - <td class='c021'>None of the △ is in the ○</td> - <td class='c008'>E</td> - <td class='c014'>None of the □ is in the ○</td> - </tr> - <tr> - <td class='c021'>Some of the △ is in the ○</td> - <td class='c008'>I</td> - <td class='c014'>Some of the □ is in the ○</td> - </tr> - <tr> - <td class='c021'>Some of the △ is not in the ○</td> - <td class='c008'>O</td> - <td class='c014'>Some of the □ is not in the ○</td> - </tr> - <tr> - <td class='c021'>All the ○ is in the △</td> - <td class='c008'>U</td> - <td class='c014'>All the ○ is in the □</td> - </tr> - <tr> - <td class='c021'>Some of the ○ is not in the △</td> - <td class='c008'>Y</td> - <td class='c014'>Some of the ○ is not in the □</td> - </tr> -</table> - -<p class='c005'>Now, taking any one of the six relations between A and X, and combining -it with either of those between B and X, we have six pairs of -premises, and the same number repeated for every different relation of -A and X. We have then thirty-six forms to consider: but, thirty -of these (namely, all but (A, A) (E, E), &c.) are half of them repetitions -of the other half. Thus, ‘Every A is X, no B is X,’ and ‘Every -B is X, no A is X,’ are of the same form, and only differ by changing -A into B and B into A. There are then only 15 + 6, or 21 distinct -<span class='pageno' id='Page_13'>13</span>forms, some of which give a necessary conclusion, while others do not. -We shall select the former of these, classifying them by their conclusions; -that is, according as the inference is of the form A, E, I, or O.</p> - -<p class='c005'>I. In what manner can a universal affirmative conclusion be -drawn; namely, that one figure is entirely contained in the other? -This we can only assert when we know that one figure is entirely -contained in the circle, which itself is entirely contained in the other -figure. Thus,</p> - -<table class='table1' summary=''> - <tr> - <td class='c019'> </td> - <td class='brt c018'>Every A is X</td> - <td class='c019'> </td> - <td class='c018'>All the △ is in the ○</td> - <td class='c019'>A</td> - </tr> - <tr> - <td class='c019'> </td> - <td class='brt c018'>Every X is B</td> - <td class='c019'> </td> - <td class='c018'>All the ○ is in the □</td> - <td class='c019'>A</td> - </tr> - <tr> - <td class='c019'>∴</td> - <td class='brt c018'>Every A is B</td> - <td class='c019'>∴</td> - <td class='c018'>All the △ is in the □</td> - <td class='c019'>A</td> - </tr> -</table> - -<p class='c010'>is the only way in which a universal affirmative conclusion can be drawn.</p> - -<p class='c005'>II. In what manner can a universal negative conclusion be drawn; -namely, that one figure is entirely exterior to the other? Only when -we are able to assert that one figure is entirely within, and the other -entirely without, the circle. Thus,</p> - -<table class='table1' summary=''> - <tr> - <td class='c019'> </td> - <td class='c018'>Every</td> - <td class='brt c018'>A is X</td> - <td class='c018'>All the △ is in the ○</td> - <td class='c019'>A</td> - </tr> - <tr> - <td class='c019'> </td> - <td class='c018'>No</td> - <td class='brt c018'>B is X</td> - <td class='c018'>None of the □ is in the ○</td> - <td class='c019'>E</td> - </tr> - <tr> - <td class='c019'>∴</td> - <td class='c018'>No</td> - <td class='brt c018'>A is B</td> - <td class='c018'>None of the △ is in the □</td> - <td class='c019'>E</td> - </tr> -</table> - -<p class='c010'>is the only way in which a universal negative conclusion can be drawn.</p> - -<p class='c005'>III. In what manner can a particular affirmative conclusion be -drawn; namely, that part or all of one figure is contained in the other? -Only when we are able to assert that the whole circle is part of one of -the figures, and that the whole, or part of the circle, is part of the other -figure. We have then two forms.</p> - -<table class='table1' summary=''> - <tr> - <td class='c019'> </td> - <td class='c018'>Every</td> - <td class='brt c018'>X is A</td> - <td class='c019'> </td> - <td class='c018'>All the ○ is in the △</td> - <td class='c019'>A</td> - </tr> - <tr> - <td class='c019'> </td> - <td class='c018'>Every</td> - <td class='brt c018'>X is B</td> - <td class='c019'> </td> - <td class='c018'>All the ○ is in the □</td> - <td class='c019'>A</td> - </tr> - <tr> - <td class='c019'>∴</td> - <td class='c018'>Some</td> - <td class='brt c018'>A is B</td> - <td class='c019'>∴</td> - <td class='c018'>Some of the △ is in the □</td> - <td class='c019'>I</td> - </tr> - <tr> - <td class='c019' colspan='6'> </td> - </tr> - <tr> - <td class='c019'> </td> - <td class='c018'>Every</td> - <td class='brt c018'>X is A</td> - <td class='c019'> </td> - <td class='c018'>All the ○ is in the △</td> - <td class='c019'>A</td> - </tr> - <tr> - <td class='c019'> </td> - <td class='c018'>Some</td> - <td class='brt c018'>X is B</td> - <td class='c019'> </td> - <td class='c018'>Some of the ○ is in the □</td> - <td class='c019'>I</td> - </tr> - <tr> - <td class='c019'> </td> - <td class='c018'>Some</td> - <td class='brt c018'>A is B</td> - <td class='c019'> </td> - <td class='c018'>Some of the △ is in the □</td> - <td class='c019'>I</td> - </tr> -</table> - -<p class='c010'><span class='pageno' id='Page_14'>14</span>The second of these contains all that is strictly necessary to the conclusion, -and the first may be omitted. That which follows when an -assertion can be made as to some, must follow when the same assertion -can be made of all.</p> - -<p class='c005'>IV. How can a particular negative proposition be inferred; -namely, that part, or all of one figure, is not contained in the other? -It would seem at first sight, whenever we are able to assert that part -or all of one figure is in the circle, and that part or all of the other -figure is not. The weakest syllogism from which such an inference can -be drawn would then seem to be as follows.</p> - -<table class='table1' summary=''> - <tr> - <td class='c018'> </td> - <td class='brt c019'>Some A is X</td> - <td class='c019'> </td> - <td class='c018'>Some of the △ is in the ○</td> - </tr> - <tr> - <td class='c018'> </td> - <td class='brt c019'>Some B is not X</td> - <td class='c019'> </td> - <td class='c018'>Some of the □ is not in the ○</td> - </tr> - <tr> - <td class='c018'>∴</td> - <td class='brt c019'>Some B is not A</td> - <td class='c019'>∴</td> - <td class='c018'>Some of the △ is not in the □</td> - </tr> -</table> - -<p class='c005'>But here it will appear, on a little consideration, that the conclusion -is only thus far true; that those As which are Xs cannot be <em>those</em> Bs -which are not Xs; but they may be <em>other</em> Bs, about which nothing is -asserted when we say that <em>some</em> Bs are not Xs. And further consideration -will make it evident, that a conclusion of this form can only be -arrived at when one of the figures is entirely within the circle, and -the whole or part of the other without; or else when the whole of one -of the figures is without the circle, and the whole or part of the other -within; or lastly, when the circle lies entirely within one of the figures, -and not entirely within the other. That is, the following are the distinct -forms which allow of a particular negative conclusion, in which it -should be remembered that a particular proposition in the premises -may always be changed into a universal one, without affecting the -conclusion. For that which necessarily follows from “some,” follows -from “all.”</p> - -<table class='table1' summary=''> - <tr> - <td class='c018'> </td> - <td class='brt c019'>Every A is X</td> - <td class='c019'> </td> - <td class='c018'>All the △ is in the ○</td> - <td class='c018'>A</td> - </tr> - <tr> - <td class='c018'> </td> - <td class='brt c019'>Some B is not X</td> - <td class='c019'> </td> - <td class='c018'>Some of the □ is not in the ○</td> - <td class='c018'>O</td> - </tr> - <tr> - <td class='c018'>∴</td> - <td class='brt c019'>Some B is not A</td> - <td class='c019'> </td> - <td class='c018'>Some of the □ is not in the △</td> - <td class='c018'>O</td> - </tr> - <tr> - <td class='c018' colspan='5'><span class='pageno' id='Page_15'>15</span> </td> - </tr> - <tr> - <td class='c018'> </td> - <td class='brt c019'>No A Is X</td> - <td class='c019'> </td> - <td class='c018'>None of the △ is in the ○</td> - <td class='c018'>E</td> - </tr> - <tr> - <td class='c018'> </td> - <td class='brt c019'>Some B is X</td> - <td class='c019'> </td> - <td class='c018'>Some of the □ is in the ○</td> - <td class='c018'>I</td> - </tr> - <tr> - <td class='c018'>∴</td> - <td class='brt c019'>Some B is not A</td> - <td class='c019'> </td> - <td class='c018'>Some of the □ is not in the △</td> - <td class='c018'>O</td> - </tr> - <tr> - <td class='c018' colspan='5'> </td> - </tr> - <tr> - <td class='c018'> </td> - <td class='brt c019'>Every X is A</td> - <td class='c019'> </td> - <td class='c018'>All the ○ is in the △</td> - <td class='c018'>A</td> - </tr> - <tr> - <td class='c018'> </td> - <td class='brt c019'>Some X is not B</td> - <td class='c019'> </td> - <td class='c018'>Some of the ○ is not in the □</td> - <td class='c018'>O</td> - </tr> - <tr> - <td class='c018'>∴</td> - <td class='brt c019'>Some A is not B</td> - <td class='c019'> </td> - <td class='c018'>Some of the △ is not in the □</td> - <td class='c018'>O</td> - </tr> -</table> - -<p class='c005'>It appears, then, that there are but six distinct syllogisms. All others -are made from them by strengthening one of the premises, or converting -one or both of the premises, where such conversion is allowable; -or else by first making the conversion, and then strengthening one of -the premises. And the following arrangement will shew that two of -them are universal, three of the others being derived from them by -weakening one of the premises in a manner which does not destroy, but -only weakens, the conclusion.</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'>1.</td> - <td class='c021'>Every A is X</td> - <td class='c008'> </td> - <td class='c022'>3.</td> - <td class='c021' colspan='2'>Every A is X</td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'>Every X is B</td> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021' colspan='2'>No B is X</td> - <td class='c008'> </td> - <td class='c009'>.........</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021' colspan='2'><hr /></td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'>Every A is B</td> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021' colspan='2'>No A Is B</td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'>│</td> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c008'>│</td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'>│</td> - <td class='c008' colspan='4'><hr /></td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>2.</td> - <td class='c021'>Some A is X</td> - <td class='c008'>4.</td> - <td class='c021'>Some A is X</td> - <td class='c008'>5.</td> - <td class='c021'>Every A is X</td> - <td class='c008'>6.</td> - <td class='c014'>Every X is A</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'>Every X is B</td> - <td class='c008'> </td> - <td class='c021'>No B is X</td> - <td class='c008'> </td> - <td class='c021'>Some B is not X</td> - <td class='c008'> </td> - <td class='c014'>Some X is not B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'>Some A is B</td> - <td class='c008'> </td> - <td class='c021'>Some A is not B</td> - <td class='c008'> </td> - <td class='c021'>Some B is not A</td> - <td class='c008'> </td> - <td class='c014'>Some A is not B</td> - </tr> -</table> - -<p class='c005'>We may see how it arises that one of the partial syllogisms is not -immediately derived, like the others, from a universal one. In the -preceding, AEE may be considered as derived from AAA, by changing -the term in which X enters universally into its contrary. If this be -done with the other term instead, we have</p> - -<table class='table1' summary=''> - <tr> - <td class='c018'>No</td> - <td class='brt c018'>A is X</td> - <td class='c012' rowspan='2'>from which universal premises we cannot deduce a universal conclusion, but only Some B is not A.</td> - </tr> - <tr> - <td class='c018'>Every</td> - <td class='brt c018'>X is B</td> - - </tr> -</table> - -<p class='c005'>If we weaken one and the other of these premises, as they stand, -we obtain</p> - -<table class='table0' summary=''> - <tr> - <td class='c021'>Some A is not X</td> - <td class='c008'> </td> - <td class='c014'>No A is X</td> - </tr> - <tr> - <td class='c021'>Every X is B</td> - <td class='c008'>and</td> - <td class='c014'>Some X is B</td> - </tr> - <tr> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c021'>No conclusion</td> - <td class='c008'> </td> - <td class='c014'>Some B is not A</td> - </tr> -</table> - -<p class='c010'><span class='pageno' id='Page_16'>16</span>equivalent to the fourth of the preceding: but if we convert the first -premiss, and proceed in the same manner,</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'> </td> - <td class='c021'>No X is A</td> - <td class='c008'> </td> - <td class='c014'>Some X is not A</td> - </tr> - <tr> - <td class='c008'>From</td> - <td class='c021'>Every X is B</td> - <td class='c008'>we obtain</td> - <td class='c014'>Every X is B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'>Some B is not A</td> - <td class='c008'> </td> - <td class='c014'>Some B is not A</td> - </tr> -</table> - -<p class='c010'>which is legitimate, and is the same as the last of the preceding list, -with A and B interchanged.</p> - -<p class='c005'>Before proceeding to shew that all the usual forms are contained in -the preceding, let the reader remark the following rules, which may be -proved either by collecting them from the preceding cases, or by independent -reasoning.</p> - -<p class='c005'>1. The middle term must enter universally into one or the other -premiss. If it were not so, the one premiss might speak of one part -of the middle term, and the other of the other; so that there would, -in fact, be no middle term. Thus, ‘Every A is X, Every B is X,’ -gives no conclusion: it may be thus stated;</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>All the As make up <em>a part</em> of the Xs</div> - <div class='line'>All the Bs make up <em>a part</em> of the Xs</div> - </div> - </div> -</div> - -<p class='c010'>And, before we can know that there is any common term of comparison -at all, we must have some means of shewing that the two parts are the -same; or the preceding premises by themselves are inconclusive.</p> - -<p class='c005'>2. No term must enter the conclusion more generally than it is -found in the premises; thus, if A be spoken of partially in the premises, -it must enter partially into the conclusion. This is obvious, since the -conclusion must assert no more than the premises imply.</p> - -<p class='c005'>3. From premises both negative no conclusion can be drawn. For -it is obvious, that the mere assertion of disagreement between each of -two things and a third, can be no reason for inferring either agreement -or disagreement between these two things. It will not be difficult to -reduce any case which falls under this rule to a breach of the first rule: -thus, No A is X, No B is X, gives</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Every A is (something which is not X)</div> - <div class='line'>Every B is (something which is not X)</div> - </div> - </div> -</div> - -<p class='c010'><span class='pageno' id='Page_17'>17</span>in which the middle term is not spoken of universally in either. Again, -‘No X is A, Some X is not B,’ may be converted into</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Every A is (a thing which is not X)</div> - <div class='line'>Some (thing which is not B) is X</div> - </div> - </div> -</div> - -<p class='c010'>in which there is no middle term.</p> - -<p class='c005'>4. From premises both particular no conclusion can be drawn. This -is sufficiently obvious when the first or second rule is broken, as in -‘Some A is X, Some B is X.’ But it is not immediately obvious -when the middle term enters one of the premises universally. The -following reasoning will serve for exercise in the preceding results. -Since both premises are particular in form, the middle term can only -enter one of them universally by being the predicate of a negative -proposition; consequently (Rule 3) the other premiss must be -affirmative, and, being particular, neither of its terms is universal. -Consequently both the terms as to which the conclusion is to be drawn -enter partially, and the conclusion (Rule 2) can only be a particular -<em>affirmative</em> proposition. But if one of the premises be negative, the -conclusion must be <em>negative</em> (as we shall immediately see). This contradiction -shews that the supposition of particular premises producing -a legitimate result is inadmissible.</p> - -<p class='c005'>5. If one premiss be negative, the conclusion, if any, must be negative. -If one term agree with a second and disagree with a third, no -agreement can be inferred between the second and third.</p> - -<p class='c005'>6. If one premiss be particular, the conclusion must be particular. -This is not very obvious, since the middle term may be universally -spoken of in a particular proposition, as in Some B is not X. But this -requires one negative proposition, whence (Rule 3) the other must -be affirmative. Again, since the conclusion must be negative (Rule 5) -its predicate is spoken of universally, and, therefore, must enter universally; -the other term A must enter, then, in a universal affirmative -proposition, which is against the supposition.</p> - -<p class='c005'>In the preceding set of syllogisms we observe one form only which -produces A, or E, or I, but three which produce O.</p> - -<p class='c005'><span class='pageno' id='Page_18'>18</span>Let an assertion be said to be weakened when it is reduced from -universal to particular, and strengthened in the contrary case. Thus, -‘Every A is B’ is called stronger than ‘Some A is B.’</p> - -<p class='c005'>Every form of syllogism which can give a legitimate result is either -one of the preceding six, or another formed from one of the six, either -by changing one of the assertions into its converse, if that be allowable, -or by strengthening one of the premises without altering the conclusion, -or both. Thus,</p> - -<table class='table0' summary=''> - <tr> - <td class='c023'>Some A is X</td> - <td class='c024' rowspan='2'>may be written</td> - <td class='c025'>Some X is A</td> - </tr> - <tr> - <td class='c023'>Every X is B</td> - - <td class='c025'>Every X is B</td> - </tr> - <tr> - <td class='c023'> </td> - <td class='c024'> </td> - <td class='c025'> </td> - </tr> - <tr> - <td class='c023' colspan='2' rowspan='2'>What follows will still follow from</td> - <td class='c025'><em>Every</em> X is A</td> - </tr> - <tr> - - <td class='c025'>Every X is B</td> - </tr> -</table> - -<p class='c010'>for all which is true when ‘Some X is A,’ is not less true when ‘Every -X is A.’</p> - -<p class='c005'>It would be possible also to form a legitimate syllogism by weakening -the conclusion, when it is universal, since that which is true of all is -true of some. Thus, ‘Every A is X, Every X is B,’ which yields -‘Every A is B,’ also yields ‘Some A is B.’ But writers on logic have -always considered these syllogisms as useless, conceiving it better to -draw from any premises their strongest conclusion. In this they were -undoubtedly right; and the only question is, whether it would not have -been advisable to make the premises as weak as possible, and not to -admit any syllogisms in which more appeared than was absolutely -necessary to the conclusion. If such had been the practice, then</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Every X is A, Every X is B, therefore Some A is B</div> - </div> - </div> -</div> - -<p class='c010'>would have been considered as formed by a spurious and unnecessary -excess of assertion. The minimum of assertion would be contained in -either of the following,</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Every X is A, Some X is B, therefore Some A is B</div> - <div class='line'>Some X is A, Every X is B, therefore Some A is B</div> - </div> - </div> -</div> - -<p class='c005'>In this tract, syllogisms have been divided into two classes: first, -<span class='pageno' id='Page_19'>19</span>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.</p> - -<table class='table0' summary=''> - <tr> - <th class='c008'></th> - <th class='c008'>Universal.</th> - <th class='c022'> </th> - <th class='c008'>Particular.</th> - <th class='c009'> </th> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'> </td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every A is X</td> - <td class='c022'> </td> - <td class='c021'>Some A is X</td> - <td class='c009'>I</td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every X is B</td> - <td class='c022'>──────</td> - <td class='c021'>Every X is B</td> - <td class='c009'>A</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c022'> </td> - <td class='c021'><hr /></td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every A is B</td> - <td class='c022'> </td> - <td class='c021'>Some A is B</td> - <td class='c009'>I</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'> </td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'>Some A is X</td> - <td class='c009'>I</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'>No X is B</td> - <td class='c009'>E</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'> </td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'>┌</td> - <td class='c021'><hr /></td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every A is X</td> - <td class='c022'>│</td> - <td class='c021'>Some A is not B</td> - <td class='c009'>O</td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No X is B</td> - <td class='c022'>─────┼</td> - <td class='c021'>Every A is X</td> - <td class='c009'>A</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c022'>│</td> - <td class='c021'> </td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No A is B</td> - <td class='c022'>│</td> - <td class='c021'>Some B is not X</td> - <td class='c009'>O</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'>└</td> - <td class='c021'><hr /></td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'>Some B is not A</td> - <td class='c009'>O</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'> </td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'>Every X is A</td> - <td class='c009'>A</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'>......</td> - <td class='c022'> </td> - <td class='c021'>Some X is not B</td> - <td class='c009'>O</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'><hr /></td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'> </td> - <td class='c021'>Some A is not B</td> - <td class='c009'>O</td> - </tr> -</table> - -<p class='c005'>In all works on logic, it is customary to write that premiss first -which contains the predicate of the conclusion. Thus,</p> - -<table class='table0' summary=''> - <tr> - <td class='c021'>Every X is B</td> - <td class='c008'> </td> - <td class='c014'>Every A is X</td> - </tr> - <tr> - <td class='c021'>Every A is X</td> - <td class='c008'>would be written, and not</td> - <td class='c014'>Every X is B</td> - </tr> - <tr> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c021'>Every A is B</td> - <td class='c008'> </td> - <td class='c014'>Every A is B</td> - </tr> -</table> - -<p class='c010'>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—</p> - -<table class='table0' summary=''> - <tr><td class='c015' colspan='4'><span class='pageno' id='Page_20'>20</span></td></tr> - <tr> - <td class='c008'>XB</td> - <td class='c008'>BX</td> - <td class='c008'>XB</td> - <td class='c009'>BX</td> - </tr> - <tr> - <td class='c008'>AX</td> - <td class='c008'>AX</td> - <td class='c008'>XA</td> - <td class='c009'>XA</td> - </tr> - <tr> - <td class='c008'><hr /></td> - <td class='c008'><hr /></td> - <td class='c008'><hr /></td> - <td class='c009'><hr /></td> - </tr> - <tr> - <td class='c008'>AB</td> - <td class='c008'>AB</td> - <td class='c008'>AB</td> - <td class='c009'>AB</td> - </tr> -</table> - -<p class='c010'>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,</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every X is B</td> - <td class='c008'> </td> - <td class='c008'>A</td> - <td class='c021'>Every X is B</td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>I</td> - <td class='c021'>Some A is X</td> - <td class='c008'>becomes</td> - <td class='c008'>A</td> - <td class='c021'>Every X is A:</td> - <td class='c014'>(Co +)</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>I</td> - <td class='c021'>Some A is B</td> - <td class='c008'> </td> - <td class='c008'>I</td> - <td class='c021'>Some A is B</td> - <td class='c014'> </td> - </tr> -</table> - -<p class='c010'>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.</p> - -<table class='table0' summary=''> - <tr><td class='c015' colspan='6'><em>First Figure.</em></td></tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every X is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>A</td> - <td class='c014'>Every X is B</td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every A is X</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>I</td> - <td class='c014'>Some A is X</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every A is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>I</td> - <td class='c014'>Some A is B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No X is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>E</td> - <td class='c014'>No X is B</td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every A is X</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>I</td> - <td class='c014'>Some A is X</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No A is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>O</td> - <td class='c014'>Some A is not B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr><td class='c015' colspan='6'><em>Second Figure.</em></td></tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No B is X</td> - <td class='c021'>(Co)</td> - <td class='c021'> </td> - <td class='c008'>E</td> - <td class='c014'>No B is X (Co)</td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every A is X</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>I</td> - <td class='c014'>Some A is X</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No A is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>O</td> - <td class='c014'>Some A is not B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every B is X</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>A</td> - <td class='c014'>Every B is X</td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No A is X</td> - <td class='c021'>(Co)</td> - <td class='c021'> </td> - <td class='c008'>O</td> - <td class='c014'>Some A is not X</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No A is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>O</td> - <td class='c014'>Some A is not B</td> - </tr> - <tr> - <td class='c008'><span class='pageno' id='Page_21'>21</span> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr><td class='c015' colspan='6'><em>Third Figure.</em></td></tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every X is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>E</td> - <td class='c014'>No X is B</td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every X is A</td> - <td class='c021'>(Co+)</td> - <td class='c021'> </td> - <td class='c008'>A</td> - <td class='c014'>Every X is A (Co+)</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>I</td> - <td class='c021'>Some A is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>O</td> - <td class='c014'>Some A is not B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>I</td> - <td class='c021'>Some X is B</td> - <td class='c021'>(Co)</td> - <td class='c021'> </td> - <td class='c008'>O</td> - <td class='c014'>Some X is not B</td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every X is A</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>A</td> - <td class='c014'>Every X is A</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>I</td> - <td class='c021'>Some A is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>O</td> - <td class='c014'>Some A is not B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every X is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>E</td> - <td class='c014'>No X is B</td> - </tr> - <tr> - <td class='c008'>I</td> - <td class='c021'>Some X is A</td> - <td class='c021'>(Co)</td> - <td class='c021'> </td> - <td class='c008'>I</td> - <td class='c014'>Some X is A (Co)</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>I</td> - <td class='c021'>Some A is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>O</td> - <td class='c014'>Some A is not B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr><td class='c015' colspan='6'><em>Fourth Figure.</em></td></tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every B is X</td> - <td class='c021'>(+)</td> - <td class='c021'> </td> - <td class='c008'>I</td> - <td class='c014'>Some B is X</td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every X is A</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>A</td> - <td class='c014'>Every X is A</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>I</td> - <td class='c021'>Some A is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>I</td> - <td class='c014'>Some B is A</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>A</td> - <td class='c021'>Every B is X</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>E</td> - <td class='c014'>No B is X (Co)</td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No X is A</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>A</td> - <td class='c014'>Every X is A (Co+)</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No A is B</td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'>O</td> - <td class='c014'>Some A is not B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'>E</td> - <td class='c021'>No B is X (Co)</td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'>I</td> - <td class='c021'>Some X is A (Co)</td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'>O</td> - <td class='c021'>Some A is not B</td> - <td class='c008'> </td> - <td class='c014'> </td> - </tr> -</table> - -<p class='c005'>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:</p> - -<table class='table1' summary=''> - <tr><td class='c015' colspan='4'><span class='pageno' id='Page_22'>22</span></td></tr> - <tr> - <td class='c018'>Every A is X</td> - <td class='brt c018'>Every B is X</td> - <td class='c018'>Some A is X</td> - <td class='c018'>Some B is X</td> - </tr> - <tr> - <td class='c018'>Every X is B</td> - <td class='brt c018'>Every X is A</td> - <td class='c018'>No X is B</td> - <td class='c018'>No X is A</td> - </tr> - <tr> - <td class='c018'><hr /></td> - <td class='brt c018'><hr /></td> - <td class='c018'><hr /></td> - <td class='c018'><hr /></td> - </tr> - <tr> - <td class='c018'>Every A is B</td> - <td class='brt c018'>Every B is A</td> - <td class='c018'>Some A is not B</td> - <td class='c018'>Some B is not A</td> - </tr> - <tr> - <td class='c018'> </td> - <td class='brt c018'> </td> - <td class='c018'> </td> - <td class='c018'> </td> - </tr> - <tr> - <td class='c018'>Every A is X</td> - <td class='brt c018'>Every B is X</td> - <td class='c018'>Every A is X</td> - <td class='c018'>Every B is X</td> - </tr> - <tr> - <td class='c018'>No X is B</td> - <td class='brt c018'>No X is A</td> - <td class='c018'>Some B is not X</td> - <td class='c018'>Some A is not X</td> - </tr> - <tr> - <td class='c018'><hr /></td> - <td class='brt c018'><hr /></td> - <td class='c018'><hr /></td> - <td class='c018'><hr /></td> - </tr> - <tr> - <td class='c018'>No A is B</td> - <td class='brt c018'>No B is A</td> - <td class='c018'>Some B is not A</td> - <td class='c018'>Some A is not B</td> - </tr> - <tr> - <td class='c018'> </td> - <td class='brt c018'> </td> - <td class='c018'> </td> - <td class='c018'> </td> - </tr> - <tr> - <td class='c018'>Some A is X</td> - <td class='brt c018'>Some B is X</td> - <td class='c018'>Every X is A</td> - <td class='c018'>Every X is B</td> - </tr> - <tr> - <td class='c018'>Every X is B</td> - <td class='brt c018'>Every X is A</td> - <td class='c018'>Some X is not B</td> - <td class='c018'>Some X is not A</td> - </tr> - <tr> - <td class='c018'><hr /></td> - <td class='brt c018'><hr /></td> - <td class='c018'><hr /></td> - <td class='c018'><hr /></td> - </tr> - <tr> - <td class='c018'>Some A is B</td> - <td class='brt c018'>Some B is A</td> - <td class='c018'>Some A is not B</td> - <td class='c018'>Some B is not A</td> - </tr> -</table> - -<p class='c005'>In the list of page <a href='#Page_19'>19</a>, 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 possible -<em>weakest</em> forms 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 page <a href='#Page_10'>10</a>.</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'>A</td> - <td class='c021'>Every A is X</td> - <td class='c021'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'>U</td> - <td class='c021'>Every X is B</td> - <td class='c021'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c022'>A</td> - <td class='c021'>Every A is B</td> - <td class='c021'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c022'>I</td> - <td class='c021'>Some A is X</td> - <td class='c022'>I</td> - <td class='c021'>Some B is X</td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c022'>U</td> - <td class='c021'>Every X is B</td> - <td class='c022'>U</td> - <td class='c021'>Every X is A</td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'><hr /></td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c022'>I</td> - <td class='c021'>Some A is B</td> - <td class='c022'>I</td> - <td class='c021'>Some A is B</td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c022'>A</td> - <td class='c021'>Every A is X</td> - <td class='c022'>A</td> - <td class='c021'>Every B is X</td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c022'>E</td> - <td class='c021'>No B is X</td> - <td class='c022'>E</td> - <td class='c021'>No A is X</td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'><hr /></td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c022'>E</td> - <td class='c021'>No A is B</td> - <td class='c022'>E</td> - <td class='c021'>No A is B</td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c021'> </td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c008'>I</td> - <td class='c021'>Some A is X</td> - <td class='c022'>A</td> - <td class='c021'>Every B is X</td> - <td class='c022'>U</td> - <td class='c014'>Every X is A</td> - </tr> - <tr> - <td class='c008'>E</td> - <td class='c021'>No B is X</td> - <td class='c022'>O</td> - <td class='c021'>Some A is not X</td> - <td class='c022'>Y</td> - <td class='c014'>Some X is not B</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c021'><hr /></td> - <td class='c021'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>O</td> - <td class='c021'>Some A is not B</td> - <td class='c022'>O</td> - <td class='c021'>Some A is not B</td> - <td class='c022'>O</td> - <td class='c014'>Some A is not B</td> - </tr> -</table> - -<p class='c005'><span class='pageno' id='Page_23'>23</span>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:</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line in8'>‘That race’ is ‘a race of Asiatic origin:’</div> - <div class='line in8'>Every ‘race of Asiatic origin’ is ‘a race which must have possessed some of the arts of life:’</div> - <div class='line'>Therefore, That race <em>is</em> a race which must have possessed some of the arts of life.</div> - </div> - </div> -</div> - -<p class='c005'>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.’</p> - -<p class='c007'>I shall now give a few detached illustrations of what precedes.</p> - -<p class='c005'>“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,</p> - -<table class='table0' summary=''> - <tr><td class='c015' colspan='3'><span class='pageno' id='Page_24'>24</span></td></tr> - <tr> - <td class='c008'> </td> - <td class='c008'>All weak princes</td> - <td class='c014'>are prone to favourites</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'>He</td> - <td class='c014'>was prone to favourites</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'><hr /></td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>Therefore</td> - <td class='c008'>He</td> - <td class='c014'>was a weak prince</td> - </tr> -</table> - -<p class='c010'>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.</p> - -<p class='c005'>‘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,</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'> </td> - <td class='c014'>A is (a magnitude less than B)</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c014'>B is (a magnitude less than C)</td> - </tr> - <tr> - <td class='c008'>Therefore</td> - <td class='c014'>A is (a magnitude less than C)</td> - </tr> -</table> - -<p class='c010'>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,</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c021'>A is less than B</td> - <td class='c026'> </td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'>Less than</td> - <td class='c021'>B is less than C:</td> - <td class='c026'>following from B is less than C.</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'><hr /></td> - <td class='c021'><hr /></td> - <td class='c026'> </td> - </tr> - <tr> - <td class='c008'>Therefore</td> - <td class='c008'> </td> - <td class='c021'>A is less than C</td> - <td class='c026'> </td> - </tr> -</table> - -<p class='c010'>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 syllogism -<span class='pageno' id='Page_25'>25</span>can 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,</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'> </td> - <td class='c014'>Every A is B, and there are Bs which are not As</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c014'>Every B is C, and there are Cs which are not Bs</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>Therefore</td> - <td class='c014'>Every A is C, and there are Cs which are not As</td> - </tr> -</table> - -<p class='c010'>Or thus:</p> - -<table class='table0' summary=''> - <tr> - <td class='c021'>The Bs contain all the As, and more</td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c021'>The Cs contain all the Bs, and more</td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c021'><hr /></td> - <td class='c014'> </td> - </tr> - <tr> - <td class='c021'>The Cs contain all the As, and more</td> - <td class='c014'> </td> - </tr> -</table> - -<p class='c010'>The most technical form, however, is,</p> - -<table class='table0' summary=''> - <tr> - <td class='c021'>From</td> - <td class='c014'>Every A is B; [Some B is not A]</td> - </tr> - <tr> - <td class='c021'> </td> - <td class='c014'>Every B is C; [Some C is not B]</td> - </tr> - <tr> - <td class='c021'>Follows</td> - <td class='c014'>Every A is C; [Some C is not A]</td> - </tr> -</table> - -<p class='c005'>This sort of argument is called <i><span lang="fr" xml:lang="fr">à fortiori</span></i> argument, 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, <i><span lang="fr" xml:lang="fr">à fortiori</span></i>, 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 it <i><span lang="fr" xml:lang="fr">à fortiori</span></i>.</p> - -<p class='c005'>The argument <i><span lang="fr" xml:lang="fr">à fortiori</span></i>, may then be defined as a universally -<span class='pageno' id='Page_26'>26</span>affirmative 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,</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>All the As make up part (and part only) of the Xs</div> - <div class='line'>Every X is B;</div> - </div> - </div> -</div> - -<p class='c010'>then we are certain that</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>All the As make up part (and part only) of the Bs.</div> - </div> - </div> -</div> - -<p class='c010'>But if we are at liberty further to say that</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>All the As make up part (and part only) of the Xs</div> - <div class='line'>All the Xs make up part (and part only) of the Bs</div> - </div> - </div> -</div> - -<p class='c010'>then we conclude that</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>All the As make up <em>part of part</em> (only) of the Bs</div> - </div> - </div> -</div> - -<p class='c010'>and the words in Italics mark that quality of the conclusion from which -the argument is called <i><span lang="fr" xml:lang="fr">à fortiori</span></i>.</p> - -<p class='c005'>Most syllogisms which give an affirmative conclusion are generally -meant to imply <i><span lang="fr" xml:lang="fr">à fortiori</span></i> arguments, 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.’</p> - -<p class='c005'>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 an <em>hypothetical</em> syllogism. 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</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>When A is B, it follows that P is Q;</div> - </div> - </div> -</div> - -<p class='c010'><span class='pageno' id='Page_27'>27</span>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 Q <em>only</em> when 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,</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'> </td> - <td class='c008'>Every</td> - <td class='c021'>K is Z</td> - <td class='c008'>Every</td> - <td class='c014'>K is Z</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c021'>L is a K</td> - <td class='c008'> </td> - <td class='c014'>L is not a K</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c021'><hr /></td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008'>Therefore</td> - <td class='c008'> </td> - <td class='c021'>L is a Z</td> - <td class='c008'> </td> - <td class='c014'>No conclusion</td> - </tr> -</table> - -<p class='c010'>Similarly, according as a particular case (M) is or is not Z, we have</p> - -<table class='table0' summary=''> - <tr> - <td class='c008'>Every</td> - <td class='c021'>K is Z</td> - <td class='c008'>Every</td> - <td class='c014'>K is Z</td> - </tr> - <tr> - <td class='c008'> </td> - <td class='c021'>M is a Z</td> - <td class='c008'> </td> - <td class='c014'>M is not a Z</td> - </tr> - <tr> - <td class='c008' colspan='2'><hr /></td> - <td class='c008'> </td> - <td class='c014'><hr /></td> - </tr> - <tr> - <td class='c008' colspan='2'>No conclusion</td> - <td class='c008'> </td> - <td class='c014'>M is not a K</td> - </tr> -</table> - -<p class='c010'>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.</p> - -<p class='c007'>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.</p> - -<p class='c005'>But how is any proposition to be proved false, except by proving a -<span class='pageno' id='Page_28'>28</span>contradiction 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.</p> - -<p class='c005'>(1.) If there be an equiangular triangle not equilateral, it follows -that a whole can be found which is not greater than its part.<a id='r1' /><a href='#f1' class='c027'><sup>[1]</sup></a></p> - -<div class='footnote' id='f1'> -<p class='c005'><a href='#r1'>1</a>. This is the proposition in proof of which nearly the whole of the demonstration -of Euclid is spent.</p> -</div> - -<p class='c005'>(2.) It is false that there can be any whole which is not greater than -its part (self evident).</p> - -<p class='c005'>(3.) Therefore it is false that there is any equiangular triangle -which is not equilateral; or all equiangular triangles are equilateral.</p> - -<p class='c005'>When a proposition is established by proving the truth of the matters -it contains, the demonstration is called <em>direct</em>; when by proving the -falsehood of every contradictory proposition, it is called <em>indirect</em>. 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.</p> - -<p class='c005'>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</p> - -<table class='table2' summary=''> -<colgroup> -<col width='88%' /> -<col width='11%' /> -</colgroup> - <tr> - <td class='c021'>Every A is B</td> - <td class='c009'>(1)</td> - </tr> -</table> - -<p class='c010'>may be easily made the means of deducing,</p> - -<table class='table2' summary=''> -<colgroup> -<col width='88%' /> -<col width='11%' /> -</colgroup> - <tr> - <td class='c021'>Every (thing not A) is not B</td> - <td class='c009'>(2)</td> - </tr> -</table> - -<p class='c010'>which last gives</p> - -<table class='table2' summary=''> -<colgroup> -<col width='88%' /> -<col width='11%' /> -</colgroup> - <tr> - <td class='c021'>Every B is A</td> - <td class='c009'>(3)</td> - </tr> -</table> - -<p class='c005'><span class='pageno' id='Page_29'>29</span>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.</p> - -<p class='c005'>The following proposition contains a method which is of frequent use.</p> - -<p class='c005'><span class='sc'>Hypothesis.</span>—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, <em>and one only</em>. 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</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>When A is true, P is true</div> - <div class='line'>When B is true, Q is true</div> - <div class='line'>When C is true, R is true</div> - </div> - </div> -</div> - -<p class='c010'><span class='sc'>Consequence</span>: then it follows that</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>When P is true, A is true</div> - <div class='line'>When Q is true, B is true</div> - <div class='line'>When R is true, C is true</div> - </div> - </div> -</div> - -<p class='c010'>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.</p> - -<table class='table1' summary=''> - <tr> - <td class='brt c011'>Case 1. If</td> - <td class='c018'>When P is Q,</td> - <td class='c018'>A is B</td> - </tr> - <tr> - <td class='brt c011'> </td> - <td class='c018'>When P is not Q,</td> - <td class='c018'>A is not B</td> - </tr> - <tr> - <td class='brt c011'>It follows that</td> - <td class='c018'>When A is B,</td> - <td class='c018'>P is Q</td> - </tr> - <tr> - <td class='brt c011'> </td> - <td class='c018'>When A is not B,</td> - <td class='c018'>P is not Q</td> - </tr> - <tr> - <td class='c011' colspan='3'> </td> - </tr> - <tr> - <td class='brt c011' rowspan='3'>Case 2. If</td> - <td class='c018'>When A is greater than B,</td> - <td class='c018'>P is greater than Q</td> - </tr> - <tr> - - <td class='c018'>When A is equal to B,</td> - <td class='c018'>P is equal to Q</td> - </tr> - <tr> - - <td class='c018'>When A is less than B,</td> - <td class='c018'>P is less than Q</td> - </tr> - <tr> - <td class='c011' colspan='3'><span class='pageno' id='Page_30'>30</span> </td> - </tr> - <tr> - <td class='brt c011' rowspan='3'>It follows that</td> - <td class='c018'>When P is greater than Q,</td> - <td class='c018'>A is greater than B</td> - </tr> - <tr> - - <td class='c018'>When P is equal to Q,</td> - <td class='c018'>A is equal to B</td> - </tr> - <tr> - - <td class='c018'>When P is less than Q,</td> - <td class='c018'>A is less than B</td> - </tr> -</table> - -<hr class='c028' /> - -<p class='c005'>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.</p> - -<p class='c005'>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</p> - -<div class='lg-container-b'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Is the assertion true?</div> - <div class='line'>Will a white ball be drawn?</div> - </div> - </div> -</div> - -<p class='c010'>may be such that the answer, ‘most probably,’ expresses the same degree -of likelihood in both cases.</p> - -<p class='c005'>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 we <em>have</em> found 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.</p> - -<p class='c005'><span class='pageno' id='Page_31'>31</span>Thus we have the succession of such results as in the following -table:—</p> - -<table class='table0' summary=''> - <tr> - <th class='c008'>Odds in favour of an event</th> - <th class='c008'>Probability for</th> - <th class='c009'>Probability against</th> - </tr> - <tr> - <td class='c008'> </td> - <td class='c008'> </td> - <td class='c009'> </td> - </tr> - <tr> - <td class='c008'>1 to 1</td> - <td class='c008'>½</td> - <td class='c009'>½</td> - </tr> - <tr> - <td class='c008'>2 to 1</td> - <td class='c008'>⅔</td> - <td class='c009'>⅓</td> - </tr> - <tr> - <td class='c008'>3 to 1</td> - <td class='c008'>¾</td> - <td class='c009'>¼</td> - </tr> - <tr> - <td class='c008'>3 to 2</td> - <td class='c008'>⅗</td> - <td class='c009'>⅖</td> - </tr> - <tr> - <td class='c008'>4 to 1</td> - <td class='c008'>⅘</td> - <td class='c009'>⅕</td> - </tr> - <tr> - <td class='c008'>4 to 3</td> - <td class='c008'>⁴⁄₇</td> - <td class='c009'>³⁄₇</td> - </tr> - <tr> - <td class='c008'>5 to 1</td> - <td class='c008'>⅚</td> - <td class='c009'>⅙</td> - </tr> - <tr> - <td class='c008'>&c.</td> - <td class='c008'>&c.</td> - <td class='c009'>&c.</td> - </tr> -</table> - -<p class='c005'>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 the <em>intrinsic probability</em> of 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.</p> - -<p class='c005'>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. -<span class='pageno' id='Page_32'>32</span>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.</p> - -<p class='c005'>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, <em>a + b − ab</em> is the probability of the truth of a conclusion -after the introduction of an argument of the intrinsic probability <em>b</em>, the -previous probability of the said conclusion having been <em>a</em>.</p> - -<p class='c005'>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.</p> - -<p class='c005'>When the conclusion was neither likely nor unlikely beforehand -(or had the probability ½), the shortest way of applying the preceding -rule (in which <em>a + b − ab</em> becomes ½ + ½<em>b</em>) 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 or <span class='fraction'>3 + 4<br /><span class='vincula'>2 × 4</span></span>.</p> - -<div class='nf-center-c0'> -<div class='nf-center c002'> - <div>THE END.</div> - </div> -</div> - -<div class='nf-center-c0'> -<div class='nf-center c002'> - <div><span class='small'>LONDON:—PRINTED BY JAMES MOYES,</span></div> - <div><span class='small'>Castle Street, Leicester Square.</span></div> - </div> -</div> - -<div class='pbb'> - <hr class='pb c003' /> -</div> -<div class='tnotes x-ebookmaker'> - -<div class='chapter ph2'> - -<div class='nf-center-c0'> -<div class='nf-center c029'> - <div>TRANSCRIBER’S NOTES</div> - </div> -</div> - -</div> - - <ol class='ol_1 c002'> - <li>Silently corrected obvious typographical errors and variations in spelling. - - </li> - <li>Retained archaic, non-standard, and uncertain spellings as printed. - </li> - </ol> - -</div> - -<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK FIRST NOTIONS OF LOGIC ***</div> -<div style='text-align:left'> - -<div style='display:block; margin:1em 0'> -Updated editions will replace the previous one—the old editions will -be renamed. -</div> - -<div style='display:block; margin:1em 0'> -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. 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