{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_67017/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_67017","id":"gutenberg_67017","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/67017","summary":null,"tags":["Category: Mathematics","Category: Philosophy & Ethics","Logic"],"text":"FIRST NOTIONS\n                                   OF\n                                 LOGIC\n                 (PREPARATORY TO THE STUDY OF GEOMETRY)\n\n\n                                   BY\n\n                          AUGUSTUS DE MORGAN,\n\n                     OF TRINITY COLLEGE, CAMBRIDGE,\n        PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON.\n\n  The root of all the mischief in the sciences, is this; that falsely\n  magnifying and admiring the powers of the mind, we seek not its real\n                             helps.\u2014BACON.\n\n\n                                LONDON:\n\n                     PRINTED FOR TAYLOR AND WALTON,\n\n           BOOKSELLERS AND PUBLISHERS TO UNIVERSITY COLLEGE.\n\n                         28 UPPER GOWER STREET.\n\n                             M.DCCC.XXXIX.\n\n\n\n\n\u2042 This Tract contains no more than the author has found, from\nexperience, to be much wanted by students who are commencing with\nEuclid. It will ultimately form an Appendix to his Treatise on\nArithmetic.\n\nThe author would not, by any means, in presenting the minimum necessary\nfor a particular purpose, be held to imply that he has given enough of\nthe subject for all the ends of education. He has long regretted the\nneglect of logic; a science, the study of which would shew many of its\nopponents that the light esteem in which they hold it arises from those\nhabits of inference which thrive best in its absence. He strongly\nrecommends any student to whom this tract may be the first introduction\nof the subject, to pursue it to a much greater extent.\n\n_University College, Jan, 8, 1839._\n\n\n                    LONDON:\u2014PRINTED BY JAMES MOYES,\n                    Castle Street, Leicester Square.\n\n\n\n\n                             FIRST NOTIONS\n                                   OF\n                                 LOGIC.\n\n\nWhat we here mean by Logic is the examination of that part of reasoning\nwhich depends upon the manner in which inferences are formed, and the\ninvestigation of general maxims and rules for constructing arguments, so\nthat the conclusion may contain no inaccuracy which was not previously\nasserted in the premises. It has nothing to do with the truth of the\nfacts, opinions, or presumptions, from which an inference is derived;\nbut simply takes care that the inference shall certainly be true, if the\npremises be true. Thus, when we say that all men will die, and that all\nmen are rational beings, and thence infer that some rational beings will\ndie, the _logical_ truth of this sentence is the same whether it be true\nor false that men are mortal and rational. This logical truth depends\nupon the structure of the sentence, and not on the particular matters\nspoken of. Thus,\n\n                 Instead of,                         Write,\n              All men will die.                   Every A is B.\n         All men are rational beings.             Every A is C.\n   Therefore some rational beings will die. Therefore some Cs are Bs.\n\nThe second of these is the same proposition, logically considered, as\nthe first; the consequence in both is virtually contained in, and\nrightly inferred from, the premises. Whether the premises be true or\nfalse, is not a question of logic, but of morals, philosophy, history,\nor any other knowledge to which their subject-matter belongs: the\nquestion of logic is, does the conclusion certainly follow if the\npremises be true?\n\nEvery act of reasoning must mainly consist in comparing together\ndifferent things, and either finding out, or recalling from previous\nknowledge, the points in which they resemble or differ from each other.\nThat particular part of reasoning which is called _inference_, consists\nin the comparison of several and different things with one and the same\nother thing; and ascertaining the resemblances, or differences, of the\nseveral things, by means of the points in which they resemble, or differ\nfrom, the thing with which all are compared.\n\nThere must then be some propositions already obtained before any\ninference can be drawn. All propositions are either assertions or\ndenials, and are thus divided into _affirmative_ and _negative_. Thus, A\nis B, and A is not B, are the two forms to which all propositions may be\nreduced. These are, for our present purpose, the most simple forms;\nthough it will frequently happen that much circumlocution is needed to\nreduce propositions to them. Thus, suppose the following assertion, \u2018If\nhe should come to-morrow, he will probably stay till Monday\u2019; how is\nthis to be reduced to the form A is B? There is evidently something\nspoken of, something said of it, and an affirmative connexion between\nthem. Something, if it happen, that is, the happening of something,\nmakes the happening of another something probable; or is one of the\nthings which render the happening of the second thing probable.\n\n                 A                \u2502is\u2502               B\n\n\n  The happening of his arrival    \u2502  \u2502an event from which it may be\n    to-morrow                     \u2502is\u2502  inferred as probable that he\n                                  \u2502  \u2502  will stay till Monday.\n\nThe forms of language will allow the manner of asserting to be varied in\na great number of ways; but the reduction to the preceding form is\nalways possible. Thus, \u2018so he said\u2019 is an affirmation, reducible as\nfollows:\n\n  What you have just said (or     \u2502is\u2502the thing which he said.\n    whatever else \u2018so\u2019 refers to) \u2502  \u2502\n\nBy changing \u2018is\u2019 into \u2018is not,\u2019 we make a negative proposition; but care\nmust always be taken to ascertain whether a proposition which appears\nnegative is really so. The principal danger is that of confounding a\nproposition which is negative with another which is affirmative of\nsomething requiring a negative to describe it. Thus \u2018he resembles the\nman who was not in the room,\u2019 is affirmative, and must not be confounded\nwith \u2018he does not resemble the man who was in the room.\u2019 Again, \u2018if he\nshould come to-morrow, it is probable he will not stay till Monday,\u2019\ndoes not mean the simple denial of the preceding proposition, but the\naffirmation of the directly opposite proposition. It is,\n\n                 A                \u2502is\u2502               B\n\n\n  The happening of his arrival    \u2502  \u2502an event from which it may be\n    to-morrow,                    \u2502is\u2502  inferred to be _im_probable\n                                  \u2502  \u2502  that he will stay till Monday,\n\nwhereas the following,\n\n The happening of his arrival  \u2502        \u2502an event from which it may be\n   to-morrow,                  \u2502is _not_\u2502  inferred as probable that he\n                               \u2502        \u2502  will stay till Monday,\n\nwould be expressed thus: \u2018If he should come to-morrow, that is no reason\nwhy he should stay till Monday.\u2019\n\nMoreover, the negative words not, no, &c., have two kinds of meaning\nwhich must be carefully distinguished. Sometimes they deny, and nothing\nmore: sometimes they are used to affirm the direct contrary. In cases\nwhich offer but two alternatives, one of which is necessary, these\namount to the same thing, since the denial of one, and the affirmation\nof the other, are obviously equivalent propositions. In many idioms of\nconversation, the negative implies affirmation of the contrary in cases\nwhich offer not only alternatives, but degrees of alternatives. Thus, to\nthe question, \u2018Is he tall?\u2019 the simple answer, \u2018No,\u2019 most frequently\nmeans that he is the contrary of tall, or considerably under the\naverage. But it must be remembered, that, in all logical reasoning, the\nnegation is simply negation, and nothing more, never implying\naffirmation of the contrary.\n\nThe common proposition that two negatives make an affirmative, is true\nonly upon the supposition that there are but two possible things, one of\nwhich is denied. Grant that a man must be either able or unable to do a\nparticular thing, and then _not unable_ and able are the same things.\nBut if we suppose various degrees of performance, and therefore degrees\nof ability, it is false, in the common sense of the words, that two\nnegatives make an affirmative. Thus, it would be erroneous to say, \u2018John\nis able to translate Virgil, and Thomas is not unable; therefore, what\nJohn can do Thomas can do,\u2019 for it is evident that the premises mean\nthat John is so near to the best sort of translation that an affirmation\nof his ability may be made, while Thomas is considerably lower than\nJohn, but not so near to absolute deficiency that his ability may be\naltogether denied. It will generally be found that two negatives imply\nan affirmative of a weaker degree than the positive affirmation.\n\nEach of the propositions, \u2018A is B,\u2019 and \u2018A is not B,\u2019 may be subdivided\ninto two species: the _universal_, in which every possible case is\nincluded; and the _particular_, in which it is not meant to be asserted\nthat the affirmation or negation is universal. The four species of\npropositions are then as follows, each being marked with the letter by\nwhich writers on logic have always distinguished it.\n\n               A _Universal Affirmative_  Every A is    B\n               E _Universal Negative_     No A is       B\n               I _Particular Affirmative_ Some A is     B\n               O _Particular Negative_    Some A is not B\n\nIn common conversation the affirmation of a part is meant to imply the\ndenial of the remainder. Thus, by \u2018some of the apples are ripe,\u2019 it is\nalways intended to signify that some are not ripe. This is not the case\nin logical language, but every proposition is intended to make its\namount of affirmation or denial, and no more. When we say, \u2018Some A is\nB,\u2019 or, more grammatically, \u2018Some As are Bs,\u2019 we do not mean to imply\nthat some are not: this may or may not be. Again, the word some means,\n\u2018one or more, possibly all.\u2019 The following table will shew the bearing\nof each proposition on the rest.\n\n _Every A is B_ affirms and contains _Some A is B_ and\u2502_No A is B_\n   denies                                             \u2502_Some A is not B_\n\n _No A is B_ affirms and contains _Some A is not B_   \u2502_Every A is B_\n   and denies                                         \u2502_Some A is B_\n\n _Some A is B_ does not     \u2502_Every A is B_   \u2502but denies _No A is B_\n   contradict               \u2502_Some A is not B_\u2502\n\n _Some A is not B_ does not \u2502_No A is B_      \u2502but denies _Every A is B_\n   contradict               \u2502_Some A is B_    \u2502\n\n_Contradictory_ propositions are those in which one denies _any thing_\nthat the other affirms; _contrary_ propositions are those in which one\ndenies _every thing_ which the other affirms, or affirms every thing\nwhich the other denies. The following pair are contraries.\n\n                       Every A is B and No A is B\n\nand the following are contradictories,\n\n                    Every A is B to Some A is not B\n                    No A is B    to Some A is B\n\nA contrary, therefore, is a complete and total contradictory; and a\nlittle consideration will make it appear that the decisive distinction\nbetween contraries and contradictories lies in this, that contraries may\nboth be false, but of contradictories, one must be true and the other\nfalse. We may say, \u2018Either P is true, or _something_ in contradiction of\nit is true;\u2019 but we cannot say, \u2018Either P is true, or _every thing_ in\ncontradiction of it is true.\u2019 It is a very common mistake to imagine\nthat the _denial_ of a proposition gives a right to _affirm_ the\ncontrary; whereas it should be, that the _affirmation_ of a proposition\ngives a right to _deny_ the contrary. Thus, if we deny that Every A is\nB, we do not affirm that No A is B, but only that Some A is not B;\nwhile, if we affirm that Every A is B, we deny No A is B, and also Some\nA is not B.\n\nBut, as to contradictories, affirmation of one is a denial of the other,\nand denial of one is affirmation of the other. Thus, either Every A is\nB, or Some A is not B: affirmation of either is denial of the other, and\n_vice vers\u00e1_.\n\nLet the student now endeavour to satisfy himself of the following.\nTaking the four preceding propositions, A, E, I, O, let the simple\nletter signify the affirmation, the same letter in parentheses the\ndenial, and the absence of the letter, that there is neither affirmation\nnor denial.\n\n          From A follow\u2502(E), I, (O)\u2502From (A) follow         O\n          From E       \u2502(A), (I), O\u2502From (E)                I\n          From I       \u2502(E)        \u2502From (I)        (A), E, O\n          From O       \u2502(A)        \u2502From (O)        A, (E), I\n\nThese may be thus summed up: The affirmation of a universal proposition,\nand the denial of a particular one, enable us to affirm or deny all the\nother three; but the denial of a universal proposition, and the\naffirmation of a particular one, leave us unable to affirm or deny two\nof the others.\n\nIn such propositions as \u2018Every A is B,\u2019 \u2018Some A is not B,\u2019 &c., A is\ncalled the _subject_, and B the _predicate_, while the verb \u2018is\u2019 or \u2018is\nnot,\u2019 is called the _copula_. It is obvious that the words of the\nproposition point out whether the subject is spoken of universally or\npartially, but not so of the predicate, which it is therefore important\nto examine. Logical writers generally give the name of _distributed_\nsubjects or predicates to those which are spoken of universally; but as\nthis word is rather technical, I shall say that a subject or predicate\nenters wholly or partially, according as it is universally or\nparticularly spoken of.\n\n1. In A, or \u2018Every A is B,\u2019 the subject enters wholly, but the predicate\nonly partially. For it obviously says, \u2018Among the Bs are all the As,\u2019\n\u2018Every A is part of the collection of Bs, so that all the As make a part\nof the Bs, the whole it _may_ be.\u2019 Thus, \u2018Every horse is an animal,\u2019\ndoes not speak of all animals, but states that all the horses make up a\nportion of the animals.\n\n2. In E, or \u2018No A is B,\u2019 both subject and predicate enter wholly. \u2018No A\nwhatsoever is any one out of all the Bs;\u2019 \u2018search the whole collection\nof Bs, and _every_ B shall be found to be something which is not A.\u2019\n\n3. In I, or \u2018Some A is B,\u2019 both subject and predicate enter partially.\n\u2018Some of the As are found among the Bs, or make up a part (the whole\npossibly, but not known from the preceding) of the Bs.\u2019\n\n4. In O, or \u2018Some A is not B,\u2019 the subject enters partially, and the\npredicate wholly. \u2018Some As are none of them any whatsoever of the Bs;\nevery B will be found to be no one out of a certain portion of the As.\u2019\n\nIt appears then that,\n\nIn affirmatives, the predicate enters partially.\n\nIn negatives, the predicate enters wholly.\n\nIn contradictory propositions, both subject and predicate enter\ndifferently in the two.\n\nThe _converse_ of a proposition is that which is made by interchanging\nthe subject and predicate, as follows:\n\n                   The proposition.   Its converse.\n                   A Every A is B    Every B is A\n                   E No A is B       No B is A\n                   I Some A is B     Some B is A\n                   O Some A is not B Some B is not A\n\nNow, it is a fundamental and self-evident proposition, that no\nconsequence must be allowed to assert more widely than its premises; so\nthat, for instance, an assertion which is only of some Bs can never lead\nto a result which is true of all Bs. But if a proposition assert\nagreement or disagreement, any other proposition which asserts the same,\nto the same extent and no further, must be a legitimate consequence; or,\nif you please, must amount to the whole, or part, of the original\nassertion in another form. Thus, the converse of A is not true: for, in\n\u2018Every A is B,\u2019 the predicate enters partially; while in \u2018Every B is A,\u2019\nthe subject enters wholly. \u2018All the As make up a part of the Bs, then a\npart of the Bs are among the As, or some B is A.\u2019 Hence, the only\n_legitimate_ converse of \u2018Every A is B\u2019 is, \u2018Some B is A.\u2019 But in \u2018No A\nis B,\u2019 both subject and predicate enter wholly, and \u2018No B is A\u2019 is, in\nfact, the same proposition as \u2018No A is B.\u2019 And \u2018Some A is B\u2019 is also the\nsame as its converse \u2018Some B is A;\u2019 here both terms enter partially. But\n\u2018Some A is not B\u2019 admits of no converse whatever; it is perfectly\nconsistent with all assertions upon B and A in which B is the subject.\nThus neither of the four following lines is inconsistent with itself.\n\n                 Some A is not B and Every B is A\n                 Some A is not B and No    B is A\n                 Some A is not B and Some  B is A\n                 Some A is not B and Some  B is not A.\n\nWe find then, including converses, which are not identical with their\ndirect propositions, _six_ different ways of asserting or denying, with\nrespect to agreement or non-agreement, total or partial, between A and,\nsay X: these we write down, designating the additional assertions by U\nand Y.\n\n                     \u2502Identical. \u2502 Identical.  \u2502\n       A Every A is X\u2502E No A is X\u2502I Some A is X\u2502O Some A is not X\n       U Every X is A\u2502\u201e No X is A\u2502\u201e Some X is A\u2502Y Some X is not A\n\nWe shall now repeat and extend the table of page 8 (A), &c., meaning, as\nbefore, the denial of A, &c.\n\n            From A or (O) follow  A,  (E),  I   (O)\n            From E or (I)        (A),  E,  (I), O,  (U),  Y\n            From I or (E)             (E)   I\n            From O or (A)        (A),            O\n            From U or (Y)             (E)   I,       U   (Y)\n            From Y or (U)                           (U)   Y\n\nHaving thus discussed the principal points connected with the simple\nassertion, we pass to the manner of making two assertions give a third.\nEvery instance of this is called a syllogism, the two assertions which\nform the basis of the third are called premises, and the third itself\nthe conclusion.\n\nIf two things both agree with a third in any particular, they agree with\neach other in the same; as, if A be of the same colour as X, and B of\nthe same colour as X, then A is of the same colour as B. Again, if A\ndiffer from X in any particular in which B agrees with X, then A and B\ndiffer in that particular. If A be not of the same colour as X, and B be\nof the same colour as X, then A is not of the colour of B. But if A and\nB both differ from X in any particular, nothing can be inferred; they\nmay either differ in the same way and to the same extent, or not. Thus,\nif A and B be both of different colours from X, it neither follows that\nthey agree, nor differ, in their own colours.\n\nThe paragraph preceding contains the essential parts of all inference,\nwhich consists in comparing two things with a third, and finding from\ntheir agreement or difference with that third, their agreement or\ndifference with one another. Thus, Every A is X, every B is X, allows us\nto infer that A and B have all those qualities in common which are\nnecessary to X. Again, from Every A is X, and \u2018No B is X,\u2019 we infer that\nA and B differ from one another in all particulars which are essential\nto X. The preceding forms, however, though they represent common\nreasoning better than the ordinary syllogism, to which we are now\ncoming, do not constitute the ultimate forms of inference. Simple\n_identity_ or _non-identity_ is the ultimate state to which every\nassertion may be reduced; and we shall, therefore, first ask, from what\nidentities, &c., can other identities, &c., be produced? Again, since we\nname objects in species, each species consisting of a number of\nindividuals, and since our assertion may include all or only part of a\nspecies, it is further necessary to ask, in every instance, to what\nextent the conclusion drawn is true, whether of all, or only of part?\n\nLet us take the simple assertion, \u2018Every living man respires;\u2019 or, every\nliving man is one of the things (however varied they may be) which\nrespire. If we were to inclose all living men in a large triangle, and\nall respiring objects in a large circle, the preceding assertion, if\ntrue, would require that the whole of the triangle should be contained\nin the circle. And in the same way we may reduce any assertion to the\nexpression of a coincidence, total or partial, between two figures.\nThus, a point in a circle may represent an individual of one species,\nand a point in a triangle an individual of another species: and we may\nexpress that the whole of one species is asserted to be contained or not\ncontained in the other by such forms as, \u2018All the \u25b3 is in the \u25cb\u2019; \u2018None\nof the \u25b3 is in the \u25cb\u2019.\n\nAny two assertions about A and B, each expressing agreement or\ndisagreement, total or partial, with or from X, and leading to a\nconclusion with respect to A or B, is called a syllogism, of which X is\ncalled the _middle term_. The plainest syllogism is the following:\u2014\n\n                   Every A is X\u2502          All the \u25b3 is in the \u25cb\n                   Every X is B\u2502          All the \u25cb is in the \u25a1\n         Therefore Every A is B\u2502Therefore All the \u25b3 is in the \u25a1\n\nIn order to find all the possible forms of syllogism, we must make a\ntable of all the elements of which they can consist; namely\u2014\n\n                       A and X           B and X\n                   Every A is X    A Every B is X\n                   No A is X       E No B is X\n                   Some A is X     I Some B is X\n                   Some A is not X O Some B is not X\n                   Every X is A    U Every X is B\n                   Some X is not A Y Some X is not B\n\nOr their synonymes,\n\n                \u25b3 and \u25cb                         \u25a1 and \u25cb\n     All the \u25b3 is in the \u25cb         A All the \u25a1 is in the \u25cb\n     None of the \u25b3 is in the \u25cb     E None of the \u25a1 is in the \u25cb\n     Some of the \u25b3 is in the \u25cb     I Some of the \u25a1 is in the \u25cb\n     Some of the \u25b3 is not in the \u25cb O Some of the \u25a1 is not in the \u25cb\n     All the \u25cb is in the \u25b3         U All the \u25cb is in the \u25a1\n     Some of the \u25cb is not in the \u25b3 Y Some of the \u25cb is not in the \u25a1\n\nNow, taking any one of the six relations between A and X, and combining\nit with either of those between B and X, we have six pairs of premises,\nand the same number repeated for every different relation of A and X. We\nhave then thirty-six forms to consider: but, thirty of these (namely,\nall but (A, A) (E, E), &c.) are half of them repetitions of the other\nhalf. Thus, \u2018Every A is X, no B is X,\u2019 and \u2018Every B is X, no A is X,\u2019\nare of the same form, and only differ by changing A into B and B into A.\nThere are then only 15 + 6, or 21 distinct forms, some of which give a\nnecessary conclusion, while others do not. We shall select the former of\nthese, classifying them by their conclusions; that is, according as the\ninference is of the form A, E, I, or O.\n\nI. In what manner can a universal affirmative conclusion be drawn;\nnamely, that one figure is entirely contained in the other? This we can\nonly assert when we know that one figure is entirely contained in the\ncircle, which itself is entirely contained in the other figure. Thus,\n\n                  Every A is X\u2502  All the \u25b3 is in the \u25cb A\n                  Every X is B\u2502  All the \u25cb is in the \u25a1 A\n                \u2234 Every A is B\u2502\u2234 All the \u25b3 is in the \u25a1 A\n\nis the only way in which a universal affirmative conclusion can be\ndrawn.\n\nII. In what manner can a universal negative conclusion be drawn; namely,\nthat one figure is entirely exterior to the other? Only when we are able\nto assert that one figure is entirely within, and the other entirely\nwithout, the circle. Thus,\n\n                   Every A is X\u2502All the \u25b3 is in the \u25cb A\n\n                   No    B is X\u2502None of the \u25a1 is in   E\n                               \u2502the \u25cb\n\n                 \u2234 No    A is B\u2502None of the \u25b3 is in   E\n                               \u2502the \u25a1\n\nis the only way in which a universal negative conclusion can be drawn.\n\nIII. In what manner can a particular affirmative conclusion be drawn;\nnamely, that part or all of one figure is contained in the other? Only\nwhen we are able to assert that the whole circle is part of one of the\nfigures, and that the whole, or part of the circle, is part of the other\nfigure. We have then two forms.\n\n                  Every X is A\u2502  All the \u25cb is in the \u25b3 A\n\n                  Every X is B\u2502  All the \u25cb is in the \u25a1 A\n\n                \u2234 Some  A is B\u2502\u2234 Some of the \u25b3 is in   I\n                              \u2502  the \u25a1\n\n\n                  Every X is A\u2502  All the \u25cb is in the \u25b3 A\n\n                  Some  X is B\u2502  Some of the \u25cb is in   I\n                              \u2502  the \u25a1\n\n                  Some  A is B\u2502  Some of the \u25b3 is in   I\n                              \u2502  the \u25a1\n\nThe second of these contains all that is strictly necessary to the\nconclusion, and the first may be omitted. That which follows when an\nassertion can be made as to some, must follow when the same assertion\ncan be made of all.\n\nIV. How can a particular negative proposition be inferred; namely, that\npart, or all of one figure, is not contained in the other? It would seem\nat first sight, whenever we are able to assert that part or all of one\nfigure is in the circle, and that part or all of the other figure is\nnot. The weakest syllogism from which such an inference can be drawn\nwould then seem to be as follows.\n\n              Some A is X  \u2502  Some of the \u25b3 is in the \u25cb\n            Some B is not X\u2502  Some of the \u25a1 is not in the \u25cb\n          \u2234 Some B is not A\u2502\u2234 Some of the \u25b3 is not in the \u25a1\n\nBut here it will appear, on a little consideration, that the conclusion\nis only thus far true; that those As which are Xs cannot be _those_ Bs\nwhich are not Xs; but they may be _other_ Bs, about which nothing is\nasserted when we say that _some_ Bs are not Xs. And further\nconsideration will make it evident, that a conclusion of this form can\nonly be arrived at when one of the figures is entirely within the\ncircle, and the whole or part of the other without; or else when the\nwhole of one of the figures is without the circle, and the whole or part\nof the other within; or lastly, when the circle lies entirely within one\nof the figures, and not entirely within the other. That is, the\nfollowing are the distinct forms which allow of a particular negative\nconclusion, in which it should be remembered that a particular\nproposition in the premises may always be changed into a universal one,\nwithout affecting the conclusion. For that which necessarily follows\nfrom \u201csome,\u201d follows from \u201call.\u201d\n\n             Every A is X  \u2502  All the \u25b3 is in the \u25cb         A\n            Some B is not X\u2502  Some of the \u25a1 is not in the \u25cb O\n          \u2234 Some B is not A\u2502  Some of the \u25a1 is not in the \u25b3 O\n\n               No A Is X   \u2502  None of the \u25b3 is in the \u25cb     E\n              Some B is X  \u2502  Some of the \u25a1 is in the \u25cb     I\n          \u2234 Some B is not A\u2502  Some of the \u25a1 is not in the \u25b3 O\n\n             Every X is A  \u2502  All the \u25cb is in the \u25b3         A\n            Some X is not B\u2502  Some of the \u25cb is not in the \u25a1 O\n          \u2234 Some A is not B\u2502  Some of the \u25b3 is not in the \u25a1 O\n\nIt appears, then, that there are but six distinct syllogisms. All others\nare made from them by strengthening one of the premises, or converting\none or both of the premises, where such conversion is allowable; or else\nby first making the conversion, and then strengthening one of the\npremises. And the following arrangement will shew that two of them are\nuniversal, three of the others being derived from them by weakening one\nof the premises in a manner which does not destroy, but only weakens,\nthe conclusion.\n\n 1. Every A is X           3. Every A is X\n    Every X is B              No    B is X                  .........\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014              \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n    Every A is B              No    A is B\n         \u2502                           \u2502\n         \u2502                 \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n         \u2502                 \u2502                   \u2502\n 2. Some  A is X 4. Some A is X     5. Every A is X     6. Every X is A\n    Every X is B    No   B is X        Some  B is not X    Some X is not B\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n    Some  A is B    Some A is not B    Some  B is not A    Some A is not B\n\nWe may see how it arises that one of the partial syllogisms is not\nimmediately derived, like the others, from a universal one. In the\npreceding, AEE may be considered as derived from AAA, by changing the\nterm in which X enters universally into its contrary. If this be done\nwith the other term instead, we have\n\n No    A is X\u2502from which universal premises we cannot deduce a universal\n             \u2502  conclusion, but only Some B is not A.\n\n Every X is B\u2502                            \u201e\n\nIf we weaken one and the other of these premises, as they stand, we\nobtain\n\n                  Some  A is not X     No   A is X\n                  Every X is B     and Some X is B\n                  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014     \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                  No conclusion        Some B is not A\n\nequivalent to the fourth of the preceding: but if we convert the first\npremiss, and proceed in the same manner,\n\n                 No    X is A               Some  X is not A\n            From Every X is B     we obtain Every X is B\n                 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014           \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                 Some  B is not A           Some  B is not A\n\nwhich is legitimate, and is the same as the last of the preceding list,\nwith A and B interchanged.\n\nBefore proceeding to shew that all the usual forms are contained in the\npreceding, let the reader remark the following rules, which may be\nproved either by collecting them from the preceding cases, or by\nindependent reasoning.\n\n1. The middle term must enter universally into one or the other premiss.\nIf it were not so, the one premiss might speak of one part of the middle\nterm, and the other of the other; so that there would, in fact, be no\nmiddle term. Thus, \u2018Every A is X, Every B is X,\u2019 gives no conclusion: it\nmay be thus stated;\n\n                 All the As make up _a part_ of the Xs\n                 All the Bs make up _a part_ of the Xs\n\nAnd, before we can know that there is any common term of comparison at\nall, we must have some means of shewing that the two parts are the same;\nor the preceding premises by themselves are inconclusive.\n\n2. No term must enter the conclusion more generally than it is found in\nthe premises; thus, if A be spoken of partially in the premises, it must\nenter partially into the conclusion. This is obvious, since the\nconclusion must assert no more than the premises imply.\n\n3. From premises both negative no conclusion can be drawn. For it is\nobvious, that the mere assertion of disagreement between each of two\nthings and a third, can be no reason for inferring either agreement or\ndisagreement between these two things. It will not be difficult to\nreduce any case which falls under this rule to a breach of the first\nrule: thus, No A is X, No B is X, gives\n\n                 Every A is (something which is not X)\n                 Every B is (something which is not X)\n\nin which the middle term is not spoken of universally in either. Again,\n\u2018No X is A, Some X is not B,\u2019 may be converted into\n\n                  Every A is (a thing which is not X)\n                  Some (thing which is not B) is X\n\nin which there is no middle term.\n\n4. From premises both particular no conclusion can be drawn. This is\nsufficiently obvious when the first or second rule is broken, as in\n\u2018Some A is X, Some B is X.\u2019 But it is not immediately obvious when the\nmiddle term enters one of the premises universally. The following\nreasoning will serve for exercise in the preceding results. Since both\npremises are particular in form, the middle term can only enter one of\nthem universally by being the predicate of a negative proposition;\nconsequently (Rule 3) the other premiss must be affirmative, and, being\nparticular, neither of its terms is universal. Consequently both the\nterms as to which the conclusion is to be drawn enter partially, and the\nconclusion (Rule 2) can only be a particular _affirmative_ proposition.\nBut if one of the premises be negative, the conclusion must be\n_negative_ (as we shall immediately see). This contradiction shews that\nthe supposition of particular premises producing a legitimate result is\ninadmissible.\n\n5. If one premiss be negative, the conclusion, if any, must be negative.\nIf one term agree with a second and disagree with a third, no agreement\ncan be inferred between the second and third.\n\n6. If one premiss be particular, the conclusion must be particular. This\nis not very obvious, since the middle term may be universally spoken of\nin a particular proposition, as in Some B is not X. But this requires\none negative proposition, whence (Rule 3) the other must be affirmative.\nAgain, since the conclusion must be negative (Rule 5) its predicate is\nspoken of universally, and, therefore, must enter universally; the other\nterm A must enter, then, in a universal affirmative proposition, which\nis against the supposition.\n\nIn the preceding set of syllogisms we observe one form only which\nproduces A, or E, or I, but three which produce O.\n\nLet an assertion be said to be weakened when it is reduced from\nuniversal to particular, and strengthened in the contrary case. Thus,\n\u2018Every A is B\u2019 is called stronger than \u2018Some A is B.\u2019\n\nEvery form of syllogism which can give a legitimate result is either one\nof the preceding six, or another formed from one of the six, either by\nchanging one of the assertions into its converse, if that be allowable,\nor by strengthening one of the premises without altering the conclusion,\nor both. Thus,\n\n          Some A is X         may be written   Some X is A\n          Every X is B              \u201e          Every X is B\n\n          What follows will still follow from  _Every_ X is A\n                           \u201e                   Every X is B\n\nfor all which is true when \u2018Some X is A,\u2019 is not less true when \u2018Every X\nis A.\u2019\n\nIt would be possible also to form a legitimate syllogism by weakening\nthe conclusion, when it is universal, since that which is true of all is\ntrue of some. Thus, \u2018Every A is X, Every X is B,\u2019 which yields \u2018Every A\nis B,\u2019 also yields \u2018Some A is B.\u2019 But writers on logic have always\nconsidered these syllogisms as useless, conceiving it better to draw\nfrom any premises their strongest conclusion. In this they were\nundoubtedly right; and the only question is, whether it would not have\nbeen advisable to make the premises as weak as possible, and not to\nadmit any syllogisms in which more appeared than was absolutely\nnecessary to the conclusion. If such had been the practice, then\n\n           Every X is A, Every X is B, therefore Some A is B\n\nwould have been considered as formed by a spurious and unnecessary\nexcess of assertion. The minimum of assertion would be contained in\neither of the following,\n\n           Every X is A, Some  X is B, therefore Some A is B\n           Some  X is A, Every X is B, therefore Some A is B\n\nIn this tract, syllogisms have been divided into two classes: first,\nthose which prove a universal conclusion; secondly, those which prove a\npartial conclusion, and which are (all but one) derived from the first\nby weakening one of the premises, in such manner as to produce a\nlegitimate but weakened conclusion. Those of the first class are placed\nin the first column, and the other in the second.\n\n                   Universal.           Particular.\n\n                A Every A is X        Some A is X     I\n                A Every X is B \u2500\u2500\u2500\u2500\u2500\u2500 Every X is B    A\n                  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014        \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                A Every A is B        Some A is B     I\n\n                                      Some A is X     I\n                                      No X is B       E\n\n                                    \u250c \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                A Every A is X      \u2502 Some A is not B O\n                E No X is B    \u2500\u2500\u2500\u2500\u2500\u253c Every A is X    A\n                  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014      \u2502\n                E No A is B         \u2502 Some B is not X O\n                                    \u2514 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                                      Some B is not A O\n\n                                      Every X is A    A\n                     ......           Some X is not B O\n                                      \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                                      Some A is not B O\n\nIn all works on logic, it is customary to write that premiss first which\ncontains the predicate of the conclusion. Thus,\n\n          Every X is B                           Every A is X\n          Every A is X would be written, and not Every X is B\n          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                           \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n          Every A is B                           Every A is B\n\nThe premises thus arranged are called major and minor; the predicate of\nthe conclusion being called the major term, and its subject the minor.\nAgain, in the preceding case we see the various subjects coming in the\norder X, B; A, X; A, B: and the number of different orders which can\nappear is four, namely\u2014\n\n                              XB BX XB BX\n                              AX AX XA XA\n                              \u2014\u2014 \u2014\u2014 \u2014\u2014 \u2014\u2014\n                              AB AB AB AB\n\nwhich are called the four figures, and every kind of syllogism in each\nfigure is called a mood. I now put down the various moods of each\nfigure, the letters of which will be a guide to find out those of the\npreceding list from which they are derived. Co means that a premiss of\nthe preceding list has been converted; + that it has been strengthened;\nCo +, that both changes have taken place. Thus,\n\n             A Every X is B         A Every X is B\n             I Some  A is X becomes A Every X is A: (Co +)\n               \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014           \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n             I Some  A is B         I Some  A is B\n\nAnd Co + abbreviates the following: If some A be X, then some X is A\n(Co); and all that is true when Some X is A, is true when Every X is A\n(+); therefore the second is legitimate, if the first be so.\n\n                            _First Figure._\n\n       A Every X is B                        A Every X is B\n       A Every A is X                        I Some  A is X\n         \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       A Every A is B                        I Some  A is B\n\n       E No    X is B                        E No    X is B\n       A Every A is X                        I Some  A is X\n         \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       E No    A is B                        O Some  A is not B\n\n\n                            _Second Figure._\n\n       E No    B is X (Co)                   E No    B is X (Co)\n       A Every A is X                        I Some  A is X\n         \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       E No    A is B                        O Some  A is not B\n\n       A Every B is X                        A Every B is X\n       E No    A is X (Co)                   O Some  A is not X\n         \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       E No    A is B                        O Some  A is not B\n\n\n                            _Third Figure._\n\n       A Every X is B                        E No    X is B\n       A Every X is A (Co+)                  A Every X is A (Co+)\n         \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       I Some  A is B                        O Some  A is not B\n\n       I Some  X is B (Co)                   O Some  X is not B\n       A Every X is A                        A Every X is A\n         \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       I Some  A is B                        O Some  A is not B\n\n       A Every X is B                        E No    X is B\n       I Some  X is A (Co)                   I Some  X is A (Co)\n         \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       I Some  A is B                        O Some  A is not B\n\n\n                            _Fourth Figure._\n\n       A Every B is X (+)                    I Some  B is X\n       A Every X is A                        A Every X is A\n         \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       I Some  A is B                        I Some  B is A\n\n       A Every B is X                        E No    B is X (Co)\n       E No    X is A                        A Every X is A (Co+)\n         \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       E No    A is B                        O Some  A is not B\n\n                          E No   B is X (Co)\n                          I Some X is A (Co)\n                            \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                          O Some A is not B\n\nThe above is the ancient method of dividing syllogisms; but, for the\npresent purpose, it will be sufficient to consider the six from which\nthe rest can be obtained. And since some of the six have A in the\npredicate of the conclusion, and not B, we shall join to them the six\nother syllogisms which are found by transposing B and A. The complete\nlist, therefore, of syllogisms with the weakest premises and the\nstrongest conclusions, in which a comparison of A and B is obtained by\ncomparison of both with X, is as follows:\n\n    Every A is X  Every B is X \u2502Some  A is X      Some  B is X\n    Every X is B  Every X is A \u2502No    X is B      No    X is A\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2502\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n    Every A is B  Every B is A \u2502Some  A is not B  Some  B is not A\n                               \u2502\n    Every A is X  Every B is X \u2502Every A is X      Every B is X\n    No    X is B  No    X is A \u2502Some  B is not X  Some  A is not X\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2502\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n    No    A is B  No    B is A \u2502Some  B is not A  Some  A is not B\n                               \u2502\n    Some  A is X  Some  B is X \u2502Every X is A      Every X is B\n    Every X is B  Every X is A \u2502Some  X is not B  Some  X is not A\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2502\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n    Some  A is B  Some  B is A \u2502Some  A is not B  Some  B is not A\n\nIn the list of page 19, there was nothing but recapitulation of forms,\neach form admitting a variation by interchanging A and B. This\ninterchange having been made, and the results collected as above, if we\ntake every case in which B is the predicate, or can be made the\npredicate by allowable conversion, we have a collection of all possible\n_weakest_ forms in which the result is one of the four \u2018Every A is B,\u2019\n\u2018No A is B,\u2019 \u2018Some A is B,\u2019 \u2018Some A is not B\u2019; as follows. The premises\nare written in what appeared the most natural order, without distinction\nof major or minor; and the letters prefixed are according to the forms\nof the several premises, as in page 10.\n\n                           A Every A is X\n                           U Every X is B\n                             \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                           A Every A is B\n\n                   I Some  A is X     I Some  B is X\n                   U Every X is B     U Every X is A\n                     \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014       \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                   I Some  A is B     I Some  A is B\n\n                   A Every A is X     A Every B is X\n                   E No    B is X     E No    A is X\n                     \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014       \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                   E No    A is B     E No    A is B\n\n        I Some  A is X     A Every B is X     U Every X is A\n        E No    B is X     O Some  A is not X Y Some  X is not B\n          \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014   \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014   \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n        O Some  A is not B O Some  A is not B O Some  A is not B\n\nEvery assertion which can be made upon two things by comparison with any\nthird, that is, every simple inference, can be reduced to one of the\npreceding forms. Generally speaking, one of the premises is omitted, as\nobvious from the conclusion; that is, one premiss being named and the\nconclusion, that premiss is implied which is necessary to make the\nconclusion good. Thus, if I say, \u201cThat race must have possessed some of\nthe arts of life, for they came from Asia,\u201d it is obviously meant to be\nasserted, that all races coming from Asia must have possessed some of\nthe arts of life. The preceding is then a syllogism, as follows:\n\n         \u2018That race\u2019 is \u2018a race of Asiatic origin:\u2019\n         Every \u2018race of Asiatic origin\u2019 is \u2018a race which must have\n            possessed some of the arts of life:\u2019\n Therefore, That race _is_ a race which must have possessed some of the\n    arts of life.\n\nA person who makes the preceding assertion either means to imply,\nantecedently to the conclusion, that all Asiatic races must have\npossessed arts, or he talks nonsense if he asserts the conclusion\npositively. \u2018A must be B, for it is X,\u2019 can only be true when \u2018Every X\nis B.\u2019 This latter proposition may be called the suppressed premiss; and\nit is in such suppressed propositions that the greatest danger of error\nlies. It is also in such propositions that men convey opinions which\nthey would not willingly express. Thus, the honest witness who said, \u2018I\nalways thought him a respectable man\u2014he kept his gig,\u2019 would probably\nnot have admitted in direct terms, \u2018Every man who keeps a gig must be\nrespectable.\u2019\n\n\nI shall now give a few detached illustrations of what precedes.\n\n\u201cHis imbecility of character might have been inferred from his proneness\nto favourites; for all weak princes have this failing.\u201d The preceding\nwould stand very well in a history, and many would pass it over as\ncontaining very good inference. Written, however, in the form of a\nsyllogism, it is,\n\n                     All weak princes are prone to favourites\n                            He        was prone to favourites\n                     \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n           Therefore        He        was a weak prince\n\nwhich is palpably wrong. (Rule 1.) The writer of such a sentence as the\npreceding might have meant to say, \u2018for all who have this failing are\nweak princes;\u2019 in which case he would have inferred rightly. Every one\nshould be aware that there is much false inference arising out of\nbadness of style, which is just as injurious to the habits of the\nuntrained reader as if the errors were mistakes of logic in the mind of\nthe writer.\n\n\u2018A is less than B; B is less than C: therefore A is less than C.\u2019 This,\nat first sight, appears to be a syllogism; but, on reducing it to the\nusual form, we find it to be,\n\n                          A is (a magnitude less than B)\n                          B is (a magnitude less than C)\n                Therefore A is (a magnitude less than C)\n\nwhich is not a syllogism, since there is no middle term. Evident as the\npreceding is, the following additional proposition must be formed before\nit can be made explicitly logical. \u2018If B be a magnitude less than C,\nthen every magnitude less than B is also less than C.\u2019 There is, then,\nbefore the preceding can be reduced to a syllogistic form, the necessity\nof a deduction from the second premiss, and the substitution of the\nresult instead of that premiss. Thus,\n\n                     A is less than B\n           Less than B is less than C: following from B is less than C.\n           \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n Therefore           A is less than C\n\nBut, if the additional argument be examined\u2014namely, if B be less than C,\nthen that which is less than B is less than C\u2014it will be found to\nrequire precisely the same considerations repeated; for the original\ninference was nothing more. In fact, it may easily be seen as follows,\nthat the proposition before us involves more than any simple syllogism\ncan express. When we say that A is less than B, we say that if A were\napplied to B, every part of A would match a part of B, and there would\nbe parts of B remaining over. But when we say, \u2018Every A is B,\u2019 meaning\nthe premiss of a common syllogism, we say that every instance of A is an\ninstance of B, without saying any thing as to whether there are or are\nnot instances of B still left, after those which are also A are taken\naway. If, then, we wish to write an ordinary syllogism in a manner which\nshall correspond with \u2018A is less than B, B is less than C, therefore A\nis less than C,\u2019 we must introduce a more definite amount of assertion\nthan was made in the preceding forms. Thus,\n\n                 Every A is B, and there are Bs which are not As\n                 Every B is C, and there are Cs which are not Bs\n                 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n       Therefore Every A is C, and there are Cs which are not As\n\nOr thus:\n\n                  The Bs contain all the As, and more\n                  The Cs contain all the Bs, and more\n                  \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                  The Cs contain all the As, and more\n\nThe most technical form, however, is,\n\n                From    Every A is B; [Some B is not A]\n                        Every B is C; [Some C is not B]\n                Follows Every A is C; [Some C is not A]\n\nThis sort of argument is called _\u00e0 fortiori_ argument, because the\npremises are more than sufficient to prove the conclusion, and the\nextent of the conclusion is thereby greater than its mere form would\nindicate. Thus, \u2018A is less than B, B is less than C, therefore, _\u00e0\nfortiori_, A is less than C,\u2019 means that the extent to which A is less\nthan C must be greater than that to which A is less than B, or B than C.\nIn the syllogism last written, either of the bracketed premises might be\nstruck out without destroying the conclusion; which last would, however,\nbe weakened. As it stands, then, the part of the conclusion, \u2018Some C is\nnot A,\u2019 follows it _\u00e0 fortiori_.\n\nThe argument _\u00e0 fortiori_, may then be defined as a universally\naffirmative syllogism, in which both of the premises are shewn to be\nless than the whole truth, or greater. Thus, in \u2018Every A is X, Every X\nis B, therefore Every A is B,\u2019 we do not certainly imply that there are\nmore Xs than As, or more Bs than Xs, so that we do not know that there\nare more Bs than As. But if we are at liberty to state the syllogism as\nfollows,\n\n           All the As make up part (and part only) of the Xs\n           Every X is B;\n\nthen we are certain that\n\n           All the As make up part (and part only) of the Bs.\n\nBut if we are at liberty further to say that\n\n           All the As make up part (and part only) of the Xs\n           All the Xs make up part (and part only) of the Bs\n\nthen we conclude that\n\n           All the As make up _part of part_ (only) of the Bs\n\nand the words in Italics mark that quality of the conclusion from which\nthe argument is called _\u00e0 fortiori_.\n\nMost syllogisms which give an affirmative conclusion are generally meant\nto imply _\u00e0 fortiori_ arguments, except only in mathematics. It is\nseldom, except in the exact sciences, that we meet with a proposition,\n\u2018Every A is B,\u2019 which we cannot immediately couple with \u2018Some Bs are not\nAs.\u2019\n\nWhen an argument is completely established, with the exception of one\nassertion only, so that the inference may be drawn as soon as that one\nassertion is established, the result is stated in a form which bears the\nname of an _hypothetical_ syllogism. The word hypothesis means nothing\nbut supposition; and the species of syllogism just mentioned first lays\ndown the assertion that a consequence will be true if a certain\ncondition be fulfilled, and then either asserts the fulfilment of the\ncondition, and thence the consequence, or else denies the consequence,\nand thence denies the fulfilment of the condition. Thus, if we know that\n\n                  When A is B, it follows that P is Q;\n\nthen, as soon as we can ascertain that A is B, we can conclude that P is\nQ; or, if we can shew that P is not Q, we know that A is not B. But if\nwe find that A is not B, we can infer nothing; for the preceding does\nnot assert that P is Q _only_ when A is B. And if we find out that P is\nQ, we can infer nothing. This conditional syllogism may be converted\ninto an ordinary syllogism, as follows. Let K be any \u2018case in which A is\nB,\u2019 and Z a \u2018case in which P is Q\u2019; then the preceding assertion amounts\nto \u2018Every K is Z.\u2019 Let L be a particular instance, the A of which may or\nmay not be B. If A be B in the instance under discussion, or if A be not\nB, we have, in the one case and the other,\n\n                        Every K is Z   Every K is Z\n                              L is a K       L is not a K\n                              \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014       \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n              Therefore       L is a Z       No conclusion\n\nSimilarly, according as a particular case (M) is or is not Z, we have\n\n                   Every     K is Z   Every K is Z\n                             M is a Z       M is not a Z\n                   \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014            \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n                   No conclusion            M is not a K\n\nThat is to say: The assertion of an hypothesis is the assertion of its\nnecessary consequence, and the denial of the necessary consequence is\nthe denial of the hypothesis; but the assertion of the necessary\nconsequence gives no right to assert the hypothesis, nor does the denial\nof the hypothesis give any right to deny the truth of that which would\n(were the hypothesis true) be its necessary consequence.\n\n\nDemonstration is of two kinds: which arises from this, that every\nproposition has a contradictory; and of these two, one must be true and\nthe other must be false. We may then either prove a proposition to be\ntrue, or its contradictory to be false. \u2018It is true that Every A is B,\u2019\nand, \u2018it is false that there are some As which are not Bs,\u2019 are the same\nproposition; and the proof of either is called the indirect proof of the\nother.\n\nBut how is any proposition to be proved false, except by proving a\ncontradiction to be true? By proving a necessary consequence of the\nproposition to be false. But this is not a complete answer, since it\ninvolves the necessity of doing the same thing; or, so far as this\nanswer goes, one proposition cannot be proved false unless by proving\nanother to be false. But it may happen, that a necessary consequence can\nbe obtained which is obviously and self-evidently false, in which case\nno further proof of the falsehood of the hypothesis is necessary. Thus\nthe proof which Euclid gives that all equiangular triangles are\nequilateral is of the following structure, logically considered.\n\n(1.) If there be an equiangular triangle not equilateral, it follows\nthat a whole can be found which is not greater than its part.[1]\n\nFootnote 1:\n\n  This is the proposition in proof of which nearly the whole of the\n  demonstration of Euclid is spent.\n\n(2.) It is false that there can be any whole which is not greater than\nits part (self evident).\n\n(3.) Therefore it is false that there is any equiangular triangle which\nis not equilateral; or all equiangular triangles are equilateral.\n\nWhen a proposition is established by proving the truth of the matters it\ncontains, the demonstration is called _direct_; when by proving the\nfalsehood of every contradictory proposition, it is called _indirect_.\nThe latter species of demonstration is as logical as the former, but not\nof so simple a kind; whence it is desirable to use the former whenever\nit can be obtained.\n\nThe use of indirect demonstration in the Elements of Euclid is almost\nentirely confined to those propositions in which the converses of simple\npropositions are proved. It frequently happens that an established\nassertion of the form\n\n                 Every A is B                     (1)\n\nmay be easily made the means of deducing,\n\n                 Every (thing not A) is not B     (2)\n\nwhich last gives\n\n                 Every B is A                     (3)\n\nThe conversion of the second proposition into the third is usually made\nby an indirect demonstration, in the following manner. If possible, let\nthere be one B which is not A, (2) being true. Then there is one thing\nwhich is not A and is B; but every thing not A is not B; therefore there\nis one thing which is B and is not B: which is absurd. It is then absurd\nthat there should be one single B which is not A; or, Every B is A.\n\nThe following proposition contains a method which is of frequent use.\n\nHYPOTHESIS.\u2014Let there be any number of propositions or assertions,\u2014three\nfor instance, A, B, and C,\u2014of which it is the property that one or the\nother must be true, _and one only_. Let there be three other\npropositions, P, Q, and R, of which it is also the property that one,\nand one only, must be true. Let it also be a connexion of those\nassertions, that\n\n                       When A is true, P is true\n                       When B is true, Q is true\n                       When C is true, R is true\n\nCONSEQUENCE: then it follows that\n\n                       When P is true, A is true\n                       When Q is true, B is true\n                       When R is true, C is true\n\nFor, when P is true, then Q and R must be false; consequently, neither B\nnor C can be true, for then Q or R would be true. But either A, B, or C\nmust be true, therefore A must be true; or, when P is true, A is true.\nIn a similar way the remaining assertions may be proved.\n\n      Case 1. If   \u2502When P is Q,              A is B\n                   \u2502When P is not Q,          A is not B\n    It follows that\u2502When A is B,              P is Q\n                   \u2502When A is not B,          P is not Q\n\n      Case 2. If   \u2502When A is greater than B, P is greater than Q\n           \u201e       \u2502When A is equal to B,     P is equal to Q\n           \u201e       \u2502When A is less than B,    P is less than Q\n\n    It follows that\u2502When P is greater than Q, A is greater than B\n           \u201e       \u2502When P is equal to Q,     A is equal to B\n           \u201e       \u2502When P is less than Q,    A is less than B\n\n                  *       *       *       *       *\n\nWe have hitherto supposed that the premises are actually true; and, in\nsuch a case, the logical conclusion is as certain as the premises. It\nremains to say a few words upon the case in which the premises are\nprobably, but not certainly, true.\n\nThe probability of an event being about to happen, and that of an\nargument being true, may be so connected that the usual method of\nmeasuring the first may be made to give an easy method of expressing the\nsecond. Suppose an urn, or lottery, with a large number of balls, black\nor white; then, if there be twelve white balls to one black, we say it\nis twelve to one that a white ball will be drawn, or that a white ball\nis twelve times as probable as a black one. A certain assertion may be\nin the same condition as to the force of probability with which it\nstrikes the mind: that is, the questions\n\n                      Is the assertion true?\n                      Will a white ball be drawn?\n\nmay be such that the answer, \u2018most probably,\u2019 expresses the same degree\nof likelihood in both cases.\n\nWe have before explained that logic has nothing to do with the truth or\nfalsehood of assertions, but only professes, supposing them true, to\ncollect and classify the legitimate methods of drawing inferences.\nSimilarly, in this part of the subject, we do not trouble ourselves with\nthe question, How are we to find the probability due to premises? but we\nask: Supposing (happen how it may) that we _have_ found the probability\nof the premises, required the probability of the conclusion. When the\nodds in favour of a conclusion are, say 6 to 1, there are, out of every\n7 possible chances, 6 in favour of the conclusion, and 1 against it.\nHence \u2076\u2044\u2087 and \u2150 will represent the proportions, for and against, of all\nthe possible cases which exist.\n\nThus we have the succession of such results as in the following table:\u2014\n\n     Odds in favour of an event Probability for Probability against\n\n               1 to 1                  \u00bd                 \u00bd\n               2 to 1                  \u2154                 \u2153\n               3 to 1                  \u00be                 \u00bc\n               3 to 2                  \u2157                 \u2156\n               4 to 1                  \u2158                 \u2155\n               4 to 3                 \u2074\u2044\u2087               \u00b3\u2044\u2087\n               5 to 1                  \u215a                 \u2159\n                &c.                   &c.               &c.\n\nLet the probability of a conclusion, as derived from the premises (that\nis on the supposition that it was never imagined to be possible till the\nargument was heard), be called the _intrinsic probability_ of the\nargument. This is found by multiplying together the probabilities of all\nthe assertions which are necessary to the argument. Thus, suppose that a\nconclusion was held to be impossible until an argument of a single\nsyllogism was produced, the premises of which have severally five to one\nand eight to one in their favour. Then \u215a \u00d7 \u2078\u2044\u2089, or \u2074\u2070\u2044\u2085\u2084, is the\nintrinsic probability of the argument, and the odds in its favour are 40\nto 14, or 20 to 7.\n\nBut this intrinsic probability is not always that of the conclusion; the\nlatter, of course, depending in some degree on the likelihood which the\nconclusion was supposed to have before the argument was produced. A\nsyllogism of 20 to 7 in its favour, advanced in favour of a conclusion\nwhich was beforehand as likely as not, produces a much more probable\nresult than if the conclusion had been thought absolutely false until\nthe argument produced a certain belief in the possibility of its being\ntrue. The change made in the probability of a conclusion by the\nintroduction of an argument (or of a new argument, if some have already\npreceded) is found by the following rule.\n\nFrom the sum of the existing probability of the conclusion and the\nintrinsic probability of the new argument, take their product; the\nremainder is the probability of the conclusion, as reinforced by the\nargument. Thus, _a + b \u2212 ab_ is the probability of the truth of a\nconclusion after the introduction of an argument of the intrinsic\nprobability _b_, the previous probability of the said conclusion having\nbeen _a_.\n\nThus, a conclusion which has at present the chance \u2154 in its favour, when\nreinforced by an argument whose intrinsic probability is \u00be, acquires the\nprobability \u2154 + \u00be \u2212 \u2154 \u00d7 \u00be or, \u2154 + \u00be \u2212 \u00bd, or \u00b9\u00b9\u2044\u2081\u2082; or, having 2 to 1 in\nits favour before, it has 11 to 1 in its favour after, the argument.\n\nWhen the conclusion was neither likely nor unlikely beforehand (or had\nthe probability \u00bd), the shortest way of applying the preceding rule (in\nwhich _a + b \u2212 ab_ becomes \u00bd + \u00bd_b_) is to divide the sum of the\nnumerator and denominator of the intrinsic probability of the argument\nby twice the denominator. Thus, an argument of which the intrinsic\nprobability is \u00be, gives to a conclusion on which no bias previously\nexisted, the probability \u215e or (3 + 4)/(2 \u00d7 4).\n\n\n                                THE END.\n\n\n                    LONDON:\u2014PRINTED BY JAMES MOYES,\n                    Castle Street, Leicester Square.\n\n------------------------------------------------------------------------\n\n\n\n\n                          TRANSCRIBER\u2019S NOTES\n\n\n 1. Silently corrected obvious typographical errors and variations in\n      spelling.\n 2. Retained archaic, non-standard, and uncertain spellings as printed.\n 3. Enclosed italics font in _underscores_.","title":"First notions of logic (preparatory to the study of geometry)","tradition":"philosophy-ethics","translated_text":null,"word_count":10673}