^L ^^W-^^L 
 
 pr^r-^ 
 
' A generation will arise in which the leaders of education will know the 
 value of logic, the value of mathematics, the value of logic in mathematics, 
 and the value of mathematics in logic. ' DE MORGAN, Syllabus, p. 44. 
 
 ' Shall we then err in regarding that as the true science of Logic, 
 which, laying down certain elementary laws, confirmed by the very testi- 
 mony of the mind, permits us thence to deduce, by uniform processes, the 
 entire chain of its secondary consequences, and furnishes for its practical 
 application methods of perfect generality. Let it be considered whether 
 in any science, viewed either as a system of truth or as the foundation of a 
 practical art, there can properly be any other test of the completeness and 
 fundamental characters of its laws, than the completeness of its system of 
 derived truths, and the generality of the methods which it serves to 
 establish.' BOOLE, Laws of Thought, p. 5. 
 
 ' It is curious to compare the properties of these quaternion sym- 
 bols with those of the Elective Symbols "of Logic, as given in Boole's 
 wonderful treatise on the Laws of Thought ; and to think that the same 
 grand science of mathematical analysis, by processes remarkably similar to 
 each other, reveals to us truths in the science of position far beyond the 
 powers of the geometer, and truths of deductive reasoning to which unaided 
 thought could never have led the logician. 'PROFESSOR TAIT, Quaternions, 
 p. 50. 
 
PRINCIPLES 
 
 OF THE 
 
 ALGEBRA OF LOGIC 
 
 WITH EXAMPLES 
 
 BY 
 
 ALEXANDER MACFARLANE 
 
 M.A., D.Sc. (EoiN.), F.R.S.E. 
 
 READ BEFORE THE ROYAL SOCIETY OF EDINBURGH 
 i6tk DECEMBER 1878 AND zotk JANUARY 1879 
 
 EDINBURGH : DAVID DOUGLAS 
 
 1879 
 
 [All rights reserved.} 
 
V 
 
 Edinburgh: Printed by Thomas and Archibald Constable 
 
 FOR 
 
 DAVID DOUGLAS. 
 
 LONDON HAMILTON, ADAMS, AND CO. 
 
 CAMBRIDGE MACMILLAN AND CO. 
 
 GLASGOW JAMES MACLEHOSE. 
 
TO THE 
 
 REV. PHILIP KELLAND, M.A., F.R.S. 
 
 PROFESSOR OF MATHEMATICS IN EDINBURGH UNIVERSITY 
 
 of tfje 3&oga1. Soctetg of 
 
 THIS WORK IS DEDICATED 
 
 AS A MARK OF RESPECT 
 
 BY 
 
 A FORMER PUPIL. 
 
 515055 
 
PREFACE. 
 
 THESE ' Principles ' were originally contributed to 
 the Royal Society of Edinburgh in a Memoir received 
 by the Secretary Qth October 1878, and in a supple- 
 mentary paper received 5th November. I had the 
 honour of reading an Abstract before the Society at 
 the meetings of i6th December and 2Oth January. 
 In the interval between the 5th November and the 
 present time I have improved several of the demon- 
 strations, introduced illustrative matter, and prepared 
 the collection of examples. The work, in its present 
 state, forms an elementary treatise on the science of 
 Formal Reasoning. 
 
 I consider it proper to state that the theory of the 
 operation of the mind in reasoning about Quality, 
 which is advanced in this work, occurred to me five 
 years ago ; and that I have directed towards its 
 development the whole of my subsequent study of 
 the Mathematical, Physical, and Natural Sciences, 
 which are embraced in the curriculum for the degree 
 of Doctor of Science (Mathematics) at the University 
 of Edinburgh. 
 
 ALEXANDER MACFARLANE. 
 
 EDINBURGH, 23^ 'January 1879. 
 
CONTENTS. 
 
 PAGE 
 
 I. The Science of Formal Logic an Algebra, . . i 
 
 II. Universe and Character, .... 5 
 
 III. The sign = , ...... 15 
 
 IV. The signs + and -, ..... 17 
 V. The signs x and -r , . . . . .20 
 
 VI. Rule of Signs, . .... 25 
 
 VII. Integral Symbols, . . . . .26 
 
 VIII. On the Equation as expressing a general proposition, 28 
 
 IX. The principle of Identity and the Axioms of Im- 
 mediate Inference, . . .36 
 X. Axioms of Mediate Inference, ... 40 
 XI. Conditions for a Character being Single, . . 42 
 XII. The signs of inequivalence > and < , . . 53 
 
 XIII. Division, ...... 54 
 
 XIV. Expansion of a function of a number of independent 
 
 symbols in terms of the primary parts into which 
 
 the universe is divided by the symbols, . . 61 
 
 XV. Definition, ...... 69 
 
 XVI. Inference from one or more equations of the form 
 
 x=tn (Categorical), ... 7 
 XVII. Inference from one or more equations of the form 
 
 xy=m (Hypothetical), 81 
 
 XVIII. On certain forms of the disjunctive equation, . 106 
 
 XIX. The Aristotelian forms of inference, . . .113 
 
x Contents. 
 
 PAGE 
 
 XX. On Probability, . . .118 
 
 XXI. Fundamental relations between the single functions 
 
 of a number of independent characters, . . 122 
 
 XXII. General Method of deducing a conclusion of a required 
 
 form from given data, . . . .124 
 
 XXIII. On Boole's General Method, . . . .131 
 
 Examples, . . . . . 135 
 
ALGEBRA OF LOGIC. 
 
I. THE SCIENCE OF FORMAL LOGIC AN 
 ALGEBRA. 
 
 1. THOUGH it is evident a priori to one who reflects on 
 the matter, that the theory of Necessity and the theory of 
 Probability are the complementary parts of one whole, it 
 is nevertheless true that the foundations of the general 
 science, of which they form the parts, were not laid until 
 quite recent times. The merit of conceiving and under- 
 taking this important unification is due in some measure 
 to De Morgan, but principally to Boole. 
 
 2. That the science of inference is capable of being 
 treated analytically, may be inferred from the fact that the 
 ordinary rules about Conversion and Syllogism are estab- 
 lished by a comparison of circles, taken to represent the 
 terms of the propositions considered. In one of the best 
 modern manuals of Logic, it is stated that the testing 
 whether a given combination of premises leads to a valid 
 inference, and the proof of the validity or invalidity, must 
 depend on the comparison of the spheres, within which, 
 according to the premises, the notions under consideration 
 find application ; and that these spheres are made apparent 
 to the senses by geometrical figures (especially by circles) 
 whose reciprocal relations agree with the relations of the 
 spheres of the notions to each other in all relations essential 
 for derronstration. ( Ueberweg's Logic, translated by Pro- 
 fessor Lindsay, p. 379.) The introduction of these dia- 
 grams is commonly attributed to Euler. 
 
 A 
 
2 The Science of 
 
 3. Corresponding to this graphical method, which con- 
 sists in the use of diagrams, there is an analytical method, 
 which consists in the use of symbols. The relative advan- 
 tages and disadvantages of the two, when applied to Quality, 
 are precisely the same as when applied to Quantity. The 
 diagram exhibits an individual case of the given data with 
 all the clearness of the concrete ; on the other hand, the 
 analytical expression separates the essential relations from 
 the accidental, with which they must be mixed up in any 
 individual example. 
 
 4. The reason why the operations of Boole's calculus 
 appear mysterious and its employment difficult, is, that the 
 calculus is not founded upon a sufficient theory of the 
 operation of the mind in reasoning about Quality. That it 
 is not all that a Logical organon ought to be, is evident 
 from what Venn says in Mind, vol. i. p. 484 : ' The dis- 
 tinctive characteristic of Boole's system is the boldness, 
 not to say audacity, with which he carries on his processes 
 through stages which have no logical or other signification 
 whatever, that is,, which admit of no possible interpretation 
 provided only they terminate in an interpretable result/ 
 Boole himself claims nothing higher for his calculus. He 
 would, however, have objected to the statement which 
 Professor Jevons makes (Principles of Science, p. 71), that 
 ' Boole imported the conditions of number into the science 
 of Logic, and produced a system which, though wonderful 
 in its results, was not a system of logic at all.' 
 
 5. It is the object of this little work to investigate the 
 foundations of the analytical method of reasoning about 
 Quality, with special reference to the principles laid down 
 by Boole as the basis of his calculus, and to the observations 
 which have been published by various philosophers con- 
 cerning these principles. I bring forward a new theory of 
 the operation of the mind in reasoning about Quality,, which 
 enables me to correct Boole's principles, and place them 
 on a clear rational basis. I endeavour to show that the 
 
Formal Logic an Algebra. 3 
 
 analytical method of reasoning about Quality is an Algebra, 
 which coincides with the Algebra of Quantity when the 
 symbols are integral, but is a generalised form of the 
 latter when the symbols are fractional. The rest of the 
 work is taken up with the investigation of problems by 
 means of this algebraic organon, especially such problems 
 as are suggested by the ordinary Logic. 
 
 6. Logic, as the Algebra of Quality, is a formal science. It 
 investigates the general properties of the symbol of Quality, 
 and by means of these properties deduces equations which 
 are true generally, or combines such equations with data 
 of given forms. It is not its province to consider how a 
 particular form of datum can in any case be asserted to be 
 true that subject of investigation being left to the Transcend- 
 ental Logic ; it is sufficient that examples of such a form 
 occur, or may occur, in the practical or theoretical activities 
 of mankind. 
 
 7. The properties of the symbol of Quality are not laws 
 of thought in the common acceptation of that term. For 
 the properties of the symbol of Quantity, on which the 
 ordinary algebra is founded, are held not to be laws of 
 thought, but to refer to the actual constitution of things ; 
 and there is no difference in the two methods, when 
 developed, which indicates the existence of such a dis- 
 tinction. If the basis of the science of Quality is subjec- 
 tive, it is so only in the same sense in which the basis of 
 the science of Quantity is subjective. There is ground for 
 believing that the true reason why the former science has 
 remained so stationary is, that there has been too much 
 introspection into the individual mind in the hope of 
 rinding laws of thought there, and too little contemplation 
 of the form and nature of the truths of Science. The 
 logician assumes that all men reason equally well about 
 Quality, fallacies being possible only by a momentary lapse 
 of attention ; but the mathematician never assumes that all 
 men reason equally well about Quantity. 
 
4 The Science of 
 
 8. Boole entitled his great work on reasoning ' An 
 Investigation of the Laws of Thought, on which are 
 founded the Mathematical Theories of Logic and Prob- 
 abilities,' and in several places he says that the Laws in 
 question are subjective in a sense in which the Laws of 
 Quantity are not. He considers 
 
 **=* 
 
 in particular to be a subjective law ; but I have endea- 
 voured to show (Art. 118) that it is a special condition, 
 which the symbol of this Algebra must satisfy in order to 
 be of a particular kind. 
 
 9. No one, I suppose, contends that the properties of the 
 Chemical Symbol, or of the Quaternionic Symbols, are laws 
 of thought. Since the corresponding properties of the 
 different symbols differ greatly among one another, it is 
 surely better in every case to consider the actual constitu- 
 tion of things as suggesting rules for thought to the mind, 
 rather than the mind imposing laws of thought upon itself. 
 
 10. Logic, as the Algebra of Quality, is a true organon. 
 It can determine whether a conclusion of a required form 
 can be deduced from data of given forms ; and if so, what 
 that conclusion is. It can manipulate complex data, as is 
 shown in the examples appended. Bacon's judgment 
 ' Syllogismus ad principia scientiarum non adhibetur, ad 
 media axiomata frustra adhibetur, quum sit subtilitati 
 naturae longe impar' however true of the scholastic ex- 
 position of the syllogism, does not apply to the Algebra of 
 Quality ; for the latter can be made to discover principles, 
 and to imitate to some extent the subtlety of Nature. It 
 may be said (to adapt a remark of De Moivre) that in- 
 numerable questions in the theory of necessary and pro- 
 bable reasoning can be solved without any manner of 
 trouble to the imagination, by the mere force of the notation 
 supplied by this Algebra. 
 
 The Algebra of Quantity is acknowledged to be the 
 weapon for the philosopher who attacks the Experimental 
 
Formal Logic an Algebra. 5 
 
 Sciences; the Algebra of Quality is the weapon for the 
 philosopher who attacks the Sciences of Observation. 
 
 11. Thus viewed, Formal Logic is not the short and dry 
 science which even Kant held it to be. Any one who has 
 studied Boole's Calculus, may well imagine that the theory 
 of reasoning was not completed by Aristotle ; and that so 
 far from any ' System ' of Logic having ever been written, 
 there is still need to consider the foundations. 
 
 II. UNIVERSE AND CHARACTER. 
 
 12. Boole in his analysis of language draws no distinc- 
 tion between Substantive and Adjective ; he considers their 
 function in reasoning to be the same. He says (Laws of 
 Thought, p. 27), 'The substantive proper and the adjective 
 may indeed be regarded as differing only in this respect, 
 that the former -expresses the substantive existence of the 
 individual thing or things to which it refers ; the latter im- 
 plies that existence. If we attach to the adjective the 
 universally understood subject " being" or " thing," it be- 
 comes virtually a substantive, and may for all the essential 
 purposes of reasoning be replaced by the substantive/ 
 Accordingly, he uses the symbol x to denote ' men' or ' good 
 things' or 'white things' or 'horned things,' as the case 
 may be. For instance : he says, if x alone stands for 
 ' white things ' and y for ' sheep,' let xy stand for ' white 
 sheep ;' and in like manner if z stands for ' horned things ' 
 and x and y retain their previous interpretations, let zxy 
 represent ' horned white sheep.' 
 
 13. Again ; when investigating the operations of the 
 mind, which are implied in the use of language as an in- 
 strument of reasoning, he finds no difference in the opera- 
 
6 Universe and Character. 
 
 \ 
 
 tion expressed by a substantive from that expressed by an 
 adjective. He says that there is a universe of discourse; 
 but this universe is not one described by a substantive. 
 ' In every discourse,' he says, ' whether of the mind con- 
 versing with its own thoughts, or of the individual in his 
 intercourse with others, there is an assumed or expressed 
 limit within which the subjects of its operations are con- 
 fined. The most unfettered discourse is that in which the 
 words we use are understood in the widest possible applica- 
 tion, and for them the limits of discourse are co-extensive 
 with those of the universe itself. But more usually we 
 confine ourselves to a less spacious field. Sometimes, in 
 discoursing of men, we imply (without expressing the limita- 
 tion) that it is of men only under certain circumstances and 
 conditions that we speak, as of civilised men, or of men in 
 the vigour of life, or of men under some other condition or 
 relation. Now, whatever may be the extent of the field 
 within which all the objects of our discourse are found, that 
 field may properly be termed the universe of discourse/ 
 Laws of Thought, p. 42. 
 
 14. From the passage just quoted, as well as from many 
 others, it appears that what Boole means by the universe of 
 discourse is not the objects denoted by a Universal Sub- 
 stantive, but a definite part of the whole realm of things 
 a limited portion of the physical universe, with all the 
 entities which are or can be imagined to be in it, whether 
 mental or physical, ponderable or imponderable, atomic or 
 complex. 
 
 15. The substantive 'men' expresses an operation of 
 election from the universe of all the beings to which the 
 term 'men' is applicable; the adjective 'good' in com- 
 bination, as 'good men,' directs us still further to elect 
 mentally from the class of men all those who possess the 
 further quality of good ; and if another adjective were pre- 
 fixed to the combination, it would direct a similar operation 
 upon 'good men.' In short, he supposes that the mind 
 
Universe and Character. 7 
 
 always proceeds along the predicamental line; whereas that 
 is only one mode of its procedure. 
 
 16. In consequence of this analysis, the subjects of 
 thought in Logic and in Arithmetic are said to be perfectly 
 distinct; and it is not of any importance to compare the 
 symbols of logic with the symbols of quantity generally. 
 Attention is directed so exclusively to an Algebra in which 
 the symbols x, y> z, etc., admit indifferently of the values 
 o and i, and of these alone, that some logicians have sup- 
 posed that the symbols can have no other value. 
 
 17. Another consequence of this analysis is, that Boole 
 is obliged to make a new and independent investigation of 
 * Secondary ' Propositions. In the case of Secondary Pro- 
 positions, the proper interpretation of the symbol i is held 
 to be * eternity,' or a part of eternity. The question is sug- 
 gested, whether in the case of Primary Propositions i does 
 not really represent space. He thinks not; because the 
 sign of identity = connecting the members of the corre- 
 sponding equation, implies that the things which they repre- 
 sent are identical, not simply that they are found in the 
 same portion of space. The reason why the symbol i in 
 Secondary Propositions represents not the universe of events, 
 but the eternity in whose successive moments and periods 
 they are evolved, is, that the same sign of identity connect- 
 ing the logical members of the corresponding equations im- 
 plies, not that the events which those members represent 
 are identical, but that the times of their occurrence are the 
 same. Laws of Thought, p. 176. 
 
 18. The principles of the Calculus of Identity become 
 much clearer, and their application greatly facilitated, by 
 taking into consideration the difference of the functions of the 
 Substantive, and of the Adjective used in an attributive sense. 
 The objects expressed by the common noun, or rather 
 universal term of a proposition, constitute the universe 
 of the proposition the actual whole considered by the 
 mind in forming the judgment. The attributive adjectives, 
 
8 Universe and Character. 
 
 whether one or more, which appear in the proposition, refer 
 to that subject, and not to things in general. In thinking 
 of ' sheep that are white and horned/ I do not consider 
 1 white things' or 'horned things.' It is even questionable 
 whether the mind can consider some adjectives as denoting 
 classes of things. Can we consider < small things ' or ' wise 
 things ' or ' primary things '? Boole remarks, with reference 
 to this very attribute ' wise,' that, before denoting it by a 
 symbol, we. must consider whether it is to be used in its 
 absolute sense or only relatively. But * small' has no 
 absolute sense. ' Nothing by itself,' Aristotle lays down in 
 The Categories, ' is described as great or small. A moun- 
 tain, for instance, may be said to be " little," and a millet 
 seed "large," from the fact of the one being greater, and 
 the other less, in respect of things of the same nature.' It 
 is this reference to things of the same nature that I wish to 
 draw attention to. 
 
 19. As quantities have a certain abstract meaning in 
 themselves, but no definite meaning unless with reference 
 to a given unit ; so qualities have a certain abstract meaning 
 in themselves, but no definite meaning unless when referred 
 to a given universe. 
 
 20. Let the particular kind or collection of objects 
 considered in any judgment or series of judgments be 
 denoted by a capital letter U a symbol used in an analo- 
 gous sense by De Morgan. When the same kind or collec- 
 tion of objects is the subject of all the judgments con- 
 sidered, U need not be expressed, but is to be understood. 
 Let an attribute, character, or quality, be denoted by a 
 small letter, as x. 
 
 21. This modification of Boole's notation brings out the 
 contrast between the Substantive and the Adjective; which 
 is indeed only one form of the general contrast between 
 that which is the subject of the operations of thought and 
 the operations themselves. Another common form of the 
 contrast is that, made prominent in the Theory of Prob- 
 
Universe and Character. g 
 
 ability, between the ' event ' and ' the way in which it can 
 happen.' 
 
 If U denote the ' Members of the House of Commons at 
 the present time/ x may denote ' Liberal ' or * Conservative.' 
 Or if U denote ' triangles/ x may denote ' isosceles ' or 
 ' equilateral.' 
 
 22. Boole, in his investigation, generally considers a com- 
 bination of a Substantive with an Adjective prefixed. Some 
 languages, however, show, by a difference of. position 
 before or after the Substantive that the Adjective may be 
 used in two senses, viz., as forming part of the Substantive 
 merely, or as equivalent to a relative phrase. In the 
 English language, where the adjective is commonly placed 
 before the substantive, the distinction referred to is brought 
 out by emphasis. The prefixed adjective, when emphasised, 
 is equivalent to a relative phrase ; when not emphasised, it 
 is part of the subject of thought. 
 
 23. For example : I find on looking up a Polyglot Bible 
 that the proposition of Proverbs xv. 20 is expressed as 
 follows : 
 
 A wise son maketh a glad father. 
 
 Filius sapiens laetificat patrem. 
 
 Ein weiser Sohn erfreuet den Vater. 
 
 L'enfant sage rejouit son pere. 
 
 II figliuol savio rallegra il padre. 
 
 El hijo sabio alegra al padre. 
 
 Here the subject of discourse is 'Sons'; and it is to be 
 observed that while the two Teutonic languages place the 
 conditioning attribute before the subject, the others put it 
 after. 
 
 24. Boole holds that Primary Propositions refer to things, 
 and Secondary to facts ; and that the idea of time is in- 
 volved in the Secondary. Now there are propositions 
 relating to facts which do not involve time, or a collection 
 of portions of time, as the underlying subject; for example, 
 
io Universe and Character. 
 
 those which refer to place or a collection of places. We 
 have not only the relative ' when,' but the relatives ' where ' 
 and 'who.' Hence if fact-propositions, which relate to the 
 identity of portions of time, required a special investigation, 
 those which relate to the identity of portions of space would 
 also require a special investigation. If, however, we draw 
 a contrast between the subject and its characters, one 
 investigation suffices for all the different kinds of subject. 
 Instead of two I's, of which the one means the actually 
 existent universe, and the other eternity, there is an infinite 
 number of 7's, any one of which may be the subject of 
 discursive thought. 
 
 25. This view of an essential difference in the functions 
 of the Common Noun and Adjective is supported by the 
 results of philological research. According to Max Miiller 
 (Lectures on the Science of Language, vol. i. p. 291), the com- 
 ponent elements of language, which remain at the end of a 
 complete grammatical analysis, are of two kinds, namely, 
 roots predicative and roots demonstrative. In such a 
 language as the Chinese, where the predicative root may 
 by itself be used as a noun or a verb or an adjective, the 
 noun is still distinguished from the verb by its collocation 
 in the sentence. In the Aryan languages no predicative 
 root can by itself form a word ; in order to have a substan- 
 tive it is necessary to add a demonstrative root, this forming 
 the general subject of which the meaning contained in the 
 root is to be predicated. If Boole's view of the operation 
 of the mind were correct, we should have only predicative 
 roots. 
 
 26. Aristotle, in his discussion of the Categories, draws 
 a strong contrast between Substance (oiVta) and Quality 
 (TTOIOV) ; and between Primary and Secondary Substances. 
 By Substance is meant a particular thing (TOO rt) ; by 
 Quality that which is in a subject. In the case of the 
 Primary Substance, the thing signified is individual and one 
 in number ; in the case of the Secondary, the thing signified 
 
Universe and Character. 1 1 
 
 involves a quality, but so as to denote a particular kind of 
 substance. It is the characteristic of Substance that being 
 one and the same in number it can receive contraries ; 
 while it is the characteristic of Quality to be that with 
 respect to which things are said to be like or unlike. 
 
 27. The Primitive judgment is so called because it does 
 not refer to an exact subject, but to the whole external 
 universe as one substance having all the physical changes 
 which occur for accidents. For example, the judgment 
 
 It rains 
 
 refers to the states of a portion of the Physical Universe, 
 and of these equates the ' present ' with ' raining.' 
 
 28. The ordinary Eulerian Diagrams do not represent the 
 whole of thought, but leave it indefinite ; unless we suppose 
 it to be represented by the finite sheet, on which the attri- 
 butes are represented by circles. 
 
 29. U is in general made up of a type and certain finite 
 limitations. The type corresponds to the predicative part 
 of the Noun, of which Professor Max Miiller speaks, and 
 the limitations to the demonstrative part. The most 
 common limitations are those of Space and Time ; which, 
 in this aspect, may be looked upon as logical variables. 
 We may suppose the Time to be constant, and consider all 
 the U's throughout a given region ; or we may suppose the 
 individual to be constant, and consider its successive states 
 within a given portion of time. The zoologist, when he 
 compares the members of a genus, takes them in the adult 
 state ; when he considers the life-history of a particular 
 form, he follows an individual through its cycle of states. 
 
 30. It is with the notion of the type that questions about 
 Essence and Abstract Ideas are more properly concerned. 
 Berkeley draws a distinction between two kinds of abstract 
 ideas. 'As the mind frames to itself abstract ideas of 
 qualities or modes, so does it by the same prescission or 
 mental separation attain abstract ideas of the more com- 
 pounded beings which include several co-existent qualities.' 
 
1 2 Universe and Character. 
 
 Professor Eraser's Selections from Berkeley, p. 16. He says 
 also that there are two kinds of abstraction to correspond. 
 
 The distinction considered is correlative to that between 
 Universe and Character. 
 
 31. Arithmetical value of U. Since U signifies a definite 
 collection of individuals of a given type, its arithmetical 
 value must be an integer. The integer is in general plural, 
 but may be singular or infinite. It is infinite when the 
 individual parts are not discrete but continuous. Grammar 
 recognises two of these cases. 
 
 It is interesting to consider how the subject of thought 
 has naturally an integral value, while the operation of 
 thought has naturally a fractional value, how the relation 
 between the symbols is mirrored in the relation between 
 their kinds of quantity. 
 
 32. U also may be either real or imaginary. For 
 instance, the judgment 
 
 TpaytXa(f>os lorn ACVKOS 
 
 The goat- stag is white 
 
 refers to an imaginary universe of goat-stags. When U is 
 imaginary, it appears proper to indicate that fact by saying 
 that the arithmetical value of U is o ; which is therefore, in 
 this aspect, an integer. 
 
 33. The Universe holds the same position in the Algebra 
 of Quality that the Unit does in the Algebra of Quantity. 
 It may be said to be a generalised unit. 
 
 34. When U is used to denote the subject of thought, 
 and #, y, etc., to denote operations on it, the symbols 
 x, y, etc., have a definite arithmetical value ; and as their 
 meaning is supposed to be fixed throughout a discourse, 
 their arithmetical value must also be supposed fixed. If 
 x denotes a single positive attribute, its value is a fraction 
 lying between o and i ; but if it is negative, its value lies 
 between o and i. Suppose that we have a complex 
 character as xy; being compounded of two characters 
 x and y } which are in their statement independent of one 
 
Universe and Character. 1 3 
 
 another. It is then necessary to suppose that the arith- 
 metical values of x and y are preserved independently of 
 the combination ; for these symbols depend on U only. 
 But if x refers to U", and y not to U but to Ux, then the 
 meaning of y is not independent of x ; and y may have 
 several arithmetical values according to the several orders 
 it has in combination. 
 
 35. Thus i denotes ' all ' or ' the whole' ; while o denotes 
 ' none.' i and o are to be considered as operating symbols 
 of the same kind as x. Some is an indefinite operating 
 symbol ; but it generally carries the additional meaning of 
 having an arithmetical value which is greater than nought. 
 
 36. It is very frequently necessary to express the arith- 
 metical value of x. A convenient notation is x. Boole 
 uses for this purpose the circumlocution ' Prob. x'. 
 
 37. The mind, when reasoning on matters such as are 
 discussed in the Theory of Probability, considers a particular 
 class or kind of things ; as has been well shown by Venn in 
 his Logic of Chance. The /'s and tfs are in their first 
 signification selective symbols, with arithmetical values 
 lying between o and i. A dependent event involves 
 another event as a presupposition; and its arithmetical 
 value depends on that connection a circumstance which 
 also shows us that attributes which are independent in their 
 statement must be conceived as operating upon the uni- 
 verse directly. . 
 
 38. The Algebra of Quality is the more general method. 
 It discusses the relations of the characters of a Universe, 
 whether that universe comprise one, several, or an infinite 
 number of parts, and whether the characters change or are 
 independent of time ; whereas the Theory of Probability as 
 commonly stated (see Venn's Logic of Chance, p. 5), 
 supposes the universe to comprise a very large or infinite 
 number of individuals, and the proper arithmetical value 
 of/ to be a certain limiting ratio, to which the actual value 
 of / is continually approaching the greater the number of 
 individuals in the universe. 
 
14 Universe and Character. 
 
 39. There is an important distinction among adjectives 
 according as they involve being or having. The contrary of 
 the former is formed by prefixing not, that of the latter by 
 prefixing without. Let us consider a collection of balls. 
 The attributes which, in the Theory of Probability, such 
 balls are commonly supposed to have are such as ' white,' 
 ' red,' ' black.' These are attributes of being, since 
 they apply to the whole of the ball, or at least to the 
 whole of its surface. But we may consider a ball, which 
 has stripes of different colours upon its surface ; such attri- 
 butes as ' having a white stripe ' or * having a red stripe ' 
 are adjectives of having. The adjective 'with a white 
 stripe ' applies to the ball as a whole, but ' white ' applies 
 only to the stripe. The former denote qualities proper, the 
 latter what are called characters in the Natural Sciences. 
 As the term character is the more general of the two, I 
 prefer to use it instead of quality. 
 
 40. The ordinary doctrine of Subject and Predicate is 
 often departed from in Manuals of Logic without the writer, 
 apparently, being aware of it. For instance, Ueberweg, by 
 contraposition, converts 
 
 Every regular figure may be inscribed in a circle, 
 into 
 
 Every figure which cannot be inscribed in a circle is not 
 regular, 
 
 thus leaving ' figure ' the subject of thought unchanged, and 
 manipulating only the characters * regular ' and ' inscribable 
 in a circle.' 
 
 41. De Morgan brought the idea of the Universe into 
 prominence. In his Syllabus (section 122) he defines it as 
 meaning the whole extent of matter of thought under con- 
 sideration. I do not consider this definition as sufficiently 
 exact as indicating precisely that which in the Algebra of 
 Quality corresponds to the Unit of the Algebra of Quantity. 
 With him the term does not denote a definite subject of 
 
The Sign = 15 
 
 operations, but rather limits within which elective, not 
 selective, operations may be made. Hence his symbols are 
 all integral. Instead of separating between U and x, as in 
 Ux, he denotes the result by X. 
 
 III. THE SIGN = 
 
 42. By U{x=y} 
 
 is meant, that those 7's which have the character x are 
 identical with those which have the character y ; or, in 
 another aspect, that within the universe (/the character x 
 is equivalent to the character y. 
 
 It follows that the equation is true arithmetically. If 
 
 x = y, 
 
 then x = y ; 
 
 the equivalence of two characters involves the equality of 
 their arithmetical values. 
 
 43. Though the 7's, which have the respective characters, 
 are asserted to be identical with one another, the characters 
 may not be identical. Their relation appears to be best 
 expressed by the term Equivalence. If the proposition is 
 real, the characters derive their equivalence from nature ; 
 but if it is verbal, they derive their equivalence from con- 
 vention. Two characters may be equivalent in themselves, 
 or by definition, or by reality. For example : let U denote 
 triangles ; x, equilateral ; y, equiangular ; then 
 
 U{ X =y} 
 
 does not assert the identity, that is, the undistinguishable 
 sameness of the characters ' equilateral* and ' equiangular,' 
 but only their logical equivalence. 
 
 44. We may also have an equation of the form 
 
 U{x = x} 
 
 where x denotes the arithmetical value of x, and is sup- 
 posed known. This equation also expresses the logical 
 
1 6 The Sign = 
 
 equivalence of the members ; but the second member is 
 indefinite as regards identity. For example : let U denote 
 ' an infinite series of throws with a die/ and x denote * ace,' 
 then U{x = J}. 
 
 This is an example of a form of equation which bulks 
 largely in the Theory of Probability. 
 
 45. Since x and y, when regarded as operating on a 
 definite universe, acquire an unambiguous and exact arith- 
 metical value, no valid objection can be raised against ex- 
 pressing a judgment of the kind considered as an equation. 
 It has, indeed, been rightly maintained by the opponents 
 of the doctrine of the Quantification of the Predicate, that it 
 is by their attributes that we think of things ; but I hope to 
 show that the supposed conclusion does not follow that, 
 therefore, the proposition cannot be expressed by means of 
 an equation. The theory of reasoning about quality, ad- 
 vanced in these Principles is, that the basis of the judgment 
 alone is denotative ; and that the members of the judgment, viz., 
 the antecedent and the consequent, are both attributive. The 
 contrast comes out in the fact that the arithmetical value of 
 the subject is integral, while that of the antecedent and of 
 the consequent is generally fractional. 
 
 46. It appears to be of the essence of an Equation, that its 
 members be equal arithmetically. This holds of the Chemical 
 Equation. For example, in 
 
 Alcohol becomes Water and Olefiant Gas. 
 
 the arithmetical value of the right-hand member is equal to 
 the arithmetical value of the left-hand member. Also in 
 the Quaternion Equation 
 
 a + j3 = y, 
 the arithmetical value of a + /? is equal to that ot y. 
 
The Signs + and 
 
 3 4 
 
 IV. THE SIGNS + AND - 
 
 47. By U{x+y} 
 is meant 
 
 7's which are x together with 7's which arej>/ 
 
 'that is, the result of the addition of the selective operation 
 
 x upon U to the selective operation y upon U. Read with 
 
 respect to a single U, the expression means 
 
 A U which is either x or y. 
 
 48. Suppose that we have a collection of cubes, and 
 that some of them have a dot on their upper face, and 
 some have a dash. Let the operation 
 
 x denote the separation of those having 
 a dot ; and the operation y of those 
 having a dash. Suppose x first per- 
 formed ; the collection is broken up 
 into two parts, I. and II., viz., those 
 which have a dot, and those which are 
 without a dot. The subsequent opera- 
 tion y consists of two parts, viz., a selec- 
 tion of those having a dash, first among 
 those having a dot, and secondly, among 
 those without a dot ; and therefore gives as a result four 
 parts, i, 2, 3, 4, of the Figure, x -\- y denotes the addition 
 of those having a dot to those having a dash, whether the 
 first division, which consists of those having a dot and hav- 
 ing a dash, vanish or does not vanish. 
 
 49. Since the sign is by definition the sign of the 
 operation which is the reverse of that denoted by +; and 
 since the sign + has its ordinary algebraic meaning, the 
 only difference consisting in the fact that the symbols joined 
 involve identity ; the sign must have its ordinary algebraic 
 meaning, but modified by the above-mentioned circum- 
 stance. 
 
 B 
 
 1 
 
 
 
 
 
 1 
 
 
 1 
 
 
 
 
 
 1 
 
 
 1 
 
 
 
 
 
 1 
 
 
 1 
 
 
 
 
 
 1 
 
 
 1 
 
 
 
 
 
 1 
 
 
 v , 
 
 II. 
 
 I. 
 
 FIG. i. 
 
1 8 The Signs 4- and 
 
 U{x-y] 
 means 
 
 tf's which are x minus Z7*s which are y. 
 x and y destroy one another, so far as they coincide ; and 
 the result in general consists of a positive and a negative 
 part. 
 
 50. Consider the collection of cubes represented by fig. i. 
 x y denotes that those 'having a dot' are to be separated 
 positively, and those 'having a dash' negatively. Those 
 cubes which are both x and j> (division i) will be manipu- 
 lated, but left out finally ; the result of the operation being 
 that division 2 of the figure is taken positively, and division 
 3 negatively. 
 
 51. If the operations x and y are formally independent, 
 as in the above example, the arithmetical value, of x-\-y 
 cannot be affected by any real relation existing between x 
 and y. It is simply the sum of the arithmetical values of 
 x andjy. The operation jy (fig. i), though subsequent to x, 
 applies to the whole of the universe ; hence 
 
 x+y=x+y. 
 
 =|+f in the case represented. 
 
 The above is not the view commonly taken. De Morgan 
 says (Syllabus, section 131), 'The more classes aggregated, 
 so long as each class has something not contained in any 
 of the others, the greater the extension of the aggregate 
 term.' 
 
 52. If it were necessary for x and y in x-\-y to be 
 mutually exclusive, a restriction would be placed upon -f- 
 which would prove fatal to the development of an Algebra of 
 Quality. But the imposing of the restriction would be a 
 mistake, for the basis of our operations is not a Unit but a 
 Universe, x+y does not necessarily denote a single char- 
 acter ; but what it in general denotes is equally intelligible, 
 a summation of two characters. 
 
 Similarly, it is not necessary for x in x y to be inclusive 
 ofy, or for y to be inclusive of x. In the Algebra of Quan- 
 
The Signs -f and 19 
 
 tity these conditions hold ; they do not hold in the Algebra 
 of Quality, because the Universe is a generalised Unit. The 
 Universe becomes a Unit when it contains only one indi- 
 vidual. 
 
 53. Definition of Single. A symbol x is said to be single 
 when it does not select any member of the universe more 
 than once, and always selects with the same sign. 
 
 Observation. This embraces exclusion, and both positive 
 and negative inclusion. 
 
 54. If we have z=x+y, 
 
 and z is a single character, then x and y must be mutually 
 exclusive. It is in this way that the Algebra imposes the 
 restriction when it is requisite. Similarly, if 
 
 z=xy, 
 and z be single, then the x must be inclusive of the y. 
 
 55. To express not x in terms of x. 
 
 By not x is meant the whole excepting x; hence it is 
 expressed by i x. 
 
 The arithmetical value of not x is i x. In the example 
 of Art. 48, ix denotes ' not having a dot,' and forms part 
 II. of the figure ; while i y denotes ' not having a dash,' 
 and forms parts 2 and 4. 
 
 56. If x is positive and single ; then ix is positive and 
 single. 
 
 For x being included in i, and i being single and 
 positive, the result of i x is single and positive. For an 
 analytical proof, see Art. 120. 
 
 57. When x and y are in their statement independent of 
 one another, their combination by + and is subject to 
 the formal laws 
 
 (i) 
 (2) 
 x-y=-y+x. (3) 
 
 58. The truth of the formal law (i) may be seen by a 
 consideration of the particular instance of the cubes with 
 
20 
 
 The Signs + and 
 
 different marks. If we first separate those having a dot, 
 then separate those having a dash, and add the two results 
 together ; we shall get the same cubes in the final result as 
 when we commence by separating those having a dash, then 
 separate those having a dot, and add the two results 
 together. 
 
 The formal law (2) shows that independent characters, 
 which are not necessarily single, are subject to the same 
 law of combination as independent characters, which are 
 single. 
 
 The formal law (3) shows that as a symbol of opera- 
 tion of the mind is quite as independent as -f 
 
 V. THE SIGNS X AND -f 
 
 59. By Uxy 
 is meant 
 
 U's which are both x and y. 
 
 In common language ' both ' is frequently dropped. Ac- 
 cording to Boole + expresses the signification of ' and ' ; 
 while X properly expresses sequence. It is, however, an 
 additional argument in favour of a difference of notation for 
 the Universe and its characters, and generally between 
 operations which are subordinate and operations which are 
 co-ordinate ; that, when it is done, + expresses ' or ' and 
 X expresses ' and.' 
 
 60. Consider, as before, a collection of cubes, Art. 48 ; 
 
 but we shall now suppose that we have 
 a great number. As before, let x denote 
 ' having a dot,' and y ' having a dash.' 
 Let x separate to the left hand (fig. 2) ; 
 and then let y separate to the bottom. 
 The operation xy is a consequence, 
 being the separation of the cubes, which 
 both have a dot and also have a dash. 
 
 I 
 
 I 
 
 FIG. 2. 
 
The Signs x and -r- 21 
 
 61. When the symbols x andjy are independent that is, 
 when each refers to U simply the compound symbol xy 
 has for any given U a definite arithmetical value. This 
 value, however, is not determinable from those of x and y ; 
 but they give limits to the value. Consider the example of 
 the cubes (fig. 2). It is evident that xy cannot be greater 
 (include more) than x, or greater than y ; and that it 
 cannot be less than (must include) x-\-y i, or o. Hence 
 its arithmetical value cannot be greater than x or y, and 
 cannot be less than x+y i, or o. 
 
 62. It is important to draw a distinction between the 
 independence of the operations x andj>, and the relation' of 
 these to one another as characters. The operation y is 
 independent of x, when it is not confined by its expression to 
 Ux, but applies to U. In the Theory of Probability it is 
 assumed that, when x and y are independent, 
 
 But this is an assumption resting on the supposition that the 
 ratio of the Uy's in the x part to the x part is the same as the 
 ratio of the Uy's in the not x part to the not x part. It is 
 assumed that the two straight lines, which together separate 
 the y in fig. 2, form one straight line. It is, no doubt, the 
 . best assumption to make, when we do not know how the two 
 portions of the line lie ; and is of the same nature as the 
 doctrine that the values of two characters not being known, 
 they are to be assumed to be equal. We have supposed 
 that the Universe contains a great number of cubes, in 
 order that each portion of the y line may be supposed to be 
 straight. 
 
 63. Venn shows the true nature of the assumption in his 
 Logic of Chance, p. 157 : 'For the establishment of the 
 rule under discussion, it is both necessary and sufficient 
 that the division into classes caused by each of the above 
 distinctions, should sub-divide each of the classes created 
 by the other distinction in the same ratio in which it sub- 
 divides the whole. If the independence be granted and so 
 
22 The Signs x and '-f- 
 
 defined as to mean this, the rule of course will stand, but, 
 without especial attention being drawn to the point, it does 
 not seem that the word would naturally be so understood.' 
 The distinction maybe called that between formal and real 
 independence. 
 
 64. But one character may in its statement involve 
 another character, so as to be formally dependent on the 
 latter. Let U denote the universe of objects, a any 
 character, and x a character which is formally dependent 
 on a. Then Ua x denotes 
 
 /'s which are a and of these such as are x ; 
 or Z7's which have a which have x. 
 
 The symbol x operates upon Ua, not upon U. 
 
 The arithmetical value of x is measured with reference 
 to Ua, not to U. Hence 
 
 65. In this case, and this case only, does the mind pro- 
 ceed in the predicamental line. Boole supposes that it 
 always proceeds in that manner. In Uxy he considers jy to 
 operate on Ux, in fact, to mean Ux y , and supposes that it 
 can preserve the same signification in Uy. But in fact its 
 meaning and arithmetical value may both be changed. 
 
 66. In the ordinary Algebra \ \ means one-half of a 
 third or one-third of a half. The one operation is formally 
 dependent on the other. But we have shown, Art. 61, that 
 in the Algebra of Quality 
 
 II 
 
 does not mean one-half of a third or a third of a half, and 
 can be considered to be equivalent to such an expres- 
 sion only when the characters are really independent of 
 one another. Thus the Algebra of Quality is more general 
 than the Algebra of Quantity. 
 
 67. Suppose that we consider the ' Mammalia.' 
 Then an example of a x is 
 
 having red blood corpuscles 
 
 which are oval ; 
 
The Signs x and -f- 23 
 
 and an example of a.^ is 
 
 having red blood corpuscles 
 
 which are circular and discoid. 
 
 It is the function of the Relative Pronoun to denote depend 
 ence. 
 
 68. When x and y are formally independent of one 
 another 
 
 xy =yx. 
 
 Also, when x, y and z are formally independent of one 
 another, 
 
 69. Since a x is equivalent to an independent -character 
 of arithmetical value a x, the laws that are true of inde- 
 pendent characters apply to a x as a whole. 
 
 70. Also, the same laws apply to characters which are 
 co-ordinately dependent upon a common character. For 
 example, 
 
 U{a x +a y =a x+y }. 
 and U{a xy =a yx }. 
 
 71. Special laws. 
 
 U{a x =x a }. 
 
 Also, if a+b+c+...+k=i, 
 
 then a x +b x +c x +....+k x =.x. 
 
 Observation. It is supposed that x is such that it can 
 be freed of its dependence on a ; which is not possible when 
 x refers to a noun contained in a. 
 
 72. Deduction of the meaning of -f from that of X- 
 
 Let xy=xz, 
 
 then -r being the reverse of X is to 
 have such a signification that 
 
 xz 
 
 >=* 
 
 It is evident from fig. 3, which repre- 
 sents a case of 
 
 xy=.xz FlG - 3- 
 
24 The Signs x and^ 
 
 that -- is indefinite. Hence the proper reading of 
 
 00 
 
 xz 
 
 y= x 
 is 
 
 y is equivalent to an 
 x 
 
 It is true in the case of fig. 3 that 
 yz=x, 
 
 hence z=, 
 
 which means 
 
 z is equivalent to an . 
 
 y 
 
 For there are evidently several 2*5 which can satisfy the 
 condition 
 
 yz=x, 
 although x and y be kept constant. 
 
 COR. is not in general = i . 
 x 
 
 For if were necessarily = i, the truth of the equation 
 
 xy=xz 
 
 would necessarily involve that of 
 y=z. 
 
 But an inspection of the figure shows that this is not the 
 case. 
 
 73. The condition y=z is always satisfied by the fractions 
 of ordinary Algebra ; but it is not satisfied by the fractions of 
 the Algebra of Quality, unless as a special case. This pecu- 
 liarity I not necessarily equal to i ) was clearly pointed 
 
 out by Boole. It is commonly supposed, and I think that 
 Boole himself supposed, that it is the only peculiarity of the 
 symbols of this Algebra. For example, Professor Robertson 
 Smith says : ' There is one limitation only to our right to 
 
The Signs x and-^r 25 
 
 manipulate logical and mathematical identities by the same 
 rules. From xy=xz we cannot infer y=*z. 9 In Transla- 
 tion of Ueberweg, p. 569. Now the fact is that each sign 
 has a generalised meaning. 
 
 74. Boole interprets to mean an indefinite class 
 
 o 
 
 symbol ; but this interpretation, he says, cannot except 
 upon the ground of analogy be deduced from its arith- 
 metical properties, but must be established experimentally. 
 Laws of Thought, p. 92. In Art. 166 I. show that under a 
 
 certain condition must be single within a certain part of 
 o 
 
 the universe. When no condition is imposed upon it, its 
 meaning is quite indeterminate. 
 
 75. Definition of the Index sign. 
 
 x m , where m is an integral symbol, is defined to mean 
 the selective operation x repeated m times. For instance, 
 
 X 2 = XX. 
 
 It is to be noted that x may be single or complex, and 
 positive or negative or both. 
 
 76. Law of Indices. 
 
 x m x n = 
 
 Also 
 
 x m x n 
 
 x m 
 
 But is not in general =x m ~ n . 
 x n 
 
 For if it were, 2- would be = x ; which is not necessarily 
 
 oc 
 
 true. 
 
 VI. RULE OF SIGNS. 
 
 77. If + 2 be defined to be equivalent to +, it follows 
 by the usual proof, that 
 
 + - = -j 
 
 -+ = -; 
 
 and = +. 
 
26 Rule of Signs. 
 
 Observation. It is, of course, immaterial in what con- 
 nection -\ occurs; whether in +( x), or + x( y), or 
 i x 
 The superposition of the two mental operations can 
 
 be considered apart from the accidents with which the 
 operations are mixed up in use. 
 
 VII. INTEGRAL SYMBOLS. 
 
 78. To find the meaning of , where m and n are each 
 
 n 
 
 integral. 
 
 Let w=. 
 
 n 
 
 Then nw=-m. (Art. 72); for every 
 
 definition applies conversely. 
 
 Since m and n are each integral (that is, the whole re- 
 peated a number of times), it is evident that the latter 
 equation cannot be satisfied unless n is a divisor of 
 
 m. Then w = quotient. Thus is impossible unless n 
 
 n 
 
 divides m. 
 
 COR. i. i= i. |=2.. 
 
 COR. 2. \ is impossible. 
 
 But \x will be possible, when x = 2y. For then we have 
 fjy / which is =y. 
 
 Observation. | does not mean the same as J. For \ 
 means the whole upon two wholes ; whereas simply means 
 one-half. 
 
 79. To find the meaning of , and ; where n is any 
 
 n o 
 
 integral symbol. 
 
 First; for -. 
 
Integral Symbols. 2 7 
 
 Let w= 
 
 n 
 
 Then nw=o (by definition). 
 
 From this latter equation it is evident that-w must 
 be=o. 
 
 COR. =o. 
 
 i 
 
 Second; for . 
 
 o 
 
 Let w =-2. 
 
 O 
 
 Then ow=n (by definition). 
 
 Suppose w to be integral; the latter equation is then 
 evidently impossible. 
 
 Suppose it to be fractional and single; the equation is 
 then also impossible. So also, when w is supposed frac- 
 tional and complex. 
 
 Hence is an impossible selective operation, 
 o 
 
 COR. Ifx= y, and x is possible ; then y must be o. 
 
 For ox=y t 
 
 but x being a possible selective symbol, ox=o ; 
 .-. o=y. 
 
 80. To find the meaning of -- 
 
 Let w = - 
 
 o 
 
 Then ow = o. 
 
 The latter equation will be satisfied when w is an integral 
 symbol; also when w is any fractional symbol, whether 
 
 single or complex. Hence - is quite indeterminate. 
 
 o 
 
 81. Since the symbols and the signs of the Algebra of 
 Quality have a meaning, which is a generalised form of 
 their meaning in the Algebra of Quantity, every theorem in 
 
28 
 
 On the Equation 
 
 the latter is true in the former, provided that any special 
 conditions which have been introduced are removed. For 
 example : 
 
 be a cP 
 ca b ft 
 ab c <? 
 
 is not = 
 
 but it is = 
 
 abc 
 
 i a 2 a s 
 
 i b^ ffl 
 
 I ^" $ 
 
 i a 2 a* 
 
 i c 2 <* 
 
 The former equation can be deduced from the latter in 
 the Algebra of Quantity, because in it 
 
 a -^- = i always. 
 abc 
 
 82. The propositions contained in the Theory of Deter- 
 minants^ when so generalised, are of great interest and 
 importance in the analysis of Quality. The ordinary de- 
 monstration of these propositions does not necessarily 
 depend upon the peculiarities of the symbol of Quantity. 
 It, however, proceeds on the assumption that the symbols 
 are independent of one another. 
 
 83. In the preceding investigation of the fundamental 
 definitions of this Algebra and their immediate conse- 
 quences, I have endeavoured to imitate the clear logical 
 manner in which the fundamental definitions of the Calculus 
 of Quaternions are laid down by Professor Tait in his 
 Treatise on Quaternions. The definitions given are perfectly 
 exact, and in this respect stand in marked contrast to the 
 vague indefinite statements about + and X which are to 
 be found in even the best manuals of Logic. 
 
 VIII. ON THE EQUATION AS EXPRESSING 
 A GENERAL PROPOSITION. 
 
 84. Any relation between the characters of a definite 
 universe can be expressed by means of an equation between 
 
as expressing a General Proposition. 29 
 
 the symbols which represent the characters. The logical 
 equation is an analytical statement of a truth, which may- 
 require for its expression in language one or more general 
 propositions a statement which, from its brevity, clear- 
 ness, and exactness, is of the greatest aid to the mind when 
 reasoning from complex data. Whether other logical rela- 
 tions besides those of Quality can be stated more fully and 
 concisely by means of symbols and the equation than by 
 means of words and the proposition, is a different question. 
 
 85. Every general proposition refers to a definite universe ; 
 which is the subject of the judgment, and, it may be, of a 
 series of judgments. For example; 
 
 All men are mortal, 
 refers to the universe ' men.' 
 
 No men are perfect, 
 refers to the same universe. 
 
 86. It is often said, for instance by Boole (Laws of 
 T/wught, chap, iv.), that 
 
 All men are mortal beings, 
 is a more complete statement of the judgment 
 
 All men are mortal, 
 
 or, at all events, expresses the same judgment of the mind. 
 Now, in the one case, the subject of discursive thought is 
 the class 'beings;' in the other, the class 'men/ Can 
 judgments, which apply to different universes of objects, be 
 the same? In considering whether, in the universe of 
 ' men,' the attribute ' mortal ' is equivalent to ' the whole,' 
 I do not require to consider beings in general. 
 
 87. Let U denote men, and x mortal ; then 
 
 All men are mortal 
 is expressed by 
 
 V{i=x}. 
 But if the judgment is 
 
 Beings which are human are mortal ; 
 let U denote beings, x mortal, y human ; then 
 U{y=x-w\ 
 
30 On the Equation 
 
 (where w denotes an unknown character, the complement 
 of y to x) expresses the judgment considered. 
 Another form of the equation is 
 U{y=yx}. 
 Again ; to express 
 
 No men are perfect ; 
 
 U{o=p}. 
 And to express 
 
 Some men are wise ; 
 
 U{v=w} 9 
 where v is a proper fraction, and greater than o. 
 
 88. We may write these equations, dispensing with 
 abbreviations, which is indeed a very good test of the 
 soundness of a notation. 
 
 Men {all = mortal}. 
 Men { none = perfect}. 
 Men { some = wise}. 
 
 These are precise and undistorted representations of the 
 judgments actually expressed by 
 
 All men are mortal. 
 No men are perfect. 
 Some men are wise. 
 
 89. We may consider the analytical expression of those 
 examples of general judgments, which Aristotle discusses in 
 his book on Interpretation. 
 
 7TCIS !<TTtV av6p(D7T 
 
 OV TTOS lo-TlV av6p(j)TTO<S 
 
 Tras TTIV av6pd)7ro<s ov Si/catos. 
 ov Tras 4<mv ai/0/30>7ros ov SI'KCUOS. 
 The common universe of these propositions is 
 and the quality considered is St'/ccuos. The equations are 
 respectively 
 
 i=a. (i) 
 
 i v=S. (2) 
 
 i = i-& (3) 
 
 1-^=1-8. (4) 
 
as expressing a General Proposition. 3 1 
 
 i v denotes all minus some, some being greater than nottght ; 
 which is equivalent to not all. 
 
 90. It is evident that the grammatical subject of the pro- 
 position does not form one member of the equation. The 
 members of the equation may be the quantity and the 
 predicate; as in the above examples, and in judgments 
 such as 
 
 Men {^-= shortsighted}. 
 
 Or, they may be the antecedent and consequent ; as in 
 Triangles which are equilateral are equiangular, 
 
 T{xy w}. 
 
 91. Since a universe may contain one, several, or an 
 infinite number of individual parts, and since a judgment 
 always refers to a universe, it is evident that the judgment 
 will be modified in an important manner according to the 
 number of individual parts in its universe. The strictly 
 singular judgment refers to a universe, which has only one 
 individual part \ as, for example, 
 
 A given man at a given time is sleeping. 
 It is to be observed that, in order to produce a singular 
 universe, it is not enough that the substance be fixed ; the 
 time must also be fixed. 
 
 92. When the judgment refers to a singular universe, the 
 only values which any character can assume are i and o. 
 The former is required to make an affirmation, and the 
 latter to make a negation. 
 
 The given man at the given time is sleeping. 
 
 is not sleeping. 
 
 is not-sleeping. 
 
 is not not-sleeping, 
 
 are expressed respectively thus 
 
 (O 
 
 o=,. (a) 
 
 i = i-*. (3) 
 
 0=1- s. (4) 
 
32 On the Equation 
 
 93. Aristotle remarks that here equation (3) follows 
 from equation (2) but that in the more general case (Art. 
 89) when U is plural, equation (3) does not follow from 
 equation (2). It is only judgments of the kind considered 
 in the preceding article that are affirmative or negative in 
 the strict sense of these words. 
 
 94. There are singular judgments of another kind. For 
 example ; 
 
 Of all the planets Jupiter is the greatest ; 
 the analytical expression of which is 
 
 Planets { Jupiter = the greatest}. 
 
 ' Jupiter' is a singular character, so is 'the greatest;' that 
 is, they can apply to only one of the individual parts of the 
 universe. Hence the singularity of this kind of judgments 
 does not consist in the universe of the judgment being an 
 individual, but in the characters being singular. 
 
 95. It is also important to consider the special nature of 
 the judgment which relates to a universe having an infinite 
 number of parts. For, questions arise as to how it is 
 possible in such a case to assert relations among the 
 characters. 
 
 96. As the relations of the characters of the universe may 
 be given in an infinite variety of ways ; the logical equation, 
 in order to be adequate, must be capable of assuming an 
 infinite variety of forms. The forms treated in manuals of 
 Logic are but a few of the most common. Not only ought 
 we to be able to state explicitly what is thought implicitly, 
 but we ought to be able to state in one equation any one 
 condition, which a number of characters satisfy; even though 
 it take a great number of sentences to express that con- 
 dition. 
 
 97. The reason why Formal Logic has so long been 
 unable to cope with the subtlety of nature, is that too much 
 attention has been given to pictorial notations. Arithmetic 
 could never be developed by means of the Roman system 
 
as expressing a General Proposition. 33 
 
 of notation ; and Formal Logic cannot be developed so 
 long as Barbara is represented by 
 
 or even by the simpler spicular notation of De Morgan. 
 We cannot manipulate data so crudely expressed ; because 
 the nature of the symbols has not been investigated, and 
 laws of manipulation derived from their general properties. 
 
 98. A consideration of the nature of propositions about 
 abstract numbers is of importance here, as throwing light 
 upon the nature of the logical equation. Kant appears to 
 be the only logician who has discussed them at all 
 thoroughly. He says : ' The self-evident propositions as 
 to the relation of numbers are certainly synthetical, but not 
 universal, like those of geometry, and for this reason cannot 
 be called axioms, but numerical formulae. That 
 
 7+5 = 12 
 
 is not an analytical proposition. For neither in the repre- 
 sentation of seven, nor of five, nor of the composition of 
 the two numbers do I cogitate the number twelve. But 
 although the proposition is synthetical, it is nevertheless 
 only a singular proposition. In so far as regard is here had 
 merely to the synthesis of the homogeneous (the units), it 
 cannot take place except in one manner, although our use 
 of these numbers is afterwards general.' Critique of Pure 
 Reason. 
 
 99. The full expression of the example discussed is 
 
 where U may denote any unit or any universe. For it is 
 a logical as well as an arithmetical proposition. 
 
 Similarly #"{7x5 = 35}, 
 
 whatever 7 be. 
 
 It is evident that the source of the nexus is not convention. 
 The peculiarity is, that the equivalence is true inde- 
 pendently of the nature of U. 
 
 100. It is also of importance to consider the nature of the 
 c 
 
34 On the Equation 
 
 chemical symbols and of the chemical equation. The chemist 
 considers a universe of material molecules ; and such sym- 
 bols as H and Cl denote a distinctive character or set of 
 characters possessed by some of the molecules. It is a 
 property of these characters, that they are exclusive of each 
 other. HCl does not denote matter which has the pro- 
 perty H and also the property Cl ; but matter which has 
 an entirely distinct property HCL The meaning of HCl 
 is ' formed by the combination of H and Cl? The symbols 
 have fixed fractional values that is, fixed proportions ; in 
 this respect differing from the symbols of Quality. 
 
 101. The chemical equation expresses how the distinc- 
 tive characters of a particular set of molecules are changed 
 into other distinctive characters, and what the latter are. 
 For example ; 
 
 Sulphate of potassium and chloride of barium become 
 
 sulphate of barium and chloride of potassium, 
 expresses that in a particular quantity of matter the sum 
 of distinctive characters 
 
 becomes changed into the sum of distinctive characters 
 
 The symbol = involves becoming. (Professor Crum Brown's 
 Manual of Chemistry, p. 46.) Hence the chemical equation 
 cannot be reversed without reversing the meaning of the 
 copula =. In De Morgan's language, the copula here is 
 inconvertible. 
 
 102. The above equation expresses completely the change 
 to which it refers. Its right-hand member is as definite as 
 its left-hand member, and the two are equal arithmetically. 
 In particular cases we may require only the incomplete 
 equation 
 
 just as instead of the complete logical equation 
 
as expressing a General Proposition. 35 
 we may in particular circumstances consider only 
 
 There is an important difference between what is requisite 
 to give a complete scientific expression of a change or of a 
 relation, and what is requisite to give an expression of such 
 a part of that change or relation as may be necessary in a 
 particular process of reasoning. 
 
 103. The equations of Natural History are examples of 
 logical equations. The dental equation of Man is written 
 
 .2 2 I I 2 2 1 ^ 
 
 -/ -- - c - ; pm -- ; * 3 -=32. 
 
 2-2 I I 2-2 3-3 
 
 But this truth may be written as a logical equation ; thus 
 
 ( 1=1 c pm m ") 
 Man} 2 ~ 2 Z2 2 ~ 2 3-3 t ; 
 
 ( 2 2 T I 2 2 3 3) 
 
 that is, considering the universe of men in the adult state, 
 the whole is identical with those having incisors which are 
 
 in number and arrangement 2 -^ 2 -, having canines which 
 
 are in number and arrangement - , having premolars 
 which are in number and arrangement - , having molars 
 
 which are in number and arrangement 3 ?. Each of the 
 
 3~3 
 
 '2-2 S-I 
 
 characters --- . - , etc., must be equal to i. In 
 
 2 2 I I 
 
 *y *y *} _____ 2 
 
 , the character is dependent with respect to t\ and 
 
 is itself composed of four co-ordinate characters. 
 
 104. If we consider the universe of Mammals in the 
 adult state ; to what is the character 
 
 2 2 I I 2 2 3 
 
 equivalent ? We have, I think, 
 
36 The Principle of Identity and 
 
 Mammalia j bimana + ^ uad 
 ="',. a <i-i* w a- 
 
 2 2 I I 2 2 3 3 
 
 and, if BO, we have a statement which is as complete and 
 exact as any chemical equation. 
 
 105. Professor Huxley, in his argument against the 
 assertion that Biology differs from the Physico-chemical 
 and Mathematical sciences in being 'inexact/ holds that 
 the statements 
 
 A vertebrated animal has jaws which never open side- 
 ways, but always up and down, 
 and 
 
 An annulose animal has jaws which always open side- 
 ways, and never up and down, 
 
 are as exact as any of the propositions in Euclid. (On the 
 Value of the Natural History Sciences. Lay Sermons, 
 Essays and Reviews, p. 79.) 
 
 The above statements are exact, in the sense of being 
 capable of expression by exact equations ; thus 
 
 Animals / ^brated = having jaws 
 
 never opening sideways X opening up and down. J 
 
 Animals / annulose = having jaws 
 
 opening sideways X never opening up and down. J 
 
 IX. THE PRINCIPLE OF IDENTITY AND THE 
 AXIOMS OF IMMEDIATE INFERENCE. 
 
 106. Let U denote any universe; and x any character, 
 positive or negative, integral or fractional ; then 
 
 U{x = x}. 
 
the Axioms of Immediate Inference. 37 
 
 The Principle of Identity, when put into the above form, 
 does not appear to merit the abuse which has been heaped 
 upon 
 
 A is A 
 and 
 
 Ens est Ens. 
 
 Another form of this principle is 
 
 X X = O, 
 
 which involves the meaning of the sign . 
 
 107. An inference is said to be immediate, when it is 
 deduced from an equivalent equation and an identical 
 equation (or the former repeated) by means of the axioms 
 of the science. 
 
 It has been a matter of perplexity to logicians, how a 
 conclusion can be drawn from one premise. Ueberweg 
 says : c The immediateness in the so-called " immediate " 
 inference is relative. It implies that this kind of inference 
 does not require, as mediate inference does, the addition 
 of a second datum to the first, but at once and of itself 
 yields the derived judgment, which^ is nevertheless another 
 judgment, and not merely another verbal expression.' 
 (Lindsay's translation of Ueberweg, p. 226.) 
 
 It is not very evident, how one equation can at once and 
 of itself yield another, which has any claim to be called 
 different. But if we consider that an identical equation 
 can be formed, and that the axioms of the science, which 
 depend upon what is essential in the symbols, allow us to 
 combine these equations ; a great part, if not the whole, 
 of the mystery disappears. 
 
 108. The general axioms of immediate inference are : 
 
 I. If an equation and an identical equation be added 
 together ; the resulting equation is true. 
 
 Thus if oc=y \ 
 
 since z=z, 
 
 x+z=y+z. 
 
 II. If an equation and an identical equation be sub- 
 
38 The Principle of Identity and 
 
 tracted the one from the- other ; the resulting equation is 
 true. 
 
 Thus if x =y; 
 
 since z=z, 
 
 xz=yz. 
 
 III. If an equation and an identical equation be mul- 
 tiplied together ; the resulting equation is true. 
 
 Thus if x=y- 
 
 since z=z 9 
 
 xz=yz. 
 
 IV. If an equation and an identical equation be divided 
 the one by the other ; the resulting equation is true. 
 
 Thus if x=y- 
 
 since z=z, 
 
 j = Z. 
 z z 
 
 109. A common form of application of Axiom II. consists 
 in subtracting the given equation from the identical equation 
 
 1 = 1. 
 For example ; the universe being Triangles, 
 
 equilateral = equiangular ; 
 . . i equilateral = i equiangular. 
 This is one form of the process of Contraposition. 
 HO. An equation, when multiplied by a fractional symbol, 
 is still true (Axiom III.) ; but it is not so general as before. 
 The multiplication of an equation by such a symbol means 
 the introduction of an hypothesis. A proposition, which is 
 true independently of an hypothesis, is true under the hypo- 
 thesis; but not conversely. 
 111. The axiom that 
 
 When the signs of the terms of an equation are changed 
 into their respective converse signs, the resulting 
 equation is true, 
 
 reduces Axioms II. and IV. to cases of Axioms I. and III. 
 respectively. 
 
the Axioms of Immediate Inference. 39 
 
 Thus if x=y; 
 
 then #= y, 
 
 where + is changed into its converse ; and also 
 i __ i 
 
 * 7 
 
 where X is changed into its converse -r . 
 
 112. Given the principle of Identity 
 
 x is equivalent to x, 
 to prove the principle 
 
 x is equivalent to not not x. 
 Since x=x, 
 
 and = 1 1, 
 
 x= i i +x. II. Axiom of Immediate Inference. 
 = i ( i x) by Rule of Signs. 
 = not not x. (Art. 55.) 
 
 COR. i #=i {i (i #)}. 
 
 Generally. In the case of x, we have an even number 
 of i's alternately positive and negative ; and in the case of 
 i x, an odd number of I's alternately positive and nega- 
 tive. It is customary with grammarians to point out, that 
 an adjective, with an even number of nets before it, is 
 equivalent to the simple adjective. 
 
 Observation. x=i (ix), 
 
 is true whatever U be ; and also without x being restricted, 
 as above, to be single. 
 
 When U is singular, x can only be i or o (Art. 92) j and 
 
 x=i (i x) 
 
 then becomes what is called by some the Principle of Con- 
 tradiction. 
 
 113. Given the principle of Identity 
 
 x is equivalent to x, 
 to prove the principle of Excluded Middle 
 
 x + not x is equivalent to the whole. 
 x=x, 
 
 .'. XX = Oj 
 
 but 1 = 1, 
 
 /. x-\- 1 x= i. I. Axiom of Immediate Inference. 
 
40 The Principle of Identity, etc. 
 
 That is, x -f not x is equivalent to the whole, whatever 
 U be. U may be singular, plural or infinite. 
 
 The principle of Excluded Third, as enunciated by many 
 logicians, presupposes that U is singular. For example ; 
 Ueberweg states it thus (Translation, p. 260) 
 
 ' Judgments opposed as contradictories (such as A is B, 
 and A is not B) can neither both be false nor can admit 
 the truth of a third or middle judgment, but the one or the 
 other must be true.' 
 
 Let 7 be singular, then 
 
 U{x+i-x=i}. 
 
 Now when 7 is singular; x can only be i or o; and in 
 consequence i x can only be o or i. Hence either 
 
 U{ X =i] 
 
 or else 7{.r=o} 
 
 must be true. 
 
 When U is plural, it is of course not the case that either 
 
 /{*=!} 
 
 or else U{x=o] 
 
 must be true. 
 
 Observation. x + 1 x = i 
 
 does not necessarily presuppose any restriction upon the 
 signification of x, such as to be single. 
 
 X. AXIOMS OF MEDIATE INFERENCE. 
 
 114. A mediate inference may be defined as, an inference 
 from two or more independent equations by means of the 
 axioms of the science. 
 
 115. The general axioms of mediate inference are : 
 
 I. If the equations be added, the resulting equation is 
 true. For example j if 
 
 x=y, and a=.b; 
 then x-\-ay-\-b. 
 
Axioms of Mediate Inference. 41 
 
 II. If two equations be subtracted the one from the 
 other ; the resulting equation is true. For example if 
 
 x=y, and a=b; 
 then xa-=yb. 
 
 III. If two equations be multiplied together ; the result- 
 ing equation is true. 
 
 x=y, and<z=; 
 then ax=by. 
 
 IV. If two equations be divided the one by the other ; 
 the resulting equation is true. 
 
 x=y, and a=Z>; 
 
 then =>. 
 
 a b 
 
 116. It is not necessary, that the two equations should 
 contain a common character ; but it is necessary that both 
 should refer to the same universe : just as the ordinary 
 algebraic equations in a given investigation must be sup- 
 posed to refer to the same unit. In the latter sense only is 
 it true, that no reasoning is possible without a middle term. 
 
 117. Kant would not consider the above to be true 
 axioms. He says (Critique of Pure Reason, Meiklejohn's 
 translation, p. 124) : 'The propositions 
 
 If equals be added to equals, the wholes are equal, 
 and 
 If equals be taken from equals, the remainders are equal, 
 
 are analytical, because, I am immediately conscious of the 
 identity of the production of the one quantity with the pro- 
 duction of the other ; whereas axioms must be a priori 
 synthetical propositions.' But these propositions are not 
 analytical, in the sense of depending upon convention for 
 their truth ; they are true of the symbols of Quantity, as 
 symbols of Quantity ; and only in that sense may be said 
 to be analytical. An axiom appears to be precisely such a 
 proposition. 
 
42 Conditions for a Character being single. 
 
 XL CONDITIONS FOR A CHARACTER BEING 
 SINGLE. 
 
 118. If x is single and positive, then 
 
 x z =x; 
 and if x is single and negative, then 
 
 and conversely. 
 
 For the definition of ' single,' see Art. 53. 
 
 First ; when x is positive. 
 Let y be another single and positive 
 symbol, and let y be wholly included 
 in x (fig. 4). Then 
 
 xy=y. 
 
 Let>- expand until y=x ; 
 then xx x, 
 
 FIG. 4. that is, x 2 =x. 
 
 Second ; when x is negative. 
 
 Let x= i x' ; x' being positive. 
 Then * 2 = -i 2 x' 2 . 
 
 = i 2 x' by first part. 
 =x' by Rule of Signs. 
 
 ^^ "~"X, 
 
 Hence, if x is single ; 
 
 x* 
 
 By a consideration of the figure it is also evident that if 
 
 X 2 = X, 
 
 x must be single and positive ; 
 and that if x* =x, 
 
 x must be single and negative. 
 
 Hence # 2 =dbr 
 
 is the necessary and sufficient condition for x representing 
 a single character. 
 
 119. More generally : If x is single and positive, 
 
 x m =.x ; 
 and if x is single and negative, 
 
 x m =(i) m ~ 1 x. 
 
Conditions for a Character being single. 43 
 
 120. If x is single positively, then i-x is single 
 positively. 
 
 For (i -*) 2 = i - 2x+x*. (Art. 68.) 
 
 = i-* + (**-*), 
 = i x by given condition ; 
 
 /. i x is single positively. (Art. 118.) 
 It is when i x is single positively, that it means the same 
 as not x. 
 
 121. Given that U{x^=x] 
 
 to prove that U{x ( i x) = o } . 
 
 X 2 = X, 
 
 and x 2 =x 2 , Principle of Identity, 
 
 therefore x 2 x 2 =x x 2 ; II. Axiom of Immediate Infer- 
 ence ; 
 
 that is, Q=X x 2 , 
 
 therefore o=x (i x). Distributive Law, Art. 68. 
 
 Another Proof. 
 
 x -f i x = i , Principle of Excluded Middle . 
 
 x=x, Principle of Identity, 
 
 therefore x (x+i-x)=x; III. Axiom of Immediate In- 
 ference. 
 
 therefore # 2 + x (ix)=x; Distributive Law. 
 but x"=x, given. 
 
 therefore #(i x)=o. II. Axiom of Immediate Infer- 
 ence. 
 
 122. Meaning of these equations. 
 
 U{x*=x} 
 means 
 7*s which are x and x are identical with those which are x. 
 
 It is true only when x is positively single with reference 
 to U. 
 
 Since x is single positively, i x must be single positively 
 (Art. 120). Hence i x means 'not x.' Hence 
 
 ^{^(,-^ = 0} 
 
 means 
 
 6^s which are x and not x are none. 
 
44 Conditions for a Character being single. 
 
 Since U{x* 
 
 is the condition for x being single, relatively to the uni- 
 verse U ; 
 
 U{ X (i*)=o} 
 
 is the general form of the other mode of stating the con- 
 dition. 
 
 123. When 7 is singular, any fractional character x must 
 be single (Art. 92). Hence, when that condition is satis- 
 fied, 
 
 and jr 
 
 are true always. The latter is then the exact analytical 
 expression of the Principle 
 
 It is impossible for an object at a given time to have 
 a quality and the contrary quality. 
 
 This is what is commonly called the Principle of Contra- 
 diction; but the most general form of the Principle is that 
 enuntiated in Art. 122, where U may be singular, plural, or 
 infinite. 
 
 124. As the deduction and interpretation of 
 
 is of great importance in the Algebra of Quality, I shall 
 quote the original proof given by Boole. (Laws of 
 Thought, p. 49.) 
 
 ' That axiom of metaphysicians which is termed the prin- 
 ciple of contradiction, and which affirms that it is impossible 
 for any being to possess a quality, and at the same time not to 
 possess it, is a consequence of the fundamental law of thought, 
 whose expression is x 2 =x.' 
 
 Let us write this equation in the form 
 
 x-x*=o 
 whence we have 
 
 x (ix) =o, 
 
 both these transformations being justified by the axiomatic 
 laws of combination and transposition. Let us, for sim- 
 plicity of conception, give to the symbol x the particular 
 
Conditions for a Character being single. 45 
 
 interpretation of men, then i x will represent the class of 
 ' not men.' Now the formal product of the expressions of 
 two classes represents that class of individuals which is 
 common to them both. Hence x (i x) will represent the 
 class whose members are at once 'men' and 'not men/ 
 Now the formal product of the expressions of two classes re- 
 presents that class of individuals which is common to them 
 both. Hence x (i x) will represent the class whose 
 members are at once ' men ' and ' not men/ and the latter 
 equation thus express the principle, that a class whose mem- 
 bers are at the same time men and not men does not exist. In 
 other words, that it is impossible for the same individual to 
 be at the same time a man and not a man. 
 
 125. With reference to the above deduction, the Rev. 
 Robert Harley observes as follows (British Association 
 Report for 1866) : 'From the logical equation x 2 =x the 
 equation x x 2 =o is derived by subtracting x 2 from both 
 members, and the result is put under the form x (i x) = o 
 by the law of distribution. It is to be observed, however, 
 that at every step of the process the principle of identity 
 x=x is assumed, and in Boole's interpretation of the final 
 result the same principle is used, for it is implied that the 
 x without the brackets is identical with the x within. 
 Further, in the final interpretation not only is the principle 
 of contradiction employed, but the principle of excluded 
 middle is also employed. For in interpreting i x to 
 mean not x, it is tacitly assumed that every one of the 
 things of which the universe represented by unity is made 
 up is either x or not x. It would thus appear that these 
 three principles of identity, contradiction, and excluded 
 middle are incapable of being reduced to more elementary 
 truths. They are axiomatic, and Boole made use of them 
 unconsciously in framing his laws of logical interpretation.' 
 
 126. Boole's investigation can be defended from the 
 above criticism; but the defence requires that the definitions 
 and axioms of the science be laid down in an exact manner ; 
 
46 Conditions for a Character being single. 
 
 as I have endeavoured to do in these ' Principles.' Hence 
 I shall consider how the above remarks affect my own 
 investigation, and shall assume that x denotes a character 
 referred to a definite universe. 
 
 127. It is true that the Principle of Identity 
 
 x=x 
 
 is assumed, in the sense assigned to it in Art. 106. Mr. 
 Harley considers it to mean that the one x as a symbol is 
 identical with the other x as a symbol ; whereas it must 
 mean that the result of the operation denoted by the one 
 symbol is 'identical with the result of the operation denoted 
 by the other symbol. 
 
 In the proof that, under the given condition of 
 
 x*=x, 
 
 ix means 'not x' (Art. 120), the Principle of Excluded 
 Middle brought in is 
 
 x-\-i x=i ; 
 which is only another form of 
 
 x=x. (Art. 113.) 
 
 It is to be observed that the Principle of Excluded Middle 
 quoted in the second proof of Art. 121, is that general form 
 of the Principle which is mentioned at the end of Art. 113. 
 
 The Principle of Contradiction is not brought in. The 
 proof follows from the Distributive Law, and the given 
 condition put into the form x* x=o by means of the 
 Principle of Identity and the Second Axiom of Immediate 
 Inference. 
 
 128. What Mr. Harley seems to mean by the Principle of 
 Contradiction is the truth expressed by 
 
 x=i (i x). 
 That principle has been deduced from 
 
 x=x (Art. 112) ; 
 
 and is distinct from 
 
 x (i x)=o, 
 the one being an identity, and the other a condition. 
 
 129. With reference to the same deduction, Venn 
 
Conditions for a Character being single. 47 
 
 observes (Mind, vol. i. p. 491): 'Boole says that the 
 axiom which is termed the Principle of Contradiction, and 
 which " has been commonly regarded as the fundamental 
 axiom of metaphysics, is but a consequence of a law of 
 thought mathematical in its form," viz. the law whose ex- 
 pression is x*=x. This is doubtless a very elementary 
 truth, but to regard it as the source of the Law of Contradic- 
 tion surely argues a strange inversion of order. However 
 that law be regarded, nothing can well be considered more 
 ultimate. We could not distinguish one thing from another 
 without it ; we could not, even to go no further than these 
 symbols, distinguish x from what is not x without making 
 use of it. And yet Boole gives a demonstration of their 
 dependence, a demonstration every step of which demands 
 the law several times over.' 
 
 130. However well this criticism may avail against 
 Boole's doctrine, where no distinction is drawn between the 
 subject and the operations of thought ; I think that it is 
 avoided by the investigation in these ' Principles/ But is 
 that, by which we distinguish one thing from another, the 
 Principle of Contradiction ? It does not appear to be the 
 principle considered in Art. 122. 
 
 131. Boole, as above said, considers x*=x to be a law of 
 thought. He also directs attention to the circumstance, 
 that the equation which expresses that fundamental law of 
 thought is of the second degree. (Laws of Thought, p. 50.) 
 He ventures to assert, that, if that circumstance had not 
 existed, the whole procedure of the understanding would 
 have been different from what it is. In reply to a possible 
 objection, that the existence of the equation x*=x neces- 
 sitates the existence of the equation x*=x, which is of the 
 third degree, he answers : 
 
 1 The equation x 5 =x is not interpretable in the system 
 of logic. For writing it in either of the forms 
 
 x (ix) ( i 
 
48 Conditions for a Character being single. 
 
 we see that its interpretation, if possible at all, must involve 
 that of the factor i+x, or of the factor i x. The 
 former t is not interpretable, because we cannot conceive of 
 the addition of any class x to the universe i ; the latter is 
 not interpretable, because the symbol i is not subject to 
 the law x (i x) = o, to which all class symbols are 
 subject.' 
 
 132. Our position is, that x*=x does not express a law of 
 thought ; as the mind can comprehend with equal ease a 
 selective symbol x, which does not satisfy that condition. 
 It is simply the condition for the character being single and 
 positive. If x is single negatively ; then the equation 
 
 X 3 = X, 
 
 is actually true. The factor i-{-x, considered as a com- 
 pound character, is as interpretable as x; and though i 
 does not satisfy the condition 
 
 x z =x, 
 it satisfies the condition 
 
 #2= X. 
 
 The source, whence it arises that x z =x is the necessary 
 and sufficient condition for x being single, is not the mind, 
 but the actual constitution of things ; so that, being a law 
 of things, it ought to be made a rule for the intellect. 
 
 133. Such, it appears to me, are the relations existing 
 between the properties of the symbol of Quality, the 
 Axioms of Operation, the forms of the Principle of Identity, 
 and the forms of the condition which the symbol of Quality 
 must satisfy in order to be single. A consideration of these 
 relations is of great speculative importance ; for they clearly 
 show that the language of Mathematical Analysis is that 
 scientific language whose existence has been dreamt of 
 by some philosophers. 
 
 134. Leslie Ellis remarked that x 2 =x is not true, when 
 x denotes a relative term, such as ' father.' The arguments, 
 which accompany this statement, appear to show the exist- 
 ence of a supposition, that the organon for reasoning about 
 
Conditions for a Character being single. 49 
 
 the characters of a definite universe can be applied to all 
 departments of reasoning. It is thus of the greatest im- 
 portance clearly to define the nature of that operation, 
 which the Algebra of Quality denotes by x ; as it is to such, 
 and such only, that its method applies. 
 
 135. When a logical equation is given which involves x, 
 we can eliminate x from the equation, if we know that 
 x 2 =x, or x 2 = x. (Art. 286.) But this must be looked 
 upon as another equation as expressing another relation 
 which the symbols must satisfy. Hence, it is no doubt more 
 accurate to say that the elimination of n symbols requires 
 n-\-i independent equations to be given, than to say (with 
 Boole) that no relation whatever can be proved to obtain be- 
 tween the number of terms to be eliminated and the number 
 of propositions from which the elimination is to be effected. 
 
 136. Given that x and y are each positive and single, to 
 find the condition that x-\-y may be single. 
 
 (x+y)*=x+y, (Art. 118.) 
 
 that is, x-\-2xy-\'y =x-\-y, by given conditions ; 
 
 .-. 2xy=o, 
 
 that is, xy=o. (Art. 79.) 
 
 See fig. 5, Art. 148. 
 
 137. Given that x and y are each positive and single ; to 
 find the condition (i) that xy may be single and positive ; 
 and (2) that xy may be single and negative. 
 (i.) (x-y)*=x-y, (Art. 118.) 
 
 .*. x 2 +y 2 2xy=x y. 
 
 that is, x-\-y 2xy=xy^ by given conditions ; 
 that is, 2 (y xy) = o. 
 that is, y (ix)=o. 
 (2.) (*-j)*= _(*_,),, (Art. 118.) 
 
 The truth of these results can be seen from fig. 5. 
 
 D 
 
5<D Conditions for a Character being single. 
 
 138. If each of the factors of a product is single, the product 
 must be single. 
 
 Consider the term xy ; where x z = x, axidy*=y. 
 Then (*y) f =#y. 
 
 =zb^d=y, by given data. 
 
 = #7, by Rule of Signs. 
 
 Hence xy must be single. 
 
 COR. If x and y are each single and positive, x (iy) 
 must be single. 
 
 For i y is single (Art. 120). 
 
 139. If xy is single, and x single ; y is not necessarily 
 single, 
 
 (xyfxy, (i) and# 2 =a;. (2) 
 
 From (i) x*y*=xy 9 
 
 .'. from (2) xy*=xy, 
 
 x 
 
 but is not necessarily = i. (Art. 72.) 
 
 oc 
 
 Hence y* is not necessarily equal to y. 
 
 COR. The same is true, when y= . Boole assumes 
 
 o 
 
 that in such a case can be single only. It must 
 o 
 
 be single within x. 
 
 140. If x and y are each single and positive, then 
 x m x 
 
 For x m -x, andy=j> (Art. 119) ; 
 
 x m 
 
 
 
 y y 
 
 , . x x 
 therefore 
 
 COR. i. Letj/=#. 
 
 Then = 
 
 X X 
 
 Hence in the Algebra of Quality 
 
 x 
 
 ^not in general =x m ~ M . 
 
Conditions for a Character being single. 5 1 
 
 COR. 2. Since i is single and positive, and o single and 
 positive, 
 
 i m i 
 
 Also = 
 
 Also =-. 
 
 0* O 
 
 Observation. -={-}*. See Art 167. 
 
 x \ / x \ 2 
 
 ion. =(). 
 
 jr \y) 
 
 141. If x, y, z are each positive and single ; to find (i) 
 the condition for (x+y) z being single ; (2) the condition for 
 (xy) z being single and positive ; (3) the condition for (x y]z 
 being single and negative. 
 
 (i.) (x+y)* ** = (x+y) z, (Art 118.) 
 
 that is, x 2 z 2 -{-2xyz 2 -{-y z z z =xz-\-yz, 
 that is, xz-\-2xyz-\-yz?=xz+yz. 
 
 .'. xyz=o. 
 
 Observation. The condition xyz=o does not necessarily 
 involve xy=o. 
 
 (2.) (x-y)* * = (x-y) z. (Art. 118.) 
 
 /. xz2xyz+yzxzyz. 
 
 .'.yz(ix)=o. 
 
 (3.) (x-yY z * = -(x-y) z. (Art. 118.) 
 
 . '. xz 2xyz-\-yz= xz-\-yz. 
 .'. xz (iy) = o. 
 
 142. ffx, y, z, u are each positive and single ; to find the 
 conditions for (x+y) (z+u) being single. 
 
 (x+yY (z+uY = (*+y) (*+)- (Art 118.) 
 that is, xz+xu-\-yz+yu+2{xzu+yzu+zxy+uxy} 
 
 =xz+xtt+yz+yu, 
 that is, xzu-}-yzu-\-zxy-{-uxy=o. 
 
 Hence xzu=o yzu=o zxy=o uxy=o' } 
 since no term can be negative. 
 
52 Conditions for a Ctiaracter being single. 
 
 143. If XD x^ x s are each single and positive ; to find the 
 condition for x^-\-x^-\-x z being single and positive. 
 
 (x 1 +x 2 +x s ) z =x 1 +x 2 -}-x 3 (Art. 118.) 
 
 144. If x is positive and not negative ; then 
 
 X 2 X = W, 
 
 where w denotes a quantity which is positive and not negative. 
 Suppose that x consists of two positive single parts 
 
 #! and x 2 . 
 
 Then (x l +xtf-(x l +x z ) = 2XiX, i ; 
 and 2x l x z is positive and not negative. (Art. 138.) 
 
 Similarly it may be shown that when x consists of n 
 positive single parts, the expression x* x is positive and 
 not negative. 
 
 145. Jfx=Xi-}-x z -\-x 3 -{-x 4) where x is single, and x l} x^ 
 x s , x each positive and not negative; then x^ x 2 , x 3 , x+ are 
 each single ; and every sum of two, and every sum of three^ is 
 also single 
 
 x is single, and since it is equivalent to the sum of a 
 number of positive terms, it must be positive ; therefore 
 
 4. (Art. 118.) 
 
 1 . 2 .8-r-4. 
 
 Now Xi 2 =Xi+Wi, where w\ is positive and not negative 
 (Art. 144), 
 and x^x^-\-w^ 
 
 But each of these terms is positive only ; hence each must 
 be=o. 
 
 Hence w l = o, and therefore x^=Xi ; that is, x^ is single. 
 So are x.^ x z , x+. 
 
Conditions for a Character being single. 53 
 
 Also x l x 2 = o' } therefore xi-{-x 2 is single. (Art. 136.) 
 
 Also x l x 2 -{-x l x z +x z x z = o' ) 
 therefore ^1+^2+^3 is single. (Art. 143.) 
 
 The proposition of this Article enables us to draw deduc- 
 tions from that form of equation which Boole discusses 
 very frequently 
 
 
 XII. THE SIGNS OF INEQUIVALENCE > AND <. 
 
 146. Definition of greater than and less than. 
 
 x>y 
 
 denotes that x y is positive and positive only. Hence 
 y<x denotes that y x is negative and negative only. 
 COR. If x and y are each single and positive, 
 
 then x>y 
 means that the x includes the y ; 
 
 and y<x 
 means that the y is included in the x. 
 
 147. Axioms. 
 
 I. If x>y, and a denote any character; then 
 
 x+a>y+a. 
 
 For x+a(y+a)=x-y, 
 
 which is given to be positive only. 
 Similarly ; it follows that 
 
 x a>y a. 
 
 II. If x>y, and a denote any symbol which is positive 
 and positive only ; then 
 
 ax>ay. 
 
 For axay=a (x y) 
 
 but x y is positive only, and a is positive only, therefore 
 
 a(x y) is positive only. (Art. 77.) 
 
 If a is negative and negative only ; then 
 ax<ay. 
 
54 The Signs of Inequivalence > and < . 
 
 For ax ay = a (x y) \ 
 
 but x y is positive only, and a negative only; hence 
 a (xy) is negative only. (Art. 77.) 
 
 Note. In what follows, unless the contrary is stated, 
 the fundamental symbols employed are supposed to be 
 single and positive. 
 
 XIII. DIVISION. 
 
 148. To find the primary parts into which a universe is 
 divided by any number of independent selective operations 
 x, y, z, etc. 
 
 Suppose we have two symbols x 
 
 rcoj 
 
 Now 
 and 
 multiply these two equations together, 
 
 FIG. 5. 
 
 Since x and y are each positive, the 
 parts 
 
 . 
 
 are each positive (Art. 120). Also their sum is equivalent 
 to the whole. 
 
 Since each of the terms is positive, and their sum single, 
 they must be exclusive of one another (Art. 145) ; that is, 
 each term, each sum of two, each sum of three is single. 
 
 These then are the primary parts of i, when we have two 
 independent operations x and y. Fig. 5. 
 
 149. There are other sets of positive parts, the sum of 
 which is equivalent to the whole ; but they each consist of 
 fewer terms than the primary set. For example : since 
 xy+x(iy)=x; 
 
 where we have three positive parts. 
 
 150. Suppose that we have three symbols, x, y, z. 
 
Multiply xy+x(i 
 by 
 
 Division. 55 
 
 -x)y+(i-x)(i-y) = i 
 -y)(i-z)+(i-x)yz 
 
 It may be shown, as before, that xyz, xy (12), etc., are 
 each positive ; and that they are exclusive of one another. 
 Hence they are the primary parts, into which the universe 
 is divided by the three independent operations x, y, z. 
 
 And so for any number of independent symbols. 
 
 151. The mode, in which three independent symbols 
 divide a universe, can be well illustrated by means of a 
 cube. The cube is sup- 
 posed to contain a large 
 number of individuals. 
 The selection of the 
 units according to the 
 character x breaks up 
 the cube into two parts, 
 which are represented in 
 fig. 6, as separated from 
 one another by a single 
 plane. The selection 
 according to y again breaks up each of these two portions 
 into two portions ; the separating partition consists of two 
 parts, which are not necessarily in one plane. Though 
 the value of y is fixed (that of x being supposed to be fixed 
 already) the two parts of the y partition are still free 
 to vary, provided they cut off a sum equal to y. When 
 in addition the value of one of the four parts, as xy, is 
 given, the values of the other three follow necessarily. 
 The selection according to the third character z breaks up 
 the whole into eight compartments. The partition separat- 
 ing the z from the not z consists of four parts. When the 
 value of z is fixed and of xz, yz, and xyz, then the eight 
 parts are fully determined. The value of U is the remain- 
 ing condition required to make up the eight. 
 
 FIG. 6. 
 
56 Division. 
 
 152. The number of parts, into which a universe is primarily 
 divided by n independent operations, is 2 n . 
 
 For the parts, as above shown, are obtained by multi- 
 pliers, each of which consists of two terms ; and there is 
 one multiplier for each symbol; hence when there are n 
 symbols, there are 2 n parts. 
 
 153. Any single positive symbol, a, may be divided in the 
 same manner as i. 
 
 For a(x+i x)=a, 
 
 and 0(y+ 1 y) a > 
 
 therefore a*{xy+x(i y)+(i x)y+(ix)(i y)} =a 2 ; 
 that is, a{xy+x(i y)+(i x)y+(i x)(i y)}=a. 
 
 154. The tree of Independent Division may be repre- 
 sented thus. 
 
 U 
 
 | 
 
 x ix 
 
 I I 
 
 '7' f 
 
 Z I Z Z IZ Z IZ Z IZ 
 
 and so on. This form is exact and much more general 
 than the tree of Porphyry. Porphyry's tree is a division of 
 the particular universe of Substances, and is a mixture of 
 the formal and the material. Only the first two parallel 
 rows are given ; which correspond to 
 
 U 
 
 ix 
 
 i-y 
 
 iz 
 
 the remaining part of the tree being left undeveloped. 
 
 155. Formal Division, when thus treated, is not so un- 
 important as it is commonly supposed to be. Were it kept 
 in mind by men of science when classifying a given group of 
 objects, according to a number of independent characters 
 
Division. 5 7 
 
 Natural Science would gain 'in clearness and exactness. 
 The formal division ought first to be made ; then such parts, 
 as are equal to o, can be left out of the material division. 
 
 156. It is important to remember what can, and what 
 can not, be done by the Algebra of Quality. By it we can 
 show that 
 
 but only observation can tell us what, in any given case, 
 the value of xy is ; or enable us to state any relation 
 between the terms. But there is nothing to prevent the 
 formal logician from supposing certain relations to be given 
 as data, and then proceeding to show what necessarily 
 follows. 
 
 157. Consider any succession of throws with two coins. 
 Let h^ denote having the first head up, h% having the second 
 head up ; / x having the first tail up, and / 2 having the 
 second tail up. 
 
 Then h l -\-t l = i, from the nature of the throw, 
 
 and ^2+^2 = i, ,, 
 
 Multiply together, 
 
 then ^2+^/2+^2+^/2 = 1. 
 
 The four characters h-^h^ h-^t^ t-^h^ t-^t^ are thus subject 
 to the condition, that their sum is equivalent to the whole. 
 To determine them completely, three independent data 
 must be added. 
 
 It is to be observed that ' having the head of the first 
 coin up ' and ' having the head of the second coin up ' are 
 totally distinct characters, and therefore must be denoted 
 by different symbols. The reason why they are denoted 
 by the same symbol h in the Theory of Probability, is, 
 because their ultimate arithmetical values, when U is suffi- 
 ciently great, are equal. 
 
 158. When extra conditions, sufficient to determine 
 the parts necessarily, are not given, we may consider 
 what are the best assumptions to make what are the 
 assumptions which will lead us least astray. In making 
 
58 Division. 
 
 these assumptions we are guided by our knowledge of the 
 nature of 7 and its characters h^ h z , etc. 
 
 The arithmetical values of the parts may be capable of 
 determination a priori, when U is indefinitely large. In 
 the above example, each of the parts will then be = J. 
 
 159. In the Linnaean Classification of Plants we have an 
 example of a formal division by means of two characters 
 and their subordinate characters. Linnaeus considered the 
 universe of 
 
 Plants having flowers 
 
 having stamens of equal length x having pistils. 
 Then 
 
 Monandria + Diandria -f Triandria + . . . 
 
 . . . + Dodecandria -f- Polyandria = i ( i ) ; 
 and 
 
 Monogy nia + Digynia + Trigynia + 
 
 ... + Dodecagynia-f-Porygynia=i (2). 
 By multiplying these two equations together we get the 
 several formal classes. 
 
 If we consider the universe of 
 
 Plants having flowers ; 
 
 then ' having stamens ' and * having pistils ' are two inde- 
 pendent characters ; and * one in number,' * two in num- 
 ber,' etc., are dependent characters. The terms of (i) are 
 necessarily exclusive of one another. They would not be 
 so, if they meant 
 
 having stamens , having stamens , etc. 
 
 one at least two at least 
 
 160. Definition of Contrary. Let x denote a single posi- 
 tive selective operation; then i x is its formal contrary. 
 
 Observation. If y is such that with respect to the given 
 universe 
 
 y=i-x 
 
 in reality ; then y is the material, but not the formal con- 
 trary of x. For example ; in the case of the succession of 
 throws with a couple of coins, considered in Art. 157, 
 
Division. 59 
 
 That is, ' having the tail of the first coin up ' is identical 
 with 'not having the head of the first coin up;' and it is 
 therefore the material contrary of * having the head of the 
 first coin up.' 
 
 161. Consider the parts of the division 
 
 x(iy) agrees with xy in the factor x, but has the contrary 
 factor as regards y. Hence x(iy) may be said to be a 
 contradictory of the first degree of xy ; and xy t a contradic- 
 tory of the first degree of x(iy). Again; (i x)(i y) is 
 contrary to xy, both with respect to x, and with respect to 
 y : hence it is a contradictory of the second degree of xy. 
 
 Definition. A term, which is a product of factors each 
 of the form x or i x, is said to be of the n th order, when it 
 contains n such factors. 
 
 Definition. Two terms of the n th order are said to involve 
 with respect to one another a contradiction of the r tk degree, 
 when r factors of the one are contrary to r factors of the 
 other. 
 
 162. Definition. Two terms of the n th order are said to 
 be opposites, when they involve a contradiction of the- n th 
 degree. 
 
 For example; xy and (i #)(i y) in the one diagonal 
 of the square (fig. 5) are opposites : so also x(iy) and 
 (i x)y in the other diagonal. The two terms, which lie 
 in any one of the four diagonals of the cube (fig. 6), are 
 opposite to one another. 
 
 Since two opposite terms have r=n, we can speak of an 
 opposition of the #** degree. 
 
 COR. An opposition of the first degree is a contrariety. 
 
 163. If it is objected that the above definitions are not 
 in strict accordance with usage, I answer that they are at 
 least exact and equal to the complexity of nature. It is 
 impossible to use words in strict accordance with a usage 
 
60 Division. 
 
 which is contradictory. What some have denoted by con- 
 trary, others have denoted by contradictory. 
 
 The importance of an exact definition of these terms will 
 be seen from a consideration of the use De Morgan makes 
 of contrary. He says what is equivalent to the assertion 
 that (ix)(i y) is the contrary of x-\-y. Now 
 (i x)(iy) = i -xy+xy, 
 
 provided xy=o. 
 
 But xy is not necessarily =o. Hence if i x is defined 
 to be the contrary of x ; (ix)(iy) is not the contrary 
 of x-\-y. 
 
 164. In a division by n independent selective symbols, each 
 primary term has 
 
 i . 2...r 
 contradictories of the r* h degree. 
 
 The primary terms are formed by multiplying together 
 n expressions, each of which is the sum of a symbol taken 
 directly and the same symbol taken contrarily (Art. 148). 
 Let a denote directly and b contrarily ; then the expansion of 
 
 by the Binomial Theorem, groups together the contradic- 
 tories of the different degrees. 
 
 a n denotes the particular term considered ; 
 
 a n ~ l b a contradictory of the first degree ; 
 
 a n ~W second 
 
 Now 
 
 Hence the number of contradictories of the r th degree with 
 respect to the given term is 
 
 n(n-i)...(n r+i) 
 i. 2 ... r 
 
Expansion of a Function, etc. 61 
 
 COR. i. A term has but one opposite. 
 For when r=n, 
 
 i. 2 ... r 
 
 COR. 2. When there are n symbols, there are 2 n ~ l pairs 
 of opposites. 
 
 For each term has one opposite \ and there are 2 H terms 
 (Art. 152). But only one-half of these 2" pairs will be 
 different ; hence the number of pairs of opposites is 
 
 -! 
 
 XIV. EXPANSION OF A FUNCTION OF A 
 NUMBER OF INDEPENDENT SINGLE AND 
 POSITIVE OPERATIONS IN TERMS OF THE 
 PRIMARY PARTS INTO WHICH THE UNI- 
 VERSE IS DIVIDED BY THE OPERATIONS. 
 
 165. To expand in terms of x and i x, which are the 
 
 00 
 
 parts into which the universe is divided by the selective opera- 
 tion x. 
 
 a and b being independent of x. 
 
 The symbol x, being single and positive, can assume the 
 value i and the value o. Hence the above equation, being 
 an identity, must be true when either of these values is 
 substituted for x. 
 
 Let x=i. Then i x=o-, and o also =o, for b is an 
 absolute constant. Hence a=i. 
 
 Let#=o. Then 00=0, because a is also an absolute 
 
 constant. Hence b= . 
 o 
 
 Thus -=*+-(i-*). 
 
 X O 
 
62 Expansion of a Function, etc. 
 
 Observation i. The above is the expansion for , whether 
 occur by itself or as a factor in an expression. If it 
 
 00 
 
 occur by itself, and is limited to being a possible operation 
 
 then i x, being the factor of , must be =o (Art. 79). 
 
 o 
 
 Hence x=i. 
 
 Observation 2. The general expansion for x is 
 
 x=x+o(ix), 
 which is true whether x occur by itself or as a factor. 
 
 166. To expand - . 
 x 
 
 x=x+o(ix), 
 I = *+l-(i-*). 
 
 x(ix)-\ x(i x)-\ 0(1 
 
 and - = *+ 
 
 .-. x=x z 
 
 Now x z =x, given condition. 
 
 .-.(i -*) = !-*; 
 and x(i x)=o. 
 Hence the third term consists of an absolute constant 
 
 multiplied by o ; it therefore vanishes. The fourth 
 o 
 
 term has the constant multiplied by a constant o ; its 
 o 
 
 co-efficient is therefore . 
 
 Thus =*+(!-*). 
 
 x o v 
 
 This expansion, of course, can be obtained directly. 
 
 COR. i. is always possible. For no impossible co- 
 efficients occur in its expansion. 
 
 COR. 2. is not always single. 
 
 oc 
 
Expansion of a Function, etc. 
 
 For example ; suppose that we consider 
 
 x=x(y+z), 
 
 where y and z denote the circles of 
 fig. 7, and x the shaded parts of these 
 circles. 
 
 , 
 oc 
 
 for x is single and positive. 
 
 Fig. 7. 
 
 X 
 
 Now y+z recurs ; hence is not necessarily single. 
 
 x 
 
 X 
 
 In the case of represented by the figure, 
 
 OC 
 
 COR. 3.. When is single; is restricted to being 
 
 single within i x. 
 
 For each of the terms of the expansion is positive only ; 
 
 hence each must be single (Art. 145). Therefore must 
 
 o 
 
 be single within i x (Art. 139). 
 
 X f X \ 2 X* 
 
 167. When is taken indefinitely, I 1 = - 
 
 X \X J X 
 
 Now 
 
 = x+(\ (i x) ; for x* is given =x. 
 
 f J cannot be more indeterminate than 
 \o) o 
 
 
 .'(I taken generally =^ + (i x). 
 
 (Art. 80.) 
 
64 Expansion of a Function, etc. 
 
 But = , for x*=x 
 x z x' 
 
 =*+(!-*).. 
 
 (X \ 2 X* 
 
 j taken generally = 
 
 168. To expand x+y. 
 
 ~Letx+y=axy+0x(i-y)+<:(i-x)y+<t(i-x)(i-y). 
 
 Since x and y are each positive, they can each assume 
 the values i and o. If they are not restricted by any con- 
 dition, there are four sets of singular values, viz. : 
 
 x=i y=i (i) 
 
 *=i y= (2) 
 
 x=o y=i (3) 
 
 x=o y=o (4) 
 
 The above equation, being an identity, is true when any 
 one of these sets of values is substituted for x and y. Put 
 in the first set, then 
 
 Since b, c, d, do not involve any variable; 
 = 0, <ro=o, do=Q. 
 
 Hence a=2. 
 
 Similarly by means of the other sets of values, we get 
 
 ^=i <r=i d=Q. 
 Therefore x+y= 2xy~\-x(i y) +( i x)y+o(i x)( i y). 
 
 Observation i. The above is the general expansion for 
 x+y, that is, the expansion which is true, whether x+y 
 occur by itself or as a factor. In the former case the 
 equation can be reduced to 
 
 x+y= 2xy+x(i -^)+(i x)y. 
 
 Observation 2. If x+y be restricted to be single, then 
 each of the terms must be single. (Art. 145.) Hence 2xy 
 must be single ; which necessitates 
 
Expansion of a Function, etc. 65 
 
 Suppose it is given that x-\-y is single; we can expand 
 taking account of that datum. 
 
 For then xy=o (Art. 136); hence 
 
 x+y=fa(i -y)+c(i x)y+d(i x)(i -y). 
 But now only three sets of singular values are possible, 
 viz. : 
 
 x=i jy=o 
 
 X=Q y=i 
 
 x=o y=o> 
 
 These, however, suffice to determine the three constants b,c,d. 
 
 169. To expand xy. 
 
 Since there are two independent symbols, there are four 
 co-efficients ; and we can form four equations to determine 
 the four co-efficients, as in the case of x+y. We get 
 
 a=o, #=i, = i, d=o. 
 
 Hence xy=oxy-\-x(iy) ( ix)y-}-o(ix) (iy). 
 When xy occurs by itself, the expansion can be re- 
 duced to 
 
 x-y=x(i-y)-(i-x)y. 
 
 170. To expand -. 
 
 Since x=x-\-o(i x), 
 
 and L =y+ L (l _ y)i (Art. 165.) 
 
 multiply together, then 
 
 for Xo= , when the factor o is not a part but a co- 
 o o 
 
 efficient. 
 
 COR. i. When is an expression by itself, the third 
 term may be cancelled. 
 
66 Expansion of a Function, etc. 
 
 COR. 2. When in addition is possible, the second 
 
 term may be cancelled. 
 
 For the expansion must be possible, and it contains only 
 positive terms ; hence each term must be possible. There- 
 fore x (i-y) = o (Art. 79). 
 
 Now is an absolute constant, since we are supposing 
 o 
 
 to occur by itself; hence the x o formed by the second 
 y o 
 
 term is =p. 
 
 The truth of this last statement may be seen better by 
 means of the following considerations. 
 
 When is restricted to be possible, y cannot take the 
 
 value o without x taking the value o simultaneously. Hence 
 we have only three sets of singular values, viz. : 
 x=i y=i 
 
 x=o y=o. 
 
 These ought to be sufficient to determine the expansion ; 
 but they will be so, only if we write 
 x 
 
 where the second term of the general expression is omitted. 
 Hence = 
 
 COR. 3. If in addition be restricted to be single, then 
 the fourth term of the expansion is single (Art. 145). 
 171. To expand with respect to y alone. 
 
 Since -l=j,+-I(i--y). 
 
Expansion of a Function, etc. 67 
 
 If is possible, then the factor multiplying must be 
 
 = o. Hence x ( i y) = o. 
 
 x o 
 
 .-. X y-\ 
 
 y ^o 
 
 We are not entitled to say that the which occurs here 
 
 o 
 
 is = o ; because the o of the numerator depends upon the 
 
 co-efficient as well as upon i y. 
 o 
 
 If in addition is single ; then = (ixy). 
 y o o v 
 
 But xy=x, for x(iy) = o ; 
 
 x 
 
 This expansion differs from that obtained by expanding 
 with respect to both x and y in not having the second term 
 multiplied by i y. It is therefore less definite. 
 
 172. To expand . 
 x+y 
 
 By applying the method of Art. 168 we obtain 
 i 
 
 x+y 2 '; ' o x 
 
 This is the general expansion for - 
 
 When we consider by itself, and restrict it to be 
 x+y 
 
 possible ; then the constituent multiplying the co-efficient 
 must be =o (Art. 79). Also the constituent multiplying 
 
 the co-efficient must be =o; for xy does not contain 2 
 as a factor (Art. 78). 
 
 173. To expand - 
 
 x+y 
 
 (Art. 168.) 
 
68 Expansion of a Function, etc. 
 
 and ^ = 7^+^ I -^)+( I -^+^( I -^)( 1 -^)^ 
 
 (Art. 172.) 
 multiply together, then 
 
 for all the other terms vanish, on account of their consti- 
 tuents being =o j 
 
 but-j=i (Art. 78), 
 
 This result may be easily verified by expanding directly. 
 I have deduced it by means of the results of Arts. 168 and 
 172, for the purpose of showing that when the function 
 expanded is a factor of an expression, its true expansion 
 consists of all the terms, even though some of these have 
 impossible or zero co-efficients. 
 
 174. To expand x 
 
 y+z 
 Since x=x+o(i x), (Art. 165.) 
 
 and ^g=^y z +J f ( l --*)+( I -y) z +^( 1 -y)(*-*) 
 
 (Art. 172.) 
 
 When -^ occurs by itself or multiplied by an absolute 
 y+z 
 
 constant, the fifth, sixth, and seventh terms vanish. (Art. 79.) 
 
 When in addition -^ is possible, the constituents of the 
 y+z 
 
 first and fourth terms must each be equal to o ; and there- 
 fore the terms themselves equal to o. Then 
 
Definition. 69 
 
 together with the conditions xyz=o, and^(i y)(i z) = o. 
 
 If in additio 
 must be single. 
 
 If in addition -^ is single, then (ix)(iy)(iz) 
 y+z o 
 
 XV. DEFINITION. 
 
 175. Formal definition is the establishment by conven- 
 tion of an equivalence between a complex expression and a 
 simple expression. 
 
 176. For instance; it may be said 
 
 Let ix=x'. 
 
 Here an equivalence is established between the significa- 
 tion of the complex character i x and the simple character 
 x'. Such a convention, if adopted in a particular course of 
 reasoning, must be adhered to throughout ; but it is some- 
 times established not for one course of reasoning only, but 
 to hold in general. 
 
 The equivalence is established for the sake of gaining 
 shortness in writing, and also for the purpose of expressing 
 individually (as it were) the result of the complex expres- 
 sion. 
 
 177. We may also say 
 
 Let x y =x'. 
 
 Here we denote by one symbol the result of the product of 
 the dependent character y upon the character x, on which 
 it depends. 
 
 The kind of conventional equation, usually considered 
 by logicians under the head of ' Definition,' is related to the 
 preceding form. Its type is 
 
 Let Ux= U'. 
 Here U' is the individual symbol denoting the universe Ux. 
 
70 Inference from one or more Equations 
 
 Conventional equations of this kind are usually of a per- 
 manent nature. 
 
 178 As an example of 
 
 U 1 Ux, 
 
 I take the definition of ' Vertebrata ' given by Nicholson 
 (Manual of 'Zoology ', p. 383). When expressed analytically, 
 it is as follows : 
 
 Vertebrata 
 
 . I f having a body composed of a number of definite segments 
 
 \ arranged along a longitudinal axis 
 
 X having a nervous system 
 
 in its main masses dorsal X completely shut off by a partition from 
 
 the haemal region of the body 
 X having limbs 
 
 not more than four in number X turned away from the neural 
 
 aspect of the body 
 
 ( having a bony axis known as the vertebral column+without a ) 
 I vertebral column X having a notochord j ' 
 
 Here x is of the form a x b yz "(,-> {*/+( i d)e] ; and 
 affords a good example of the import of dependent cha- 
 racters. 
 
 XVI. INFERENCE FROM ONE OR MORE 
 EQUATIONS OF THE FORM 
 
 x=m ; 
 THAT IS, FKOM CATEGORICAL JUDGMENTS. 
 
 179. Form of the equation. An unknown single character 
 of the first order and with unity for its co-efficient is 
 equated to a known character, or else to a known arith- 
 metical value. These are written 
 x=m, 
 and x=m, 
 respectively. The former is read 
 
 Z7's which are x are identical with 7*8 which are m ; 
 and the latter 
 
 f/'s which are x have the arithmetical value of U's 
 which are m. 
 
of the form xm. ( Categorical. ) 7 1 
 
 It is generally sufficient, when an investigation has to be 
 made, to make it for the former kind of equation only ; as 
 the result obtained is generally true foi^the latter kind of 
 equation, provided m be changed into m. 
 
 180. The primary categorical forms comprise four special 
 cases of the equation 
 
 x=m. 
 
 When 
 m=i , the equation becomes the universal affirmative. 
 
 = o; universal negative. 
 
 =v (a fraction greater than o) particular affirmative. 
 
 = i v; particular negative. 
 
 Hence x=m may_be called the general form of the cate- 
 gorical judgment, m is the definite some considered in the 
 Theory of Probability. 
 
 181. The term categorical is frequently ascribed to judg- 
 ments which differ in an important respect from those to 
 which the term is primarily defined to apply. The forms 
 
 xy=i, xy=o, xy=v, xy=iv 
 are not distinguished from the forms 
 
 tf=I, # = 0, X = V, X=IV. 
 
 For instance ; one of the examples of a universal negative 
 categorical judgment given by Ueberweg is (System of Logic > 
 p. 308) 
 
 No innocent person is unhappy. 
 But this is converted into 
 
 No unhappy person is innocent. 
 
 In the new form of the statement ' person ' remains where 
 it was before ; hence ' persons ' and not * innocent persons ' 
 is the true subject of thought. The analytical expressions 
 for these judgments are 
 
 Persons {innocent unhappy =o}, 
 and Persons {unhappy innocent =0} ; 
 
 both of which are of the form 
 
72 Inference from one or more Equations 
 
 The above equation contains a term of the second order, 
 and is therefore very different from one which contains a 
 term of the first order only. The equation 
 
 where the term on the left-hand side is of the second order, 
 may be called the general form of the hypothetical equation 
 of the first degree. 
 
 182. Immediate inference by Contraposition. 
 
 If x=m, 
 then i #=1 m. (Art. 108.) 
 
 In the same manner, 
 
 if x=m, 
 then i #=1 m. 
 
 The latter is a well-known proposition in the Theory of 
 Probability, and is one of the logical theorems which are 
 constantly assumed in stating arithmetical problems. 
 
 COR. i. Let#=i. Then i x=o. 
 
 COR. 2. Let x=v, where v is greater than o. 
 Then i x=i v ; where iv must be less than i and 
 may be o. 
 
 It is said that no inference can be drawn by contraposi- 
 tion from 
 
 xv. 
 
 But this is not correct. It follows necessarily that 
 
 IX=IV, 
 
 what does not follow, is, that i z/ must be greater than o 
 when v is greater than o. 
 183. Conversion ofx=m. 
 The only way in which the equation 
 
 x=m 
 
 admits of conversion, consists in exchanging the position of 
 its members thus 
 
 m=x. 
 
 It is only to such equations, as involve two unknowns x and 
 
of the form xm. ( Categorical.) 73 
 
 y, that a process of conversion in the proper sense of the 
 term applies. For instance 
 
 xy=m 
 may be changed into yx=m ; 
 
 and x=y w 
 
 into y=x-\-w. 
 
 184. An attempt, however, is frequently made to convert 
 the categorical form after the manner of an hypothetical of 
 the first degree. It is, for instance, said that 
 
 All men are fallible ; 
 can be converted into 
 
 Some fallibles are men. 
 
 But several logicians have observed that such a procedure 
 confounds Accident with Substance. The universe of the 
 judgment is confounded with the characters of the judg- 
 ment ; selective symbols are changed into elective symbols. 
 The result in the above or any other instance does not re- 
 commend the process. The given judgment really means 
 
 Men {i=fallible} ; 
 which can be written 
 
 Men{fallible=i}. 
 The inference by contraposition is 
 
 Men (o=not fallible}. 
 
 185. It is important to consider the differences in the 
 signification of 
 
 #=i, 
 
 according to the nature of U as regards variables. If Per- 
 sonality (which is the logical Space) is constant, and State 
 (which is the logical Time) varies ; then 
 
 x=i 
 
 expresses that in every state of the given individual the 
 character x is present. 
 
 Similarly, when the State is fixed, and the Personality 
 varies; x=i 
 
 expresses that in every individual in the given state, the 
 character x is present. 
 
74 Inference from one or more Equations 
 
 When both Personality and State vary ; 
 
 #=1 
 
 expresses that in every individual in every state, the cha- 
 racter x is present. 
 
 Examples of these respective cases are 
 
 This crow is always black. 
 
 All crows at a given age are black. 
 
 All crows are always black. 
 
 186. The formal treatment of the equation 
 
 x=i 
 
 is the same, whether the equation express an essential pro- 
 perty, a general property, or an inseparable accident (for 
 the given universe). The Algebra of Logic makes no dif- 
 ference in this respect between the necessary truth 
 
 Triangles { equilateral = equiangular} , 
 and the observed fact 
 
 (having condyles on the occipital bone X possessing red blood corpuscles 
 two in number non-nucleated , 
 
 =suckle their young, j 
 
 187. Given x = m and y = n ; 
 
 to find what can be concluded about xy, x(iy), (ix)y, 
 and (i x) (iy). 
 First, for xy. 
 
 Since (i x) (i y)=i x y-\-xy, 
 
 xy=x+yi+(ix) (iy) ; 
 
 but (i x) (iy) is positive only; (Arts. 120 and 77.) 
 therefore xy is greater than x+y i ; 
 
 that is,xy>x+yi. (Art. 146.) 
 Again ; since x (i y)=x xy, 
 
 xy=x-x (iy); 
 
 but x(iy) is positive only; (Art. 120.) 
 
 therefore xy is less than x ; 
 
 that is, xy<x. (Art. 146.) 
 
 Similarly it may be shown that xy <y. 
 
of the form xm. ( Categorical. ) 75 
 
 Second, for x(i y). 
 
 Since xyx-\-y i + (i x)(i jv), 
 
 let y be changed into i y ; 
 
 then x(iy)=xy+(i x)y; 
 
 but (i x)y is positive only ; 
 
 therefore x (i y) >x y. 
 And in a similar manner it may be shown that 
 
 and <i y. 
 Third Jor(ix)y. 
 
 (i x)y is the same as x(i y), excepting that x 
 have replaced one another. Hence 
 
 Fourth, for ( i x) ( i y). 
 
 Change x into i x, and y into i y, in the limits for xy. 
 Then 
 
 188. Hence by putting in the given data 
 
 x=m, and^=, 
 we obtain 
 
 xy>m+n i ; 
 
 (i y)>m n; 
 
76 Inference from one or more Equations 
 
 Observation. When the data are of the form 
 
 x=m, y=n, 
 
 the above inequalities are still true, provided the right-hand 
 members be stroked with a vinculum. 
 
 189. From the above we infer; that any two terms, 
 which involve two degrees of contradiction, have the sum 
 of their minor limits =o ; and the sum of their correspond- 
 ing major limits =i. Also that any two terms, which in- 
 volve but one degree of contradiction, have the sum of their 
 
 minor limits ==h ! 2J^ ) i land the sum of their corre- 
 
 \n 
 spending major limits = 2 4 * _ 
 
 [i- 
 
 190. Any one of the equations 
 
 xy=x+y-i + (i-x)(i- 
 
 xyx x(iy), 
 
 xy=y-y(ix) 9 
 
 suffices to give xy exactly, provided each one of the terms 
 of the right-hand member is given as a datum. 
 
 191. Example. Suppose that of the persons on board a 
 ship which was wrecked, the passengers formed two-thirds ; 
 and those that were saved in the wreck three-fourths. How 
 many passengers must have been saved,, how many lost ; 
 how many of the crew must have been saved, how many 
 lost? 
 
 Let/ denote the passengers ; c the crew ; s saved ; / lost. 
 
 Then/-f<r=i ; .-. c\p. 
 
 Also s+l= i ; .-./=! s. 
 
 And/=|; j=f. 
 
 Now /*>/-}-.? i, 
 
 The passengers saved formed at least five-twelfths. 
 
of the form x=m. ( Categorical. ) 7 7 
 
 Since the limit of (i /)(i s) is the same as that for ps 
 but with a contrary sign ; 
 
 (I -/)(!-,)> -A. 
 
 /<?j/ may have been none. 
 Again. ( I p) s> /+j, 
 
 -- >-*+!, 
 
 7% rrae; saved formed at least one-twelfth. 
 
 Since the limit for p(is) is the same as that for (i p)s 
 but taken with the contrary sign, 
 
 /(I -*)>-_!. 
 
 The passengers lost may have been none. 
 
 192. Suppose the additional information given : that 
 the crew lost formed one-sixth. 
 
 Then(i-/)(i-,)=J. 
 Now /*=/+* i+(i /)(!-*), (Ait. 190.) 
 
 -^) = i-i-(i-j)(i-/), (Art. 190.) 
 =f-f, 
 =A- 
 Also(i />=i/ (i-/)(i-j), (Art. 190.) 
 
 Suppose in addition that the persons on board were 1 20 
 in number. 
 
 Then 7= 120. 
 
 And Ups='jo. (i) 
 
 Up(i-s)=io. (2) 
 
 *7(i-/).= 2o. (3) 
 
 tf(i-/)(i-.0 = 20. ( 4 ) 
 
 Also, when the data of Art. 191 only are given ; 
 
 (i) 
 10. (2) 
 
7 8 Inference from one or more Equations 
 
 193. Given 
 
 x=m; y n; z=p ; 
 to find what can be concluded about xyz. 
 
 Since xy=x-\-y i + (i x)(i-y)', 
 
 multiply by z, 
 
 then xyz xz +yz z-\-(i x)(i y)z. (Art. 108.) 
 But xz=x+z i + i --#i 2 
 
 + (!-*)(! -y),. (i) 
 
 Now (i #)(i s) + (i--y)(i 3)-|-(i ^)(I-JF>, being 
 a sum of terms each of which cannot be negative, cannot 
 itself be negative ; 
 
 .'. xyz>x+y+z 2. (Art. 146.) 
 
 Again, xy=xx(i y) ; 
 
 multiply by z, 
 
 then xyz=xz x(iy)z; (2) 
 =x x(i z)x(iy)z 
 <x. 
 
 For the remainder is minus the sum of two terms each of 
 which can be positive only. (Art. 146.) 
 Similarly it can be proved that 
 
 xyz<y, and <z. 
 
 Hence, by putting in the given values, 
 xyz > m + n -\-p 2. 
 
 194. Other two equations may be derived from (i) by 
 cyclical change of the symbols ; but, as the terms of the 
 first order enter similarly into (i), the derived equations 
 will give the same limit as (i) gives. 
 
 Also, two other equations may be derived from (2) 
 by cyclical change of the symbols. Each of these yields 
 two forms; but only three limits of the kind considered 
 can be derived. 
 
of the form x^m. ( Categorical. ) 79 
 
 195. The limits for any of the other terms of the third 
 order can be deduced from those for xyz. 
 
 For example; to find those for x(iy}(iz). 
 
 i-y=i-n, 
 and i z= \p. 
 .-. x(i y)(i z)>m+i fl+i p 2; 
 
 >mnp. 
 Also. x(i jy)(i z]<m. 
 
 196. Generally 
 
 
 Suppose it true for n; and multiply by x n+l . 
 Then 
 
 (Art. 187.) 
 
 .+x n +x n+l (n+i)+i. 
 Hence if the inequality is true for n, it is also true for 
 
 +i. 
 
 t it is true for 2 ; hence it is true generally. 
 COR. If r of the factors are direct, and n r contrary; 
 e minor limit becomes 
 
 Xi+Xt + ...+x r x r+l x r+z ...x n r+i. 
 For the minor limit is 
 
 = ,, -r+i. 
 
 197. De Morgan makes some investigations upon the sub- 
 ject of the preceding ten articles under the name of Numeri- 
 cally Definite Syllogisms. The fundamental data with 
 which he deals are inequalities, and he always supposes m 
 to be arithmetical. But there is nothing to prevent us from 
 supposing that the fundamental data are equations, and 
 investigating what follows. A slight reflection upon the 
 nature of statistical data will serve to show the importance 
 of such an investigation. 
 
8o Inference from one or more Equations 
 
 198. Given a x =m, a y =n, a=p ; to find limits to a xy . 
 
 a*y = ax+y-\ + (\- X ) (!->)> 
 
 =a x +a y -a+a (l - x ) (l _y>. (Art. 70.) 
 
 a xy >a x +a y a. 
 Also a xy a x - x (^- y ^ 
 
 =a x -a x(l - y} . (Art. 70.) 
 
 a xy <a x . 
 
 Similarly a xy <a y . 
 Hence a xy >m+n p. 
 but < m. 
 and <n. 
 COR. Let a= i. 
 Then we get the results of Art. 188. 
 
 199. To find limits to , when it is single. 
 
 Since is given to be single. 
 
 where (i x) (i y) is single. (Art. 170, COR. 3.) 
 o 
 
 Hence (i x) (i y) < i x and < i y ; 
 
 o H 
 
 also xy<x and <j ; 
 
 .\<x+ix,t.e.<i. (i) 
 
 (2) 
 
 (3) 
 
 <i. (4) 
 
 But as xy=x (Art. 170), the expansion can be reduced to 
 
 (5) 
 
 therefore, the important limits are 
 
 , x 
 and >x. 
 
of the form xm. ( Categorical. ) 8 1 
 
 COR. i. Letj=i. 
 
 Then equation (5) becomes =x. 
 COR. 2. Let#=i andjy =i. 
 
 Then equation (5) becomes =i. 
 
 XVII. INFERENCE FROM ONE OR MORE 
 EQUATIONS OF THE FORM 
 
 xym; 
 THAT IS, FROM HYPOTHETICAL JUDGMENTS. 
 
 200. Form of the equation. We have an unknown single 
 character of the second order equated to a known symbol, 
 which may either be a symbol of identity or express arith- 
 metical value. 
 
 xym 
 means that 
 t7's which are x and y are identical with 7's which are m. 
 
 xym 
 means that 
 
 ' 7's which are x and y have an arithmetical value equal 
 to that of Urn ; or, simply, that xy has an arithmetical value 
 m. 
 
 201.' A particular case is 
 
 xy=x; 
 
 which is an equation universal with respect to x, and means 
 that 
 
 7's which are x are y. 
 
 xy=y 
 is the other universal form. 
 
 xy=o 
 means that 
 
 /'s which are x and y are none. 
 
 F 
 
82 Inference from one or more Equations 
 
 202. Varieties of the -equation xy=.m. 
 There are four varieties 
 
 xy=m. (i) 
 
 x(iy) = m. (2) 
 
 (ix)y=m. (3) 
 
 (ix)(iy) = m. (4) 
 
 Each of these has two universal forms, viz. 
 
 (1) xy=x; xy=y. 
 
 (2) x(L-y)=x; x(i-y)=i-y. 
 
 (3) (L-x)y=i-x; (i-x)y=y. 
 
 (4) (i-x)(i-y)=i-x; (i-x) (i-j') = i-j. 
 
 203. The right-hand member, m y may assume a great 
 variety of forms. 
 
 For example 
 
 xy=\ x. 
 X y=i-y. 
 xy x-\-yi. 
 Or we may have 
 
 xy=ix. 
 xy=i-y. 
 xy=x-\-yi. 
 
 The latter examples differ from the former in that the 
 equivalence asserted does not involve identity. 
 Observation. The equation 
 
 xy=x+y-i 
 is an equation of Condition ; whereas the equation 
 
 xy=x+y i + (i x) (i y) 
 is an Identity. 
 
 204. Jfxy=m, then xy=m; btit not conversely. 
 
 For xym expresses the identity of the 7's which are x 
 and y with those which are m. Hence the arithmetical value 
 of xy must be equal to that of m ; that is, 
 
 xy~m. 
 But suppose it is given that 
 
 xy=m. 
 
of the form xy m. ( Hypothetical. ) 83 
 
 Since xy may in general have the quantitative value m in 
 a plurality of ways, and 
 
 xy=m 
 is only one of these ways ; it does not follow that 
 
 xy=m. 
 
 COR. If 'xy=x, then xy=-x. 
 
 For xy can have the quantitative value x in only one 
 way ; and xy=-x 
 
 is one way. 
 
 205. There are certain forms of the equation 
 
 xy=m, 
 
 which can be satisfied only by one or both of the symbols having 
 a singular value. 
 
 First example ; xy=ix. 
 
 Then xy+x=i, 
 
 .*. xy=o, (Art. 136) for i is single. 
 .*. i #=o, by given equation. 
 
 .. x=i j 
 and .:y=o. 
 Second example ; xy=x -\-y. 
 
 Then xy=o, (Art. 136.) 
 
 .-. x+y=o, 
 .-. x=o and^=o. 
 For both terms are positive only. 
 Third example ; xy = x ( i y). 
 Then 2xy=x t 
 .-. xy=o; for x is single. 
 
 .-. ^=o. 
 y is left undetermined. 
 
 206. The equation may express a condition, which can 
 be reduced to a simpler condition. 
 
 For example; let xy=(ix) (iy). 
 
 Then xy= i xy-\-xy ; 
 .'. 0=1 xy; 
 
84 Inference from one or more Equations 
 
 207. Given xy m; 
 
 to fatdx(i y), (ix)y, (ix) (iy) in terms of x, y, 
 and m. 
 
 First, x ( i y) =x xy, 
 
 =x m. 
 
 Second, (ix)y =y m. 
 
 Third, (i *) (i -.>') = ixy+xy, 
 
 The last equation, which gives the expression for the 
 opposite of the given term, is commonly called the inference 
 obtained by Contraposition. The other two equations give 
 expressions for those terms, which involve only one degree 
 of contradiction. 
 COR. i. Let xy=x. 
 
 Then*(i-jy) = o. (i) 
 
 (ix)y=y-x. (a) 
 
 (i-x)(i-y) = i-y. (3) 
 
 COR. 2. Let xy=x-\-y i. 
 
 Then^(i j) = i- y. (i) 
 
 (i-x)y=i-x. (a) 
 
 (i-x)(i-y) = o. (3) 
 
 COR. 3. Let xy=so. 
 
 Then^(i y)=x. (i) 
 
 (!-x)y=y. (a) 
 
 (i-x)(i-y)=i-x-y. (3) 
 
 208. Let xyv^ where v is greater than o. 
 
 Then #(i 7)=.% z>; 
 
 but v, though restricted to being greater than o, may be 
 equivalent to x. Hence x(i y) is not necessarily greater 
 than o. 
 
 It may be shown in a similar manner that (ix)y is not 
 necessarily greater than o. 
 
 Again. (i x)(i ^)=i x y+v; 
 
 but v may be equivalent to x-\-y 1 
 
 .-. (i x)(i y) is not necessarily greater than o. 
 
of the form xy m. (Hypothetical. ) 85 
 
 The last statement contains what is meant by the pro- 
 position, that the particular affirmative (hypothetical) judg- 
 ment does not admit of Contraposition. 
 
 209. The Conversion of an Equation consists in making 
 two symbols replace one another as far as possible. 
 
 As applied to equations of the form 
 
 xy=m 
 the Conversion may be Simple, or per Accidens. 
 
 210. The Conversion is Simple, when x and y enter 
 similarly into the equation. 
 
 For example : if xy=o ; 
 
 211. The Conversion is per Accidens, when ^andjy do not 
 enter similarly into the equation. 
 For example : let xy=x. 
 
 but y > x, 
 
 .: y x is in general greater than o. 
 Hence the equation yx=y 
 
 does not in general follow ; the additional condition required 
 being 
 
 y=x. 
 
 212. Given xy=.m, and yz-=n; 
 
 to find minor limits to xz, x(\z),(\ x)z, and (ix)(iz) 
 expressed in terms of m, n, x, y, z. 
 First for xz. 
 
 Since xz=x+z i-{-(ix)(iz) ; 
 multiply byy, 
 
 then xyz=xy+yzy+(ix)y(iz], 
 
 .'. xyz>xy-\-yzy, 
 for (i x)y(i z) is positive only. 
 
 But xz>xyz, 
 .'. xz>xy+yzy. 
 
 Hence, by putting in the given data, 
 xz>m-\-n y. 
 
86 Inference from one or more Eqitations 
 
 Similarly; since 
 
 x(i-y)z>x(i-y)+(i-y)z-(i-y); 
 and xz>x( iy)z; 
 
 that is, >xy yz i -f x +y 4- z, 
 that is, > m n i-}-x -\-y-\-z. 
 Second for x(i z). 
 
 Since zy=n, 
 
 (iz)y=yn. 
 
 Thus z and 12 replace one another in one of the given 
 conditions, and so also n and yn. The results, when 
 similarly transformed, will be true. 
 
 Hence x(i z)>m-\-y n y, 
 
 that is, >m n. 
 Also #(i z)>m (/) i +x+y+ (i 2), 
 
 that is > m+n+x z. 
 In the same manner it may be shown that 
 (i x)z> m+n; 
 
 and >mnx-\-z. 
 Also that 
 
 (i x)(iz) >m n+y ; 
 
 and >;//+;/+! xy z. 
 COR. Let y= i. 
 
 Then xz>m-\-n i, and >o. 
 
 ^r(i z)>mn, and >o.. 
 (i ^)^> m-{-n, and >o. 
 (i ^)(i z)> w ^+i, and >o. 
 These are the results of Art. 188. 
 
 The second limit ought in each case to become o ; for 
 when y= i, i j=o. 
 
 213. The sum of the two limits for any one of the terms is 
 independent of m and n. 
 
 Since xyz>m-\-n-y, 
 and #(i y)z> m n i -\-x-\-y -{-z, 
 add, then xz> i+x-\-z; 
 
 which is independent of m and n. 
 
of the form xy m. (Hypothetical. ) 87 
 
 By adding together the two limits of any other term, it 
 will be found that the result is independent of m and n. 
 
 Hence; when each of the two limits is >o, each limit 
 must be < \-\-x-\-z. 
 
 When each of the two limits is <o, each must be 
 
 When one is >o, and the other <o; the former is 
 > i +x-\-z, and the latter is < i -\-x-\-z. 
 
 214. Given xy = m, and (iy)z=n; to find minor limits 
 to xz. 
 
 Here the common symbol occurs contrarily in the two 
 data; whereas in the former case (Art. 212) it occurred 
 similarly. 
 
 Since (i y)z=n, 
 
 yz-=zn. 
 
 Hence, if we make n and zn replace one another in 
 the results of the former case, these results become true 
 for the present case. 
 
 Thus xz>m4-(z n} y, 
 that is. >m ny+z. 
 Also xz> m (z n) i -\-x-\-y-\-z, 
 that is, > m-\-n i-\-x+y. 
 Similarly for the other terms. 
 
 215. The sum of the corresponding limits of any two terms, 
 which are opposite to each other, is equal to o. 
 
 Consider for example those limits of xz and of 
 
 (i x) (iz) which involve y. 
 They are obtained from the equation 
 
 y{xz (i #)(i z}=x-\-z i}, 
 by supposing (ix)y(i z) = o and xyz=o respectively. 
 
 Hence they are equivalent but of opposite sign; and 
 therefore their sum is =o. 
 
 216. A limit, when it assumes either of the forms 
 
 m+x, m+y, n+y, 
 cannot be negative. 
 
88 Inference from one or more Equations 
 
 Consider the first form. 
 
 .m=xy, by given equation. 
 
 and xy<x, (Art. 187.) 
 
 .-. m<x, 
 that is, xm cannot be negative. 
 
 The same proof applies to the other forms. 
 Example of application. 
 
 Given xym (i); z(iy)=i y (2). 
 
 From (i) x(i y)=x m; 
 
 .'. X z>xm + iy(iy), (Art 212.) 
 
 that is, xz>x m; 
 but xm cannot be negative, 
 /. the limit for xz cannot be negative. 
 217. A limit, which is of either of the forms 
 
 m-xy+i, nyz+i, 
 
 cannot be negative. 
 
 Consider the first form. 
 
 m=xy, by given equation. 
 
 and xy>x+y i, (Art. 187.) 
 
 that is, mxy-\-i cannot be negative. 
 It can be shown in a similar manner that n y z-\-i 
 cannot be negative. 
 Example of application. 
 
 Given xy=m (i) ; yz=z (2). 
 
 From (i) (ix)(iy)=i xy+m; 
 
 from (2) ( I _ JV )( I _^) ==I _ J; 
 
 /. (i x)(i z)>i x y+m. (Art. 212.) 
 .-. the limit for (i ^)(i z) cannot be negative. 
 218. A limit, which is of either of the forms 
 
 ;;/, , 
 
 cannot be negative. 
 
 For m=xy, and xy cannot be negative. 
 Example of application. 
 
 Given xy=m (i); yz=y (2). 
 
 therefore xz > m +y y, 
 
of the form xy = m. {Hypothetical. ) 89 
 
 Hence the limit for xz cannot be negative. 
 
 219. When the miner limit is equal to a factor of the term 
 of which it is a limit, the inequivalence becomes an equivalence. 
 
 For any term, which can form the left-hand member of 
 the inequivalences considered, cannot be greater than any 
 of its factors. 
 
 Example of application. 
 
 Letxy=x (i) ; z(iy)=z (2). 
 
 Then from ( 2 ) ( i z}y y ; 
 
 therefore x( i z) > x -\-y y, 
 
 >x ; 
 but x is a factor of x(i- z), 
 
 therefore x(i z)=x. 
 
 This corresponds to what is called a universal conclusion. 
 Nothing can be more elegant than the manner in which 
 the general analytical conclusion becomes universal, when 
 the data necessitate the conclusion to be universal. 
 
 220. When the limit is equal to the common symbol, or its 
 contrary, the conclusion does not necessarily become universal. 
 
 For neither y nor i y is a factor of the left-hand mem- 
 bers of the inequivalences considered, viz. xz, x(i z), etc. 
 Example of application. 
 
 Letxy=y (i) ; yz=y (2). 
 
 then xz>y-\-yy, 
 
 >y> 
 
 but y is not a factor of xz, 
 
 .'. xz=y is not necessarily true. 
 Observation. If we consider the limits of xyz, then 
 
 xyz>y 
 becomes xyz=y. 
 
 221. To find the condition, to which the sum or difference 
 (as the case may be) of m and n must be subject, in order that 
 a given limit may be positive only. 
 
 First ; when m and n each enter positively into the limit. 
 Example : m-\-ny. 
 
 This expression cannot be negative, if m-\-n>y; that is, 
 if m-\-n includes y. 
 
QO Inference from one or more Equations 
 
 Second ; when m and n each enter negatively into the 
 limit. 
 
 Example : -m n i -\-x-\-y+z. 
 
 This expression cannot be negative, if m-\-n <x-\-y-\-z i ; 
 that is, if m-\-n is included in x-\-y+z i. 
 
 Third ; when m and n enter, the one positively, and the 
 other negatively. 
 
 Exam pie : m n y + z. 
 
 This expression cannot be negative, if mn>y z ; that 
 is, \im-n include y z. 
 
 Another form of writing this condition is 
 ;/ m<z y, 
 
 where the signs of the members have been changed, and in 
 consequence the sign of inequivalence also changed. 
 222. Suppose, for example, that 
 
 xy=m ; yz=.n. 
 
 Then the limit for xyz is positive only ; if m-\-n >y. 
 
 x(iy)z 
 
 xy(i-z) m 
 
 x (iy)(iz) <xz. 
 
 (i x\yz <o. 
 
 (ix)(iy)z >xz. 
 
 
 
 (ix)(iy)(iz) >x+y+zi. 
 
 Again ; suppose that 
 
 xy=m; (i}>)z=tn. 
 
 Then the limit for xyz is positive only ; if m n>yz. 
 x(\-y)z 
 xy(i-z) m+n>z. 
 
 
 
 
 
 
 (i-x)y(i-z) m-n<y 
 
of the form xy m. (Hypothetical. ) 9 1 
 
 223. The above investigation applies to equations of 
 the form 
 
 xy=m ; yz = n. 
 
 Let x=fv -, y=ft> ; z=^ 
 
 Then maximum value of m--n=.. 
 
 
 
 mnmum 
 
 ,, maximum 
 
 minimum 
 
 The limit for xyz is positive ; 
 
 x(iy)z 
 xy(iz) 
 
 ,, X (I }') (l Z) ,, 
 
 (i x)yz 
 
 (i x (i y ) z ,, 
 
 Suppose that the given equations are of the form 
 
 and let #, j, 2 have the same arithmetical values as before. 
 
 Then maximum value of m-\-n=-f u . 
 minimum _ _=fv_ 
 maximum mn= ^. 
 minimum = ft. 
 
 The limit for xyz is positive; if m n> f^. 
 x(iy)z __"_< A- 
 
 . xy(iz) 
 
 x(iy) iz) 
 
 (1-^)72 
 
 (i-x)(iy)z 
 
 5) 
 
 > T V 
 
92 Inferencejrom one or more Equations 
 
 224. 
 
 xz m+n Q| 3 | 
 
 *(i 2) mn i >ci 
 
 (i-x)z m-n 
 
 (i-x)(i-z) m+n Q| C3 
 
 xym ; yzn. 
 
 T^;.>'=A; 
 
 FIG. 8. 
 
 xz mn J [ 
 X (i-z) m+n CH f 
 
 (i- x )z m + n c^f) 3+ 
 
 xy=m; (iy)z=n. 
 
 *=A^=ft; *=TDF- 
 FIG. 9. 
 
 These relations are represented graphically in figs. 8 
 and 9. The values of m+n are represented by distances 
 from o along the top line of fig. 8. Its possible values, 
 under the given conditions 
 
 X=*,}>=TU,Z=T<>> 
 
 lie between the straight strokes. Its minimum value, when 
 the limit for xyz is positive, is 'represented by the distance 
 from o to a stroke pointing in the positive direction ; and 
 its maximum value, when x(iy)z is positive, is repre- 
 sented by the distance from o to the stroke pointing in the 
 negative direction. 
 
of the form xy m. ( Hypothetical. ) 93 
 
 Similarly, the values of m n are represented by lengths 
 from o along the second line ; when positive by a length to 
 the right, when negative by a length to the left. Its 
 possible values, under the given conditions, lie between the 
 tall strokes. Its values, when the limit for xy(\ z) is 
 positive, are represented by distances cut off by the stroke 
 pointing to the right ; and its values, when the limit for 
 x (i y) (i z) is positive, by distances cut off by the stroke 
 pointing to the left. 
 
 And so on for the others. 
 
 225. When m-\-n (or mn) has a value inside [ ], then 
 the two limits are simultaneously positive; when inside 
 ] [, the two limits are simultaneously negative ; when 
 outside [ ] or ] [, the one limit is positive and the other 
 negative. 
 
 226. Given xy=m, yzn; 
 
 to find what relations necessarily exist between m-\-n or mn 
 and x, y, 2. 
 
 Write each of the equations in its four equivalent forms, 
 as follows 
 
 xy=m. (i) 
 
 (i-x)y=y-m. (2) 
 
 x(i-y)=x-m. (3) 
 
 (i-x)(i-y) = i-x-y+m. (4) 
 
 yz=n. (5) 
 
 (i y)z=z n. (6) 
 
 y(i-z)=yn. (7) 
 
 (8) 
 
 From (i) and (6) xy (i y)z>m+z n i ; (Art. 188) 
 (i y) = o, 
 
94 Inference from one or more Equations 
 
 From (i) and (8) we obtain by similar reasoning 
 o>m-\-i y z-\-n i ; 
 
 By this method we obtain 
 
 From (i) and (6), m n<iz. 
 
 (8), m+n<y+z. 
 
 (2) (6), >y+z-i. 
 (8), mn>z. 
 
 (3) (5), >-(i-^ 
 % (7), 
 
 (4) (5) 
 
 (7), m n<x. 
 
 227. G 1 /^^ xy=m, (iy)z=n; 
 
 to find what relations necessarily exist between m-\-n or mn 
 and x, y, z. 
 
 Write each of the equations in its four forms as follows 
 
 xy=m. (i) 
 
 (i-x)y=y-m. (a) 
 
 x(i-y)=x-m. (3) 
 
 (i-x)(i-y)=i-x-y+m. (4) 
 
 yz=zn. (5) 
 
 (i-^). (6) 
 
 y(i-z)=y-z+n. (7) 
 
 (!->) (l-0)=I-^-. (8) 
 
 Then by applying the method of the preceding Article 
 we obtain 
 
 From (i) and (6), m + n<i. 
 (8), mn<y. 
 
 (2) (6), m - n >-(i-y). 
 (8), +>o. 
 
 (3) (5)> 
 (7), 
 
 (4) (5)i 
 
 (7) m+n<x+z. 
 
of the form xy m. ( Hypothetical. ) 95 
 
 Observation. The merit of this investigation is its ex- 
 haustiveness. The truth of any of the individual results 
 can be established directly. Consider, for example, the 
 first of the above relations. 
 
 xy <y and (i y) z<i y, 
 
 that is, m-\-n<i. 
 
 228. To find expressions for the limits independently of one 
 of the two data m, n. 
 
 First ; when the datum, which is to be eliminated, enters 
 positively. 
 
 Example. If xym^ (i) and (i y)z=n- } (2) then it 
 can be shown (Art. 214) that 
 
 (i -^-x)y(i z) > m+n+y z. 
 
 From (2) n> y+z and >o, (Art. 217.) 
 
 .'. (i-#)y(i -*)>-;; (3) 
 
 and > m-\-y z. (4) 
 
 Second ; when the datum, which is to be eliminated, 
 enters negatively. 
 
 Example. Under the same conditions as above, 
 
 (i x)(i-y)(i z)>m n+i x y. 
 From (2) ;z<i jy and <2, 
 
 ... (i-#)(i_j)(i _*)>--#; (5) 
 
 and >m-}-i x y z. (6) 
 
 Observation i. In each case one of the pair of expres- 
 sions obtained cannot be positive. 
 
 The expression (3) cannot be positive ; for m is positive 
 only. 
 
 The expression (5) cannot be positive j for m cannot 
 include more that x. 
 
 Observation 2. The truth of the other two results may 
 be verified as follows : 
 
 The inequality (4) asserts that 
 
 that is, (i x)y (i z)>y(i x) z, 
 
 that is, z>y(i x)z - } 
 which is evidently true. (Art. 193.) 
 
96 Inference from one or more Equations 
 The inequivalence (6) asserts that 
 
 i-x-y-z-, 
 that is, >(i-x)(i-y)z, 
 
 that is, z>(i x)(i y)z' } 
 which is evidently true. (Art. 193.) 
 
 229. 71? find the condition which m or n must satisfy, in 
 order that it may be impossible for the limit of a given term to 
 be negative. 
 
 When the condition for m is required, ;/ must be elim- 
 inated from the general form of the limit in the manner 
 shown in the preceding Article. One of the results obtained 
 is necessarily negative ; hence not more than one condition 
 can be found. If m enters positively into the other result ; 
 then the required condition is, that m include the remainder 
 of the expression with its sign changed. If it enters nega- 
 tively ; then it must be included in the remainder. 
 
 For-example; m>x-\-y-^z i, 
 
 is the condition which m must satisfy under the data of 
 Article 226, in order that it may be impossible for the limit 
 of (i x)(i y)(i z) to be negative. 
 
 Similarly, if m <y z, 
 
 the limit for (i x)y(i z) cannot be negative. 
 
 230. Two MIDDLE TERMS. 
 
 Given xy=m, yz=n, zu=p ; 
 
 to find those mijior limits to the terms of the fourth order having 
 x and u as factors, which can be expressed in terms of m, n, 
 p, x, y, z or u. 
 
 xyz>m+ny, (Art. 212.) 
 
 and zup, 
 
 .'. xyzu~>m+ny-\-pz; (Art. 212.) 
 that is, >m-\-n-\-p yz. (i) 
 
 Also, x(i-y}z> m ni+x+y+z, (Art. 212.) 
 
 and zu=.p, 
 
 .'. x (17) zu> m n \-\-x-\-y-\-z-\-p z ; 
 that is, > m n+p i+x+y. (2) 
 
of the form xy m. {Hypothetical. ) 97 
 
 Again xy (i 
 
 and (iz)u=ufl, 
 .'. xy(iz)u>mn-}-up(iz); 
 
 that is, >m np 1+^+ u. (3) 
 
 Also, x(i y)(i z)> m+n+x z, (Art. 212.) 
 
 and (i z) u=up, 
 
 :. x(i y)(i z)u> m+n p i+x+u. (4) 
 Now there are four and only four terms of the second 
 order which can be formed by means of the middle symbols 
 y and z, viz. 
 
 y*> (i -y) z > y (* -*) (* -y) ( l - z \ 
 
 Hence the above are all the terms of the fourth order 
 which contain x and u as factors. 
 
 COR. Each of the four limits obtained is a limit to xu. 
 
 For xu is greater than each of the terms. 
 
 231. The limits for the terms of the fourth order, which 
 have x and i , or i x and , or ix and i u as 
 factors, can be found in a similar manner, or deduced from 
 the above results by making the appropriate transforma- 
 tions. 
 
 For example ; to deduce the limit for 
 
 (l-*)(l-j) (!-*)(!-*). 
 
 xyzu>m-\-n-\-pyz ; 
 and (ix) (iy) = ix 
 (i-y)(i-z) = i- 
 (iz) (i u)=i 
 
 232. The Sorites is the simplest example possible of a 
 conclusion derived from three equations of the kind con- 
 sidered. The premises are 
 
 CTs which are x arejy; xy=x. (i) 
 
 C^s which are y are z ; yz=y. (2) 
 
 7's which are z are u ; zu=z. (3) 
 G 
 
98 Inference from one or more Equations 
 
 Now xyzu >m+n +/ y z, (Art. 224. ) 
 .-. >x+y-\-z y z, by premises, 
 
 that is, >x, 
 
 =x. (Art. 219.) 
 
 The conclusion is read 
 
 U's which are x are u and y and z. 
 What is commonly called the conclusion is 
 
 U's which are x are u ; 
 
 which is evidently only a part of the full conclusion. 
 We have shown in the preceding Article that 
 (i x) (i j>) (12) (i^v)>m+n+f+ixyg*-K 
 Hence by the premises >x+y-\-z+ i x y z u, 
 
 >iu. 
 
 U*s which are not u are not x and noty and not z. 
 This conclusion is complementary to the other. 
 Observation. The investigation of the two preceding 
 Articles shows how much truth there is in the statement 
 that no conclusion can be drawn when more than one of 
 the premises is not universal. 
 
 233. Given the conclusions of Art. 230 ; to deduce minor 
 limits to the terms of the third order having x and u as 
 factors. 
 
 It has been shown that 
 
 xyzu>m-\-n-\-p y z. (i) 
 
 x (iLy)zu> m n-\-pi -\-x-{-y. (2) 
 
 xy(i z}u>m n p i+z+u. (3) 
 
 x(i j(i z) u> m -\-np\ -\-x-\-u. (4) 
 
 By adding (i) to (3) ; xyu>2m i y+u. (5) 
 
 (2) (4); x(iy)u> 2m 2 + 2x+y+u. 
 
 (6) 
 
 (i) (2); XZU>2p-I+X-Z. (7) 
 
 (3) (4)j X(IZ)U> 2p2+X+Z+2U. 
 
 (8) 
 
 Now j, i j, z, iz 
 
 comprise all the different expressions which can form the 
 third factor of a term of the third order having x and u 
 
of the form xy^m. ( Hypothetical. ) 99 
 
 constant factors. Hence the above comprise all the re- 
 quired limits. 
 
 COR. i.. Each of these four limits is a limit to xu. 
 COR. 2. Each of these limits is less than the one which 
 can be obtained directly. 
 For example ; consider (5). 
 
 xy=m, and u=u, 
 
 .-. xyu>m+ui. (Art. 188.) 
 
 Now (2m i y-\-u) (m-\-u i)=m y, 
 
 but m y is necessarily negative, 
 /. the limit (5) is less than the one obtained directly. 
 Observation. The integer 2 appears in each of these 
 limits. Its presence indicates that the conclusion is a 
 fortiori. 
 
 234. To deduce a minor limit to the term of the second 
 order having x and u as factors. 
 
 Add (5) and (6) ; 
 
 then xu > 3 -f- 2x-\-2u. (9) 
 
 By adding (7) and (8) we get the same expression. 
 COR. i. The total number of limits for xu which can be 
 deduced by the method under consideration is 3*. 
 For it is = 2 s + 2-2 + i, 
 
 .-. = 3'- 
 
 COR. 2. The sum of the primary limits for xu is inde- 
 pendent of m, n, and/. 
 
 For the sum of (i), (2), (3), (4) is 3 + 2*4-22*. 
 
 235. To prove that 
 
 xy>nx-\-ny2n-\-i, 
 where n denotes any integral symbol. 
 Suppose it is true for any one n. 
 
 Then xy>nx-\-ny 2/2+1, 
 
 and xy>x+y i. (Art. 187.) 
 
 .-. x z y' i >nx -\-ny- 2n + i+x+y i i ; (Art. 187.) 
 that is, xy>(n + i)x+ (n+i)y2 ( + i) + i. 
 Hence, if it is true for any n, it is true for /2+ 1. But it 
 is true for i ; hence for 2, and hence generally. 
 
ioo Inference from one or more Equations 
 
 236. Given xy=m ; yz=n ; zu-=.p ; 
 to find the relations which necessarily exist between 
 m-\-np, mn+p, mnp, and x, y, z, u. 
 Write the equations as follows : 
 
 xy=m. (i) 
 
 x(i-y)=x-m. (2) 
 
 (i-x)y=y-m. (3) 
 
 (i-x)(i-y)=i-x-y+m. (4) 
 
 yz=n. (5) 
 
 y(i-z)=y-n. (6) 
 
 (i-y) z=z-n. (7) 
 
 (i-^)(i-^) = i-^- 2 +. (8) 
 
 zu=p. (9) 
 
 z(i-u)=z-j>. (10) 
 
 (iz)u = ufl. (n) 
 
 (i-^)(i u)=izu+J>. (12) 
 
 From(i), (5), (11)^^(1 z)u>m+n p+u 2. (Art. 193.) 
 
 but s (i z)=o, 
 
 /. Q>m-\-np-\-u2, 
 
 that is, m-\-np<2 #. 
 
 The truth of this result may be verified as follows : 
 It asserts that o > xy +yz zu+u2, 
 
 that is, 2>xy+{yz+u(i z)} ; 
 
 r\owyz-\-u (i z)<i, and xy<\\ hence their sum is <2. 
 By the above method it may be shown 
 from (i) (5) (n) that m+n p<2 u 
 
 (12) m+n+p^z-L+z+u 
 (6) (9) m -n+p<2-y 
 
 (10) mnp<2yz 
 
 (?) (9) m-n+p<2-z 
 
 (10) m -n-p<2-2z 
 (u) mnp<2zu 
 
 (12) mn+p<i+u 
 
 (8) (9) m+n+j><i+y+z 
 
 (10) m+np<i+y 
 
 (n) m+n jKi+y+z-u 
 
 (12) m+n+p<y+2Z+u 
 
of the form xy ^: '> {potHettial. ) * i o i 
 
 By taking (2) instead of (i) inequivalences involving > 
 will be obtained. And so on. 
 
 237. To find the condition or conditions which the sum or 
 difference (as the case may be) of any two of the three m, n, p, 
 must satisfy in order that it may be impossible for a given 
 limit to be negative. 
 
 Suppose that m and n are the known data. If p occurs 
 positively in the given limit, then either of its minor limits 
 is to be substituted ; if it occurs negatively, then either of 
 its major limits is to be substituted. 
 Example of first case. 
 
 When xy = m,yz= n, zu =/, 
 
 xyzu > m -f n -\-p y z. 
 Now/ >z+u i and >o, 
 .*. xyzu>m-\-n y-\-u i ; 
 and also >m+n y z. 
 Hence the required conditions are 
 
 and m+n>y+z. 
 
 The latter of these conditions is impossible. 
 Example of second case. 
 Under the same conditions as above, 
 
 xy( i z)u > m n p i + z + u. 
 
 Now/<2 and <#, 
 .'. xy(iz)u>m n \-\-u\ 
 and also >mn i-f-z. 
 Hence the required conditions are 
 m n>i u, 
 and mn>iz. 
 Both of these are possible. 
 
 238. To find the condition which any one of the three 
 m, n, p, must satisfy, in order that it may be impossible for 
 a given limit to be negative. 
 Consider the condition for m. 
 
 Suppose first that n and / each enter positively into the 
 given limit. 
 
IO2 'Trifert'ttee'fr'Jm &&&, or more Equations 
 
 Example. When xy=m,yz=n, zu=.p, 
 xyzu>m-\-n-\-pyz. 
 
 Now n>y-\-z 
 
 hence n+p>y+z 
 
 and >o, 
 and >o; 
 
 +*+-!. (l) 
 
 >y+z- . (2) 
 
 >z+u- . (3) 
 
 >o. (4) 
 
 Therefore the required conditions are 
 
 2-j-i- u (i) 
 
 (2) 
 
 m>y+iu (3) 
 
 m>y+z (4) 
 
 The conditions (2), (3), (4) are evidently impossible. 
 Second ; when n and/ enter with opposite signs. 
 Example. Under the same condition's as above 
 x(i y)zu>mn-\-p i -\-x-\-y. 
 Now n<y and<2, 
 
 i and >o; 
 +u-i. (i) 
 
 >-!. (3) 
 
 >-. (4) 
 
 Therefore the required conditions are 
 
 m<x-\-z-\-u 2, (i) 
 
 <A7 I. (2) 
 
 2 . (3) 
 
 i. (4) 
 
 The condition (2) is evidently impossible. 
 The condition (3) is impossible ; because 
 
 m>x+y i, 
 .-. m>x+y i (i u). 
 Similarly m > x +y iz. 
 Third ; when n and /each enter negatively. 
 Example. Under the same conditions as above, 
 
of the form xy m. \ (Hypothetical. ) : i o 3 
 
 xy(iz)u>mnp i-\-z-\-u. 
 Now n<y and <z, 
 p<z and <u; 
 
 ,-. n +p<y+z (i) 
 
 <y+u. (2) 
 
 <*+* (3) 
 
 <z+u. (4) 
 
 Hence the required conditions are 
 
 m>y-\-i u. (i) 
 
 (2) 
 (3) 
 
 >i. (4) 
 
 The conditions (i), (2), (4) are impossible. 
 239. If m, n,fl, be all eliminated from a limit in the above 
 manner, only one of the resulting expressions is such as can be 
 positive. 
 
 Consider the third example of the preceding Article 
 xy(\ z)u> m n p i -\-z-\-u. 
 Now m>x+y i and >o, 
 n <y and <z, 
 
 p<z and <u, 
 
 .'. m np>x\z (i) 
 
 >x i u (2) 
 
 >x+y-i-2z (3) 
 
 >x+y \-z-u (4) 
 
 >-y-* (5) 
 
 >-y-u (6) 
 
 >-2Z ( 7 ) 
 
 >zu. (8) 
 
 Hence xy(iz)u>x+u2 (i) 
 
 >x+z-2 (a) 
 
 ^+^ 2 (3) 
 
 - 2 (4) 
 
 i (5) 
 
 i (6) 
 
 i (7) 
 
 (8) 
 
IO4 Inference from prte or more Equations 
 
 Only (3) can be positive. It is the limit obtained by 
 means of the proposition of Art. 196. 
 
 240. Given a system of n-\-i equations of the form 
 
 where there are n middle terms ; then there are 2 n limits 
 of the first kind, n2 n ~ l of the second kind, and generally 
 
 ofthe r , h k 
 
 The number of primary terms formed by n independent 
 symbols is 2 n . (Art. 152.) Hence there are 2" limits of 
 the first kind. 
 
 A limit of the second kind is formed by adding any pair 
 of terms which involve only one degree of contradiction. 
 
 But a term of the n th order has n contradictories of the 
 first degree. (Art. 164.) So for each of the 2" terms ; but 
 only one-half of the sums so obtained are different. Hence 
 there are ti2 n ~ l limits of the second kind. 
 
 A limit of the third kind is formed by adding any 
 four terms, which differ with respect to two symbols only. 
 For example ; 
 
 Now each term has n ( n ~ I ) contradictories of the second 
 
 I . 2 
 
 degree (Art. 164). Hence there are H ( n ~ I ' 2 n sums; but 
 
 
 of these only one-fourth are different. Therefore there are 
 
 I . 2 
 
 Generally for the limits of the r* h kind. 
 They are obtained by adding terms, which differ in r i 
 factors. But there are to each term 
 
 2) contradictories of the r _j* degree . 
 
 r i 
 
 (Art. 164.) 
 
of the form xy m. (Hypothetical. ) 105 
 But of the sums obtained only L. are different; hence 
 
 there are 
 
 n(n-i)...(n-r+2) 2n _ r+l 
 
 til 
 COR. The total sum of the limits is 3". 
 
 For it J S =2-^-i...^" I " 
 
 241. ONE MIDDLE TERM OF THE SECOND ORDER. 
 Given xyz = m, yzu = n ; 
 
 to find limits to xyzu, etc., in terms of m, n, x,y, z. 
 x yz=m yz>u-=n, 
 .-. xyzu>m-\-n yz, (Art. 188.) 
 
 >m+n-y. (i) 
 
 and >m-\-n z. (2) 
 
 Again. x( i yz) xm and #( i yz) = u , 
 
 .-. ^(i JF^>^ w+ n (i ^'^), (Art. 188.) 
 that is > mn i +x+u+yz. 
 Now yz>y+z i and >o, 
 .'. x(i yz)u >mn 2 +x+y+z+u. (3) 
 and also > m n i+x+u. (4) 
 
 COR. xu is greater than any one of the above limits. 
 For it is greater than xyzu, or x(iyz)u. 
 
 242. Given xyz=m andy(i z)u=n-, 
 
 to find minor limits to xyzu, etc., in terms of m, n, x,y, z, u. 
 y(iz)u=n, 
 
 .'. yzu=-yun; 
 and xyz=.m, 
 
 .-. xyzu>mn+yuyz. (Art. 188.) 
 
 Novryu yz>y-\-u i y. (i) 
 
 >y+u i z. (2) 
 
 >-y- (3) 
 
 >o-z. (4) 
 
io6 On certain forms of the 
 
 Therefore xyzu >mni-\-u. ( i ) 
 >m n \-\-y z-\-u. (2) 
 
 >m-n-y. (3) 
 
 >mnz. (4) 
 
 The limits for any of the other terms of the fourth order 
 can be reduced in a similar manner. 
 
 XVIII. ON CERTAIN FORMS OF THE 
 DISJUNCTIVE EQUATION. 
 
 243. One species of disjunctive equation consists of a 
 single positive character equated to the sum of a number of 
 positive characters. The general form of the equation is 
 given in Art. 145, and several properties are there deduced. 
 Its chief function is to express the characters which a given 
 character comprises. 
 
 The simplest case is when the disjunct member consists 
 of two terms only, as 
 
 xy-\-z; 
 the reading of which is 
 
 7's which are x are identical with those that are y 
 together with those that are z. 
 
 244. Another species of disjunctive equation consists of 
 a single positive character x equated to the sum of a num- 
 ber of terms multiplied by the character x. The principal 
 forms are 
 
 x=x{y(i-z)+z(i-y)}. (i) 
 
 and x=x{yz+y (i -z)+z (i -y)}. (2) 
 We have already discussed 
 
 x=xyz ; 
 and the remaining one of the four 
 
 x=x {yz-\-y(i-z) 
 is an identity. (Art. 153.) 
 
Disjunctive Equation. 107 
 
 Equation (i) is read 
 
 7's which are x are either y and not z or z and not y. 
 
 Equation (2) is read 
 
 It's which are x are either y and z or y and not z or 
 z and notjy. 
 
 COR. Since yz+y (**) =y, 
 
 equation (2) can be reduced to 
 
 that is, Z7's which are x are either y or not 7 and z. 
 
 Similarly x=x{z-\-(iz)y}. 
 
 Observation. Here x is a conditioning character. When 
 it is put equal to i, the equation ceases to be conditional. 
 
 245. The equation 
 
 may take the form 
 
 x-y^-w, 
 
 where w is indefinite, but single and positive. From the 
 nature of the equation, it follows that 
 
 wy=-o. 
 The equation x=y+w is read 
 
 Only 7's which are x can be y ; 
 
 and it is evidently the equational mode of stating the in- 
 equality, 
 
 x>y. 
 
 246. Conversion of the disjunctive equation 
 
 x=y+z. 
 
 By the Conversion of a disjunctive equation is meant its 
 transformation into an equation having one of the terms of 
 the disjunct member as the new left-hand member. 
 
 If *=?+*, 
 then y=x z ; 
 that is, 
 
 U's which are y are identical with those which are x 
 excepting those which are z. 
 
1 08 On certain forms of the 
 
 =^~ w ; 
 that is, 
 
 7's which are y must be x. 
 If the original equation is written 
 
 then the form deduced by conversion is 
 
 y<x. 
 247. Contraposition of the disjunctive equation 
 
 x=y+z. 
 
 The contraposition of a disjunctive equation consists in 
 deducing the expression for the contrary of the left-hand 
 member. 
 
 then i x=i y z ; 
 that is, 
 
 7's which are not x comprise all excepting those which 
 are y and excepting those which are z. 
 
 If x=y-\-w, 
 then i x=i y w ; 
 that is, 
 
 7's which are not x are not j>. 
 Observation. If x=y+w 
 
 is expressed by x >y ; 
 
 then i x=i y w 
 
 is expressed by i x< i y. 
 
 248. If x-=.y-\-z ; then xy=y, and xz=z: but not con- 
 versely. 
 
 x=y+z t 
 multiply by y, 
 
 then xy=y*+yz 
 
 =y. (Art. 145.) 
 
 Similarly xz=z. 
 
 Given xyy and xz=z; 
 
 then xy+xz=y+z, (Art. 291.) 
 
 y+z 
 
Disjunctive Equation. 109 
 
 But is not necessarily =y+z ; (Art. 173.) 
 
 hence it does not follow that xy-\-z. 
 
 249. If x-=.y+w, where w has the meaning assigned to it 
 in Art. 245 : then the derived equation 
 
 xy=y 
 is equivalent to the original. 
 
 xy=y, 
 
 Now (i y) is single, since x is single (Art. 145), and 
 o 
 
 it is such that 
 
 and it is not restricted in any other manner. Hence it is 
 equivalent to w. 
 
 Observation. x=y+w, x>y, and xy=y, express the 
 same truth. To express the truth considered by means of 
 an equation, either three symbols must be used, or else a 
 term of the second order introduced. 
 
 250. Inference from two disjunctive equations having a 
 common character. 
 
 First; when the character occurs oppositely in the two 
 equations. 
 
 x=y+z. (i) x'=i-y+z'. (2) 
 Add; x+x'=i+z+z', 
 
 that is, x i x' +z+z'. 
 
 Second; when the character occurs similarly in the two 
 equations. 
 
 x=y+z. (i) x'=y+z f . (2) 
 Subtract ; x x' = z z', 
 
 that is, x=x'+z z r . 
 
1 10 On certain forms of the 
 
 Observation. When z and z' are each indefinite, the 
 former conclusion amounts to 
 
 Only 7's which are x are not x'. 
 
 The latter conclusion under the same conditions amounts 
 to 
 
 Some 7's which are x are x. 
 For x>z, by equation (i). 
 
 251. If x=x(y+z), then x=x{y(i z)+z(i y)}- } and 
 conversely. 
 
 that is, x{i+y-\-z 2y2z -\-2yz} =*>, 
 
 that is, x{i-y(i-z)-z(i-y)}=o, 
 
 that is, x=x{y(i z)+z(i y)}. 
 
 As this proof can be reversed, the converse proposition 
 must be true. 
 
 Observation. The process here exemplified of squaring 
 an expression which is equal to o is of great importance." 
 The expression 
 
 x{i-y(i-z)-z(i-y)} 
 is of a form which cannot be negative ; while the expression 
 
 x{i-y-z} 
 is of a form which may be negative. See Art. 289. 
 
 252. 71? convert the equation 
 
 .-. xy=x(\z\ 
 
 x . . 
 
 ->' = -(!-*), 
 
 COR. i. Multiply by x y 
 
 then xyx(iz) : 
 
Disjunctive Equation. i.i 1 
 
 which involves as a part of its truth 
 xy=xy(i-z), 
 
 i.e. t7's which are x and y are not z. 
 COR. 2. Multiply by iy, 
 
 then o=#(i-jO(i-*)4~(i -#)(!-.>); 
 .-. x(iy)(iz) = o, for neither term can be negative ; 
 
 .-. x(i-y)=x(i-y)z; 
 i.e. U's which are x and not y are z. 
 Similarly it may be shown that 
 
 xz=x(i-y); 
 and x(iz)=x(iz)y. 
 253. To convert the equation 
 
 x=x{y+(i-y)z}. 
 Now x=xy-}-x(iy) z } 
 .\xy(i-z)=x(i-z), 
 
 This expression differs from the corresponding one ob- 
 tained from 
 
 x=x (y+z) 
 
 in having xz in addition. 
 o 
 
 COR. i. Multiply by x, 
 
 then xy=x (i z) -j xz ; 
 
 hence xy is not, as previously, identical with x (iz), but 
 includes it. 
 
 COR. 2. Multiply by i y, 
 
U2 On certain forms of the 
 
 therefore, since none of the terms of the right-hand member 
 can be negative, 
 
 that is, x (i y)=x (i y) z. 
 Similarly z = x ( i y) -\ ( i x) -\ xy. 
 
 254. If x=x(y+z}', 
 
 then xy=xy(iz), and x (i y)=x (i y) z; and con- 
 versely. 
 
 xx (y+z), 
 .-. x=x{y (i-z}+z (i-y)}, (Art. 251.) 
 
 Multiply by y ; then 
 
 xy=xy(i-z). (0 
 
 Multiply by i y ; then 
 
 x(i-y}=x(i-y)z. (2) 
 
 To prove the converse. 
 
 From (i) xyz=-o. 
 
 From (2) x (i y) (i z)=o, 
 
 that is, x{iyz+yz}=o, 
 .-. by(i) x=x(y-\-z). 
 
 Observation. It is commonly said that the disjunctive 
 proposition 
 
 [7's are either y or z 
 
 is equivalent to the four hypothetical propositions 
 U's which are jy are not z. (i) 
 
 7's which are not y are z. (2) 
 
 7's which are z are not y. (3) 
 
 IFs which are not z arej. (4) 
 
 Now from i=y+z 
 
 there follows, by making x= i in the above equations : 
 
 y=y(i-z). (i) 
 
 i-y = (i-y)z. (2) 
 
 z=z(i-y). (3) 
 
 i-z=(i-z)y. (4) 
 
Disjunctive Equation. 1 1 3 
 
 But (i) and (2) without (3) and (4) are together equi- 
 valent to the given equation ; and so are (3) and (4) with- 
 out (i) and (2). 
 
 255. If x=x{y+(i-y)z}; then x (i-y)=x(i-y) z : 
 and conversely. 
 
 x=x{y+(i-y)z}, 
 =x{yz+y (i -z)+(i y) z}, 
 =*{i-(i _,)(!-*)}, (Art. 148.) 
 
 that is, x (i y]=x (i y) z. 
 
 Since the proof can be reversed, the converse proposition 
 must be true. 
 
 XIX. THE ARISTOTELIAN FORMS OF 
 INFERENCE. 
 
 256. I shall assume that the Aristotelian moods do not 
 consist of the mere application of a general proposition to 
 an individual case, but of the combination of two general 
 propositions. I shall further assume that the data are 
 equations not of the first but of the second order ; in other 
 words, that the premises each involve one hypothesis. 
 
 They are mere cases of the general form of inference 
 discussed in Article 212, viz. 
 
 If xy = m, and yz = n, 
 then xyz >m-\-n y ; 
 and a fortiori xz>m -\-n-y. 
 
 The analytical reductions, which are sometimes required 
 to fit the premises of a mood for being put into the above 
 formula, correspond to. the reductions indicated by the 
 significant letters in the name of the mood. 
 
 Observation. The fact that the number of Aristotelian 
 moods is a prime number suffices to throw doubt upon 
 the completeness of the scheme. 
 
 H 
 
1 1 4 The A ristotelian forms of Inference. 
 
 257. The First Figure. 
 Barbara. 7's which are y are z ; 
 
 7's which are x sney. 
 y=yz. (i) x=xy. (2) 
 
 /. xz>x+y y ; 
 
 x 
 
 that is, 7's which are x are z. 
 Celarent. /"'s which are y are not z ; 
 
 7's which are x are 7. 
 ^=7(1 s). (i) x=xy. (2) 
 
 x; 
 
 that is, 7's which are # are not z. 
 Darii. 7's which are y are z ; 
 
 7's which are # and y are some. 
 
 (Art. 219), 
 (i) 
 
 w 
 
 (2) 
 
 /. xz>v+yy; 
 
 that is, 
 
 Ferio. 
 
 7's which are a; and z are some. 
 
 7's which are y are not z ; (i) 
 
 7's which are x and^ are some. (2) 
 y=y(i-z). (i) xy=v. (2) 
 
 .'. x(i-z)>v+y-y; 
 
 that is, 6^'s which are x and not z are some. 
 
 258. 7%^ Second Figure. 
 
 Cesare. The premises are 
 
 z=z(i y). (i) and x=xy.. (2) 
 
 Since the first premise contains i y, and the second y, 
 one of the two must be transformed. In this case it is the 
 transformation of the first premise which leads to the 
 ordinary conclusion. 
 
 From (i) zy=o, 
 
The Aristotelian forms of Inference. 1 1 5 
 
 Hence x(\ z)>x+y y, 
 
 =x. 
 
 The transformation here made corresponds to that 
 indicated by the letter s in the name of the mood. 
 
 Observation. The conclusion in the above case is 
 deduced more shortly by multiplying together the two 
 data. 
 
 Camestres. z~zy. (i) and x=x(iy). (2) 
 
 From (2) y=y(i x) ', 
 
 .'. (i-x)z>y+z-y, 
 
 =z'> 
 
 .'. x(iz)=x. 
 
 Here both equation (2) and the immediate conclusion 
 are transformed by a process which corresponds to simple 
 conversion. 
 
 Festino. z=z(iy). (i) and xy=v. (2) 
 From(i) (i-z)y=y, 
 
 Baroco. z=zy. (i) and x(i y)v. (2) 
 
 It is necessary to transform (i) so as to contain iy. 
 
 Now ( i z) ( i y) = i y z +yz always, 
 
 = i-y by(i) 
 
 Hence x(i z)>v+i y (i y) 
 
 >v. 
 
 The method of reduction here employed corresponds 
 to the contraposition of the major premise. The ana- 
 lytical investigation shows the reason why the mood 
 gave so much trouble to the Scholastic logicians, and 
 also shows how readily problems, which are difficult 
 to the unaided mind, may be solved with the help of a 
 true organon. 
 
 The ductio per impossibile is as follows. 
 
 Ifx(i-z) is not greater than o, it must be equal to o; 
 for it cannot be negative. 
 
1 1 6 The A ristotelian forms of Inference. 
 
 Let it be equal to o. 
 
 Then xz==x. 
 
 Multiply (i) by x } then xz=xzy. 
 
 Hence x=xy 
 
 that is x(iy) = o. 
 
 But x(iy)=v by (2) 
 
 hence x(iz)not=o 
 
 therefore x(i'z) is greater than o. 
 
 Observation. Here iy is a factor of the left-hand 
 member of the conclusion. In the rest of the moods y is 
 the corresponding factor. 
 
 259. The Third Figure. 
 
 Darapti. y=yz. (i) andy=yx. (2) 
 
 Hence xz>y+y y, 
 
 >y. 
 The conclusion is commonly said to be 
 
 7's which are x and z are some ; 
 but it evidently amounts to more, viz., 
 
 7's which are x and z include those which are y. 
 Felapton. y=y(iz). (i) a.ndy=yx. (2) 
 Hence x(iz) >y +y y, 
 
 Disamis. yz=v. (i) Bndy=yx. (2) 
 Hence xz>v+y y, 
 
 >v.' 
 
 DatisL y=yz. (i) and xyv. (2) 
 Hence xz > v. 
 
 Bocardo. y(iz)=v. (i) andjy=>a. (2) 
 Hence x( i z) > v. 
 
 Ferison. y=y(iz). (i) and xy=v'. (2) 
 Hence x(iz)>v. 
 
 260. The Fourth Figure. 
 
 Bamalip. z=zy. (i) and^=j^:. (2) 
 
 Hence xz>y-\-z y, 
 =z. 
 
The A ristotelian forms 'of Inference. 1 1 7 
 
 The conclusion is commonly said to be 
 
 7's which are x and z are some. 
 But it really amounts to 
 
 7's which are x include those that are z, 
 or, in other words, 
 
 Only 7's which are x can be z. 
 
 The character x is a conditio sine qua non with respect to 
 the character z. 
 
 Calemes. z=zy. (i) and 7=7 (i x). (2) 
 
 Hence (i x) z>y+z y, 
 
 = 2. 
 
 The conclusion in this form means 
 
 Only C/"'s which are not x can be z ; 
 but it can be converted into 
 
 x=x(i-z), 
 that is, 
 
 7's which are x are not z. 
 Dimatis. zy=v. (i) and y=yx. (2) 
 
 Hence xz>v. 
 
 Fesapo. z=z(iy). (i) audy=yz. (2) 
 From (i) y=y(i-z), 
 
 ' x(i-z)>y+y-y, 
 
 >y. 
 
 This conclusion is not particular merely, but asserts that 
 the character x (i z) is a conditio sine qud non with respect 
 to the character y. 
 
 Fresison. z=z(iy). (i) and xy=v. (2) 
 
 From (i) y=y(i-z), 
 
 .-. x(i z)>v+y y, 
 
 >v. 
 
 Here the conclusion is merely particular. For an 
 investigation of De Morgan's Syllogistic forms see the 
 Examples, page 135. 
 
n8 On Probability. 
 
 XX. ON PROBABILITY. 
 
 261. It is important to draw a sharp distinction between 
 the combining of two general propositions, and the appli- 
 cation of a general proposition to an individual case. The 
 former of these processes is investigated by the Theory of 
 Necessity ; the latter by the Theory of Probability. Any 
 one who clearly perceives this distinction, is in no danger 
 of falling into the absurdity of maintaining that true 
 syllogistic reasoning involves the fallacy of the Circle. 
 
 262. The Probability that a given member of a universe 
 has a character x is measured by the arithmetical value of x 
 referred to that universe. 
 
 The probability here spoken of, or at least its measure, 
 is entirely objective. The value of this objective proba- 
 bility may not be fully known in which case we consider 
 its most probable value. The latter depends upon the 
 state of our knowledge concerning the objective proba- 
 bility, and in consequence varies with that state. 
 
 263. If x is equivalent to a function of a number of charac- 
 ters, the probability of an individual U being x is equal to 
 that function taken arithmetically. 
 
 For every logical equation is true arithmetically. As an 
 example, 
 
 let *=, 
 
 (i-*) (Art. 199.) 
 
 Hence the results of the preceding Articles may be made 
 to yield theorems in Probability by taking the right-hand 
 members of the equations arithmetically, and supposing U 
 to denote an individual member of the universe. 
 
On Probability. 1 19 
 
 264. The probability of a U which is x being y is measured 
 by the arithmetical value of y in Ux y . 
 Let Ux=U'. 
 
 The probability of a U" being y is measured by the 
 arithmetical value of y in U'y. (Art. 262.) 
 
 But Ux is identical with U' ; therefore the probability of 
 a U which is x being y is measured by the arithmetical 
 value of y in Ux y . 
 
 COR. The probability of a U which is x being y is equal to 
 
 For it is equal to that of y in xy ; 
 
 .-. equal to ^ (Art. 64.) 
 
 
 .-. equal to. 
 
 x 
 
 265. If the probability of a U which is x being y is equal 
 to the probability of a U which is not x being y, then either is 
 equal to the probability of a U being y. 
 
 xy (ix 
 
 Any one of the three equations 
 
 * i-x 
 
 expresses the condition for x and y being really indepen- 
 dent of one another. (Art. 62.) 
 
 266. Required the probability of a U which is x being y, 
 having given that 
 
 (i) U{x=y+z}; 
 
 or (2) U{xyz\. 
 
I2O On Probability. 
 
 First Case. Probability required =? (Art. 264. COR.) 
 
 Second case. Probability required = -4- 
 
 ,. by(2) . = ?=* =I . 
 
 
 
 267. Given x 
 
 required the probability of a U which is x being either x l 
 or x a . 
 
 Probability required -_*(*'+*') (Art. 264), 
 
 *- ' (Art. 145.) 
 
 268. Given x=y+zu, 
 
 required (i) the probability that a U which is x is y, and (2) 
 the probability that it is not y. 
 
 Since zu=z(iu)u(iz), 
 
 Now x cannot be negative, and z(iu) and u(i z) are 
 exclusive of one another ; therefore y must include u(i z). 
 
 Hence xy{i u(i z)}+z(i u). 
 
 (i.) Probability that a U which is x isjy =-4- 
 
 oc 
 
 .y<- 1 u \ I _ z )>, from given equation 
 
 y+zu 
 
On Probability. 1 2 1 
 
 Suppose that uz is not known. Then we affirm that 
 
 (2.) Probability that a U which is x is not y = _ 
 
 zzu 
 
 Hence if zu is not known, we can affirm that 
 prob. < 
 
 269. GVw //fo probability that a U is x, and the pro- 
 bability that a U which is x is y ; to find the probability 
 that a U is y. 
 
 The data are x=p, and x y pq. It is required to find y. 
 
 , 
 
 oc 
 
 Hence j/>/^and <pq+ip. 
 
 The problem can also be investigated as follows : 
 
 (i-x)(i-y). (i) 
 
 . (2) 
 
 Hence y< \-x-\-xy and >jpy. 
 
 Hence y < i p +pq and >pq. 
 
122 Fundamental relations, etc. 
 
 XXI. FUNDAMENTAL RELATIONS BETWEEN 
 THE SINGLE FUNCTIONS OF A NUMBER 
 OF INDEPENDENT CHARACTERS. 
 
 270. By the fundamental relations between the single 
 functions of the characters x^ x^...x n is meant the different 
 forms of the relation connecting the part x l x z ...x n with 
 the other parts into which the universe is divided by the 
 characters. 
 
 The relations in the case of two independent characters 
 are given in Article 190. They suffice to combine any 
 number of premises, which are linked together by common 
 characters after the manner shown in Article 240. As, how- 
 ever, the data of a problem are not necessarily of that 
 nature, it is important to consider all the fundamental 
 relations which exist in a given case. 
 
 271. To find the fundamental relations, when there are 
 three characters x, y, z. 
 
 Expand the term which is the opposite of xyz, 
 (i-x)(iy)(iz)=ix-y-z+yz+zx+xy-xyz. (i) 
 Expand a term which involves two degrees of contradic- 
 tion, 
 
 (i x)(i y)z=z xz yz-\-xyz. (2) 
 
 Expand a term which involves one degree of contradic- 
 tion, 
 
 (i-x)yz=yz-xyz. (3) 
 
 There are other two equations similar to (2), and other 
 two similar to (3) ; hence, if 
 
 xyz=o -\-xyz 
 is also counted, there are 2 3 altogether. 
 
 The other relations which exist may be derived by 
 transforming these fundamental relations. For example : 
 change z into i z, then (2) becomes 
 (i -x)(i j;)(i -z)= i -z-x(i -z)-y(i - z)+xy(i -z). 
 
Fundamental relations, etc. 123 
 
 272. When there are n characters, there are 2 H fundamental 
 relations. 
 
 For each primary part into which the universe is divided 
 by the n characters yields a fundamental relation ; and there 
 are 2* such parts. (Art. 152.) 
 
 273. Consider the relation 
 
 (i x) (i y) (i z^+x+y+zyz zxxy -\-xyz i. 
 
 If the arithmetical values of any seven of the eight terms 
 are known, then that of the remaining term can be found. 
 If the arithmetical values of any six of them are known, 
 then that of the remainder can be found ; and so on. 
 
 Suppose that the remainder can be reduced to two parts 
 /! and / 2 having the same sign. 
 
 Then A+/ 2 =; 
 .-. t l = n t z . 
 
 Hence ^ and / 2 must each be less than n. 
 Suppose that the two parts / x and / 2 , to which the re- 
 mainder is reduced, occur with opposite signs. 
 
 Then t l t z =7t;_ 
 .'. /!>, and t*>n. 
 
 274. The following example illustrates the manner in 
 which these relations can be applied to solve questions in 
 Probability. 
 
 Given x=d ; y=b ; xz=dp ; yz=-bq ; 
 
 and (i x) (17)2=0 ; 
 required z. 
 
 By (2) (Art 271) (i *) (*y) z=z xz yz-\-xyz. 
 
 .'. zxz-\-yzxyz by last datum ; 
 .-. z<xz-\-yz x yz+i. (i) (Art. 188.) 
 <xz+yz y xz+i. (2) 
 <xz+yz. _ (3) 
 
 .-. z< i a+dp. (i) 
 
 (2) 
 (3) 
 
124 Fundamental relations ', etc. 
 
 By relation (3) (Art. 271) (\x)yz-=yzxyz. 
 
 .: (ix)yz=yz-\-zxzyz, by the last datum. 
 .-. z=xz+(ix)yz. 
 
 .'. z>dp. 
 It may be shown in a similar manner that 
 
 z>bq. 
 Boole deduces these results by a much longer process. 
 
 XXII. GENERAL METHOD OF DEDUCING A 
 CONCLUSION OF A REQUIRED FORM FROM 
 GIVEN DATA. 
 
 275. When we know a necessary relation existing be- 
 tween the forms of the given data and of the required con- 
 clusion, and between these forms alone, we can find the 
 conclusion by simply substituting the data in the necessary 
 relation. If the known relation involves other forms in 
 addition, it may be made to yield limits to the required 
 conclusion. If, however, all the forms of the given data do 
 not exist in the relation, the limits deduced may not coin- 
 cide with the actual limits necessitated by the data. 
 
 276. I have discovered a method, by means of which it 
 is easy to find the necessary relation which exists between 
 the forms of any given data and the form of the required 
 conclusion. The forms are supposed to be functions of 
 single (Art. 118) symbols. 
 
 Equate the form of the conclusion to the sum of the forms 
 of the data each multiplied by a constant, and with a con- 
 stant added ; then substitute each of the sets of singular 
 values which the characters can assume. The equations 
 obtained express the relations which necessarily .exist be- 
 tween tJie constants ; they may suffice to determine each of the 
 
General method, etc. 125 
 
 constants exactly, or they may require one or more of the 
 constants to have indefinite values. 
 
 It is supposed that the functions do not contain any 
 inverse factors. 
 
 277. Given x=m, x(iy) = n, (ix)y=p; required xy, 
 
 Let xy=a+bx-\-cx(iy)+d(ix)y. 
 Since x and y are given to be single and (we may sup- 
 pose) positive, each can assume the values i and o. Hence 
 
 from x=i, y=i, i=a-\-b. (i) 
 
 from #=1,^=0, o=a+Z>-{-c. (2) 
 
 from #=0, y=i, o = a -\-d. (3) 
 
 from x=o, jy = o, o=a. (4) 
 
 The determinant of the right-hand members of these 
 equations is 
 
 i i o o 
 i i i o 
 i o o i 
 1000 
 
 which is equal to i. Hence a definite relation exists 
 between the form of the conclusion and the forms of the 
 given data. Definite values can be found for the constants, 
 viz. 
 
 a=o, &=i, c= i, do. 
 Hence xy=x x (i y), 
 
 =.mn. 
 
 The problem here discussed can of course be solved by 
 much simpler methods. Its solution by the general method 
 is introduced to illustrate that method. 
 
 278. Given y=m, xy=n ; required x. 
 
 Let x= a -\-by-\-cxy. 
 
 Then i=a+b+c. (i) 
 
 i=*. (a) 
 
 o=a+l> (3) 
 
 o=a. (4) 
 
1 2 6 General method of deducing a Conclusion 
 The determinant of the right-hand members is 
 
 I I I O 
 
 I O O O 
 
 I I O O 
 
 I O O O 
 
 Since the elements of the fourth column vanish, the 
 value of the determinant is o. Hence there is no definite 
 relation between the forms considered. 
 
 The fourth row is identical with the second; but the 
 left-hand member of (4) is not identical with that of (2). 
 
 The equations can be made identical by putting in 
 
 o 
 
 instead of i and of o, and restricting within the limits 
 
 o 
 
 i and o. Then we have 
 
 o 
 
 =0. 
 
 o 
 
 The determinant of these equations is 
 iii 
 i o o 
 
 I I O 
 
 the value of which is i. And 
 
 o o 
 
 =-,*=-_,,=,. 
 
 Hence x = y + xy, 
 
 o o 
 
 Since lies between i and o, 
 o 
 
 x>n and <i m-\-n. 
 
 (3) 
 
of a required form from given data. 127 
 Observation. The fundamental relations 
 
 and (i y) x=xxy, 
 readily show that 
 
 x<i y-\-xy and >xy. 
 
 279. The following problem is an instance of the general 
 problem investigated in the preceding Article : 
 
 The probability that it thunders upon a given day is /, 
 the probability that it both thunders and hails is q y but of 
 the connection of the two phenomena of thunder and hail, 
 nothing further is supposed to be known. Required the 
 probability that it hails on the proposed day. Boole, 
 Laws of Thought, p. 276. 
 
 Let U a succession of states of the atmosphere at a 
 given place, an individual state being of the length of a 
 day. 
 x= containing a hailstorm; y containing a thunderstorm. 
 
 Then j=/, xy=q. 
 Hence, by the preceding Article, 
 
 Hence x cannot be greater than ^+ 1 p, and cannot be 
 less than q. 
 
 280. Let us take the three data 
 
 y = m, xy=n, i-^=/ 5 
 
 which are equivalent to two independent data, and inquire 
 how this is detected by the method, when we seek for the 
 relation connecting x with the given functions. 
 
 Let x-=a-\-by-\-cxy-\-d(\ y). 
 
 i=#++r. (i) 
 
 i=a +d. (2) 
 
 o=a +d. (4) 
 
128 General method of deducing a Conclusion 
 
 The determinant of these equations is 
 
 Now this determinant has the sum of its second and 
 fourth columns identical with its first column ; hence its 
 value is o. Thus the analytical reason why the deter- 
 minant vanishes indicates the real reason of the insuffi- 
 ciency of the data. 
 
 281. Given xy+x (i 
 
 (i) 
 (2) 
 
 (3) 
 (4) 
 
 to find x (i y) -\-y (ix). 
 Letx(i-y)+y(i-x)=a+l>{x+(i-x)y} 
 
 Then o=a+& 
 
 The determinant is 
 
 i i o 
 iii 
 iii 
 
 i o i 
 
 in which the second and third rows are identical. But the 
 equations (2) and (3) have the same left-hand member. 
 Hence one of them may be struck out. 
 
 Then 
 
 = 1 
 
 and a= i, =i, <r=i. 
 Hence x (i y)+y (i x)= i 
 The above is the general solution of the following pro- 
 blem : 
 
of a required form from given data. 129 
 
 The probability that one or both of two events happen is 
 /, that one or both of them fail is q. What is the proba- 
 bility that only one of these happens 1 
 
 It is i-\-p-\-q. 
 
 282. Given x, y, xz, yz, and (i x) (i y) z ; required z. 
 Let zax+by -\-cxz-\-dyz-\-e (i x) (i y) z. 
 
 Then by means of the eight sets of singular values, 
 which the three characters can assume, we obtain the equa- 
 tions 
 
 i=a + b+c+d. (i) 
 
 i=a +*" 
 
 o=a. 
 
 i= b - 
 
 o= b. 
 
 = 0. 
 
 The determinant expression is 
 
 (3) 
 (4) 
 (5) 
 (6) 
 (7) 
 (8) 
 
 Since the determinant contains five columns and seven 
 rows, two of the equations are identical or must be made 
 identical. Now the sum of the third and fifth rows is 
 identical with the first, but the values of the corresponding 
 equations are 2 and i respectively. Denote this value by 
 
 , subject to the restriction that it cannot be greater than 
 o 
 
 2 nor less than i. Also the sum of the fourth and sixth 
 rows is identical with the second, and the values of the cor- 
 
 j 
 
130 General method, etc. 
 
 responding equations are equal. Hence the system of 
 equations reduces to 
 
 ~=a+b+c+d. (i) 
 
 i=a +c. (2) 
 
 o=. (3) 
 
 o= b. (4) 
 
 (5) 
 From these equations we obtain 
 
 * j 
 0=0, #=o, c=i t a= 1, e=i. 
 
 Since the limits of are 2 and i, the limits of i 
 o o 
 
 are i and o ; hence d , where cannot be greater 
 
 o o 
 
 than i nor less than o. 
 
 Thus z=xz+ yz+(ix) (i -y) z. 
 
 o 
 
 Observation. This is a more general form of the pro- 
 blem considered in Article 274. 
 
 283. It is evident from the manner in which the equa- 
 tions are derived, that when an additional datum is given, 
 the only alteration made in the equations obtained for the 
 fewer data is the introduction of the new constant into some 
 of the equations. 
 
 284. Suppose that the forms considered involve three 
 characters x, y, z, and that each has z as a factor. Then 
 the term independent of the characters may be dispensed 
 with ; and four of the eight equations reduce to the form 
 
 = 0. 
 
 For, z being a factor of each of the terms, the four sets of 
 singular values in which z=o cause each side of the 
 identity to become o. 
 
On Boole s General Method. 1 3 1 
 
 XXIII. ON BOOLE'S GENERAL METHOD. 
 
 285. The method of the preceding chapter, while ad- 
 mirably adapted for problems in which the data are what 
 may be called explicit equations, is not adapted for pro- 
 blems in which the data are implicit equations. For the 
 latter we require the special method invented by Boole. I 
 propose to consider the elements of that method in the 
 light of the principles of this work. 
 
 286. To eliminate x from an equation f(x) = o, when it 
 is given that x (ix)=o. 
 
 Letf(x)=ax+&(ix). 
 
 Since x(ix) = o, x can assume the value i and the 
 value o. By putting these singular values of x into the 
 above identity, we get/(i)=#, and/(o)=. 
 Hence /(i) .+/(o) (i -#) = o. 
 
 .-. {/(!)_/ ( )}*=-/(o). (i) 
 
 and {/(i)-/(o)} (i -*)=/(i). (2) 
 Multiply together (i) and (2), 
 then {/(i)-/(o)} * (i-*)= -/(i)/(o) j 
 but x(i x) = o, 
 
 ../(i)/(o)=o. 
 
 Observation i. The condition /(i)/(o) = o will be satis- 
 fied independently of x(ix) = o, if /(i) /(o) = o ; that 
 is, if the given function does not contain x. It may also 
 be satisfied independently of either of these conditions; for 
 f(i)f(o] generally involves fractional symbols. 
 
 Observation 2. The equation /(i)/(o) = o is true, when 
 
 x=i, and when x = o; therefore when x= , where 
 
 o o 
 
 denotes any value lying between i and o. The contradic- 
 tory values of x are united into one in the same manner as 
 those of a in Art. 278. Hence 
 
132 On Bootes General Method. 
 
 expresses a relation among certain characters of the universe 
 7", which holds true whatever the value of the character x. 
 Observation 3. If the given equation isf(x)=m; it can 
 be shown that the result of the elimination of x from the 
 equation is 
 
 287. To eliminate x and y from an equation f(x, y) = o ; 
 x and y being such that x (i -x)=o, andy(i y) = o. 
 
 As x and y are independent, x can be eliminated first, 
 and thenj'. The elimination of x by the preceding Article 
 gives 
 
 By a double application of the same process with respect 
 to j>, we get 
 
 /(I, !)/(!, 0)/(0, I)/(0, 0) = 0. 
 
 288. The method is illustrated by the following simple 
 example. 
 
 Let the equation be xy+ z. 
 To eliminate #, write the equation in the form 
 f(x)=x-y-z=o. 
 
 that is 72=0, 
 
 provided y and z as well as x are known to be single and 
 
 positive. 
 
 To eliminate y, write the equation in the form 
 
 Then (1+2 x)(z-x) = o, 
 
 that is, z(i x) = o. 
 
 289. The second power of any expression which is a func- 
 tion of single symbols cannot be negative. 
 
 For the expression can be expanded in terms of the 
 primary parts into which the universe is divided by the 
 symbols. The second power of the expansion is equal to 
 the sum of the second powers of the terms of the expansion, 
 as the products all vanish. The second power of each term 
 
On Boole s General Method. 1 33 
 
 is composed of the second power of the co-efficient multi- 
 plied by the second power of the part. The former cannot 
 be negative (Art. 77) and the latter cannot be negative 
 (Art. 138) ; hence their product cannot be negative. But 
 the sum of a number of terms, each of which cannot be 
 negative, cannot itself be negative. 
 
 290. To prove that x-\-y+z2xy2xz-\-2yz cannot be 
 negative. 
 
 For it is equal to {xyz} 2 . 
 It is evident that the expression will be as negative as 
 possible, when x=i,y=i. Its value is then z. 
 
 291. If each equation of the system 
 
 x 1 = o, # 2 = o, . . . x n =o, 
 
 has its left-hand member of a form which cannot be negative ; 
 then the equation 
 
 *i+* a + ...+#=o 
 is equivalent to the system of equations. 
 
 When the equations of the system are true, the com- 
 pound equation is true, for it is formed by adding the 
 equations of the system. (Art. 115.) 
 
 When the compound equation is true, each equation of 
 the system must be true ; because the left-hand member of 
 the compound equation is a sum of functions each of which 
 cannot be negative. 
 
 292. If any one equation of the above system, x r =o, 
 has a left-hand member x r of a form which can be negative ; 
 then x r 2 =o and not x r =o must be put into the compound 
 equation. 
 
 If x r were put into the compound equation, then the 
 truth of each of the original equations could not be inferred 
 necessarily from the truth of the compound equation. 
 But x? cannot be negative (Art 289). Hence when x r 2 =o 
 is put into the compound equation, its truth and the truth 
 of the others can be inferred from the truth of the com- 
 pound equation. Now from x r 2 =o we can deduce x r =o, 
 whether x r is or is not of a form which can be negative. 
 
134 On Boole s General Method. 
 
 Hence the equations of the system can be inferred back 
 from the truth of the compound equation, provided x r be 
 squared. 
 
 293. Example. 
 
 The equation x=y gives x y=o. Now x y is of a 
 form which can be negative. Hence it is (x /) 2 =o 
 which must be put into an equation, which is to be equiva- 
 lent to x=y together with other equations. 
 
 It is curious to observe that (x y) 2 =o gives 
 
 that is, x(iy) = o, andj(i x) = o ; 
 
 that is, x=xy, andy=yx. 
 
 Thus the single equation xy must for the purpose con- 
 sidered be resolved into the two equations x=xy and 
 
EXAMPLES. 
 
 AT page 56 of his Syllabus, De Morgan gives the follow- 
 ing twenty-four forms of inference, under the name of 
 Metaphysical Syllogisms : 
 
 1. Dependent of dependent is dependent. 
 
 2. Essential of independent is independent. 
 
 3. Independent of essential is independent. 
 
 4. Essential of essential is essential. 
 
 5. Dependent of inessential is inessential. 
 
 6. Inessential of dependent is inessential. 
 
 7. Repugnant of alternative is dependent. 
 
 8. Repugnant of independent is inalternative. 
 
 9. Independent of alternative is irrepugnant. 
 
 10. Alternative of repugnant is essential. 
 
 11. Alternative of inessential is irrepugnant. 
 
 12. Inessential of repugnant is inalternative. 
 
 13. Dependent of repugnant is repugnant. 
 
 14. Essential of irrepugnant is irrepugnant. 
 
 15. Irrepugnant of repugnant is independent. 
 
 16. Essential of alternative is alternative. 
 
 17. Dependent of inalternative is inalternative. 
 
 18. Inalternative of alternative is inessential. 
 
 19. Repugnant of essential is repugnant. 
 
 20. Repugnant of irrepugnant is inessential. 
 
 21. Irrepugnant of dependent is irrepugnant. 
 
 22. Alternative of dependent is alternative. 
 
 23. Alternative of inalternative is independent. 
 
 24. Inalternative of essential is inalternative. 
 
 In each case three characters of a given universe are 
 considered. Let the first-mentioned character be denoted 
 by x, the second by y, the third by z. The first term in 
 the proposition expresses the relation existing between x 
 
136 Examples. 
 
 and y ; the second that existing between y and z ; and the 
 third the relation which exists in consequence between x 
 and z. 
 
 By x being dependent on y is meant x (i y) = o. 
 
 independent of ,, x ( i y) = v. 
 
 essential to (i x)y=o. 
 
 ,, inessential to (i x)y=v. 
 
 repugnant to xy=o. 
 
 irrepugnant to ,, xy=v. 
 
 alternative to (ix)(iy) = o. 
 
 inalternative to ,, (i x) (i y) = v. 
 
 Some of these syllogisms coincide with Aristotelian forms, 
 if the latter have the meaning assigned to them in Articles 
 256-260. The truth of the conclusions can be proved by 
 means of the theorem enuntiated in Article 256 excepting 
 those of the sixteenth and twenty-second forms, which are 
 not true as enuntiated. 
 
 Consider the sixteenth. The data are 
 
 xy=y (i) and y+(iy)z=i. (2). 
 
 Substitute xy for y in (2), 
 then xy-\-(ixy) z=i, 
 
 that is, (i xy) (iz) = o. 
 
 Hence xy, but not y, is alternative to z with respect to i. 
 It is evident that x is alternative to z with respect to y. 
 The conclusion of the twenty-second form must be corrected 
 by introducing the same condition. 
 
 25. For every man in the house there is a person who 
 is aged ; some of the men are not aged. It follows that 
 some persons in the house are not men. De Morgan, 
 Syllabus, p. 29. 
 
 Let U= people in the house ; 
 
 w= man, and =aged : 
 
 Then m<d (i). and m(\d) = v. (2). 
 
 From (i) i m>i a; 
 
 but from (2) i a is greater than o. 
 
 .'. i m is greater than o. 
 
 
Examples. 137 
 
 26. Most men in a certain company have coats ; most 
 men in the same company have waistcoats ; therefore some 
 in the company have coats and waistcoats. De Morgan. 
 
 27. Of the inhabitants of a certain country those having 
 black hair form one-tenth, and those who are short-sighted 
 one-twelfth. What proportion must be (i) black-haired 
 and short-sighted, (2) black-haired and not short-sighted, 
 (3) not black-haired and short-sighted, (4) neither black- 
 haired nor short-sighted ? 
 
 28. Eighteen out of twenty-one F's are X; fifteen out 
 of twenty-one Y's are Z; therefore twelve Z's are X. De 
 Morgan. 
 
 Here the F's form the universe, and the arithmetical 
 value of the universe is given. 
 
 29. For every Z there is an X which is Y ; some Z's are 
 not F's. De Morgan, Syllabus, p. 29. 
 
 The two conclusions which De Morgan deduces can be 
 proved in the following manner : 
 
 The data are xy>z (i). and z(i y)=v. (2). 
 From (2) (iz)y=y+vz, 
 
 .*. x (i z)>z-\>y-\-v z y 
 
 >v. 
 
 Again, from (i) (T x) (i y) > i xy+z, 
 ,\. (i x) z>ixy+z-{-vi+y. 
 >z+vx. 
 
 30. Of a certain community, which comprises 12,000 
 persons, three-fourths are Presbyterian, two-thirds Liberal, 
 and one-sixth on the electoral roll. What is the least and 
 what the greatest possible strength of each of the twelve 
 classes formed by taking two of the characters; as, for 
 example, Presbyterian and Liberal, Presbyterian and not 
 Liberal, etc. ? Also what is the greatest and what the least 
 possible strength of each of the eight classes formed by 
 taking three characters ; as, for example, not Presbyterian 
 and not Liberal and not on the electoral roll ? 
 
 31. In a series of twelve throws with two dice, ace and 
 
138 Examples. 
 
 two occurred together in one-half of the throws, ace and 
 not-three occurred together in one-third, and ace occurred 
 in two-thirds. How often, at least, must two and three 
 have occurred together? 
 
 32. In a series of throws with two dice, ace and two 
 occurred together in one-sixth ; ace and four in one-twelfth. 
 How often, at least, must two and not four have occurred 
 together ? 
 
 33. Of the matriculated students of a certain University, 
 two-thirds are medical ; three-fourths are under the age of 
 2 1 ; and five-sixths are natives of the British Islands. Also 
 the medical under the age of 21 form one-half; and the 
 colonial or foreign under the same age one-twelfth. The 
 number of students is 2400. What is the least, and what 
 the greatest possible strength of each of the remaining ten 
 classes of the second order ; also of each of the eight classes 
 of the third order ? 
 
 34. From P follows Q, and -from R follows S; but Q 
 and cannot both be true ; therefore Pand R cannot both 
 be true. 
 
 Let U = the collection of times considered. / = when 
 P is true, q = when Q is true, etc. 
 Then the data are 
 
 P-M ( x )- r=rs ( 2 ) ^=- (3)- 
 
 Multiply together (i) and (2), 
 
 then pr=pqrs, 
 
 = o. by (3). 
 
 35. Every X is one only of the two, POT Q; every Fis 
 both P and Q, except when P is M, and then it is neither ; 
 therefore no X is K De Morgan, Formal Logic, p. 124. 
 
 Let U denote the subject of which x, j,/, q are charac- 
 ters. Then m is not a character of U, but of Up. Hence 
 the data are 
 
 x=x{p(i-g)+q (i-p)} (i) 
 
 Multiply together the two equations ; then each of the 
 
Examples. 139 
 
 four terms of the right-hand member will vanish, because 
 
 Hence xy=o. 
 
 36. Every X is either P, Q, or R ; but every P is J/, 
 every Q is Af, every R is M ; therefore every X is M. De 
 Morgan. 
 
 37. Dilemma. What is ^4 is either B or C; what is D 
 is neither ^ nor C. Therefore what is D is not A. 
 
 38. Constructive Dilemma. t7's which are x are y ; 7's 
 which are z are 7; and 7's are either # or not x and 0. 
 Therefore 7's are j. 
 
 The data are x=xy (i) 3=29; (2) 
 
 a.ndx+(ix)z=i (3) 
 
 Substitute for # and z in (3) by means of (i) and (2). 
 Then xy+zy xz=i, 
 
 \-\-xz 
 
 >=IHT 
 
 =xz-}-x(i z) f -\-(i x)z-\ (i x)(i z\ 
 
 = i. 
 Hence 7's are y. 
 
 39. The Method of Difference, which bulks so largely in 
 some Systems of Logic, can be disposed of in a few lines. 
 
 In the states of a certain substance when the property A 
 is present, the property B is also present ; and when the 
 property A is absent, the property B is also absent. 
 What follows ? 
 
 The data are 
 
 a=ab (i) . (i a) = (i a)(i^&). (2) 
 
 From (2) i a=ia b-\-ab, 
 
 .'. b=a from (i). 
 
 Hence the states in which the property A is present are 
 identical with the states in which the property B is present. 
 
140 Examples. 
 
 40. The members of a board were all of them either 
 bondholders or shareholders, but not both ; and the bond- 
 holders, as it happened, were all on the board. What con- 
 clusion can be drawn ? Venn, Mind, vol. i. p. 487, discussed 
 by Jevons, Principles of Science, p. 90. 
 Let U= members of the company. 
 
 d= director b= bondholder .$= shareholder. 
 Then d=d{6(i-s)+s(i-6)}. (i) 
 
 and b=bd. (2) 
 
 Put (i) into the form 
 
 </{i-J(i-f)-j(i-J)} = 0> 
 and (2) into the form 
 
 b(i-d) = o- 
 
 then their left-hand members consist of functions which 
 cannot be negative. Hence these two equations are 
 together equivalent to 
 
 d{i-b(i-s)-s(i-b)}+b(i-d) = o. (Art. 291.) 
 
 By means of this equation any one of the three d, b, s, 
 can be expressed in terms of the remaining two. For 
 example 
 
 . 
 
 - 
 
 -b) (I - J). 
 
 =b(i j) + (i b)s, from the nature of d, 
 o 
 
 with the condition fo=o, 
 
 =^+5 by taking in bs=o. 
 
 o 
 
 For the sake of illustrating its application I have here 
 employed Boole's Method of dealing with implicit equa- 
 tions. The conclusions can be deduced by a simpler 
 method. Thus 
 
 d=d(b+s) (i). (Art. 251.) 
 
 and db=b. (2). 
 
Examples. 141 
 
 Hence d 
 
 which is equivalent to d=b-\ s. 
 
 o 
 
 Multiply by s, 
 then ds=bs+ds, 
 
 41. It has been observed that In a certain class of sub- 
 stances, where the properties A and B are present, the pro- 
 perty C is present ; and where B and C are present, A is 
 present. Does it follow that B is present where C and A 
 are present ? 
 
 Let 7= the class of substances. 
 a= having the property A, etc. 
 The data are ab=abc and bc=bca. 
 Hence ab(ic)+bc(La) = o. (Art. 291.) 
 
 Now -=i. (Art. 78.) 
 
 
 And is impossible. (Art. 78.) Therefore ab (i <r) = o 
 
 and fc(i 0) = o; which are the given equations taken 
 each by itself. The sixth term is cancelled. (Art. 79.) 
 
 Add together the terms whose co-efficient is , 
 
 then ca abc-\ ( i b). 
 
 Hence it does not follow that B is present where C and 
 A are present. 
 
142 Examples, 
 
 42. Suppose that an analysis of the properties of a par- 
 ticular class of substances has led to the following general 
 conclusions, viz. : 
 
 ist, That wherever the properties A and B are combined, 
 either the property C, or the property Z>, is present also ; 
 but they are not jointly present. 
 
 2d, That wherever the properties B and C are combined, 
 the properties A and D are either both present with them, 
 or both absent. 
 
 3d, That wherever the properties A and B are both 
 absent, the properties C and D are both absent also ; and 
 vice versa, where the properties C and D are both absent, 
 A and B are both absent also. 
 
 Let it then be required from the above to determine 
 what may be concluded in any particular instance from the 
 presence of the property A with respect to the presence or 
 absence of the properties B and C, paying no regard to the 
 property D. Boole, Laws of Thought, p. 1 1 8. 
 
 Here the universe under consideration is a particular 
 class of Substances. Let a= having the property A ; 
 a'= without the property A ; and so on. 
 
 Then ab=ab (c+d). (i) 
 
 bc=bc(ad+a'd'). (2) 
 
 a'b'=c'd'. (3) 
 
 Since the first equation, when put into the form 
 
 contains a function which can be negative, it must be 
 squared before being put into the equivalent equation. 
 (Art. 292.) This squaring amounts to writing the equation 
 in the form 
 
 ab-ab{cd'-\-dc'}. 
 
 Similarly (3) must be squared. Hence the equivalent 
 equation is 
 
 ab{i-cd'-dc'}+b<:{i-ad-a'd'}+a'b'(i-c'd') 
 
 +c'd'(i-a'{>') = o. 
 As d enters into this equation, and is not to enter into 
 
Examples. 143 
 
 the required conclusion, it must be eliminated by Art. 286. 
 Now 
 
 f(i)=abc+bca'+a'b', 
 and f (>)== abc' -\-bca-\- a' b'c-\-c'(\-~ a' b')) 
 
 is true whatever value d may have, consistent with being 
 positive and single. 
 
 Hence a= 
 
 ^'+- 
 o o 
 
 Hence the substances, which have the property A, are 
 identical with those which have the property C and are 
 without the property JB, together with an indefinite part of 
 those which are without the property C. 
 
 The conclusion can be deduced very simply by the fol- 
 lowing method 
 
 ab=ab(c+d). (i) 
 
 bc=bc{ad+(i-a}(i-d)}. (2) 
 
 (i- fl )(i_J) = (i-,)(i-</). (3) 
 
 Multiply together (i) and (2), 
 then abc-=.abc(acd-\-ad\ 
 
 = 2 abed ; 
 but abc is single, 
 
 .'. abc=o. (4) 
 
 Again, multiply (3) by c, 
 
 then c(i a)(i ^)=o, 
 
 .-. ^i ^) = o, by (4) 
 
 .-. a= 
 
 This expression for a coincides with that obtained above. 
 
 43. Find also what may be concluded from the presence 
 of the property C with reference to the properties A and B. 
 
 44. In a certain class of substances, those which have 
 
144 Examples. 
 
 the property A and the property X, together with those 
 which have the property B and are without the property X, 
 are identical with those which have the property C. Also 
 those which have the property D and the property X, 
 together with those which have the property E and are 
 without the property X, are identical with those which have 
 the property F. What relation necessarily exists between 
 the properties A, B, C, D, E, F? 
 The data are 
 
 x)=c. (i) 
 
 From (i) *= t- 
 
 a b 
 
 From (2) *=^. 
 
 a e 
 
 Hence, as the equations are true simultaneously, 
 c-b_f-e. 
 a -b~~ d-e' 
 
 45. In a certain succession of throws with six dice, those 
 having a two and an ace, together with those having a three 
 and a six, were identical with the whole ; and those having 
 a four and an ace, together with those having a five and a 
 six, were identical with the whole. What relations existed 
 between having a two, having a three, having a four, and 
 having a five ? 
 
 ' Having a two ' here means ' having one two at least.' 
 The equations are 
 
 i. (i) 
 
 i* (2) 
 
 Multiply (i) by d and (2) by , and subtract ; 
 then (adbc)x=db; 
 _ d-b 
 
 * ^ ; 7" 
 
 ad be 
 
Examples. 145 
 
 Expand ~ in terms of the primary parts into which 
 ad be 
 
 the universe is divided by , , t, d. Then since x is pos- 
 
 sible, each of the co-efficients of must be o. We thus 
 
 o 
 
 obtain 
 
 0(i-r)(i-^) = o (i) (i-a)b(i-c)(i-d) = o (2) 
 
 (i-a)(i_fy*=o (3) (i-)(i-*)(i-*X=o (4) 
 
 The equations (i) and (2) can be combined into the 
 equivalent equation 
 
 b(*-c)(i-d)= . (5) 
 
 Similarly (3) and (4) into (i 0)(i b}d=o. (6) 
 
 Equation (5) can be put into the form 
 
 that is, the throws having a three had a four or, if not a 
 four, a five. 
 
 Similarly d=d{a+(i a)b}. 
 
 From the expansion for y other two relations can be 
 obtained, viz. 
 
 and <:=<:{# +(i a)b}. 
 
 46. Boole solves the following highly complex problem : 
 
 Let the observations of a class of natural productions be 
 supposed to have led to the following general results : 
 
 ist, That in whichsoever of these productions the pro- 
 perties A and C are missing, the property E is found, 
 together with one of the properties B and Z>, but not with 
 both. 
 
 2d, That wherever the properties A and D are found 
 while E is missing, the properties B and C will either both 
 be found, or both be missing. 
 
 3d, That wherever the property A is found in conjunction 
 with either B or J t or both of them, there either the pro- 
 perty C or the property D will be found, but not both of 
 them. And conversely, wherever the property C or D is 
 
 K 
 
1 46 Examples. 
 
 found singly, there the property ^4 will be found in conjunc- 
 tion with either B or E, or both of them. 
 
 Let it then be required to ascertain, first, what in any 
 particular instance may be concluded from the ascertained 
 presence of the property A, with reference to the properties 
 B, C, and D; also whether any relations exist indepen- 
 dently among the properties B, C > and D. Secondly, what 
 may be concluded in like manner respecting the property B> 
 and the properties A, C, and D. Laws of Thought, p. 146. 
 
 The equations are 
 
 ad(i-e)=ad(i-e){bc+(i-b) (i-c)} (2) 
 
 ab+ae(i-b) = d(i-c)+c(i.-d) (3) 
 
 The function of (3) must be squared before being put 
 into the single equation. (Art. 292.) Then e can be 
 eliminated by means of Article 286. By expanding the func- 
 tion for a in terms of b, <r, d, we get 
 
 = * (!-<*) + <* (I -*) + (!-*) (I-*) (l-rf). (l) 
 
 By taking the function for b in terms of a, c, d and ex- 
 panding it, we obtain 
 
 and acd=Q (3), (i a) <r(i <f) = o, (4) 
 
 <i-)(x-*)*=o. (5) 
 
 By means of (3), (4), and (5) equation (2) can be reduced 
 to 
 
 47. A certain philosopher has observed that his library 
 satisfies the following conditions : 
 
 (i.) Languages, which contain Dictionaries and Poetry, 
 contain Mathematics. 
 
 (2.) Languages, which contain Novels and Mathematics, 
 contain Natural Philosophy. 
 
Examples. 147 
 
 (3.) Languages, which contain Natural Philosophy, con- 
 tain either Mathematics or Dictionaries, but not both. 
 
 (4.) Languages, which contain no Novels, contain either 
 Poetry or no Natural Philosophy. 
 
 (5.) Languages, which contain Novels, Dictionaries, and 
 no Mathematics, contain either Poetry and no Natural 
 Philosophy, or Natural Philosophy and no Poetry. 
 
 (6.) Languages, which contain Poetry and Natural 
 Philosophy, contain Dictionaries and Mathematics, or no 
 Dictionaries. 
 
 Are these data all independent of one another ? Is there 
 any special relation between two only of the subjects? 
 What are the relations among the subjects independently of 
 Natural Philosophy ? 
 
 48. To find (i # y)* in terms of x+y, xy t and xy. 
 
 Let (i-x-yY=a+b(x+.y)+c(x-y)+dxy. 
 
 Then 1=0+ 2^ +d. 
 
 o=a-\-& -\-c. 
 
 o=a-\-b~ c. 
 
 i=a. 
 
 The value of the determinant of the right-hand members 
 is 2. When the equations are solved by the method of 
 determinants, we get 
 
 22 0.4 
 
 a=-, 1,= --, c=-, d=-. 
 
 Hence by Articles 78 and 79 
 
 and (i x yY = i 
 
 I introduce this example to illustrate the truth and power 
 of the method by indeterminate co-efficients. 
 
 49. There are three bags, one containing 2 white and 3 
 red balls, the second containing 5 white and 2 red balls, the 
 third containing 4 white and 7 red balls ; and all the bags 
 are equally likely to be drawn from. A white ball has 
 been drawn ; what is the chance that it was drawn from 
 the first bag? Gross, Algebra, p. 225. 
 
148 Examples. 
 
 Let U = succession of drawings of a ball. 
 
 a = from the first bag ; b = from the second bag ; 
 
 c = from the third bag. 
 <z w =from the first bag and white, 
 a r and 
 
 The data are 
 
 Now 
 
 .. w=a w -{-b^-\-c w . (Art. 71.) 
 
 Probability required =^ 
 
 _ W Sj 
 
 *+*+* 
 
 50. Suppose that in the drawing of balls from an urn 
 attention had only been paid to those cases in which the 
 balls drawn were either of a particular colour, * white,' or of 
 a particular composition, ' marble,' or were marked by both 
 these characters, no record having been kept of those cases 
 in which a ball that was neither white nor of marble had 
 been drawn. Let it then have been found, that whenever 
 the supposed condition was satisfied, there was a probability 
 p that a white ball would be drawn, and a probability q that 
 a marble ball would be drawn : and from these data alone 
 let it be required to find the probability that in the next 
 drawing, without reference at all to the condition above men- 
 tioned, a white ball will be drawn ; also the probability that a 
 marble ball will be drawn. Boole, Laws of Thought, p. 262. 
 
 U long succession of drawings of a ball from the urn ; 
 
 x = producing a white ball, 
 
 y = producing a marble ball. 
 
Examples. 149 
 
 Let x+(i x)y=a. 
 
 Then a x =afi t and a y -=zaq. 
 
 It is required to find x and y. 
 Now ^={^+(1 ^}^, 
 
 =3*. 
 
 Similarly a y =y- 
 
 Hence /= _ , r * ^ (i) (Art. 64.) 
 
 x+(i-x)y 
 
 and g= _ . .- - r (2) 
 
 x+(i-x)y 
 
 From (i) and (2) py=.qx. (3) 
 
 From (i) ^=/{^+(i ^}, 
 
 hence from (3) . 
 
 Boole proceeds further on the assumption that xy=xy. 
 That is only the most probable assumption among an 
 infinite number, and the value for x deduced by its means 
 is only the most probable value of the probability of a 
 given drawing producing a white ball. Let the assumption 
 be introduced, 
 
 then 
 
 And similarly 
 
 ~~ 
 
 51. The probability that a witness A speaks the truth is 
 , the probability that another witness B speaks the truth 
 and the probability that they disagree in a statement 
 is ps. What is the probability that if they agree, their 
 statement is true ? Boole, Laws of Thought, p. 279. 
 Let U = Testimonies by A and by B about a fact ; 
 x = in which A told the truth, 
 y = in which B told the truth. 
 
1 50 Examples. 
 
 Then the data are 
 
 *=A (0 y=P* (2) 
 
 and x(i-y}+y(i-x}=fr. (3) 
 
 It is required to find the value of the dependent xy in 
 
 Required value = = xy -. (Art. 64.) 
 
 Now from (3) 2xy=x+y p z , 
 
 =A +A -A- from (i ) and ( 2) 
 Also from (3) tfy+ (i .#)(i _y) == i / 3 . 
 
 Hence required value = |A_+A-A 
 
 I~/l 
 
 COR. The value of the dependent (i #)(i j) in 
 
 g 
 
 To deduce these conclusions Boole employs a long and 
 elaborate process. He does not see that xy can be deduced 
 by very simple algebraic operations from the given data. 
 At the end of his investigation he remarks that the number of 
 the data exceeds that of the simple events which they involve. 
 The considerations of Article 151 show that the statement 
 is not correct that the data are just sufficient to determine 
 all the single functions of x and y necessarily. 
 
 52. A says that B says that a certain event took place ; 
 required the probability that the event did take place, 
 /! and / 2 being A's and l?s respective probabilities of 
 speaking the truth. Mathematics from the Educational 
 TimeS) vol. xxvii. 
 
 We must suppose that a succession of statements of A 
 about /?'s statements about an event taking place have been 
 observed. It is required to deduce the probability that 
 one of these statements refers to an event which actually 
 took place. 
 
Examples. 1 5 1 
 
 Let U= statements of A about B's statements about an 
 event taking place. 
 
 x =s which truly reported a statement made by B ; 
 
 y = which truly reported the event. 
 The data are 
 
 *=A (i) *y=AA- (2) 
 
 and y is required. 
 Now xy=x y1 
 
 Hence 
 and 
 
 There has been much discussion about the true answer 
 to the above question. No fewer than four different solu- 
 tions are given in the Reprint, viz. 
 
 Todhunter's Algebra AA+C 1 "" AX 1 A)- 
 
 Artemas Martin A {AA+(i -A)(i ~A)}- 
 
 American Mathematicians and Woolhouse A A- 
 
 Cayley /^,+^(i -A)(i -/.)+*(i -(i -/i)- 
 
 The meaning of /? and K in Cayley's solution is explained 
 by the following paragraph : 
 
 ' B told A that the event happened, or he did not tell A 
 this ; the only evidence is A's statement that B told him 
 that the event happened; and the chances are p and 
 i /i. But, in the latter case, either B told A that the 
 event did not happen, or he did not tell him at all ; the 
 chances (on the supposition of the incorrectness of A's 
 statement) are /? and i /? ; and the chances of the three 
 cases are thus p l9 /?(i /i), and (i /*)(i P\}> On the 
 suppositions of the first and second cases respectively, the 
 
152 Examples. 
 
 chances for the event having happened are fa and i A ; 
 on the supposition of the third case (viz., here there is no 
 information as to' the event) the chance is /c, the antecedent 
 probability ; and the whole chance in favour of the event is 
 
 Todhunter assumes that is equal to i A ; and Wool- 
 o 
 
 house assumes that it is equal to o. Cayley's solution 
 expresses in terms of the unknown quantities /3 and K. 
 
 The next example contains the solution of the general 
 problem, when there are n persons. 
 
 53. A! says that A t says that A* says . . . that A n 
 says that a certain event took place. The probabilities 
 of A ly A 2 , A s , . . . A n respectively speaking the truth are 
 A> A> A> Pn> Required the probability that the event 
 took place. 
 
 Let U = succession of statements of A l about A % saying 
 etc. 
 
 x l = which reported truly a statement of A^ 
 
 X z = ,, A 3 
 
 #3 = A* 
 
 
 
 #_!= A H 
 
 x n = which reported truly the event. 
 Let #1 x 2 denote that # a is formally dependent on x l ; 
 that is, x lXt 
 
 Then x l -x^x i . . . x n ^ - x n =p^p z p^ - - A-i^ 
 and Xt'XfX, . . . x ft - l 
 
 Now x =**'**** ' ' * 
 
 (i 
 
 COR. i. Suppose that each of the persons always reports 
 truly. 
 
Examples. 153 
 
 Then/u/2? etc., each equal to i ; 
 
 and therefore x u i+xo=i. 
 
 o 
 
 COR. 2. Suppose that each person excepting A n always 
 reports truly. 
 
 Then x n =p n . 
 
 COR. 3. Suppose that A n always reports falsely. 
 
 Then A,= 
 
 COR. 4. Suppose that any other of the persons excepting 
 A n always reports falsely. 
 
 Then x n =-< 
 
 o 
 
 COR. 5. Suppose that each reports falsely as often as 
 truly. 
 
 Then each p is equal to -i. 
 
 Hence x n 
 
 When n is infinitely great, the range between the limits 
 is i ; that is, there is complete uncertainty. 
 
 54. A goes to hall p times in a consecutive days and sees 
 B there q times. What is the most probable number of 
 times that B was in hall in the a days ? Whitworth, Choice 
 and Chance, p. 239. 
 
 Let U = the consecutive days ; 
 x = on which A went to hall, 
 y = on which B went to hall. 
 
 The data are 
 
 and it is required to find Uy. 
 
 Now y=xy+ (i x), 
 
 o 
 
 a p. 
 
154 Examples. 
 
 We cannot proceed further without assuming a condition 
 which is not given. 
 
 Assume that B is as likely to go to hall when A does not 
 go, as when A does go ; then 
 
 _ 
 P 
 
 55. The probabilities of two causes AI and A 2 are a^ and 
 # 2 respectively. The probability that if the cause A l pre- 
 sent itself, an event E will accompany it (whether as a 
 consequence of the cause A l or not) is p ly and the proba- 
 bility that if the cause A z present itself, that event E will 
 accompany it, whether as a consequence of it or not, is / 2 . 
 Moreover, the event E cannot appear in the absence of 
 both the causes A l and A z . Required the probability of 
 the event E. Boole, Laws of Thought^ p. 321. 
 
 The causes A l and A t and the event E are characters of 
 a definite universe. Let them be denoted by x, y, z respec- 
 tively. 
 
 The data are 
 
 x=dt (i) y=d t (2) 
 
 xz=d l p l (3) yz=difa (4) 
 
 and (i-*)(i-j)s=o. (5) 
 
 Article 282 shows that the probability required cannot be 
 determined exactly from the data. The limits, between 
 which it must lie, are given in Article 274. 
 
 Boole gives the relations which exist among the data. 
 These relations can be determined very readily by means 
 of the method of Article 226. Thus 
 
 iAr=i d l and xz=d l p l , 
 
 that is, 
 
 Similarly 
 
 Again x(i z)=x 
 
 and 2 
 
Examples. 155 
 
 
 that is, 
 
 Similarly d^p^<\ 
 
 And there is no other way in which the process can be 
 applied to the given data ; hence these are all the relations 
 which exist among them. The solution of this problem 
 was discussed by Cayley, Boole, and Wilbraham in the 
 Philosophical Magazine, 4th series, vol. vi. p. 259 ; vol. vii. 
 p. 29 and p. 465; vol. viii. p. 87 and p. 431. Wilbraham 
 gives as the solution 
 
 *=<*!/, +^>, 
 
 where f is necessarily less than either a 1 ^> l or a z p z . He 
 maintains that we can get no further in the solution without 
 further assumptions or data, and remarks that the disadvan- 
 tage of Boole's method in such cases is, that it does not 
 show us whether the problem is really determinate. 
 
 The method by indeterminate co-efficients gives a solu- 
 tion which agrees with Wilbraham's (Art. 282), and in every 
 case supplies the desiderated information as to whether the 
 proposed problem is determinate. 
 
 What is given by Boole's solution is not the mathematical 
 probability of the event E, but the most probable value of 
 the probability which can be deduced from the given data. 
 
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