,,first ,,notions ,,of ,,logic 7,,preparatory ,,to ,,the ,,study ,,of ,,geometry7 ,,by ,,augustus ,,de ,,morgan1 ,,of ,,trinity ,,college1 ,,cambridge1 ,,professor ,,of ,,mathematics ,,in ,,university ,,college1 ,,london4 ,the root of all the mischief in the sciences1 is this2 that falsely magnifying and admiring the powers of the mind1 we seek not its real helps4--,,bacon4 ,,london3 ,,printed ,,for ,,taylor ,,and ,,walton1 ,,booksellers ,,and ,,publishers ,,to ,,university ,,college4 #bh ,,upper ,,gower ,,street4 ,m4,,dccc4,,xxxix4 ,this ,tract contains no more than the author has found1 from experience1 to be much wanted by students who are commencing with ,euclid4 ,it will ultimately form an ,appendix to his ,treatise on ,arithmetic4 ,the author would not1 by any means1 in presenting the minimum necessary for a particular purpose1 be held to imply that he has given enough of the subject for all the ends of education4 ,he has long regretted the neglect of logic2 a science1 the study of which would shew many of its opponents that the light esteem in which they hold it arises from those habits of inference which thrive best in its absence4 ,he strongly recommends any student to whom this tract may be the first introduction of the subject1 to pursue it to a much greater extent4 ,university ,college1 ,jan1 #h1 #ahci4 ,,london3--,,printed ,,by ,,james ,,moyes1 ,castle ,street1 ,leicester ,square4 ,,first ,,notions ,,of ,,logic4 ,what we here mean by ,logic is the examination of that part of reasoning which depends upon the manner in which inferences are formed1 and the investigation of general maxims and rules for constructing arguments1 so that the conclusion may contain no inaccuracy which was not previously asserted in the premises4 ,it has nothing to do with the truth of the facts1 opinions1 or presumptions1 from which an inference is derived2 but simply takes care that the inference shall certainly be true1 if the premises be true4 ,thus1 when we say that all men will die1 and that all men are rational beings1 and thence infer that some rational beings will die1 the logical truth of this sentence is the same whether it be true or false that men are mortal and rational4 ,this logical truth depends upon the structure of the sentence1 and not on the particular matters spoken of4 ,thus1 ,instead of1 ,write1 ,all men will die4 ,every ,a is ,b4 ,all men are rational beings4 ,every ,a is ,c4 ,therefore some rational beings will die4 ,therefore some ,cs are ,bs4 ,the second of these is the same proposition1 logically considered1 as the first2 the consequence in both is virtually contained in1 and rightly inferred from1 the premises4 ,whether the premises be true or false1 is not a question of logic1 but of morals1 philosophy1 history1 or any other knowledge to which their subject-matter belongs3 the question of logic is1 does the conclusion certainly follow if the premises be true8 ,every act of reasoning must mainly consist in comparing together different things1 and either finding out1 or recalling from previous knowledge1 the points in which they resemble or differ from each other4 ,that particular part of reasoning which is called inference1 consists in the comparison of several and different things with one and the same other thing2 and ascertaining the resemblances1 or differences1 of the several things1 by means of the points in which they resemble1 or differ from1 the thing with which all are compared4 ,there must then be some propositions already obtained before any inference can be drawn4 ,all propositions are either assertions or denials1 and are thus divided into affirmative and negative4 ,thus1 ,a is ,b1 and ,a is not ,b1 are the two forms to which all propositions may be reduced4 ,these are1 for our present purpose1 the most simple forms2 though it will frequently happen that much circumlocution is needed to reduce propositions to them4 ,thus1 suppose the following assertion1 #8,if he should come to-morrow1 he will probably stay till ,monday'2 how is this to be reduced to the form ,a is ,b8 ,there is evidently something spoken of1 something said of it1 and an affirmative connexion between them4 ,something1 if it happen1 that is1 the happening of something1 makes the happening of another something probable2 or is one of the things which render the happening of the second thing probable4 ,a is ,b ,the happening of his arrival an event from which it may be to-morrow is inferred as probable that he will stay till ,monday4 ,the forms of language will allow the manner of asserting to be varied in a great number of ways2 but the reduction to the preceding form is always possible4 ,thus1 #8so he said' is an affirmation1 reducible as follows3 ,what you have just said 7or isthe thing which he said4 whatever else #8so' refers to7 ,by changing #8is' into #8is not1' we make a negative proposition2 but care must always be taken to ascertain whether a proposition which appears negative is really so4 ,the principal danger is that of confounding a proposition which is negative with another which is affirmative of something requiring a negative to describe it4 ,thus #8he resembles the man who was not in the room1' is affirmative1 and must not be confounded with #8he does not resemble the man who was in the room4' ,again1 #8if he should come to-morrow1 it is probable he will not stay till ,monday1' does not mean the simple denial of the preceding proposition1 but the affirmation of the directly opposite proposition4 ,it is1 ,a is ,b ,the happening of his arrival an event from which it may be to-morrow1 is inferred to be improbable that he will stay till ,monday1 whereas the following1 ,the happening of his arrival an event from which it may be to-morrow1 is not inferred as probable that he will stay till ,monday1 would be expressed thus3 #8,if he should come to-morrow1 that is no reason why he should stay till ,monday4' ,moreover1 the negative words not1 no1 c41 have two kinds of meaning which must be carefully distinguished4 ,sometimes they deny1 and nothing more3 sometimes they are used to affirm the direct contrary4 ,in cases which offer but two alternatives1 one of which is necessary1 these amount to the same thing1 since the denial of one1 and the affirmation of the other1 are obviously equivalent propositions4 ,in many idioms of conversation1 the negative implies affirmation of the contrary in cases which offer not only alternatives1 but degrees of alternatives4 ,thus1 to the question1 #8,is he tall8' the simple answer1 #8,no1' most frequently means that he is the contrary of tall1 or considerably under the average4 ,but it must be remembered1 that1 in all logical reasoning1 the negation is simply negation1 and nothing more1 never implying affirmation of the contrary4 ,the common proposition that two negatives make an affirmative1 is true only upon the supposition that there are but two possible things1 one of which is denied4 ,grant that a man must be either able or unable to do a particular thing1 and then not unable and able are the same things4 ,but if we suppose various degrees of performance1 and therefore degrees of ability1 it is false1 in the common sense of the words1 that two negatives make an affirmative4 ,thus1 it would be erroneous to say1 #8,john is able to translate ,virgil1 and ,thomas is not unable2 therefore1 what ,john can do ,thomas can do1' for it is evident that the premises mean that ,john is so near to the best sort of translation that an affirmation of his ability may be made1 while ,thomas is considerably lower than ,john1 but not so near to absolute deficiency that his ability may be altogether denied4 ,it will generally be found that two negatives imply an affirmative of a weaker degree than the positive affirmation4 ,each of the propositions1 #8,a is ,b1' and #8,a is not ,b1' may be subdivided into two species3 the universal1 in which every possible case is included2 and the particular1 in which it is not meant to be asserted that the affirmation or negation is universal4 ,the four species of propositions are then as follows1 each being marked with the letter by which writers on logic have always distinguished it4 ,a ,universal ,affirmative ,every ,a is ,b ,e ,universal ,negative ,no ,a is ,b ,i ,particular ,affirmative ,some ,a is ,b ,o ,particular ,negative ,some ,a is not ,b ,in common conversation the affirmation of a part is meant to imply the denial of the remainder4 ,thus1 by #8some of the apples are ripe1' it is always intended to signify that some are not ripe4 ,this is not the case in logical language1 but every proposition is intended to make its amount of affirmation or denial1 and no more4 ,when we say1 #8,some ,a is ,b1' or1 more grammatically1 #8,some ,as are ,bs1' we do not mean to imply that some are not3 this may or may not be4 ,again1 the word some means1 #8one or more1 possibly all4' ,the following table will shew the bearing of each proposition on the rest4 ,every ,a is ,b affirms and contains ,some ,a is ,b and,no ,a is ,b denies ,some ,a is not ,b ,no ,a is ,b affirms and contains ,some ,a is not ,b ,every ,a is ,b and denies ,some ,a is ,b ,some ,a is ,b does not ,every ,a is ,b but denies ,no ,a is ,b contradict ,some ,a is not ,b ,some ,a is not ,b does not ,no ,a is ,b but denies ,every ,a is ,b contradict ,some ,a is ,b ,contradictory propositions are those in which one denies any thing that the other affirms2 contrary propositions are those in which one denies every thing which the other affirms1 or affirms every thing which the other denies4 ,the following pair are contraries4 ,every ,a is ,b and ,no ,a is ,b and the following are contradictories1 ,every ,a is ,b to ,some ,a is not ,b ,no ,a is ,b to ,some ,a is ,b ,a contrary1 therefore1 is a complete and total contradictory2 and a little consideration will make it appear that the decisive distinction between contraries and contradictories lies in this1 that contraries may both be false1 but of contradictories1 one must be true and the other false4 ,we may say1 #8,either ,p is true1 or something in contradiction of it is true2' but we cannot say1 #8,either ,p is true1 or every thing in contradiction of it is true4' ,it is a very common mistake to imagine that the denial of a proposition gives a right to affirm the contrary2 whereas it should be1 that the affirmation of a proposition gives a right to deny the contrary4 ,thus1 if we deny that ,every ,a is ,b1 we do not affirm that ,no ,a is ,b1 but only that ,some ,a is not ,b2 while1 if we affirm that ,every ,a is ,b1 we deny ,no ,a is ,b1 and also ,some ,a is not ,b4 ,but1 as to contradictories1 affirmation of one is a denial of the other1 and denial of one is affirmation of the other4 ,thus1 either ,every ,a is ,b1 or ,some ,a is not ,b3 affirmation of either is denial of the other1 and vice vers4 ,let the student now endeavour to satisfy himself of the following4 ,taking the four preceding propositions1 ,a1 ,e1 ,i1 ,o1 let the simple letter signify the affirmation1 the same letter in parentheses the denial1 and the absence of the letter1 that there is neither affirmation nor denial4 ,from ,a follow7,e71 ,i1 7,o7,from 7,a7 follow ,o ,from ,e 7,a71 7,i71 ,o,from 7,e7 ,i ,from ,i 7,e7 ,from 7,i7 7,a71 ,e1 ,o ,from ,o 7,a7 ,from 7,o7 ,a1 7,e71 ,i ,these may be thus summed up3 ,the affirmation of a universal proposition1 and the denial of a particular one1 enable us to affirm or deny all the other three2 but the denial of a universal proposition1 and the affirmation of a particular one1 leave us unable to affirm or deny two of the others4 ,in such propositions as #8,every ,a is ,b1' #8,some ,a is not ,b1' c41 ,a is called the subject1 and ,b the predicate1 while the verb #8is' or #8is not1' is called the copula4 ,it is obvious that the words of the proposition point out whether the subject is spoken of universally or partially1 but not so of the predicate1 which it is therefore important to examine4 ,logical writers generally give the name of distributed subjects or predicates to those which are spoken of universally2 but as this word is rather technical1 ,i shall say that a subject or predicate enters wholly or partially1 according as it is universally or particularly spoken of4 #a4 ,in ,a1 or #8,every ,a is ,b1' the subject enters wholly1 but the predicate only partially4 ,for it obviously says1 #8,among the ,bs are all the ,as1' #8,every ,a is part of the collection of ,bs1 so that all the ,as make a part of the ,bs1 the whole it may be4' ,thus1 #8,every horse is an animal1' does not speak of all animals1 but states that all the horses make up a portion of the animals4 #b4 ,in ,e1 or #8,no ,a is ,b1' both subject and predicate enter wholly4 #8,no ,a whatsoever is any one out of all the ,bs2' #8search the whole collection of ,bs1 and every ,b shall be found to be something which is not ,a4' #c4 ,in ,i1 or #8,some ,a is ,b1' both subject and predicate enter partially4 #8,some of the ,as are found among the ,bs1 or make up a part 7the whole possibly1 but not known from the preceding7 of the ,bs4' #d4 ,in ,o1 or #8,some ,a is not ,b1' the subject enters partially1 and the predicate wholly4 #8,some ,as are none of them any whatsoever of the ,bs2 every ,b will be found to be no one out of a certain portion of the ,as4' ,it appears then that1 ,in affirmatives1 the predicate enters partially4 ,in negatives1 the predicate enters wholly4 ,in contradictory propositions1 both subject and predicate enter differently in the two4 ,the converse of a proposition is that which is made by interchanging the subject and predicate1 as follows3 ,the proposition4 ,its converse4 ,a ,every ,a is ,b ,every ,b is ,a ,e ,no ,a is ,b ,no ,b is ,a ,i ,some ,a is ,b ,some ,b is ,a ,o ,some ,a is not ,b ,some ,b is not ,a ,now1 it is a fundamental and self-evident proposition1 that no consequence must be allowed to assert more widely than its premises2 so that1 for instance1 an assertion which is only of some ,bs can never lead to a result which is true of all ,bs4 ,but if a proposition assert agreement or disagreement1 any other proposition which asserts the same1 to the same extent and no further1 must be a legitimate consequence2 or1 if you please1 must amount to the whole1 or part1 of the original assertion in another form4 ,thus1 the converse of ,a is not true3 for1 in #8,every ,a is ,b1' the predicate enters partially2 while in #8,every ,b is ,a1' the subject enters wholly4 #8,all the ,as make up a part of the ,bs1 then a part of the ,bs are among the ,as1 or some ,b is ,a4' ,hence1 the only legitimate converse of #8,every ,a is ,b' is1 #8,some ,b is ,a4' ,but in #8,no ,a is ,b1' both subject and predicate enter wholly1 and #8,no ,b is ,a' is1 in fact1 the same proposition as #8,no ,a is ,b4' ,and #8,some ,a is ,b' is also the same as its converse #8,some ,b is ,a2' here both terms enter partially4 ,but #8,some ,a is not ,b' admits of no converse whatever2 it is perfectly consistent with all assertions upon ,b and ,a in which ,b is the subject4 ,thus neither of the four following lines is inconsistent with itself4 ,some ,a is not ,b and ,every ,b is ,a ,some ,a is not ,b and ,no ,b is ,a ,some ,a is not ,b and ,some ,b is ,a ,some ,a is not ,b and ,some ,b is not ,a4 ,we find then1 including converses1 which are not identical with their direct propositions1 six different ways of asserting or denying1 with respect to agreement or non-agreement1 total or partial1 between ,a and1 say ,x3 these we write down1 designating the additional assertions by ,u and ,y4 ,identical4 ,identical4 ,a ,every ,a is ,x,e ,no ,a is ,x,i ,some ,a is ,x,o ,some ,a is not ,x ,u ,every ,x is ,a ,no ,x is ,a ,some ,x is ,a,y ,some ,x is not ,a ,we shall now repeat and extend the table of page #h 7,a71 c41 meaning1 as before1 the denial of ,a1 c4 ,from ,a or 7,o7 follow ,a1 7,e71 ,i 7,o7 ,from ,e or 7,i7 7,a71 ,e1 7,i71 ,o1 7,u71 ,y ,from ,i or 7,e7 7,e7 ,i ,from ,o or 7,a7 7,a71 ,o ,from ,u or 7,y7 7,e7 ,i1 ,u 7,y7 ,from ,y or 7,u7 7,u7 ,y ,having thus discussed the principal points connected with the simple assertion1 we pass to the manner of making two assertions give a third4 ,every instance of this is called a syllogism1 the two assertions which form the basis of the third are called premises1 and the third itself the conclusion4 ,if two things both agree with a third in any particular1 they agree with each other in the same2 as1 if ,a be of the same colour as ,x1 and ,b of the same colour as ,x1 then ,a is of the same colour as ,b4 ,again1 if ,a differ from ,x in any particular in which ,b agrees with ,x1 then ,a and ,b differ in that particular4 ,if ,a be not of the same colour as ,x1 and ,b be of the same colour as ,x1 then ,a is not of the colour of ,b4 ,but if ,a and ,b both differ from ,x in any particular1 nothing can be inferred2 they may either differ in the same way and to the same extent1 or not4 ,thus1 if ,a and ,b be both of different colours from ,x1 it neither follows that they agree1 nor differ1 in their own colours4 ,the paragraph preceding contains the essential parts of all inference1 which consists in comparing two things with a third1 and finding from their agreement or difference with that third1 their agreement or difference with one another4 ,thus1 ,every ,a is ,x1 every ,b is ,x1 allows us to infer that ,a and ,b have all those qualities in common which are necessary to ,x4 ,again1 from ,every ,a is ,x1 and #8,no ,b is ,x1' we infer that ,a and ,b differ from one another in all particulars which are essential to ,x4 ,the preceding forms1 however1 though they represent common reasoning better than the ordinary syllogism1 to which we are now coming1 do not constitute the ultimate forms of inference4 ,simple identity or non-identity is the ultimate state to which every assertion may be reduced2 and we shall1 therefore1 first ask1 from what identities1 c41 can other identities1 c41 be produced8 ,again1 since we name objects in species1 each species consisting of a number of individuals1 and since our assertion may include all or only part of a species1 it is further necessary to ask1 in every instance1 to what extent the conclusion drawn is true1 whether of all1 or only of part8 ,let us take the simple assertion1 #8,every living man respires2' or1 every living man is one of the things 7however varied they may be7 which respire4 ,if we were to inclose all living men in a large triangle1 and all respiring objects in a large circle1 the preceding assertion1 if true1 would require that the whole of the triangle should be contained in the circle4 ,and in the same way we may reduce any assertion to the expression of a coincidence1 total or partial1 between two figures4 ,thus1 a point in a circle may represent an individual of one species1 and a point in a triangle an individual of another species3 and we may express that the whole of one species is asserted to be contained or not contained in the other by such forms as1 #8,all the is in the '2 #8,none of the is in the '4 ,any two assertions about ,a and ,b1 each expressing agreement or disagreement1 total or partial1 with or from ,x1 and leading to a conclusion with respect to ,a or ,b1 is called a syllogism1 of which ,x is called the middle term4 ,the plainest syllogism is the following3-- ,every ,a is ,x ,all the is in the ,every ,x is ,b ,all the is in the ,therefore ,every ,a is ,b,therefore ,all the is in the ,in order to find all the possible forms of syllogism1 we must make a table of all the elements of which they can consist2 namely-- ,a and ,x ,b and ,x ,every ,a is ,x ,a ,every ,b is ,x ,no ,a is ,x ,e ,no ,b is ,x ,some ,a is ,x ,i ,some ,b is ,x ,some ,a is not ,x ,o ,some ,b is not ,x ,every ,x is ,a ,u ,every ,x is ,b ,some ,x is not ,a ,y ,some ,x is not ,b ,or their synonymes1 and and ,all the is in the ,a ,all the is in the ,none of the is in the ,e ,none of the is in the ,some of the is in the ,i ,some of the is in the ,some of the is not in the ,o ,some of the is not in the ,all the is in the ,u ,all the is in the ,some of the is not in the ,y ,some of the is not in the ,now1 taking any one of the six relations between ,a and ,x1 and combining it with either of those between ,b and ,x1 we have six pairs of premises1 and the same number repeated for every different relation of ,a and ,x4 ,we have then thirty-six forms to consider3 but1 thirty of these 7namely1 all but 7,a1 ,a7 7,e1 ,e71 c47 are half of them repetitions of the other half4 ,thus1 #8,every ,a is ,x1 no ,b is ,x1' and #8,every ,b is ,x1 no ,a is ,x1' are of the same form1 and only differ by changing ,a into ,b and ,b into ,a4 ,there are then only #ae #f1 or #ba distinct forms1 some of which give a necessary conclusion1 while others do not4 ,we shall select the former of these1 classifying them by their conclusions2 that is1 according as the inference is of the form ,a1 ,e1 ,i1 or ,o4 ,i4 ,in what manner can a universal affirmative conclusion be drawn2 namely1 that one figure is entirely contained in the other8 ,this we can only assert when we know that one figure is entirely contained in the circle1 which itself is entirely contained in the other figure4 ,thus1 ,every ,a is ,x ,all the is in the ,a ,every ,x is ,b ,all the is in the ,a ,every ,a is ,b ,all the is in the ,a is the only way in which a universal affirmative conclusion can be drawn4 ,,ii4 ,in what manner can a universal negative conclusion be drawn2 namely1 that one figure is entirely exterior to the other8 ,only when we are able to assert that one figure is entirely within1 and the other entirely without1 the circle4 ,thus1 ,every ,a is ,x,all the is in the ,a ,no ,b is ,x,none of the is in ,e the ,no ,a is ,b,none of the is in ,e the is the only way in which a universal negative conclusion can be drawn4 ,,iii4 ,in what manner can a particular affirmative conclusion be drawn2 namely1 that part or all of one figure is contained in the other8 ,only when we are able to assert that the whole circle is part of one of the figures1 and that the whole1 or part of the circle1 is part of the other figure4 ,we have then two forms4 ,every ,x is ,a ,all the is in the ,a ,every ,x is ,b ,all the is in the ,a ,some ,a is ,b ,some of the is in ,i the ,every ,x is ,a ,all the is in the ,a ,some ,x is ,b ,some of the is in ,i the ,some ,a is ,b ,some of the is in ,i the ,the second of these contains all that is strictly necessary to the conclusion1 and the first may be omitted4 ,that which follows when an assertion can be made as to some1 must follow when the same assertion can be made of all4 ,,iv4 ,how can a particular negative proposition be inferred2 namely1 that part1 or all of one figure1 is not contained in the other8 ,it would seem at first sight1 whenever we are able to assert that part or all of one figure is in the circle1 and that part or all of the other figure is not4 ,the weakest syllogism from which such an inference can be drawn would then seem to be as follows4 ,some ,a is ,x ,some of the is in the ,some ,b is not ,x ,some of the is not in the ,some ,b is not ,a ,some of the is not in the ,but here it will appear1 on a little consideration1 that the conclusion is only thus far true2 that those ,as which are ,xs cannot be those ,bs which are not ,xs2 but they may be other ,bs1 about which nothing is asserted when we say that some ,bs are not ,xs4 ,and further consideration will make it evident1 that a conclusion of this form can only be arrived at when one of the figures is entirely within the circle1 and the whole or part of the other without2 or else when the whole of one of the figures is without the circle1 and the whole or part of the other within2 or lastly1 when the circle lies entirely within one of the figures1 and not entirely within the other4 ,that is1 the following are the distinct forms which allow of a particular negative conclusion1 in which it should be remembered that a particular proposition in the premises may always be changed into a universal one1 without affecting the conclusion4 ,for that which necessarily follows from 8some10 follows from 8all40 ,every ,a is ,x ,all the is in the ,a ,some ,b is not ,x ,some of the is not in the ,o ,some ,b is not ,a ,some of the is not in the ,o ,no ,a ,is ,x ,none of the is in the ,e ,some ,b is ,x ,some of the is in the ,i ,some ,b is not ,a ,some of the is not in the ,o ,every ,x is ,a ,all the is in the ,a ,some ,x is not ,b ,some of the is not in the ,o ,some ,a is not ,b ,some of the is not in the ,o ,it appears1 then1 that there are but six distinct syllogisms4 ,all others are made from them by strengthening one of the premises1 or converting one or both of the premises1 where such conversion is allowable2 or else by first making the conversion1 and then strengthening one of the premises4 ,and the following arrangement will shew that two of them are universal1 three of the others being derived from them by weakening one of the premises in a manner which does not destroy1 but only weakens1 the conclusion4 #a4 ,every ,a is ,x #c4 ,every ,a is ,x ,every ,x is ,b ,no ,b is ,x 444444444 ------------------------ ------------------------ ,every ,a is ,b ,no ,a is ,b #b4 ,some ,a is ,x #d4 ,some ,a is ,x #e4 ,every ,a is ,x #f4 ,every ,x is ,a ,every ,x is ,b ,no ,b is ,x ,some ,b is not ,x ,some ,x is not ,b ------------------------ ------------------------------ -------------------------------- ------------------------------ ,some ,a is ,b ,some ,a is not ,b ,some ,b is not ,a ,some ,a is not ,b ,we may see how it arises that one of the partial syllogisms is not immediately derived1 like the others1 from a universal one4 ,in the preceding1 ,,aee may be considered as derived from ,,aaa1 by changing the term in which ,x enters universally into its contrary4 ,if this be done with the other term instead1 we have ,no ,a is ,xfrom which universal premises we cannot deduce a universal conclusion1 but only ,some ,b is not ,a4 ,every ,x is ,b ,if we weaken one and the other of these premises1 as they stand1 we obtain ,some ,a is not ,x ,no ,a is ,x ,every ,x is ,b and ,some ,x is ,b -------------------------------- ------------------------------ ,no conclusion ,some ,b is not ,a equivalent to the fourth of the preceding3 but if we convert the first premiss1 and proceed in the same manner1 ,no ,x is ,a ,some ,x is not ,a ,from ,every ,x is ,b we obtain ,every ,x is ,b -------------------------------- -------------------------------- ,some ,b is not ,a ,some ,b is not ,a which is legitimate1 and is the same as the last of the preceding list1 with ,a and ,b interchanged4 ,before proceeding to shew that all the usual forms are contained in the preceding1 let the reader remark the following rules1 which may be proved either by collecting them from the preceding cases1 or by independent reasoning4 #a4 ,the middle term must enter universally into one or the other premiss4 ,if it were not so1 the one premiss might speak of one part of the middle term1 and the other of the other2 so that there would1 in fact1 be no middle term4 ,thus1 #8,every ,a is ,x1 ,every ,b is ,x1' gives no conclusion3 it may be thus stated2 ,all the ,as make up a part of the ,xs ,all the ,bs make up a part of the ,xs ,and1 before we can know that there is any common term of comparison at all1 we must have some means of shewing that the two parts are the same2 or the preceding premises by themselves are inconclusive4 #b4 ,no term must enter the conclusion more generally than it is found in the premises2 thus1 if ,a be spoken of partially in the premises1 it must enter partially into the conclusion4 ,this is obvious1 since the conclusion must assert no more than the premises imply4 #c4 ,from premises both negative no conclusion can be drawn4 ,for it is obvious1 that the mere assertion of disagreement between each of two things and a third1 can be no reason for inferring either agreement or disagreement between these two things4 ,it will not be difficult to reduce any case which falls under this rule to a breach of the first rule3 thus1 ,no ,a is ,x1 ,no ,b is ,x1 gives ,every ,a is 7something which is not ,x7 ,every ,b is 7something which is not ,x7 in which the middle term is not spoken of universally in either4 ,again1 #8,no ,x is ,a1 ,some ,x is not ,b1' may be converted into ,every ,a is 7a thing which is not ,x7 ,some 7thing which is not ,b7 is ,x in which there is no middle term4 #d4 ,from premises both particular no conclusion can be drawn4 ,this is sufficiently obvious when the first or second rule is broken1 as in #8,some ,a is ,x1 ,some ,b is ,x4' ,but it is not immediately obvious when the middle term enters one of the premises universally4 ,the following reasoning will serve for exercise in the preceding results4 ,since both premises are particular in form1 the middle term can only enter one of them universally by being the predicate of a negative proposition2 consequently 7,rule #c7 the other premiss must be affirmative1 and1 being particular1 neither of its terms is universal4 ,consequently both the terms as to which the conclusion is to be drawn enter partially1 and the conclusion 7,rule #b7 can only be a particular affirmative proposition4 ,but if one of the premises be negative1 the conclusion must be negative 7as we shall immediately see74 ,this contradiction shews that the supposition of particular premises producing a legitimate result is inadmissible4 #e4 ,if one premiss be negative1 the conclusion1 if any1 must be negative4 ,if one term agree with a second and disagree with a third1 no agreement can be inferred between the second and third4 #f4 ,if one premiss be particular1 the conclusion must be particular4 ,this is not very obvious1 since the middle term may be universally spoken of in a particular proposition1 as in ,some ,b is not ,x4 ,but this requires one negative proposition1 whence 7,rule #c7 the other must be affirmative4 ,again1 since the conclusion must be negative 7,rule #e7 its predicate is spoken of universally1 and1 therefore1 must enter universally2 the other term ,a must enter1 then1 in a universal affirmative proposition1 which is against the supposition4 ,in the preceding set of syllogisms we observe one form only which produces ,a1 or ,e1 or ,i1 but three which produce ,o4 ,let an assertion be said to be weakened when it is reduced from universal to particular1 and strengthened in the contrary case4 ,thus1 #8,every ,a is ,b' is called stronger than #8,some ,a is ,b4' ,every form of syllogism which can give a legitimate result is either one of the preceding six1 or another formed from one of the six1 either by changing one of the assertions into its converse1 if that be allowable1 or by strengthening one of the premises without altering the conclusion1 or both4 ,thus1 ,some ,a is ,x may be written ,some ,x is ,a ,every ,x is ,b ,every ,x is ,b ,what follows will still follow from ,every ,x is ,a ,every ,x is ,b for all which is true when #8,some ,x is ,a1' is not less true when #8,every ,x is ,a4' ,it would be possible also to form a legitimate syllogism by weakening the conclusion1 when it is universal1 since that which is true of all is true of some4 ,thus1 #8,every ,a is ,x1 ,every ,x is ,b1' which yields #8,every ,a is ,b1' also yields #8,some ,a is ,b4' ,but writers on logic have always considered these syllogisms as useless1 conceiving it better to draw from any premises their strongest conclusion4 ,in this they were undoubtedly right2 and the only question is1 whether it would not have been advisable to make the premises as weak as possible1 and not to admit any syllogisms in which more appeared than was absolutely necessary to the conclusion4 ,if such had been the practice1 then ,every ,x is ,a1 ,every ,x is ,b1 therefore ,some ,a is ,b would have been considered as formed by a spurious and unnecessary excess of assertion4 ,the minimum of assertion would be contained in either of the following1 ,every ,x is ,a1 ,some ,x is ,b1 therefore ,some ,a is ,b ,some ,x is ,a1 ,every ,x is ,b1 therefore ,some ,a is ,b ,in this tract1 syllogisms have been divided into two classes3 first1 those which prove a universal conclusion2 secondly1 those which prove a partial conclusion1 and which are 7all but one7 derived from the first by weakening one of the premises1 in such manner as to produce a legitimate but weakened conclusion4 ,those of the first class are placed in the first column1 and the other in the second4 ,universal4 ,particular4 ,a ,every ,a is ,x ,some ,a is ,x ,i ,a ,every ,x is ,b ,every ,x is ,b ,a ------------------------ -------------------------- ,a ,every ,a is ,b ,some ,a is ,b ,i ,some ,a is ,x ,i ,no ,x is ,b ,e ------------------------------ ,a ,every ,a is ,x ,some ,a is not ,b ,o ,e ,no ,x is ,b ,every ,a is ,x ,a ------------------------ ,e ,no ,a is ,b ,some ,b is not ,x ,o ------------------------------ ,some ,b is not ,a ,o ,every ,x is ,a ,a 444444 ,some ,x is not ,b ,o ------------------------------ ,some ,a is not ,b ,o ,in all works on logic1 it is customary to write that premiss first which contains the predicate of the conclusion4 ,thus1 ,every ,x is ,b ,every ,a is ,x ,every ,a is ,x would be written1 and not ,every ,x is ,b ------------------------ ------------------------ ,every ,a is ,b ,every ,a is ,b ,the premises thus arranged are called major and minor2 the predicate of the conclusion being called the major term1 and its subject the minor4 ,again1 in the preceding case we see the various subjects coming in the order ,x1 ,b2 ,a1 ,x2 ,a1 ,b3 and the number of different orders which can appear is four1 namely-- ,,xb ,,bx ,,xb ,,bx ,,ax ,,ax ,,xa ,,xa ---- ---- ---- ---- ,,ab ,,ab ,,ab ,,ab which are called the four figures1 and every kind of syllogism in each figure is called a mood4 ,i now put down the various moods of each figure1 the letters of which will be a guide to find out those of the preceding list from which they are derived4 ,co means that a premiss of the preceding list has been converted2 that it has been strengthened2 ,co 1 that both changes have taken place4 ,thus1 ,a ,every ,x is ,b ,a ,every ,x is ,b ,i ,some ,a is ,x becomes ,a ,every ,x is ,a3 7,co 7 ------------------------ ------------------------ ,i ,some ,a is ,b ,i ,some ,a is ,b ,and ,co abbreviates the following3 ,if some ,a be ,x1 then some ,x is ,a 7,co72 and all that is true when ,some ,x is ,a1 is true when ,every ,x is ,a 772 therefore the second is legitimate1 if the first be so4 ,first ,figure4 ,a ,every ,x is ,b ,a ,every ,x is ,b ,a ,every ,a is ,x ,i ,some ,a is ,x ------------------------ ---------------------------- ,a ,every ,a is ,b ,i ,some ,a is ,b ,e ,no ,x is ,b ,e ,no ,x is ,b ,a ,every ,a is ,x ,i ,some ,a is ,x ------------------------ -------------------------------- ,e ,no ,a is ,b ,o ,some ,a is not ,b ,second ,figure4 ,e ,no ,b is ,x 7,co7 ,e ,no ,b is ,x 7,co7 ,a ,every ,a is ,x ,i ,some ,a is ,x ------------------------ ---------------------------------- ,e ,no ,a is ,b ,o ,some ,a is not ,b ,a ,every ,b is ,x ,a ,every ,b is ,x ,e ,no ,a is ,x 7,co7 ,o ,some ,a is not ,x ------------------------ -------------------------------- ,e ,no ,a is ,b ,o ,some ,a is not ,b ,third ,figure4 ,a ,every ,x is ,b ,e ,no ,x is ,b ,a ,every ,x is ,a 7,co7 ,a ,every ,x is ,a 7,co7 ------------------------ ------------------------------------ ,i ,some ,a is ,b ,o ,some ,a is not ,b ,i ,some ,x is ,b 7,co7 ,o ,some ,x is not ,b ,a ,every ,x is ,a ,a ,every ,x is ,a ------------------------ -------------------------------- ,i ,some ,a is ,b ,o ,some ,a is not ,b ,a ,every ,x is ,b ,e ,no ,x is ,b ,i ,some ,x is ,a 7,co7 ,i ,some ,x is ,a 7,co7 ------------------------ ---------------------------------- ,i ,some ,a is ,b ,o ,some ,a is not ,b ,fourth ,figure4 ,a ,every ,b is ,x 77 ,i ,some ,b is ,x ,a ,every ,x is ,a ,a ,every ,x is ,a ------------------------ ------------------------ ,i ,some ,a is ,b ,i ,some ,b is ,a ,a ,every ,b is ,x ,e ,no ,b is ,x 7,co7 ,e ,no ,x is ,a ,a ,every ,x is ,a 7,co7 ------------------------ ------------------------------------ ,e ,no ,a is ,b ,o ,some ,a is not ,b ,e ,no ,b is ,x 7,co7 ,i ,some ,x is ,a 7,co7 -------------------------------- ,o ,some ,a is not ,b ,the above is the ancient method of dividing syllogisms2 but1 for the present purpose1 it will be sufficient to consider the six from which the rest can be obtained4 ,and since some of the six have ,a in the predicate of the conclusion1 and not ,b1 we shall join to them the six other syllogisms which are found by transposing ,b and ,a4 ,the complete list1 therefore1 of syllogisms with the weakest premises and the strongest conclusions1 in which a comparison of ,a and ,b is obtained by comparison of both with ,x1 is as follows3 ,every ,a is ,x ,every ,b is ,x ,some ,a is ,x ,some ,b is ,x ,every ,x is ,b ,every ,x is ,a ,no ,x is ,b ,no ,x is ,a ------------------------ ------------------------ -------------------------------- -------------------------------- ,every ,a is ,b ,every ,b is ,a ,some ,a is not ,b ,some ,b is not ,a ,every ,a is ,x ,every ,b is ,x ,every ,a is ,x ,every ,b is ,x ,no ,x is ,b ,no ,x is ,a ,some ,b is not ,x ,some ,a is not ,x ------------------------ ------------------------ -------------------------------- -------------------------------- ,no ,a is ,b ,no ,b is ,a ,some ,b is not ,a ,some ,a is not ,b ,some ,a is ,x ,some ,b is ,x ,every ,x is ,a ,every ,x is ,b ,every ,x is ,b ,every ,x is ,a ,some ,x is not ,b ,some ,x is not ,a ------------------------ ------------------------ -------------------------------- -------------------------------- ,some ,a is ,b ,some ,b is ,a ,some ,a is not ,b ,some ,b is not ,a ,in the list of page #ai1 there was nothing but recapitulation of forms1 each form admitting a variation by interchanging ,a and ,b4 ,this interchange having been made1 and the results collected as above1 if we take every case in which ,b is the predicate1 or can be made the predicate by allowable conversion1 we have a collection of all possible weakest forms in which the result is one of the four #8,every ,a is ,b1' #8,no ,a is ,b1' #8,some ,a is ,b1' #8,some ,a is not ,b'2 as follows4 ,the premises are written in what appeared the most natural order1 without distinction of major or minor2 and the letters prefixed are according to the forms of the several premises1 as in page #aj4 ,a ,every ,a is ,x ,u ,every ,x is ,b ------------------------ ,a ,every ,a is ,b ,i ,some ,a is ,x ,i ,some ,b is ,x ,u ,every ,x is ,b ,u ,every ,x is ,a ------------------------ ------------------------ ,i ,some ,a is ,b ,i ,some ,a is ,b ,a ,every ,a is ,x ,a ,every ,b is ,x ,e ,no ,b is ,x ,e ,no ,a is ,x ------------------------ ------------------------ ,e ,no ,a is ,b ,e ,no ,a is ,b ,i ,some ,a is ,x ,a ,every ,b is ,x ,u ,every ,x is ,a ,e ,no ,b is ,x ,o ,some ,a is not ,x ,y ,some ,x is not ,b -------------------------------- -------------------------------- -------------------------------- ,o ,some ,a is not ,b ,o ,some ,a is not ,b ,o ,some ,a is not ,b ,every assertion which can be made upon two things by comparison with any third1 that is1 every simple inference1 can be reduced to one of the preceding forms4 ,generally speaking1 one of the premises is omitted1 as obvious from the conclusion2 that is1 one premiss being named and the conclusion1 that premiss is implied which is necessary to make the conclusion good4 ,thus1 if ,i say1 8,that race must have possessed some of the arts of life1 for they came from ,asia10 it is obviously meant to be asserted1 that all races coming from ,asia must have possessed some of the arts of life4 ,the preceding is then a syllogism1 as follows3 #8,that race' is #8a race of ,asiatic origin3' ,every #8race of ,asiatic origin' is #8a race which must have possessed some of the arts of life3' ,therefore1 ,that race is a race which must have possessed some of the arts of life4 ,a person who makes the preceding assertion either means to imply1 antecedently to the conclusion1 that all ,asiatic races must have possessed arts1 or he talks nonsense if he asserts the conclusion positively4 #8,a must be ,b1 for it is ,x1' can only be true when #8,every ,x is ,b4' ,this latter proposition may be called the suppressed premiss2 and it is in such suppressed propositions that the greatest danger of error lies4 ,it is also in such propositions that men convey opinions which they would not willingly express4 ,thus1 the honest witness who said1 #8,i always thought him a respectable man--he kept his gig1' would probably not have admitted in direct terms1 #8,every man who keeps a gig must be respectable4' ,i shall now give a few detached illustrations of what precedes4 8,his imbecility of character might have been inferred from his proneness to favourites2 for all weak princes have this failing40 ,the preceding would stand very well in a history1 and many would pass it over as containing very good inference4 ,written1 however1 in the form of a syllogism1 it is1 ,all weak princes are prone to favourites ,he was prone to favourites -------------------------------- ---------------------------------------- ------ ,therefore ,he was a weak prince which is palpably wrong4 7,rule #a47 ,the writer of such a sentence as the preceding might have meant to say1 #8for all who have this failing are weak princes2' in which case he would have inferred rightly4 ,every one should be aware that there is much false inference arising out of badness of style1 which is just as injurious to the habits of the untrained reader as if the errors were mistakes of logic in the mind of the writer4 #8,a is less than ,b2 ,b is less than ,c3 therefore ,a is less than ,c4' ,this1 at first sight1 appears to be a syllogism2 but1 on reducing it to the usual form1 we find it to be1 ,a is 7a magnitude less than ,b7 ,b is 7a magnitude less than ,c7 ,therefore ,a is 7a magnitude less than ,c7 which is not a syllogism1 since there is no middle term4 ,evident as the preceding is1 the following additional proposition must be formed before it can be made explicitly logical4 #8,if ,b be a magnitude less than ,c1 then every magnitude less than ,b is also less than ,c4' ,there is1 then1 before the preceding can be reduced to a syllogistic form1 the necessity of a deduction from the second premiss1 and the substitution of the result instead of that premiss4 ,thus1 ,a is less than ,b ,less than ,b is less than ,c3 following from ,b is less than ,c4 ------------------ ---------------------------------- ,therefore ,a is less than ,c ,but1 if the additional argument be examined--namely1 if ,b be less than ,c1 then that which is less than ,b is less than ,c--it will be found to require precisely the same considerations repeated2 for the original inference was nothing more4 ,in fact1 it may easily be seen as follows1 that the proposition before us involves more than any simple syllogism can express4 ,when we say that ,a is less than ,b1 we say that if ,a were applied to ,b1 every part of ,a would match a part of ,b1 and there would be parts of ,b remaining over4 ,but when we say1 #8,every ,a is ,b1' meaning the premiss of a common syllogism1 we say that every instance of ,a is an instance of ,b1 without saying any thing as to whether there are or are not instances of ,b still left1 after those which are also ,a are taken away4 ,if1 then1 we wish to write an ordinary syllogism in a manner which shall correspond with #8,a is less than ,b1 ,b is less than ,c1 therefore ,a is less than ,c1' we must introduce a more definite amount of assertion than was made in the preceding forms4 ,thus1 ,every ,a is ,b1 and there are ,bs which are not ,as ,every ,b is ,c1 and there are ,cs which are not ,bs ---------------------------------------- ---------------------------------------- -------------- ,therefore ,every ,a is ,c1 and there are ,cs which are not ,as ,or thus3 ,the ,bs contain all the ,as1 and more ,the ,cs contain all the ,bs1 and more ---------------------------------------- ------------------------------ ,the ,cs contain all the ,as1 and more ,the most technical form1 however1 is1 ,from ,every ,a is ,b2 ,some ,b is not ,a ,every ,b is ,c2 ,some ,c is not ,b ,follows ,every ,a is ,c2 ,some ,c is not ,a ,this sort of argument is called fortiori argument1 because the premises are more than sufficient to prove the conclusion1 and the extent of the conclusion is thereby greater than its mere form would indicate4 ,thus1 #8,a is less than ,b1 ,b is less than ,c1 therefore1 fortiori1 ,a is less than ,c1' means that the extent to which ,a is less than ,c must be greater than that to which ,a is less than ,b1 or ,b than ,c4 ,in the syllogism last written1 either of the bracketed premises might be struck out without destroying the conclusion2 which last would1 however1 be weakened4 ,as it stands1 then1 the part of the conclusion1 #8,some ,c is not ,a1' follows it fortiori4 ,the argument fortiori1 may then be defined as a universally affirmative syllogism1 in which both of the premises are shewn to be less than the whole truth1 or greater4 ,thus1 in #8,every ,a is ,x1 ,every ,x is ,b1 therefore ,every ,a is ,b1' we do not certainly imply that there are more ,xs than ,as1 or more ,bs than ,xs1 so that we do not know that there are more ,bs than ,as4 ,but if we are at liberty to state the syllogism as follows1 ,all the ,as make up part 7and part only7 of the ,xs ,every ,x is ,b2 then we are certain that ,all the ,as make up part 7and part only7 of the ,bs4 ,but if we are at liberty further to say that ,all the ,as make up part 7and part only7 of the ,xs ,all the ,xs make up part 7and part only7 of the ,bs then we conclude that ,all the ,as make up part of part 7only7 of the ,bs and the words in ,italics mark that quality of the conclusion from which the argument is called fortiori4 ,most syllogisms which give an affirmative conclusion are generally meant to imply fortiori arguments1 except only in mathematics4 ,it is seldom1 except in the exact sciences1 that we meet with a proposition1 #8,every ,a is ,b1' which we cannot immediately couple with #8,some ,bs are not ,as4' ,when an argument is completely established1 with the exception of one assertion only1 so that the inference may be drawn as soon as that one assertion is established1 the result is stated in a form which bears the name of an hypothetical syllogism4 ,the word hypothesis means nothing but supposition2 and the species of syllogism just mentioned first lays down the assertion that a consequence will be true if a certain condition be fulfilled1 and then either asserts the fulfilment of the condition1 and thence the consequence1 or else denies the consequence1 and thence denies the fulfilment of the condition4 ,thus1 if we know that ,when ,a is ,b1 it follows that ,p is ,q2 then1 as soon as we can ascertain that ,a is ,b1 we can conclude that ,p is ,q2 or1 if we can shew that ,p is not ,q1 we know that ,a is not ,b4 ,but if we find that ,a is not ,b1 we can infer nothing2 for the preceding does not assert that ,p is ,q only when ,a is ,b4 ,and if we find out that ,p is ,q1 we can infer nothing4 ,this conditional syllogism may be converted into an ordinary syllogism1 as follows4 ,let ,k be any #8case in which ,a is ,b1' and ,z a #8case in which ,p is ,q'2 then the preceding assertion amounts to #8,every ,k is ,z4' ,let ,l be a particular instance1 the ,a of which may or may not be ,b4 ,if ,a be ,b in the instance under discussion1 or if ,a be not ,b1 we have1 in the one case and the other1 ,every ,k is ,z ,every ,k is ,z ,l is a ,k ,l is not a ,k ---------------- -------------------------- ,therefore ,l is a ,z ,no conclusion ,similarly1 according as a particular case 7,m7 is or is not ,z1 we have ,every ,k is ,z ,every ,k is ,z ,m is a ,z ,m is not a ,z -------------------------- ------------------------ ,no conclusion ,m is not a ,k ,that is to say3 ,the assertion of an hypothesis is the assertion of its necessary consequence1 and the denial of the necessary consequence is the denial of the hypothesis2 but the assertion of the necessary consequence gives no right to assert the hypothesis1 nor does the denial of the hypothesis give any right to deny the truth of that which would 7were the hypothesis true7 be its necessary consequence4 ,demonstration is of two kinds3 which arises from this1 that every proposition has a contradictory2 and of these two1 one must be true and the other must be false4 ,we may then either prove a proposition to be true1 or its contradictory to be false4 #8,it is true that ,every ,a is ,b1' and1 #8it is false that there are some ,as which are not ,bs1' are the same proposition2 and the proof of either is called the indirect proof of the other4 ,but how is any proposition to be proved false1 except by proving a contradiction to be true8 ,by proving a necessary consequence of the proposition to be false4 ,but this is not a complete answer1 since it involves the necessity of doing the same thing2 or1 so far as this answer goes1 one proposition cannot be proved false unless by proving another to be false4 ,but it may happen1 that a necessary consequence can be obtained which is obviously and self-evidently false1 in which case no further proof of the falsehood of the hypothesis is necessary4 ,thus the proof which ,euclid gives that all equiangular triangles are equilateral is of the following structure1 logically considered4 7#a47 ,if there be an equiangular triangle not equilateral1 it follows that a whole can be found which is not greater than its part4#a ,footnote #a3 ,this is the proposition in proof of which nearly the whole of the demonstration of ,euclid is spent4 7#b47 ,it is false that there can be any whole which is not greater than its part 7self evident74 7#c47 ,therefore it is false that there is any equiangular triangle which is not equilateral2 or all equiangular triangles are equilateral4 ,when a proposition is established by proving the truth of the matters it contains1 the demonstration is called direct2 when by proving the falsehood of every contradictory proposition1 it is called indirect4 ,the latter species of demonstration is as logical as the former1 but not of so simple a kind2 whence it is desirable to use the former whenever it can be obtained4 ,the use of indirect demonstration in the ,elements of ,euclid is almost entirely confined to those propositions in which the converses of simple propositions are proved4 ,it frequently happens that an established assertion of the form ,every ,a is ,b 7#a7 may be easily made the means of deducing1 ,every 7thing not ,a7 is not ,b 7#b7 which last gives ,every ,b is ,a 7#c7 ,the conversion of the second proposition into the third is usually made by an indirect demonstration1 in the following manner4 ,if possible1 let there be one ,b which is not ,a1 7#b7 being true4 ,then there is one thing which is not ,a and is ,b2 but every thing not ,a is not ,b2 therefore there is one thing which is ,b and is not ,b3 which is absurd4 ,it is then absurd that there should be one single ,b which is not ,a2 or1 ,every ,b is ,a4 ,the following proposition contains a method which is of frequent use4 ,,hypothesis4--,let there be any number of propositions or assertions1--three for instance1 ,a1 ,b1 and ,c1--of which it is the property that one or the other must be true1 and one only4 ,let there be three other propositions1 ,p1 ,q1 and ,r1 of which it is also the property that one1 and one only1 must be true4 ,let it also be a connexion of those assertions1 that ,when ,a is true1 ,p is true ,when ,b is true1 ,q is true ,when ,c is true1 ,r is true ,,consequence3 then it follows that ,when ,p is true1 ,a is true ,when ,q is true1 ,b is true ,when ,r is true1 ,c is true ,for1 when ,p is true1 then ,q and ,r must be false2 consequently1 neither ,b nor ,c can be true1 for then ,q or ,r would be true4 ,but either ,a1 ,b1 or ,c must be true1 therefore ,a must be true2 or1 when ,p is true1 ,a is true4 ,in a similar way the remaining assertions may be proved4 ,case #a4 ,if ,when ,p is ,q1 ,a is ,b ,when ,p is not ,q1 ,a is not ,b ,it follows that,when ,a is ,b1 ,p is ,q ,when ,a is not ,b1 ,p is not ,q ,case #b4 ,if ,when ,a is greater than ,b1 ,p is greater than ,q ,when ,a is equal to ,b1 ,p is equal to ,q ,when ,a is less than ,b1 ,p is less than ,q ,it follows that,when ,p is greater than ,q1 ,a is greater than ,b ,when ,p is equal to ,q1 ,a is equal to ,b ,when ,p is less than ,q1 ,a is less than ,b 99 99 99 99 99 ,we have hitherto supposed that the premises are actually true2 and1 in such a case1 the logical conclusion is as certain as the premises4 ,it remains to say a few words upon the case in which the premises are probably1 but not certainly1 true4 ,the probability of an event being about to happen1 and that of an argument being true1 may be so connected that the usual method of measuring the first may be made to give an easy method of expressing the second4 ,suppose an urn1 or lottery1 with a large number of balls1 black or white2 then1 if there be twelve white balls to one black1 we say it is twelve to one that a white ball will be drawn1 or that a white ball is twelve times as probable as a black one4 ,a certain assertion may be in the same condition as to the force of probability with which it strikes the mind3 that is1 the questions ,is the assertion true8 ,will a white ball be drawn8 may be such that the answer1 #8most probably1' expresses the same degree of likelihood in both cases4 ,we have before explained that logic has nothing to do with the truth or falsehood of assertions1 but only professes1 supposing them true1 to collect and classify the legitimate methods of drawing inferences4 ,similarly1 in this part of the subject1 we do not trouble ourselves with the question1 ,how are we to find the probability due to premises8 but we ask3 ,supposing 7happen how it may7 that we have found the probability of the premises1 required the probability of the conclusion4 ,when the odds in favour of a conclusion are1 say #f to #a1 there are1 out of every #g possible chances1 #f in favour of the conclusion1 and #a against it4 ,hence and will represent the proportions1 for and against1 of all the possible cases which exist4 ,thus we have the succession of such results as in the following table3-- ,odds in favour of an event ,probability for ,probability against #a to #a #b to #a #c to #a #c to #b #d to #a #d to #c #e to #a c4 c4 c4 ,let the probability of a conclusion1 as derived from the premises 7that is on the supposition that it was never imagined to be possible till the argument was heard71 be called the intrinsic probability of the argument4 ,this is found by multiplying together the probabilities of all the assertions which are necessary to the argument4 ,thus1 suppose that a conclusion was held to be impossible until an argument of a single syllogism was produced1 the premises of which have severally five to one and eight to one in their favour4 ,then 1 or 1 is the intrinsic probability of the argument1 and the odds in its favour are #dj to #ad1 or #bj to #g4 ,but this intrinsic probability is not always that of the conclusion2 the latter1 of course1 depending in some degree on the likelihood which the conclusion was supposed to have before the argument was produced4 ,a syllogism of #bj to #g in its favour1 advanced in favour of a conclusion which was beforehand as likely as not1 produces a much more probable result than if the conclusion had been thought absolutely false until the argument produced a certain belief in the possibility of its being true4 ,the change made in the probability of a conclusion by the introduction of an argument 7or of a new argument1 if some have already preceded7 is found by the following rule4 ,from the sum of the existing probability of the conclusion and the intrinsic probability of the new argument1 take their product2 the remainder is the probability of the conclusion1 as reinforced by the argument4 ,thus1 a b ab is the probability of the truth of a conclusion after the introduction of an argument of the intrinsic probability b1 the previous probability of the said conclusion having been a4 ,thus1 a conclusion which has at present the chance in its favour1 when reinforced by an argument whose intrinsic probability is 1 acquires the probability or1 1 or 2 or1 having #b to #a in its favour before1 it has #aa to #a in its favour after1 the argument4 ,when the conclusion was neither likely nor unlikely beforehand 7or had the probability 71 the shortest way of applying the preceding rule 7in which a b ab becomes b7 is to divide the sum of the numerator and denominator of the intrinsic probability of the argument by twice the denominator4 ,thus1 an argument of which the intrinsic probability is 1 gives to a conclusion on which no bias previously existed1 the probability or 7#c #d7_/7b #d74 ,,the ,,end4 ,,london3--,,printed ,,by ,,james ,,moyes1 ,castle ,street1 ,leicester ,square4 ---------------------------------------- -------------------------------- ,,transcriber',s ,,notes #a4 ,silently corrected obvious typographical errors and variations in spelling4 #b4 ,retained archaic1 non-standard1 and uncertain spellings as printed4 #c4 ,enclosed italics font in underscores4